Multipartite Monogamous Relations For Entanglement And Discord

The distribution of quantum correlations in multipartite systems play a significant role in several aspects of the quantum information theory. While it is well known that these quantum correlations can not be freely distributed, the way that it is shared in multipartite system is an open problem even for small set of qubits. Based on new monogamy-like relations between entanglement and discord for n𝑛nitalic_n-partite systems, we show how these correlations are distributed in general, determining distinct equalities and inequalities to the quantum discord and the entanglement of formation for arbitrary multipartite pure states.

Entanglement of formation (EF) Bennett et al. (1996a); Horodecki et al. (2009); Cornelio et al. (2011) and quantum discord (QD) Ollivier and Zurek (2001); Modi et al. (2012) are acknowledged measures of quantum correlation for bipartite systems. They were derived from very distinct concepts, however. The entanglement of formation was first introduced by Bennett et al. Bennett et al. (1996a) and the goal was to quantify the cost of building entangled states by local operations and classical communication (LOCC). On the other hand, QD was first introduced by Olivier and Zurek Ollivier and Zurek (2001) with the aim of measuring the non-classicality of bipartite quantum states. It determines the amount of locally inaccessible information and is given by the difference between two classically equivalent forms of the mutual information Ollivier and Zurek (2001).

Despite their conceptual differences it is known that, as entanglement, QD can not be freely shared Modi et al. (2012). Indeed, the way that quantum correlations is distributed forbids it to be maximally correlated with two parts simultaneously since a monogamous relation arises. The first to explore this aspect was Coffman, Kundu and Wootters Coffman et al. (2000), obtaining, about seventeen years ago, the famous inequality for the monogamy of squared concurrence. There, a simple expression concerning how entanglement is distributed among a system composed of three qubits was deduced, which raised an exciting and wide research field.

The monogamy of entanglement has fundamental implications in several fields of quantum physics. For example, the lack of monogamy is considered a huge obstacle to the implementation of quantum cryptography, where unconditional security relies on the fact that the spy does not have the skills to correlate with the trusted parts Renes and Grassl (2006); Masanes (2009). On the other hand, the lack of monogamy also provides information that may help to understand the mysterious behaviour of black holes Almheiri et al. (2013), which appears when attempting to combine quantum mechanics with general relativity. Moreover, the monogamy of quantum correlation was essential for proving that asymptotic cloning is equivalent to state estimation Bae and Acín (2006) and making quantum key distribution secure Scarani et al. (2005).

In this way, to elucidate the way that quantum correlation is distributed in multipartite systems is certainly important for information processing and communication technologies in multi-user scenarios. Nevertheless, in spite of the enormous effort of the scientific community to understand how quantum correlation is distributed in general multi-partite systems, that remains an important open problem even in the case of small number of parts and space dimensions. It is exactly at this direction that we develop our work, presenting new monogamy-like equalities for quantum discord and entanglement of formation in arbitrary multi-partite systems. As we show, extending the conservative relation between EF and QD Fanchini et al. (2011) for multipartite systems, a general rule for the way how quantum discord is distributed emerges. Also, the way that EF and QD are distributed is shown to be deeply related in general multipartite states. As EF is a way to quantify the quantum communication needed to build a bipartite state and QD is a way to quantify the amount of information inaccessible by local operations, our results relate these two concepts in a new form. Indeed, in a multipartite system, the amount of quantum communication needed in each bipartition sums up equal to the sum of information trapped in nonlocal correlations as measured by quantum discord.

The paper is organized as follows. In section II we review some concepts and results necessary for this work. In section III we show two monogamy-like inequalities for fourpartite systems and two monogamy-like equalities for five-partite systems. In section IV, we extend the results to multipartite systems, presenting not only a new monogamy-like law between EF and QD, but also a general equality elucidating how QD is distributed in multipartite systems. We conclude our work in section V.

II Review and background

First of all, we need to define the concepts of entanglement and discord we are going to use. Nowadays, there exist many different measures of entanglement Horodecki et al. (2009) as well as different measures of quantum correlation Modi et al. (2012). In this work, we use the EF as the measure of entanglement, first defined by Bennett et al. in Ref. Bennett et al. (1996a) and the original QD first defined by Olivier and Zurek in Ref. Ollivier and Zurek (2001). Indeed, both definitions are crucial for our work, since our main results come from the Koashi and Winter relation Koashi and Winter (2004), a notorious equation that connects EF and QD.

First, we introduce the EF which is defined in the paradigm of local operations with classical communication (LOCC). In this paradigm, two spatially separated observers, usually called Alice and Bob, share many copies of a standard quantum state. They can manipulate their parts locally with arbitrary quantum operations and measurements and communicate classically with each other. In this context, we consider the problem of Alice and Bob having to build a particular mixed quantum state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT from the standard resource state, a maximally entangled state of two qubits,

|Φ⟩=12(|00⟩+|11⟩).ketΦ12ket00ket11\left|\Phi\right\rangle=\frac1\sqrt2(\left|00\right\rangle+\left|11% \right\rangle).| roman_Φ ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 00 ⟩ + | 11 ⟩ ) . The first attempt to quantify this task arises with the entanglement of formation Bennett et al. (1996a). For any pure state |ψab⟩ketsubscript𝜓𝑎𝑏\left|\psi_ab\right\rangle| italic_ψ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟩, it is easily evaluated through the von Neumann entropy of one of its parts Bennett et al. (1996b); Nielsen and Chuang (2000),

Eab(|ψab⟩)=S(ρa),subscript𝐸𝑎𝑏ketsubscript𝜓𝑎𝑏𝑆subscript𝜌𝑎E_ab(\left|\psi_ab\right\rangle)=S(\rho_a),italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( | italic_ψ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ⟩ ) = italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) , where ρasubscript𝜌𝑎\rho_aitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT is the reduced state of subsystem a𝑎aitalic_a. For a mixed state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT, EF is defined as

Eab(ρab)=minℰ∑ipiEab(,subscript𝐸𝑎𝑏subscript𝜌𝑎𝑏subscriptℰsubscript𝑖subscript𝑝𝑖subscript𝐸𝑎𝑏ketsubscript𝜑𝑖E_ab(\rho_ab)=\min_\mathcalE\Big\\sum_ip_iE_ab(\left,italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) = roman_min start_POSTSUBSCRIPT caligraphic_E end_POSTSUBSCRIPT ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( , where the minimization runs over all possible ensembles such that ρab=∑ipi|φi⟩⟨φi|subscript𝜌𝑎𝑏subscript𝑖subscript𝑝𝑖ketsubscript𝜑𝑖brasubscript𝜑𝑖\rho_ab=\sum_ip_i\left|\varphi_i\right\rangle\left\langle\varphi_i\right|italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ ⟨ italic_φ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |. The EF can be easily calculated for two qubits states through Wootters’s formula Wootters (1998), however this calculation is very difficult for higher dimensional system 111Today, it is known that the ultimate cost of building a state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT from the standard state turns out to be the entanglement cost which is the regularized entanglement of formation Bennett et al. (1996a); Horodecki et al. (2009); Hastings (2009); Cornelio et al. (2011). . Another way to think about the entangled is remembering that each standard state needs one bit of quantum communication to be formed. Therefore, EF is also a measure of the amount of quantum communication between Alice and Bob needed to build the state ρabsubscript𝜌𝑎𝑏\rho_abitalic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT Shor (2004); Hastings (2009); Cornelio et al. (2011); Horodecki et al. (2009). This interpretation will be useful in the following.

Second, we turn our attention to discord which aims to quantify the amount of information that is not accessible in (or is destroyed by) a measurement. Let us consider again a bipartite system with an arbitrary state ρacsubscript𝜌𝑎𝑐\rho_acitalic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT shared between two observers, Alice and Carol. The amount of uncertainty about subsystem a𝑎aitalic_a is given by the von Neumann entropy of this subsystem S(ρa)𝑆subscript𝜌𝑎S(\rho_a)italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ). If Carol makes a measurement on her subsystem c𝑐citalic_c and obtain a result ΠisubscriptΠ𝑖\Pi_iroman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT from a complete set of POVM ΠisubscriptΠ𝑖\\Pi_i\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , the state of subsystem a𝑎aitalic_a changes to the state ρaisuperscriptsubscript𝜌𝑎𝑖\rho_a^iitalic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT with a new uncertainty given by S(ρai)𝑆superscriptsubscript𝜌𝑎𝑖S(\rho_a^i)italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ). So the difference between S(ρa)𝑆subscript𝜌𝑎S(\rho_a)italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) and S(ρai)𝑆superscriptsubscript𝜌𝑎𝑖S(\rho_a^i)italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) is the amount of information learned by Carol about the subsystem a𝑎aitalic_a. On average, Carol learns

S(ρa)-∑ipiS(ρai),𝑆subscript𝜌𝑎subscript𝑖subscript𝑝𝑖𝑆superscriptsubscript𝜌𝑎𝑖S(\rho_a)-\sum_ip_iS(\rho_a^i),italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) , where pisubscript𝑝𝑖p_iitalic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is the probability of Carol find the result ΠisubscriptΠ𝑖\Pi_iroman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT in her measurement from a complete set ΠisubscriptΠ𝑖\\Pi_i\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT . Of course, there are many possible sets of POVM Carol can choose and she can always take the best one. So the maximum amount of information she can obtain is called classical correlation and is given by Henderson and Vedral (2001); Ollivier and Zurek (2001)

Ja|c←(ρac)=maxΠi[S(ρa)-∑ipiS(ρai)],superscriptsubscript𝐽conditional𝑎𝑐←subscript𝜌𝑎𝑐subscriptsubscriptΠ𝑖𝑆subscript𝜌𝑎subscript𝑖subscript𝑝𝑖𝑆superscriptsubscript𝜌𝑎𝑖J_c^\leftarrow(\rho_ac)=\max_\\Pi_i\\big[S(\rho_a)-\sum_ip% _iS(\rho_a^i)\big],italic_J start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) = roman_max start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT end_POSTSUBSCRIPT [ italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) - ∑ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_i end_POSTSUPERSCRIPT ) ] , (1) where the maximization runs over all possible POVM on Carol’s subsystem. We remark that Ja|c←superscriptsubscript𝐽conditional𝑎𝑐←J_c^\leftarrowitalic_J start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT is in general asymmetric being different from Jc|a←superscriptsubscript𝐽conditional𝑐𝑎←J_a^\leftarrowitalic_J start_POSTSUBSCRIPT italic_c | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT. The classical correlation measure (1) usually does not capture all the correlations contained in the quantum state ρacsubscript𝜌𝑎𝑐\rho_acitalic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT. The total amount of correlation is measured by the quantum mutual information Nielsen and Chuang (2000)

I(ρac)=S(ρa)+S(ρc)-S(ρac),𝐼subscript𝜌𝑎𝑐𝑆subscript𝜌𝑎𝑆subscript𝜌𝑐𝑆subscript𝜌𝑎𝑐I(\rho_ac)=S(\rho_a)+S(\rho_c)-S(\rho_ac),italic_I ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) = italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT ) + italic_S ( italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) - italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) , which is always bigger than Ja|c(ρac)subscript𝐽conditional𝑎𝑐subscript𝜌𝑎𝑐J_c(\rho_ac)italic_J start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ). In this context, the difference is the amount of inaccessible information which is measured by the quantum discord Ollivier and Zurek (2001)

δa|c←(ρac)=I(ρac)-Ja|c←(ρac).superscriptsubscript𝛿conditional𝑎𝑐←subscript𝜌𝑎𝑐𝐼subscript𝜌𝑎𝑐superscriptsubscript𝐽conditional𝑎𝑐←subscript𝜌𝑎𝑐\delta_c^\leftarrow(\rho_ac)=I(\rho_ac)-J_a^\leftarrow(\rho_% ac).italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) = italic_I ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) - italic_J start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) . In addition, the amount of correlations δa|c←(ρac)superscriptsubscript𝛿conditional𝑎𝑐←subscript𝜌𝑎𝑐\delta_c^\leftarrow(\rho_ac)italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) is the amount that is destroyed when the measurement ΠisubscriptΠ𝑖\\Pi_i\ roman_Π start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT is made in the Carol’s subsystem. When the discord vanishes, no correlations is destroyed by the measurement and the state does not change. In this case, the state is considered classical Ollivier and Zurek (2001), since it is a fundamental aspect of quantum mechanics to perturb the physical systems in a measurement.

The entanglement of formation and the classical correlation are directly connected in a pure tri-partite system by the Koashi-Winter (KW) relation Koashi and Winter (2004),

Eab+Ja|c←=Sa,subscript𝐸𝑎𝑏superscriptsubscript𝐽conditional𝑎𝑐←subscript𝑆𝑎E_ab+J_a^\leftarrow=S_a,italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT + italic_J start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT = italic_S start_POSTSUBSCRIPT italic_a end_POSTSUBSCRIPT , (2) which express a kind of monogamy between these distinct measures (from now on, for brevity and clarity, we omit the quantum state between parenthesis, i.e. Eab≡Eab(ρab)subscript𝐸𝑎𝑏subscript𝐸𝑎𝑏subscript𝜌𝑎𝑏E_ab\equiv E_ab(\rho_ab)italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ≡ italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ), Ja|c←≡Ja|c←(ρac)superscriptsubscript𝐽conditional𝑎𝑐←superscriptsubscript𝐽conditional𝑎𝑐←subscript𝜌𝑎𝑐J_c^\leftarrow\equiv J_a^\leftarrow(\rho_ac)italic_J start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ≡ italic_J start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ), etc.). It means that the amount of quantum correlations given by the EF between Alice and Bob, plus classical correlation between Alice and Carol, is equal to the uncertainty about the Alice’s system. Also, Eq. (2) can easily be rewritten to relate entanglement and discord as Fanchini et al. (2011)

Eab=δa|c←+Sa|c,subscript𝐸𝑎𝑏superscriptsubscript𝛿conditional𝑎𝑐←subscript𝑆conditional𝑎𝑐E_ab=\delta_a^\leftarrow+S_a,italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT , (3) where Sa|csubscript𝑆conditional𝑎𝑐S_aitalic_S start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT is the quantum conditional entropy of the part a𝑎aitalic_a given c𝑐citalic_c, Sa|c=S(ρac)-S(ρc)subscript𝑆conditional𝑎𝑐𝑆subscript𝜌𝑎𝑐𝑆subscript𝜌𝑐S_c=S(\rho_ac)-S(\rho_c)italic_S start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT = italic_S ( italic_ρ start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT ) - italic_S ( italic_ρ start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ). From cyclic permutations of Eq. (3) one can obtain Fanchini et al. (2011)

Eab+Eac=δa|b←+δa|c←.subscript𝐸𝑎𝑏subscript𝐸𝑎𝑐superscriptsubscript𝛿conditional𝑎𝑏←superscriptsubscript𝛿conditional𝑎𝑐←E_ab+E_ac=\delta_a^\leftarrow+\delta_a^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (4) Therefore, the amount of entanglement a central particle a𝑎aitalic_a shares with the other two is equal to the amount of discord it also shares with the same two particles. For this reason, it is called a conservation law between entanglement and discord. The entanglement and discord in a particular bipartition can be different, nonetheless the amount of quantum correlation distributed through the entire system is the same measured either by entanglement or discord.

Moreover, one can also write another type of conservation law between entanglement and discord involving a kind of cycle over the parts of the system Fanchini et al. (2012)

Eab+Ebc+Eca=δb|a←+δc|b←+δa|c←.subscript𝐸𝑎𝑏subscript𝐸𝑏𝑐subscript𝐸𝑐𝑎superscriptsubscript𝛿conditional𝑏𝑎←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑎𝑐←E_ab+E_bc+E_ca=\delta_b^\leftarrow+\delta_b^\leftarrow+% \delta_a^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c italic_a end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (5) The cycle appears in the measurements involving the definition of discord. We have a measurement in part a𝑎aitalic_a referring to b𝑏bitalic_b, then one in b𝑏bitalic_b to c𝑐citalic_c following by one in c𝑐citalic_c to a𝑎aitalic_a closing the cycle. Although Eq. (5) is already known, we are going to look at it with a new point of view, which is particularly more interesting for the generalizations gotten in this work. The right side is the sum of EFs, so it is the sum of quantum communications need to simulated the correlation present in each of one of bipartitions. The left side is the sum of discords in each bipartition, so the left side is the sum of in inaccessible information of each bipartition. Therefore, the sum of quantum communication needed in each bipartition is equal to the sum of information that is trapped in non-local correlations.

Moreover, Eq. (5) can be written in the opposite direction, since it is not different from a permutation of the parts. Therefore, as the entanglements are symmetric quantities, we can write a conservation law for discords only Fanchini et al. (2012)

δa|b←+δb|c←+δc|a←=δb|a←+δc|b←+δa|c←.superscriptsubscript𝛿conditional𝑎𝑏←superscriptsubscript𝛿conditional𝑏𝑐←superscriptsubscript𝛿conditional𝑐𝑎←superscriptsubscript𝛿conditional𝑏𝑎←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑎𝑐←\delta_b^\leftarrow+\delta_b^\leftarrow+\delta_c^\leftarrow=% \delta_b^\leftarrow+\delta_c^\leftarrow+\delta_a^\leftarrow.italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT = italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (6) Eq. (6) shows that the sum of locally inaccessible information is the same in the two possible directions of the cycle. The Eqs. (5) and (6) are generalized to multipartite systems in section IV.

III Entanglement and discord in four and five-partite systems

Given this brief overview, we are now in position to discuss the generalization of our previous results Fanchini et al. (2011, 2012). We start with four-partite system, however straight forward generalization gives inequalities instead of equalities found in three-partite systems, Eqs. (4), (5) and (6). These inequalities relates entanglement, discord and the strong subadditivity inequality which is interesting also, but they do not result in new conservation laws. Nevertheless, generalization to five-partite systems do result in equalities and new conservation laws. These results shed light on how to generalized conservations laws for arbitrary dimensions. As we discuss in the following, some difficult arises because there is a difference between even and odd number of parts. The generalizations are discussed in section IV.

III.1 Four-partite systems

III.1.1 Inequalities with a central particle

From three-partite systems to four-partite, one could expect to find equalities between sums of EF and QD. Nevertheless, what we found are inequalities which are closed related to the strong subadditivity of von Neumann entropy. The difficulty arises in dividing properly the parts of the system, since there is more ways to do it in four than in three partite systems. A first attempt, from Eq. (3), is to write the following equations,

Ea|bcsubscript𝐸conditional𝑎𝑏𝑐\displaystyle E_bcitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δa|d←+Sa|d,superscriptsubscript𝛿conditional𝑎𝑑←subscript𝑆conditional𝑎𝑑\displaystyle\delta_a^\leftarrow+S_d,italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT ,

Ea|cdsubscript𝐸conditional𝑎𝑐𝑑\displaystyle E_cditalic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δa|b←+Sa|b.superscriptsubscript𝛿conditional𝑎𝑏←subscript𝑆conditional𝑎𝑏\displaystyle\delta_b^\leftarrow+S_a.italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT . (7) Combining Eqs. (7), we get

Ea|bc+Ea|cd=δa|d←+δa|b←+Sa|d+Sa|b.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑎𝑐𝑑superscriptsubscript𝛿conditional𝑎𝑑←superscriptsubscript𝛿conditional𝑎𝑏←subscript𝑆conditional𝑎𝑑subscript𝑆conditional𝑎𝑏E_a+E_a=\delta_d^\leftarrow+\delta_b^\leftarrow+S_d+% S_b.italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT . (8) The sum of conditional entropies can be rewritten as a strong subadditivity inequality,

Sa|d+Sa|b=Sab-Sb+Sbc-Sabc≥0.subscript𝑆conditional𝑎𝑑subscript𝑆conditional𝑎𝑏subscript𝑆𝑎𝑏subscript𝑆𝑏subscript𝑆𝑏𝑐subscript𝑆𝑎𝑏𝑐0S_a+S_a=S_ab-S_b+S_bc-S_abc\geq 0.italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT = italic_S start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_a italic_b italic_c end_POSTSUBSCRIPT ≥ 0 . Therefore, the sum of conditional entropies does not cancel. Nevertheless, we get the following inequality for the distribution of entanglement and discord in four partite systems

Ea|bc+Ea|cd≥δa|d←+δa|b←.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑎𝑐𝑑superscriptsubscript𝛿conditional𝑎𝑑←superscriptsubscript𝛿conditional𝑎𝑏←E_bc+E_a\geq\delta_a^\leftarrow+\delta_a^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT ≥ italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (9)

Moreover, instead of Eq. (7), we can write the fundamental relations in the following form

Eadsubscript𝐸𝑎𝑑\displaystyle E_aditalic_E start_POSTSUBSCRIPT italic_a italic_d end_POSTSUBSCRIPT =\displaystyle== δa|bc←+Sa|bc.superscriptsubscript𝛿conditional𝑎𝑏𝑐←subscript𝑆conditional𝑎𝑏𝑐\displaystyle\delta_a^\leftarrow+S_bc.italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT . Therefore, with a similar reasoning, we obtain the complementary inequality

Eab+Ead≤δa|bc←+δa|cd←.subscript𝐸𝑎𝑏subscript𝐸𝑎𝑑superscriptsubscript𝛿conditional𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑎𝑐𝑑←E_ab+E_ad\leq\delta_bc^\leftarrow+\delta_cd^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_d end_POSTSUBSCRIPT ≤ italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (10) In Eqs. (9) and (10), we notice that particle a𝑎aitalic_a is present in all quantities being the central one of the relation. We also notice the change in particle c𝑐citalic_c. In Eq. (9), it is in the entanglements while, in Eq. (10), it is in the discords. That is, part c𝑐citalic_c is always in the greater side of inequality which is, in fact, reasonable, once that tracing out a quantum system always decreases entanglement and, most of times, also discord.

In addition, it is possible to obtain inequalities involving all possible entanglements with the part a𝑎aitalic_a. This can be done starting from the following fundamental relations

Ea|bcsubscript𝐸conditional𝑎𝑏𝑐\displaystyle E_aitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δa|d+Sa|d,subscript𝛿conditional𝑎𝑑subscript𝑆conditional𝑎𝑑\displaystyle\delta_d+S_a,italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT ,

Ea|bdsubscript𝐸conditional𝑎𝑏𝑑\displaystyle E_aitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_d end_POSTSUBSCRIPT =\displaystyle== δa|c+Sa|c,subscript𝛿conditional𝑎𝑐subscript𝑆conditional𝑎𝑐\displaystyle\delta_c+S_a,italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT ,

Ea|cdsubscript𝐸conditional𝑎𝑐𝑑\displaystyle E_aitalic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δa|b+Sa|b.subscript𝛿conditional𝑎𝑏subscript𝑆conditional𝑎𝑏\displaystyle\delta_b+S_a.italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT . (11) Combining Eqs. (11), we obtain

Ea|bc+Ea|cd+Ea|db=δa|b←+δa|c←+δa|d←+Sa|b+Sa|c+Sa|d.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑎𝑐𝑑subscript𝐸conditional𝑎𝑑𝑏superscriptsubscript𝛿conditional𝑎𝑏←superscriptsubscript𝛿conditional𝑎𝑐←superscriptsubscript𝛿conditional𝑎𝑑←subscript𝑆conditional𝑎𝑏subscript𝑆conditional𝑎𝑐subscript𝑆conditional𝑎𝑑E_bc+E_a+E_db\\ =\delta_a^\leftarrow+\delta_a^\leftarrow+\delta_d^\leftarrow% +S_b+S_a+S_d.start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_d italic_b end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT . end_CELL end_ROW (12) The sum of conditional entropies is always positive, since it is a combination of three strong subadditivity inequalities Nielsen and Chuang (2000). Therefore we have

Ea|bc+Ea|cd+Ea|db≥δa|b←+δa|c←+δa|d←.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑎𝑐𝑑subscript𝐸conditional𝑎𝑑𝑏superscriptsubscript𝛿conditional𝑎𝑏←superscriptsubscript𝛿conditional𝑎𝑐←superscriptsubscript𝛿conditional𝑎𝑑←E_bc+E_cd+E_db\geq\delta_a^\leftarrow+\delta_a^% \leftarrow+\delta_a^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_d italic_b end_POSTSUBSCRIPT ≥ italic_δ start_POSTSUBSCRIPT italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (13)

With a similar reasoning to the Eq. (10), we also have the complementary inequality

Eab+Eac+Ead≤δa|bc←+δa|cd←+δa|cd←.subscript𝐸𝑎𝑏subscript𝐸𝑎𝑐subscript𝐸𝑎𝑑superscriptsubscript𝛿conditional𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑎𝑐𝑑←superscriptsubscript𝛿conditional𝑎𝑐𝑑←E_ab+E_ac+E_ad\leq\delta_a^\leftarrow+\delta_a^\leftarrow+% \delta_cd^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_d end_POSTSUBSCRIPT ≤ italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (14)

These results show how distinct inequalities between EF and QD emerge when treating one of the particles as a central one, similarly to Eq. (4). In the following, we treat similar aspects, but now cycling the parts in discord side, similarly to Eq. (5).

III.1.2 Cycling inequalities

All the relations (9, 10, 13, 14) involves a central particle that is present in all quantities (particle a𝑎aitalic_a). So These inequalities are related to the generalization of the Eq. (4). Here we derive another type of inequalities which involve cycling the parts in discord side and is related to the generalization of the Eq. (5). This forms a cycle of inaccessible local information as is discussed in Ref. Fanchini et al. (2012), but now with four parts. First we consider the cycle a→b→𝑎𝑏a\rightarrow bitalic_a → italic_b, b→c→𝑏𝑐b\rightarrow citalic_b → italic_c, c→d→𝑐𝑑c\rightarrow ditalic_c → italic_d and d→a→𝑑𝑎d\rightarrow aitalic_d → italic_a of local inaccessible information or discords, writing down the following fundamental equations (3),

Eb|cdsubscript𝐸conditional𝑏𝑐𝑑\displaystyle E_cditalic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δb|a←+Sb|a,superscriptsubscript𝛿conditional𝑏𝑎←subscript𝑆conditional𝑏𝑎\displaystyle\delta_b^\leftarrow+S_b,italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT ,

Ec|dasubscript𝐸conditional𝑐𝑑𝑎\displaystyle E_citalic_E start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT =\displaystyle== δc|b←+Sc|b,superscriptsubscript𝛿conditional𝑐𝑏←subscript𝑆conditional𝑐𝑏\displaystyle\delta_c^\leftarrow+S_b,italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT ,

Ed|absubscript𝐸conditional𝑑𝑎𝑏\displaystyle E_abitalic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT =\displaystyle== δd|c←+Sd|c,superscriptsubscript𝛿conditional𝑑𝑐←subscript𝑆conditional𝑑𝑐\displaystyle\delta_c^\leftarrow+S_c,italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT ,

Ea|bcsubscript𝐸conditional𝑎𝑏𝑐\displaystyle E_aitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δa|d←+Sa|d.superscriptsubscript𝛿conditional𝑎𝑑←subscript𝑆conditional𝑎𝑑\displaystyle\delta_d^\leftarrow+S_d.italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT . (15) Combining Eqs. (15), we get

Ea|bc+Eb|cd+Ec|da+Ed|ab=δa|d←+δd|c←+δc|b←+δb|a←+Sa|d+Sb|a+Sc|b+Sd|c.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑏𝑐𝑑subscript𝐸conditional𝑐𝑑𝑎subscript𝐸conditional𝑑𝑎𝑏superscriptsubscript𝛿conditional𝑎𝑑←superscriptsubscript𝛿conditional𝑑𝑐←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑏𝑎←subscript𝑆conditional𝑎𝑑subscript𝑆conditional𝑏𝑎subscript𝑆conditional𝑐𝑏subscript𝑆conditional𝑑𝑐E_bc+E_b+E_da+E_d\\ =\delta_a^\leftarrow+\delta_c^\leftarrow+\delta_c^\leftarrow% +\delta_b^\leftarrow+S_a+S_b+S_c+S_d.start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT . end_CELL end_ROW The sum of conditionals entropies is always positive, due to the strong subadditivity inequality. This can be seen after some manipulation of the entropies,

Sa|d+Sb|a+Sc|b+Sd|c=(Sab+Sbc-Sb-Sabd)+(Sbc+Scd-Sbcd-Sc)≥0.subscript𝑆conditional𝑎𝑑subscript𝑆conditional𝑏𝑎subscript𝑆conditional𝑐𝑏subscript𝑆conditional𝑑𝑐subscript𝑆𝑎𝑏subscript𝑆𝑏𝑐subscript𝑆𝑏subscript𝑆𝑎𝑏𝑑subscript𝑆𝑏𝑐subscript𝑆𝑐𝑑subscript𝑆𝑏𝑐𝑑subscript𝑆𝑐0S_a+S_b+S_b+S_c\\ =(S_ab+S_bc-S_b-S_abd)+(S_bc+S_cd-S_bcd-S_c)\geq 0.start_ROW start_CELL italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = ( italic_S start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_b end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_a italic_b italic_d end_POSTSUBSCRIPT ) + ( italic_S start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_c italic_d end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_b italic_c italic_d end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ) ≥ 0 . end_CELL end_ROW Therefore, we are left with the following relation between entanglements and discords,

Ea|bc+Eb|cd+Ec|da+Ed|ab≥δa|d←+δd|c←+δc|b←+δb|a←.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑏𝑐𝑑subscript𝐸conditional𝑐𝑑𝑎subscript𝐸conditional𝑑𝑎𝑏superscriptsubscript𝛿conditional𝑎𝑑←superscriptsubscript𝛿conditional𝑑𝑐←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑏𝑎←E_a+E_cd+E_da+E_ab\geq\delta_d^\leftarrow+\delta_c^% \leftarrow+\delta_b^\leftarrow+\delta_b^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT ≥ italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (16)

We can also construct a cycle of LII using a different setup of parts. Let the cycle ab→c,→𝑎𝑏𝑐ab\rightarrow c,italic_a italic_b → italic_c , cb→d→𝑐𝑏𝑑cb\rightarrow ditalic_c italic_b → italic_d, cd→a→𝑐𝑑𝑎cd\rightarrow aitalic_c italic_d → italic_a and da→b→𝑑𝑎𝑏da\rightarrow bitalic_d italic_a → italic_b. This allows us to write the following relations between entanglements and discords from Eq. (3),

Ecdsubscript𝐸𝑐𝑑\displaystyle E_cditalic_E start_POSTSUBSCRIPT italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δc|ab←+Sc|ab,superscriptsubscript𝛿conditional𝑐𝑎𝑏←subscript𝑆conditional𝑐𝑎𝑏\displaystyle\delta_c^\leftarrow+S_ab,italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_a italic_b end_POSTSUBSCRIPT ,

Edasubscript𝐸𝑑𝑎\displaystyle E_daitalic_E start_POSTSUBSCRIPT italic_d italic_a end_POSTSUBSCRIPT =\displaystyle== δd|bc←+Sd|bc,superscriptsubscript𝛿conditional𝑑𝑏𝑐←subscript𝑆conditional𝑑𝑏𝑐\displaystyle\delta_bc^\leftarrow+S_d,italic_δ start_POSTSUBSCRIPT italic_d | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_b italic_c end_POSTSUBSCRIPT ,

Eabsubscript𝐸𝑎𝑏\displaystyle E_abitalic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT =\displaystyle== δa|cd←+Sa|cd,superscriptsubscript𝛿conditional𝑎𝑐𝑑←subscript𝑆conditional𝑎𝑐𝑑\displaystyle\delta_cd^\leftarrow+S_cd,italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT ,

Ebcsubscript𝐸𝑏𝑐\displaystyle E_bcitalic_E start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δb|ad←+Sb|ad.superscriptsubscript𝛿conditional𝑏𝑎𝑑←subscript𝑆conditional𝑏𝑎𝑑\displaystyle\delta_b^\leftarrow+S_ad.italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_a italic_d end_POSTSUBSCRIPT . (17) With a similar manipulation, the sum of conditional entropies in Eqs. (17) can be shown to be always positive due to the strong subadditivity inequality. Therefore, we are left with the following inequality between entanglements and discords for pure four-partite systems

Eab+Ebc+Ecd+Eda≤δa|cd←+δb|da←+δc|ab←+δd|bc←.subscript𝐸𝑎𝑏subscript𝐸𝑏𝑐subscript𝐸𝑐𝑑subscript𝐸𝑑𝑎superscriptsubscript𝛿conditional𝑎𝑐𝑑←superscriptsubscript𝛿conditional𝑏𝑑𝑎←superscriptsubscript𝛿conditional𝑐𝑎𝑏←superscriptsubscript𝛿conditional𝑑𝑏𝑐←E_ab+E_bc+E_cd+E_da\leq\delta_cd^\leftarrow+\delta_da^% \leftarrow+\delta_ab^\leftarrow+\delta_bc^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d italic_a end_POSTSUBSCRIPT ≤ italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_d italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (18)

Eqs. (17) and (18) above finalize our results considering fourpartite systems and cycling relations.

III.2 Five-partite systems

For pure five-partite systems, we return to find equalities instead of inequalities in a very similarly to what happens in the three-partite case,

III.2.1 Equalities with a central particle

From conservation law Eq. (4) we can write the following three equalities for five partite systems

Ea|bc+Ea|desubscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑎𝑑𝑒\displaystyle E_a+E_aitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_d italic_e end_POSTSUBSCRIPT =\displaystyle== δa|bc←+δa|de←,superscriptsubscript𝛿conditional𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑎𝑑𝑒←\displaystyle\delta_a^\leftarrow+\delta_de^\leftarrow,italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ,

Ea|bd+Ea|cesubscript𝐸conditional𝑎𝑏𝑑subscript𝐸conditional𝑎𝑐𝑒\displaystyle E_a+E_aitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_e end_POSTSUBSCRIPT =\displaystyle== δa|bd←+δa|ce←,superscriptsubscript𝛿conditional𝑎𝑏𝑑←superscriptsubscript𝛿conditional𝑎𝑐𝑒←\displaystyle\delta_a^\leftarrow+\delta_a^\leftarrow,italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ,

Ea|be+Ea|cdsubscript𝐸conditional𝑎𝑏𝑒subscript𝐸conditional𝑎𝑐𝑑\displaystyle E_a+E_cditalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_e end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δa|be←+δa|cd←.superscriptsubscript𝛿conditional𝑎𝑏𝑒←superscriptsubscript𝛿conditional𝑎𝑐𝑑←\displaystyle\delta_a^\leftarrow+\delta_a^\leftarrow.italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (19) So, we can write down the equality

Ea|bc+Ea|de+Ea|bd+Ea|ce+Ea|be+Ea|cd=δa|de←+δa|bc←+δa|ce←+δa|bd←+δa|cd←+δa|be←subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑎𝑑𝑒subscript𝐸conditional𝑎𝑏𝑑subscript𝐸conditional𝑎𝑐𝑒subscript𝐸conditional𝑎𝑏𝑒subscript𝐸conditional𝑎𝑐𝑑superscriptsubscript𝛿conditional𝑎𝑑𝑒←superscriptsubscript𝛿conditional𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑎𝑐𝑒←superscriptsubscript𝛿conditional𝑎𝑏𝑑←superscriptsubscript𝛿conditional𝑎𝑐𝑑←superscriptsubscript𝛿conditional𝑎𝑏𝑒←E_a+E_a+E_bd+E_a+E_be+E_a\\ =\delta_a^\leftarrow+\delta_bc^\leftarrow+\delta_a^% \leftarrow+\delta_bd^\leftarrow+\delta_a^\leftarrow+\delta_be% ^\leftarrowstart_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_d italic_e end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_e end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_e end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT italic_a | italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW (20) In Eq. (20) one can check that we have all combinations of entanglements and discords of the particle a𝑎aitalic_a with all other possible combinations of two particles. Therefore, similarly what happens in three-partite case, Eq. (4), the sum of entanglements a central particle, a𝑎aitalic_a, shares with all the others possible combinations of two particles is equal to the sum of all discords between the same bipartitions, establishing a monogamy-like conservation law of quantum correlations.

Despite this fact, it is also evident that Eq. (20) is a weaker statement than the three Eqs. (19). In this way, although it is possible to derive generalizations like Eq. (20), they follow from straight forward combinations of the three partite conservation law Eq. (4) with the appropriate combination of subsystems, Eqs. (19). Nevertheless, as we show bellow, nontrivial results emerges when considering cycling equalities in five-partite systems.

III.2.2 A cycling equality

More interesting are the equality that arises when we generalize Eq. (5) for five-partite systems, since they are not direct application of the similar conservation law from system of small number of parts, but really new conservation laws. From the fundamental Eq. (3), we can write the following equations,

Ea|bcsubscript𝐸conditional𝑎𝑏𝑐\displaystyle E_bcitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δa|ed←+Saed-Sed,superscriptsubscript𝛿conditional𝑎𝑒𝑑←subscript𝑆𝑎𝑒𝑑subscript𝑆𝑒𝑑\displaystyle\delta_ed^\leftarrow+S_aed-S_ed,italic_δ start_POSTSUBSCRIPT italic_a | italic_e italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a italic_e italic_d end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_e italic_d end_POSTSUBSCRIPT , (21)

Ec|desubscript𝐸conditional𝑐𝑑𝑒\displaystyle E_citalic_E start_POSTSUBSCRIPT italic_c | italic_d italic_e end_POSTSUBSCRIPT =\displaystyle== δc|ba←+Scba-Sba,superscriptsubscript𝛿conditional𝑐𝑏𝑎←subscript𝑆𝑐𝑏𝑎subscript𝑆𝑏𝑎\displaystyle\delta_c^\leftarrow+S_cba-S_ba,italic_δ start_POSTSUBSCRIPT italic_c | italic_b italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c italic_b italic_a end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_b italic_a end_POSTSUBSCRIPT , (22)

Ee|absubscript𝐸conditional𝑒𝑎𝑏\displaystyle E_eitalic_E start_POSTSUBSCRIPT italic_e | italic_a italic_b end_POSTSUBSCRIPT =\displaystyle== δe|dc←+Sedc-Sdc.superscriptsubscript𝛿conditional𝑒𝑑𝑐←subscript𝑆𝑒𝑑𝑐subscript𝑆𝑑𝑐\displaystyle\delta_dc^\leftarrow+S_edc-S_dc.italic_δ start_POSTSUBSCRIPT italic_e | italic_d italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_e italic_d italic_c end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_d italic_c end_POSTSUBSCRIPT . (23)

Eb|cdsubscript𝐸conditional𝑏𝑐𝑑\displaystyle E_cditalic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δb|ae←+Sbae-Sae,superscriptsubscript𝛿conditional𝑏𝑎𝑒←subscript𝑆𝑏𝑎𝑒subscript𝑆𝑎𝑒\displaystyle\delta_b^\leftarrow+S_bae-S_ae,italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b italic_a italic_e end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_a italic_e end_POSTSUBSCRIPT , (24)

Ed|easubscript𝐸conditional𝑑𝑒𝑎\displaystyle E_eaitalic_E start_POSTSUBSCRIPT italic_d | italic_e italic_a end_POSTSUBSCRIPT =\displaystyle== δd|cb←+Sdcd-Scb,superscriptsubscript𝛿conditional𝑑𝑐𝑏←subscript𝑆𝑑𝑐𝑑subscript𝑆𝑐𝑏\displaystyle\delta_d^\leftarrow+S_dcd-S_cb,italic_δ start_POSTSUBSCRIPT italic_d | italic_c italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d italic_c italic_d end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_c italic_b end_POSTSUBSCRIPT , (25) Notice that the entropy Sdesubscript𝑆𝑑𝑒S_deitalic_S start_POSTSUBSCRIPT italic_d italic_e end_POSTSUBSCRIPT from the first equation cancels with Scbasubscript𝑆𝑐𝑏𝑎S_cbaitalic_S start_POSTSUBSCRIPT italic_c italic_b italic_a end_POSTSUBSCRIPT from the second one; the entropy Sbasubscript𝑆𝑏𝑎S_baitalic_S start_POSTSUBSCRIPT italic_b italic_a end_POSTSUBSCRIPT from the second equation cancels with Secdsubscript𝑆𝑒𝑐𝑑S_ecditalic_S start_POSTSUBSCRIPT italic_e italic_c italic_d end_POSTSUBSCRIPT from third one and so on. When we sum all of the equations, the cycle closes and all the entropies cancel out. The result is the following equality

Ea|bc+Eb|cd+Ec|de+Ed|ea+Ee|ab=δa|de←+δb|ea←+δc|ab←+δd|bc←+δe|cd←.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑏𝑐𝑑subscript𝐸conditional𝑐𝑑𝑒subscript𝐸conditional𝑑𝑒𝑎subscript𝐸conditional𝑒𝑎𝑏superscriptsubscript𝛿conditional𝑎𝑑𝑒←superscriptsubscript𝛿conditional𝑏𝑒𝑎←superscriptsubscript𝛿conditional𝑐𝑎𝑏←superscriptsubscript𝛿conditional𝑑𝑏𝑐←superscriptsubscript𝛿conditional𝑒𝑐𝑑←E_a+E_b+E_de+E_d+E_ab\\ =\delta_a^\leftarrow+\delta_b^\leftarrow+\delta_c^% \leftarrow+\delta_d^\leftarrow+\delta_cd^\leftarrow.start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_e end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d | italic_e italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_e | italic_a italic_b end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT italic_a | italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_e italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_e | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . end_CELL end_ROW (26) Eq. (26) contains a cycle of LII information in the right side similarly what happens in the three-partite case Eq. (5). As we show in section IV, these results gives the direction for a generalization to the multipartite case.

IV Generalized Conservation Laws for multipartite systems

Now, with the particular results presented in the section III, we extend our results for multipartite systems.

IV.1 Generalized cycling conservation laws

The idea behind the deduction of Eq. (26) is the key idea for generalizing it to multipartite systems. Let us consider a system composed by N𝑁Nitalic_N parts, where N𝑁Nitalic_N is odd, and let n=(N-1)/2𝑛𝑁12n= icefrac(N-1)2italic_n = / start_ARG ( italic_N - 1 ) end_ARG start_ARG 2 end_ARG. So we can write down the following equations

E1|2,3,…,n+1=δ1|N,N-1,…,n+2←+S1,N,N-1,…,n+2-SN,N-1,…,n+2,En+1|n+2,n+3,…,N=δn+1|n,n-1,…,1←+Sn+1,n,…,1-Sn,n-1,…,1,EN|1,2,…,n=δN|N-1,N-2,…,n+1←+SN,N-1,…,n+1-SN-1,N-2,…,n+1,En|n+1,n+2,…,N-1=δn|n-1,n-2,…,N←+Sn,n-1,…,N-Sn-1,n-2,…,1,N,EN-1|N,1,…,n-1=δN-1|N-2,N-3,…,n←+SN-1,N-2,…,n-SN-2,N-3,…,n,⋮=⋮En+2|n+3,n+4,…,1=δn+2|n+1,n,…,2←+Sn+2,n+1,n,…,2-Sn+1,n,…,2,subscript𝐸conditional123…𝑛1superscriptsubscript𝛿conditional1𝑁𝑁1…𝑛2←subscript𝑆1𝑁𝑁1…𝑛2subscript𝑆𝑁𝑁1…𝑛2subscript𝐸𝑛conditional1𝑛2𝑛3…𝑁superscriptsubscript𝛿𝑛conditional1𝑛𝑛1…1←subscript𝑆𝑛1𝑛…1subscript𝑆𝑛𝑛1…1subscript𝐸conditional𝑁12…𝑛superscriptsubscript𝛿conditional𝑁𝑁1𝑁2…𝑛1←subscript𝑆𝑁𝑁1…𝑛1subscript𝑆𝑁1𝑁2…𝑛1subscript𝐸conditional𝑛𝑛1𝑛2…𝑁1superscriptsubscript𝛿conditional𝑛𝑛1𝑛2…𝑁←subscript𝑆𝑛𝑛1…𝑁subscript𝑆𝑛1𝑛2…1𝑁subscript𝐸𝑁conditional1𝑁1…𝑛1superscriptsubscript𝛿𝑁conditional1𝑁2𝑁3…𝑛←subscript𝑆𝑁1𝑁2…𝑛subscript𝑆𝑁2𝑁3…𝑛⋮⋮missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionsubscript𝐸𝑛conditional2𝑛3𝑛4…1superscriptsubscript𝛿𝑛conditional2𝑛1𝑛…2←subscript𝑆𝑛2𝑛1𝑛…2subscript𝑆𝑛1𝑛…2\beginarray[]rclllllE_1&=&\delta_1^% \leftarrow&+&S_1,N,N-1,\dots,n+2&-&S_N,N-1,\dots,n+2,\\ E_n+2,n+3,\dots,N&=&\delta_n,n-1,\dots,1^\leftarrow&+&S_n+1,n,% \dots,1&-&S_n,n-1,\dots,1,\\ E_N&=&\delta_N-1,N-2,\dots,n+1^\leftarrow&+&S_N,N-1,\dots% ,n+1&-&S_N-1,N-2,\dots,n+1,\\ E_n&=&\delta_n^\leftarrow&+&S_n,n-1,% \dots,N&-&S_n-1,n-2,\dots,1,N,\\ E_N,1,\dots,n-1&=&\delta_N-2,N-3,\dots,n^\leftarrow&+&S_N-1,N-2% ,\dots,n&-&S_N-2,N-3,\dots,n,\\ \vdots&=&\hskip 28.45274pt\vdots\\ E_n+3,n+4,\dots,1&=&\delta_n+1,n,\dots,2^\leftarrow&+&S_n+2,n+1% ,n,\dots,2&-&S_n+1,n,\dots,2,\endarraystart_ARRAY start_ROW start_CELL italic_E start_POSTSUBSCRIPT 1 | 2 , 3 , … , italic_n + 1 end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL italic_δ start_POSTSUBSCRIPT 1 | italic_N , italic_N - 1 , … , italic_n + 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL start_CELL + end_CELL start_CELL italic_S start_POSTSUBSCRIPT 1 , italic_N , italic_N - 1 , … , italic_n + 2 end_POSTSUBSCRIPT end_CELL start_CELL - end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_N , italic_N - 1 , … , italic_n + 2 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_n + 1 | italic_n + 2 , italic_n + 3 , … , italic_N end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL italic_δ start_POSTSUBSCRIPT italic_n + 1 | italic_n , italic_n - 1 , … , 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL start_CELL + end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_n + 1 , italic_n , … , 1 end_POSTSUBSCRIPT end_CELL start_CELL - end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_n , italic_n - 1 , … , 1 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_N | 1 , 2 , … , italic_n end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL italic_δ start_POSTSUBSCRIPT italic_N | italic_N - 1 , italic_N - 2 , … , italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL start_CELL + end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_N , italic_N - 1 , … , italic_n + 1 end_POSTSUBSCRIPT end_CELL start_CELL - end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_N - 1 , italic_N - 2 , … , italic_n + 1 end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_n | italic_n + 1 , italic_n + 2 , … , italic_N - 1 end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL italic_δ start_POSTSUBSCRIPT italic_n | italic_n - 1 , italic_n - 2 , … , italic_N end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL start_CELL + end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_n , italic_n - 1 , … , italic_N end_POSTSUBSCRIPT end_CELL start_CELL - end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_n - 1 , italic_n - 2 , … , 1 , italic_N end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_N - 1 | italic_N , 1 , … , italic_n - 1 end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL italic_δ start_POSTSUBSCRIPT italic_N - 1 | italic_N - 2 , italic_N - 3 , … , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL start_CELL + end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_N - 1 , italic_N - 2 , … , italic_n end_POSTSUBSCRIPT end_CELL start_CELL - end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_N - 2 , italic_N - 3 , … , italic_n end_POSTSUBSCRIPT , end_CELL end_ROW start_ROW start_CELL ⋮ end_CELL start_CELL = end_CELL start_CELL ⋮ end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_n + 2 | italic_n + 3 , italic_n + 4 , … , 1 end_POSTSUBSCRIPT end_CELL start_CELL = end_CELL start_CELL italic_δ start_POSTSUBSCRIPT italic_n + 2 | italic_n + 1 , italic_n , … , 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL start_CELL + end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_n + 2 , italic_n + 1 , italic_n , … , 2 end_POSTSUBSCRIPT end_CELL start_CELL - end_CELL start_CELL italic_S start_POSTSUBSCRIPT italic_n + 1 , italic_n , … , 2 end_POSTSUBSCRIPT , end_CELL end_ROW end_ARRAY (27) In the Eqs. (27), the next equations is always based on n𝑛nitalic_n subsystems at right from the previous one. We can check that, when the equations are summed up, the second entropy cancels with the first one from next equation until the cycle is closed. The result is the following conservation law

E1|2,3,…,n+1+E2|3,4,…,n+2+⋯+EN|1,2,…,n=δ1|N,N-1,…,n+2←+δ2|1,N,N-1,…,n+3←+⋯+δN|N-1,N-2,…,n+1←.subscript𝐸conditional123…𝑛1subscript𝐸conditional234…𝑛2⋯subscript𝐸conditional𝑁12…𝑛superscriptsubscript𝛿conditional1𝑁𝑁1…𝑛2←superscriptsubscript𝛿conditional21𝑁𝑁1…𝑛3←⋯superscriptsubscript𝛿conditional𝑁𝑁1𝑁2…𝑛1←E_2,3,\dots,n+1+E_2+\cdots+E_1,2,\dots,n=\delta_N,N% -1,\dots,n+2^\leftarrow+\delta_2^\leftarrow+\cdots+% \delta_N-1,N-2,\dots,n+1^\leftarrow.start_ROW start_CELL italic_E start_POSTSUBSCRIPT 1 | 2 , 3 , … , italic_n + 1 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 2 | 3 , 4 , … , italic_n + 2 end_POSTSUBSCRIPT + ⋯ + italic_E start_POSTSUBSCRIPT italic_N | 1 , 2 , … , italic_n end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT 1 | italic_N , italic_N - 1 , … , italic_n + 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 2 | 1 , italic_N , italic_N - 1 , … , italic_n + 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + ⋯ + italic_δ start_POSTSUBSCRIPT italic_N | italic_N - 1 , italic_N - 2 , … , italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . end_CELL end_ROW (28) Eq. (28) is the generalization of Eq. (5). It shows how the amount of quantum communication needed in each bipartition sums up equal to the sum of information trapped in nonlocal correlations as measured by quantum discord. In this way, in order of obtaining an equality for these two quantities, Eq. (28) shows that we must organize the sum of entanglements forming a cycle including all the bipartition in one direction and the sum of discords in the opposite direction, when the number of subsystems is odd.

At first sight, it appears that the idea behind derivation of the conservation law (28) only works for systems with an odd number of parts, since only in these cases the entropies have the right numbers of parts to cancel each other. Nevertheless, it is still possible to make them cancel each other varying the number of subsystems. Let us come back to the four-partite case, for instance. Then we write down the following sequence of Equations, alternating the number of parts in entanglements and discords,

Ea|bcsubscript𝐸conditional𝑎𝑏𝑐\displaystyle E_aitalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δa|d←+Sad-Sd,superscriptsubscript𝛿conditional𝑎𝑑←subscript𝑆𝑎𝑑subscript𝑆𝑑\displaystyle\delta_d^\leftarrow+S_ad-S_d,italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a italic_d end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_d end_POSTSUBSCRIPT ,

Ec|dsubscript𝐸conditional𝑐𝑑\displaystyle E_citalic_E start_POSTSUBSCRIPT italic_c | italic_d end_POSTSUBSCRIPT =\displaystyle== δc|ba←+Scba-Sba,superscriptsubscript𝛿conditional𝑐𝑏𝑎←subscript𝑆𝑐𝑏𝑎subscript𝑆𝑏𝑎\displaystyle\delta_c^\leftarrow+S_cba-S_ba,italic_δ start_POSTSUBSCRIPT italic_c | italic_b italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c italic_b italic_a end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_b italic_a end_POSTSUBSCRIPT ,

Ed|absubscript𝐸conditional𝑑𝑎𝑏\displaystyle E_ditalic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT =\displaystyle== δd|c←+Sdc-Sc,superscriptsubscript𝛿conditional𝑑𝑐←subscript𝑆𝑑𝑐subscript𝑆𝑐\displaystyle\delta_c^\leftarrow+S_dc-S_c,italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d italic_c end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_c end_POSTSUBSCRIPT ,

Eb|csubscript𝐸conditional𝑏𝑐\displaystyle E_citalic_E start_POSTSUBSCRIPT italic_b | italic_c end_POSTSUBSCRIPT =\displaystyle== δb|ad←+Sbad-Sad,superscriptsubscript𝛿conditional𝑏𝑎𝑑←subscript𝑆𝑏𝑎𝑑subscript𝑆𝑎𝑑\displaystyle\delta_ad^\leftarrow+S_bad-S_ad,italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b italic_a italic_d end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_a italic_d end_POSTSUBSCRIPT , One can check that again the second entropy cancels with the first entropy of the next equations, and the cycle closes at the last one. Summing up these equations results

Ea|bc+Ec|d+Ed|ab+Eb|c=δa|d←+δc|ba←+δd|c←+δb|ad←.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑐𝑑subscript𝐸conditional𝑑𝑎𝑏subscript𝐸conditional𝑏𝑐superscriptsubscript𝛿conditional𝑎𝑑←superscriptsubscript𝛿conditional𝑐𝑏𝑎←superscriptsubscript𝛿conditional𝑑𝑐←superscriptsubscript𝛿conditional𝑏𝑎𝑑←E_a+E_d+E_ab+E_b=\delta_a^\leftarrow+\delta_c^% \leftarrow+\delta_c^\leftarrow+\delta_ad^\leftarrow.italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c | italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b | italic_c end_POSTSUBSCRIPT = italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . (29) This equation is a conservation law for entanglement and discord for four-partite systems where the number of parts changes. Therefore we see that it is possible to derive conservation laws for multipartite systems composed of an even number of parts, but the number of parts in the bipartitions must vary accordingly to Eq. (29).

Nonetheless, the Eq. (29) does not show the full general rule for deriving a general conservation law for even number of parts. This appears when we write down the equations for a six-partite system,

E1|23subscript𝐸conditional123\displaystyle E_23italic_E start_POSTSUBSCRIPT 1 | 23 end_POSTSUBSCRIPT =δ1|654←+S1654-S654,absentsuperscriptsubscript𝛿conditional1654←subscript𝑆1654subscript𝑆654\displaystyle=\delta_654^\leftarrow+S_1654-S_654,= italic_δ start_POSTSUBSCRIPT 1 | 654 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 1654 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 654 end_POSTSUBSCRIPT , E4|56subscript𝐸conditional456\displaystyle E_56italic_E start_POSTSUBSCRIPT 4 | 56 end_POSTSUBSCRIPT =δ4|321←+S4321-S321,absentsuperscriptsubscript𝛿conditional4321←subscript𝑆4321subscript𝑆321\displaystyle=\delta_321^\leftarrow+S_4321-S_321,= italic_δ start_POSTSUBSCRIPT 4 | 321 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 4321 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 321 end_POSTSUBSCRIPT ,

E3|456subscript𝐸conditional3456\displaystyle E_3italic_E start_POSTSUBSCRIPT 3 | 456 end_POSTSUBSCRIPT =δ3|21←+S321-S21,absentsuperscriptsubscript𝛿conditional321←subscript𝑆321subscript𝑆21\displaystyle=\delta_21^\leftarrow+S_321-S_21,= italic_δ start_POSTSUBSCRIPT 3 | 21 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 321 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 21 end_POSTSUBSCRIPT , E6|123subscript𝐸conditional6123\displaystyle E_6italic_E start_POSTSUBSCRIPT 6 | 123 end_POSTSUBSCRIPT =δ6|54←+S654-S54,absentsuperscriptsubscript𝛿conditional654←subscript𝑆654subscript𝑆54\displaystyle=\delta_6^\leftarrow+S_654-S_54,= italic_δ start_POSTSUBSCRIPT 6 | 54 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 654 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 54 end_POSTSUBSCRIPT ,

E6|12subscript𝐸conditional612\displaystyle E_12italic_E start_POSTSUBSCRIPT 6 | 12 end_POSTSUBSCRIPT =δ6|543←+S6543-S543,absentsuperscriptsubscript𝛿conditional6543←subscript𝑆6543subscript𝑆543\displaystyle=\delta_6^\leftarrow+S_6543-S_543,= italic_δ start_POSTSUBSCRIPT 6 | 543 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 6543 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 543 end_POSTSUBSCRIPT , E3|45subscript𝐸conditional345\displaystyle E_45italic_E start_POSTSUBSCRIPT 3 | 45 end_POSTSUBSCRIPT =δ3|216←+S3216-S216,absentsuperscriptsubscript𝛿conditional3216←subscript𝑆3216subscript𝑆216\displaystyle=\delta_216^\leftarrow+S_3216-S_216,= italic_δ start_POSTSUBSCRIPT 3 | 216 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 3216 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 216 end_POSTSUBSCRIPT ,

E2|345subscript𝐸conditional2345\displaystyle E_2italic_E start_POSTSUBSCRIPT 2 | 345 end_POSTSUBSCRIPT =δ2|16←+S216-S16,absentsuperscriptsubscript𝛿conditional216←subscript𝑆216subscript𝑆16\displaystyle=\delta_16^\leftarrow+S_216-S_16,= italic_δ start_POSTSUBSCRIPT 2 | 16 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 216 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 16 end_POSTSUBSCRIPT , E5|612subscript𝐸conditional5612\displaystyle E_5italic_E start_POSTSUBSCRIPT 5 | 612 end_POSTSUBSCRIPT =δ5|43←+S543-S43,absentsuperscriptsubscript𝛿conditional543←subscript𝑆543subscript𝑆43\displaystyle=\delta_5^\leftarrow+S_543-S_43,= italic_δ start_POSTSUBSCRIPT 5 | 43 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 543 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 43 end_POSTSUBSCRIPT ,

E5|61subscript𝐸conditional561\displaystyle E_61italic_E start_POSTSUBSCRIPT 5 | 61 end_POSTSUBSCRIPT =δ5|432←+S5432-S432,absentsuperscriptsubscript𝛿conditional5432←subscript𝑆5432subscript𝑆432\displaystyle=\delta_5^\leftarrow+S_5432-S_432,= italic_δ start_POSTSUBSCRIPT 5 | 432 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 5432 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 432 end_POSTSUBSCRIPT , E2|34subscript𝐸conditional234\displaystyle E_2italic_E start_POSTSUBSCRIPT 2 | 34 end_POSTSUBSCRIPT =δ2|165←+S2165-S165,absentsuperscriptsubscript𝛿conditional2165←subscript𝑆2165subscript𝑆165\displaystyle=\delta_2^\leftarrow+S_2165-S_165,= italic_δ start_POSTSUBSCRIPT 2 | 165 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 2165 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 165 end_POSTSUBSCRIPT ,

E1|234subscript𝐸conditional1234\displaystyle E_1italic_E start_POSTSUBSCRIPT 1 | 234 end_POSTSUBSCRIPT =δ1|65←+S165-S65,absentsuperscriptsubscript𝛿conditional165←subscript𝑆165subscript𝑆65\displaystyle=\delta_1^\leftarrow+S_165-S_65,= italic_δ start_POSTSUBSCRIPT 1 | 65 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 165 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 65 end_POSTSUBSCRIPT , E4|561subscript𝐸conditional4561\displaystyle E_4italic_E start_POSTSUBSCRIPT 4 | 561 end_POSTSUBSCRIPT =δ4|32←+S432-S32.absentsuperscriptsubscript𝛿conditional432←subscript𝑆432subscript𝑆32\displaystyle=\delta_32^\leftarrow+S_432-S_32.= italic_δ start_POSTSUBSCRIPT 4 | 32 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT 432 end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT 32 end_POSTSUBSCRIPT . (30) Similarly with happens in the previous cases, when all the Eqs. (IV.1) are summed up, the negative entropies cancel with the positive one from the next equation. The cycle closes when the negative entropy from the last equation cancels with the positive one from the first. The number of parts in the bigger bipartitions alternates between n𝑛nitalic_n and n-1𝑛1n-1italic_n - 1. The result is the following conservation law:

E1|23+E1|234+E2|34+E2|345+E3|45+E3|456+E4|56+E4|561+E5|61+E5|612+E6|12+E6|123=δ1|654←+δ1|65←+δ2|165←+δ2|16←+δ3|216←+δ3|21←+δ4|321←+δ4|32←+δ5|432←+δ5|43←+δ6|543←+δ6|543←subscript𝐸conditional123subscript𝐸conditional1234subscript𝐸conditional234subscript𝐸conditional2345subscript𝐸conditional345subscript𝐸conditional3456subscript𝐸conditional456subscript𝐸conditional4561subscript𝐸conditional561subscript𝐸conditional5612subscript𝐸conditional612subscript𝐸conditional6123superscriptsubscript𝛿conditional1654←superscriptsubscript𝛿conditional165←superscriptsubscript𝛿conditional2165←superscriptsubscript𝛿conditional216←superscriptsubscript𝛿conditional3216←superscriptsubscript𝛿conditional321←superscriptsubscript𝛿conditional4321←superscriptsubscript𝛿conditional432←superscriptsubscript𝛿conditional5432←superscriptsubscript𝛿conditional543←superscriptsubscript𝛿conditional6543←superscriptsubscript𝛿conditional6543←E_23+E_234+E_2+E_345+E_45+E_3+E_4+E_4+E_% 5+E_5+E_6+E_6\\ =\delta_1^\leftarrow+\delta_1^\leftarrow+\delta_2^% \leftarrow+\delta_16^\leftarrow+\delta_3^\leftarrow+\delta_2% 1^\leftarrow+\delta_321^\leftarrow+\delta_4^\leftarrow+\delta_% 432^\leftarrow+\delta_43^\leftarrow+\delta_6^\leftarrow+% \delta_543^\leftarrowstart_ROW start_CELL italic_E start_POSTSUBSCRIPT 1 | 23 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 1 | 234 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 2 | 34 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 2 | 345 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 3 | 45 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 3 | 456 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 4 | 56 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 4 | 561 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 5 | 61 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 5 | 612 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 6 | 12 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 6 | 123 end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT 1 | 654 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 1 | 65 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 2 | 165 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 2 | 16 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 3 | 216 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 3 | 21 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 4 | 321 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 4 | 32 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 5 | 432 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 5 | 43 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 6 | 543 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 6 | 543 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW (31)

From Eqs. (IV.1) and Eq. (31) it is easy to state the general conservation law for multipartite systems with an even number of parts, N𝑁Nitalic_N,

E1|23…n+E1|23…,n+1+E2|34…n+1+E2|34…n+2+⋯+EN|12…n-1+EN|12…n=δ1|N-1,N-2,…,n+1←+δ1|N-1,N-2,…,n←+δ2|1,N-1,N-2,…,n+2←+δ2|1,N-1,N-2,…,n+1←+⋯+δN|N-1,N-2,…,n←+δN|N-1,N-2,…,n-1←subscript𝐸conditional123…𝑛subscript𝐸conditional123…𝑛1subscript𝐸conditional234…𝑛1subscript𝐸conditional234…𝑛2⋯subscript𝐸conditional𝑁12…𝑛1subscript𝐸conditional𝑁12…𝑛superscriptsubscript𝛿conditional1𝑁1𝑁2…𝑛1←superscriptsubscript𝛿conditional1𝑁1𝑁2…𝑛←superscriptsubscript𝛿conditional21𝑁1𝑁2…𝑛2←superscriptsubscript𝛿conditional21𝑁1𝑁2…𝑛1←⋯superscriptsubscript𝛿conditional𝑁𝑁1𝑁2…𝑛←superscriptsubscript𝛿conditional𝑁𝑁1𝑁2…𝑛1←E_1+E_1+E_34\dots n+1+E_34\dots n+2+\cdots+E% _N+E_12\dots n\\ =\delta_1^\leftarrow+\delta_N-1,N-2,\dots,n^% \leftarrow+\delta_2^\leftarrow+\delta_1,N-1,N-2,% \dots,n+1^\leftarrow\\ +\cdots+\delta_N^\leftarrow+\delta_N^\leftarrowstart_ROW start_CELL italic_E start_POSTSUBSCRIPT 1 | 23 … italic_n end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 1 | 23 … , italic_n + 1 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 2 | 34 … italic_n + 1 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 2 | 34 … italic_n + 2 end_POSTSUBSCRIPT + ⋯ + italic_E start_POSTSUBSCRIPT italic_N | 12 … italic_n - 1 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_N | 12 … italic_n end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT 1 | italic_N - 1 , italic_N - 2 , … , italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 1 | italic_N - 1 , italic_N - 2 , … , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 2 | 1 , italic_N - 1 , italic_N - 2 , … , italic_n + 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 2 | 1 , italic_N - 1 , italic_N - 2 , … , italic_n + 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL + ⋯ + italic_δ start_POSTSUBSCRIPT italic_N | italic_N - 1 , italic_N - 2 , … , italic_n end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_N | italic_N - 1 , italic_N - 2 , … , italic_n - 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW (32) where n=N/2𝑛𝑁2n= icefracN2italic_n = / start_ARG italic_N end_ARG start_ARG 2 end_ARG. The Eq. (28) and Eq. (32) are the natural generalization of Eq. (5) for an even number of parts. The organization of the sum of terms in Eq. (32) is more involving than in Eq. (28), however the same interpretation applies. On the left side we have a sum of entanglements which represent the amount of quantum communication needed to form the correlation in each bipartition while, in the right side, the discords represent the amount of information that is not accessible locally. These two equations show, for general multipartite pure states, how EF and QD are distributed in a simpler and intuitive expression, associating the amount of quantum communication needed to form the correlations in bipartitions with the amount of correlations which is inaccessible by local operations.

IV.2 Generalized Conservation Law for Discord

In this section, we now focus on how QD is distributed, extending Eq. (6) for multipartite systems. Here a closed expression, based purely on QD, is deduced demonstrating the way that quantum correlations is distributed for general pure states. To determine a closed form to the discord distribution in multipartite systems, we begin considering a four-partite system. Firstly, we note that, by means of the KW relation, we can write

δb|a←superscriptsubscript𝛿conditional𝑏𝑎←\displaystyle\delta_b^\leftarrowitalic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT =\displaystyle== Eb|cd-Sb|a,subscript𝐸conditional𝑏𝑐𝑑subscript𝑆conditional𝑏𝑎\displaystyle E_cd-S_a,italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT ,

δc|b←superscriptsubscript𝛿conditional𝑐𝑏←\displaystyle\delta_b^\leftarrowitalic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT =\displaystyle== Ec|da-Sc|b,subscript𝐸conditional𝑐𝑑𝑎subscript𝑆conditional𝑐𝑏\displaystyle E_c-S_c,italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT ,

δd|c←superscriptsubscript𝛿conditional𝑑𝑐←\displaystyle\delta_d^\leftarrowitalic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT =\displaystyle== Ed|ab-Sd|c,subscript𝐸conditional𝑑𝑎𝑏subscript𝑆conditional𝑑𝑐\displaystyle E_d-S_c,italic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT ,

δa|d←superscriptsubscript𝛿conditional𝑎𝑑←\displaystyle\delta_a^\leftarrowitalic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT =\displaystyle== Ea|bc-Sa|d,subscript𝐸conditional𝑎𝑏𝑐subscript𝑆conditional𝑎𝑑\displaystyle E_bc-S_a,italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT - italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT , (33) since that combining the Eqs. (33), we get

Ea|bc+Eb|cd+Ec|da+Ed|ab=δa|d←+δd|c←+δc|b←+δb|a←+Sa|d+Sb|a+Sc|b+Sd|c.subscript𝐸conditional𝑎𝑏𝑐subscript𝐸conditional𝑏𝑐𝑑subscript𝐸conditional𝑐𝑑𝑎subscript𝐸conditional𝑑𝑎𝑏superscriptsubscript𝛿conditional𝑎𝑑←superscriptsubscript𝛿conditional𝑑𝑐←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑏𝑎←subscript𝑆conditional𝑎𝑑subscript𝑆conditional𝑏𝑎subscript𝑆conditional𝑐𝑏subscript𝑆conditional𝑑𝑐E_bc+E_cd+E_c+E_d\\ =\delta_a^\leftarrow+\delta_d^\leftarrow+\delta_c^\leftarrow% +\delta_a^\leftarrow+S_a+S_a+S_b+S_c.start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT . end_CELL end_ROW (34)

The interesting aspect about the set of equations given by Eq. (33) is that it is possible to organize them, with Eq. (17), to obtain an equality between entanglement and discord even for 4-partite systems. Indeed, subtracting one to the other we see that

(Ea|bc+Eab)+(Eb|cd+Ebc)+(Ec|da+Ecd)+(Ed|ab+Eda)=(δa|cd←+δa|d←)+(δb|ad←+δb|a←)+(δc|ab←+δc|b←)+(δd|bc←+δd|c←).subscript𝐸conditional𝑎𝑏𝑐subscript𝐸𝑎𝑏subscript𝐸conditional𝑏𝑐𝑑subscript𝐸𝑏𝑐subscript𝐸conditional𝑐𝑑𝑎subscript𝐸𝑐𝑑subscript𝐸conditional𝑑𝑎𝑏subscript𝐸𝑑𝑎superscriptsubscript𝛿conditional𝑎𝑐𝑑←superscriptsubscript𝛿conditional𝑎𝑑←superscriptsubscript𝛿conditional𝑏𝑎𝑑←superscriptsubscript𝛿conditional𝑏𝑎←superscriptsubscript𝛿conditional𝑐𝑎𝑏←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑑𝑏𝑐←superscriptsubscript𝛿conditional𝑑𝑐←(E_a\!+\! Discord servers E_ab)+(E_b\!+\!E_bc)\\ +(E_da\!+\!E_cd)+(E_d\!+\!E_da)\\ =(\delta_a^\leftarrow\!+\!\delta_d^\leftarrow)+(\delta_ad^% \leftarrow\!+\!\delta_a^\leftarrow)\\ +(\delta_ab^\leftarrow\!+\!\delta_c^\leftarrow)+(\delta_bc^% \leftarrow\!+\!\delta_c^\leftarrow).start_ROW start_CELL ( italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) + ( italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ( italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c italic_d end_POSTSUBSCRIPT ) + ( italic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d italic_a end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL = ( italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) + ( italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ( italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) + ( italic_δ start_POSTSUBSCRIPT italic_d | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) . end_CELL end_ROW (35) Eq. (35) is a conservative equation between EF and QD for 4-partite systems and it shows two curious aspects: firstly, we note that bipartitions of different sizes must be considered, since that three and two parts terms appears in Eq. (35). Secondly, we note that, contrary to the conservative relation for a tripartite or 5-partite pure state, there is no symmetry between left and right sides of the equation above. It is certainly a strange aspect which induce us to search for another conservative relation, a symmetric one.

For this purpose, we consider a different set of equations. Again we focus on the entanglement between two parts, but contrarily to the set of equations given in Eq. (17), we write the KW relation for Ebasubscript𝐸𝑏𝑎E_baitalic_E start_POSTSUBSCRIPT italic_b italic_a end_POSTSUBSCRIPT, Ecbsubscript𝐸𝑐𝑏E_cbitalic_E start_POSTSUBSCRIPT italic_c italic_b end_POSTSUBSCRIPT, Edcsubscript𝐸𝑑𝑐E_dcitalic_E start_POSTSUBSCRIPT italic_d italic_c end_POSTSUBSCRIPT, and Eadsubscript𝐸𝑎𝑑E_aditalic_E start_POSTSUBSCRIPT italic_a italic_d end_POSTSUBSCRIPT. It is clear, since entanglement is a symmetric entity, that these amounts of entanglement are equivalent to that given in Eq. (17), but with this we can derive a different set of equations:

Ebasubscript𝐸𝑏𝑎\displaystyle E_baitalic_E start_POSTSUBSCRIPT italic_b italic_a end_POSTSUBSCRIPT =\displaystyle== δb|cd←+Sb|cd,superscriptsubscript𝛿conditional𝑏𝑐𝑑←subscript𝑆conditional𝑏𝑐𝑑\displaystyle\delta_b^\leftarrow+S_cd,italic_δ start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT ,

Ecbsubscript𝐸𝑐𝑏\displaystyle E_cbitalic_E start_POSTSUBSCRIPT italic_c italic_b end_POSTSUBSCRIPT =\displaystyle== δc|ad←+Sc|ad,superscriptsubscript𝛿conditional𝑐𝑎𝑑←subscript𝑆conditional𝑐𝑎𝑑\displaystyle\delta_ad^\leftarrow+S_ad,italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_a italic_d end_POSTSUBSCRIPT ,

Edcsubscript𝐸𝑑𝑐\displaystyle E_dcitalic_E start_POSTSUBSCRIPT italic_d italic_c end_POSTSUBSCRIPT =\displaystyle== δd|ab←+Sd|ab,superscriptsubscript𝛿conditional𝑑𝑎𝑏←subscript𝑆conditional𝑑𝑎𝑏\displaystyle\delta_d^\leftarrow+S_ab,italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT ,

Eadsubscript𝐸𝑎𝑑\displaystyle E_aditalic_E start_POSTSUBSCRIPT italic_a italic_d end_POSTSUBSCRIPT =\displaystyle== δa|bc←+Sa|bc.superscriptsubscript𝛿conditional𝑎𝑏𝑐←subscript𝑆conditional𝑎𝑏𝑐\displaystyle\delta_bc^\leftarrow+S_bc.italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT . (36) Again, combining the set of Eqs. (36) above with that given in Eqs. (33), we derive another conservative relation:

(Ea|bc+Eab)+(Eb|cd+Ebc)+(Ec|da+Ecd)+(Ed|ab+Eda)=(δa|bc←+δb|a←)+(δb|cd←+δc|b←)+(δc|da←+δd|c←)+(δd|ab←+δa|d←)subscript𝐸conditional𝑎𝑏𝑐subscript𝐸𝑎𝑏subscript𝐸conditional𝑏𝑐𝑑subscript𝐸𝑏𝑐subscript𝐸conditional𝑐𝑑𝑎subscript𝐸𝑐𝑑subscript𝐸conditional𝑑𝑎𝑏subscript𝐸𝑑𝑎superscriptsubscript𝛿conditional𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑏𝑎←superscriptsubscript𝛿conditional𝑏𝑐𝑑←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑐𝑑𝑎←superscriptsubscript𝛿conditional𝑑𝑐←superscriptsubscript𝛿conditional𝑑𝑎𝑏←superscriptsubscript𝛿conditional𝑎𝑑←(E_a\!+\!E_ab)+(E_cd\!+\!E_bc)\\ +(E_c\!+\!E_cd)+(E_ab\! Discord servers +\!E_da)\\ =(\delta_a^\leftarrow\!+\!\delta_b^\leftarrow)+(\delta_b^% \leftarrow\!+\!\delta_c^\leftarrow)\\ +(\delta_da^\leftarrow\!+\!\delta_d^\leftarrow)+(\delta_d^% \leftarrow\!+\!\delta_a^\leftarrow)start_ROW start_CELL ( italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT ) + ( italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ( italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c italic_d end_POSTSUBSCRIPT ) + ( italic_E start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d italic_a end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL = ( italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) + ( italic_δ start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ( italic_δ start_POSTSUBSCRIPT italic_c | italic_d italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) + ( italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) end_CELL end_ROW (37)

Contrarily to the conservative equation given by Eq. (35), Eq. (37) above is symmetric and, surprisingly, when combined we obtain a rule for the distribution of QD for a 4-partite system:

δa|bc←+δb|cd←+δc|ad←+δd|ab←=δa|cd←+δb|ad←+δc|ab←+δd|bc←.superscriptsubscript𝛿conditional𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑏𝑐𝑑←superscriptsubscript𝛿conditional𝑐𝑎𝑑←superscriptsubscript𝛿conditional𝑑𝑎𝑏←superscriptsubscript𝛿conditional𝑎𝑐𝑑←superscriptsubscript𝛿conditional𝑏𝑎𝑑←superscriptsubscript𝛿conditional𝑐𝑎𝑏←superscriptsubscript𝛿conditional𝑑𝑏𝑐←\delta_bc^\leftarrow+\delta_cd^\leftarrow+\delta_ad^% \leftarrow+\delta_d^\leftarrow\\ =\delta_a^\leftarrow+\delta_b^\leftarrow+\delta_ab^% \leftarrow+\delta_d^\leftarrow.start_ROW start_CELL italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . end_CELL end_ROW (38) This equation elucidates an interesting property about the way that quantum correlations is shared in a four-partite systems. It shows that the total inaccessible information, about individual parts, after bipartite cyclical measurements (i.e. δa|bc←+δb|cd←+δc|ad←+δd|ab←superscriptsubscript𝛿conditional𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑏𝑐𝑑←superscriptsubscript𝛿conditional𝑐𝑎𝑑←superscriptsubscript𝛿conditional𝑑𝑎𝑏←\delta_bc^\leftarrow+\delta_b^\leftarrow+\delta_ad^% \leftarrow+\delta_ab^\leftarrowitalic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT), is equivalent to the total inaccessible information after counter-cyclic measurements. Also, it is important to note that Eq. (38) could be directly obtained rearranging the set of equation given in Eq. (17) and Eq. (36), but we choose this manner, since the conservative equation between EOF and QD is also deduced.

Now, continuing our endeavor to generalize the result above for multipartite systems, we explore the 5-partite systems. As usual, we begin with three set of equations:

Ea|bcdsubscript𝐸conditional𝑎𝑏𝑐𝑑\displaystyle E_bcditalic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δa|e←+Sa|e,superscriptsubscript𝛿conditional𝑎𝑒←subscript𝑆conditional𝑎𝑒\displaystyle\delta_a^\leftarrow+S_e,italic_δ start_POSTSUBSCRIPT italic_a | italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_e end_POSTSUBSCRIPT ,

Eb|cdesubscript𝐸conditional𝑏𝑐𝑑𝑒\displaystyle E_cdeitalic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d italic_e end_POSTSUBSCRIPT =\displaystyle== δb|a←+Sb|a,superscriptsubscript𝛿conditional𝑏𝑎←subscript𝑆conditional𝑏𝑎\displaystyle\delta_a^\leftarrow+S_b,italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT ,

Ec|deasubscript𝐸conditional𝑐𝑑𝑒𝑎\displaystyle E_citalic_E start_POSTSUBSCRIPT italic_c | italic_d italic_e italic_a end_POSTSUBSCRIPT =\displaystyle== δc|b←+Sc|b,superscriptsubscript𝛿conditional𝑐𝑏←subscript𝑆conditional𝑐𝑏\displaystyle\delta_c^\leftarrow+S_b,italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT ,

Ed|eabsubscript𝐸conditional𝑑𝑒𝑎𝑏\displaystyle E_eabitalic_E start_POSTSUBSCRIPT italic_d | italic_e italic_a italic_b end_POSTSUBSCRIPT =\displaystyle== δd|c←+Sd|c,superscriptsubscript𝛿conditional𝑑𝑐←subscript𝑆conditional𝑑𝑐\displaystyle\delta_d^\leftarrow+S_d,italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT ,

Ee|abcsubscript𝐸conditional𝑒𝑎𝑏𝑐\displaystyle E_eitalic_E start_POSTSUBSCRIPT italic_e | italic_a italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δe|d←+Se|d,superscriptsubscript𝛿conditional𝑒𝑑←subscript𝑆conditional𝑒𝑑\displaystyle\delta_d^\leftarrow+S_e,italic_δ start_POSTSUBSCRIPT italic_e | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_e | italic_d end_POSTSUBSCRIPT , (39)

Eabsubscript𝐸𝑎𝑏\displaystyle E_abitalic_E start_POSTSUBSCRIPT italic_a italic_b end_POSTSUBSCRIPT =\displaystyle== δa|cde←+Sa|cde,superscriptsubscript𝛿conditional𝑎𝑐𝑑𝑒←subscript𝑆conditional𝑎𝑐𝑑𝑒\displaystyle\delta_cde^\leftarrow+S_cde,italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_c italic_d italic_e end_POSTSUBSCRIPT ,

Ebcsubscript𝐸𝑏𝑐\displaystyle E_bcitalic_E start_POSTSUBSCRIPT italic_b italic_c end_POSTSUBSCRIPT =\displaystyle== δb|ade←+Sb|ade,superscriptsubscript𝛿conditional𝑏𝑎𝑑𝑒←subscript𝑆conditional𝑏𝑎𝑑𝑒\displaystyle\delta_b^\leftarrow+S_b,italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_a italic_d italic_e end_POSTSUBSCRIPT ,

Ecdsubscript𝐸𝑐𝑑\displaystyle E_cditalic_E start_POSTSUBSCRIPT italic_c italic_d end_POSTSUBSCRIPT =\displaystyle== δc|abe←+Sc|abe,superscriptsubscript𝛿conditional𝑐𝑎𝑏𝑒←subscript𝑆conditional𝑐𝑎𝑏𝑒\displaystyle\delta_abe^\leftarrow+S_c,italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_b italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_a italic_b italic_e end_POSTSUBSCRIPT ,

Edesubscript𝐸𝑑𝑒\displaystyle E_deitalic_E start_POSTSUBSCRIPT italic_d italic_e end_POSTSUBSCRIPT =\displaystyle== δd|abc←+Sd|abc,superscriptsubscript𝛿conditional𝑑𝑎𝑏𝑐←subscript𝑆conditional𝑑𝑎𝑏𝑐\displaystyle\delta_d^\leftarrow+S_d,italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_a italic_b italic_c end_POSTSUBSCRIPT ,

Eeasubscript𝐸𝑒𝑎\displaystyle E_eaitalic_E start_POSTSUBSCRIPT italic_e italic_a end_POSTSUBSCRIPT =\displaystyle== δe|bcd←+Se|bcd,superscriptsubscript𝛿conditional𝑒𝑏𝑐𝑑←subscript𝑆conditional𝑒𝑏𝑐𝑑\displaystyle\delta_bcd^\leftarrow+S_bcd,italic_δ start_POSTSUBSCRIPT italic_e | italic_b italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_e | italic_b italic_c italic_d end_POSTSUBSCRIPT , (40) and

Ebasubscript𝐸𝑏𝑎\displaystyle E_baitalic_E start_POSTSUBSCRIPT italic_b italic_a end_POSTSUBSCRIPT =\displaystyle== δb|cde←+Sb|cde,superscriptsubscript𝛿conditional𝑏𝑐𝑑𝑒←subscript𝑆conditional𝑏𝑐𝑑𝑒\displaystyle\delta_cde^\leftarrow+S_cde,italic_δ start_POSTSUBSCRIPT italic_b | italic_c italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b | italic_c italic_d italic_e end_POSTSUBSCRIPT ,

Ecbsubscript𝐸𝑐𝑏\displaystyle E_cbitalic_E start_POSTSUBSCRIPT italic_c italic_b end_POSTSUBSCRIPT =\displaystyle== δc|ade←+Sc|ade,superscriptsubscript𝛿conditional𝑐𝑎𝑑𝑒←subscript𝑆conditional𝑐𝑎𝑑𝑒\displaystyle\delta_ade^\leftarrow+S_c,italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c | italic_a italic_d italic_e end_POSTSUBSCRIPT ,

Edcsubscript𝐸𝑑𝑐\displaystyle E_dcitalic_E start_POSTSUBSCRIPT italic_d italic_c end_POSTSUBSCRIPT =\displaystyle== δd|abe←+Sd|abe,superscriptsubscript𝛿conditional𝑑𝑎𝑏𝑒←subscript𝑆conditional𝑑𝑎𝑏𝑒\displaystyle\delta_abe^\leftarrow+S_abe,italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d | italic_a italic_b italic_e end_POSTSUBSCRIPT ,

Eedsubscript𝐸𝑒𝑑\displaystyle E_editalic_E start_POSTSUBSCRIPT italic_e italic_d end_POSTSUBSCRIPT =\displaystyle== δe|abc←+Se|abc,superscriptsubscript𝛿conditional𝑒𝑎𝑏𝑐←subscript𝑆conditional𝑒𝑎𝑏𝑐\displaystyle\delta_abc^\leftarrow+S_abc,italic_δ start_POSTSUBSCRIPT italic_e | italic_a italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_e | italic_a italic_b italic_c end_POSTSUBSCRIPT ,

Eaesubscript𝐸𝑎𝑒\displaystyle E_aeitalic_E start_POSTSUBSCRIPT italic_a italic_e end_POSTSUBSCRIPT =\displaystyle== δa|bcd←+Sa|bcd.superscriptsubscript𝛿conditional𝑎𝑏𝑐𝑑←subscript𝑆conditional𝑎𝑏𝑐𝑑\displaystyle\delta_a^\leftarrow+S_bcd.italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a | italic_b italic_c italic_d end_POSTSUBSCRIPT . (41) Combining the set of equations given by Eqs. (39) with the set of equations given by Eqs. (41), we directly obtain the conservative relation:

(Ea|bcd+Eae)+(Eb|cde+Eba)+(Ec|dea+Ecb)+(Ed|eab+Edc)+(Ee|abc+Eed)=(δa|bcd←+δa|e←)+(δb|cde←+δb|a←)+(δc|dea←+δc|b←)+(δd|eab←+δd|c←)+(δe|abc←+δe|d←).subscript𝐸conditional𝑎𝑏𝑐𝑑subscript𝐸𝑎𝑒subscript𝐸conditional𝑏𝑐𝑑𝑒subscript𝐸𝑏𝑎subscript𝐸conditional𝑐𝑑𝑒𝑎subscript𝐸𝑐𝑏subscript𝐸conditional𝑑𝑒𝑎𝑏subscript𝐸𝑑𝑐subscript𝐸conditional𝑒𝑎𝑏𝑐subscript𝐸𝑒𝑑superscriptsubscript𝛿conditional𝑎𝑏𝑐𝑑←superscriptsubscript𝛿conditional𝑎𝑒←superscriptsubscript𝛿conditional𝑏𝑐𝑑𝑒←superscriptsubscript𝛿conditional𝑏𝑎←superscriptsubscript𝛿conditional𝑐𝑑𝑒𝑎←superscriptsubscript𝛿conditional𝑐𝑏←superscriptsubscript𝛿conditional𝑑𝑒𝑎𝑏←superscriptsubscript𝛿conditional𝑑𝑐←superscriptsubscript𝛿conditional𝑒𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑒𝑑←(E_bcd+E_ae)+(E_cde+E_ba)+(E_dea+E_cb)\\ +(E_d+E_dc)+(E_abc+E_ed)\\ =(\delta_bcd^\leftarrow+\delta_e^\leftarrow)+(\delta_b^% \leftarrow+\delta_a^\leftarrow)+(\delta_dea^\leftarrow+\delta_% b^\leftarrow)\\ +(\delta_eab^\leftarrow+\delta_d^\leftarrow)+(\delta_e^% \leftarrow+\delta_e^\leftarrow).start_ROW start_CELL ( italic_E start_POSTSUBSCRIPT italic_a | italic_b italic_c italic_d end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_e end_POSTSUBSCRIPT ) + ( italic_E start_POSTSUBSCRIPT italic_b | italic_c italic_d italic_e end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_b italic_a end_POSTSUBSCRIPT ) + ( italic_E start_POSTSUBSCRIPT italic_c | italic_d italic_e italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c italic_b end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ( italic_E start_POSTSUBSCRIPT italic_d | italic_e italic_a italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d italic_c end_POSTSUBSCRIPT ) + ( italic_E start_POSTSUBSCRIPT italic_e | italic_a italic_b italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_e italic_d end_POSTSUBSCRIPT ) end_CELL end_ROW start_ROW start_CELL = ( italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a | italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) + ( italic_δ start_POSTSUBSCRIPT italic_b | italic_c italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) + ( italic_δ start_POSTSUBSCRIPT italic_c | italic_d italic_e italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) end_CELL end_ROW start_ROW start_CELL + ( italic_δ start_POSTSUBSCRIPT italic_d | italic_e italic_a italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) + ( italic_δ start_POSTSUBSCRIPT italic_e | italic_a italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_e | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT ) . end_CELL end_ROW (42) Also, combining the set of equation given by Eqs. (40) and Eqs. (41) we obtain the conservation law for discord for 5-partite systems:

δa|cde←+δb|ade←+δc|abe←+δd|abc←+δe|bcd←=δa|bcd←+δb|cde←+δc|ade←+δd|abe←+δe|abc←.superscriptsubscript𝛿conditional𝑎𝑐𝑑𝑒←superscriptsubscript𝛿conditional𝑏𝑎𝑑𝑒←superscriptsubscript𝛿conditional𝑐𝑎𝑏𝑒←superscriptsubscript𝛿conditional𝑑𝑎𝑏𝑐←superscriptsubscript𝛿conditional𝑒𝑏𝑐𝑑←superscriptsubscript𝛿conditional𝑎𝑏𝑐𝑑←superscriptsubscript𝛿conditional𝑏𝑐𝑑𝑒←superscriptsubscript𝛿conditional𝑐𝑎𝑑𝑒←superscriptsubscript𝛿conditional𝑑𝑎𝑏𝑒←superscriptsubscript𝛿conditional𝑒𝑎𝑏𝑐←\delta_cde^\leftarrow+\delta_b^\leftarrow+\delta_c^% \leftarrow+\delta_abc^\leftarrow+\delta_bcd^\leftarrow\\ =\delta_bcd^\leftarrow+\delta_b^\leftarrow+\delta_ade^% \leftarrow+\delta_abe^\leftarrow+\delta_e^\leftarrow.start_ROW start_CELL italic_δ start_POSTSUBSCRIPT italic_a | italic_c italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_a italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_b italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_e | italic_b italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT italic_a | italic_b italic_c italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_b | italic_c italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c | italic_a italic_d italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d | italic_a italic_b italic_e end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_e | italic_a italic_b italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . end_CELL end_ROW (43)

With the results presented in Eq. (38) and Eq. (43) it is straightforward to deduce one of our main results, a law to the QD distribution in arbitrary multi-partite pure states. For an arbitrary N-partite system, we can write

δ1|L1←+δ2|L2←+⋯+δN|LN←=δ1|R1←+δ2|R2←+⋯+δN|RN←superscriptsubscript𝛿conditional1subscript𝐿1←superscriptsubscript𝛿conditional2subscript𝐿2←⋯superscriptsubscript𝛿conditional𝑁subscript𝐿𝑁←superscriptsubscript𝛿conditional1subscript𝑅1←superscriptsubscript𝛿conditional2subscript𝑅2←⋯superscriptsubscript𝛿conditional𝑁subscript𝑅𝑁←\delta_L_1^\leftarrow+\delta_L_2^\leftarrow+\cdots+\delta_N^\leftarrow\\ =\delta_1^\leftarrow+\delta_2^\leftarrow+\cdots+\delta_N^\leftarrowstart_ROW start_CELL italic_δ start_POSTSUBSCRIPT 1 | italic_L start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 2 | italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + ⋯ + italic_δ start_POSTSUBSCRIPT italic_N | italic_L start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT 1 | italic_R start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT 2 | italic_R start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + ⋯ + italic_δ start_POSTSUBSCRIPT italic_N | italic_R start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT end_CELL end_ROW (44) where the notation Lisubscript𝐿𝑖L_iitalic_L start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT means all the (N-2)𝑁2(N-2)( italic_N - 2 ) parts on left of the part i𝑖iitalic_i, that is i-1,i-2,…,1,N-1,…,i+2𝑖1𝑖2…1𝑁1…𝑖2\i-1,i-2,\dots,1,N-1,\dots,i+2\ italic_i - 1 , italic_i - 2 , … , 1 , italic_N - 1 , … , italic_i + 2 . For instance, L2subscript𝐿2L_2italic_L start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT, means all the subsystems 1,N,N-1,…,31𝑁𝑁1…3\1,N,N-1,\dots,3\ 1 , italic_N , italic_N - 1 , … , 3 . Similarly, Risubscript𝑅𝑖R_iitalic_R start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT means the N-2𝑁2N-2italic_N - 2 parts on right of the part i𝑖iitalic_i. We note that, given a system with N𝑁Nitalic_N-parts, the sum of inaccessible information of each i𝑖iitalic_i-th part, given an observation on the rest, with exception of that immediately on right of i𝑖iitalic_i, is equal to the sum of inaccessible information of each i𝑖iitalic_i-th part, given an observation on the rest, with exception of that immediately on left of i𝑖iitalic_i. An intuitive illustrative scheme, is showed in Figure 1, where we show the idea exposed above for one part. There, on inset (a) we show the inaccessible information on part 1111, given an observation on the rest, with exception of that immediately on its left, i.e. part n𝑛nitalic_n. On inset (b) we show again the inaccessible information on part 1111, but now given an observation on the rest, with exception of that immediately on its right, i.e. part 2222. The monogamous equality is reached doing the same procedure for all n𝑛nitalic_n parts. These results generalizes the conservation law for QD considering multipartite systems.

IV.3 Conservation law with a measurement in one part

As a final result, we present another simple monogamy-like conservation law that can be derived generalizing the of four-partite systems. Consider the four equations:

Ebc|asubscript𝐸conditional𝑏𝑐𝑎\displaystyle E_bcitalic_E start_POSTSUBSCRIPT italic_b italic_c | italic_a end_POSTSUBSCRIPT =\displaystyle== δbc|d←+Sbc|d,superscriptsubscript𝛿conditional𝑏𝑐𝑑←subscript𝑆conditional𝑏𝑐𝑑\displaystyle\delta_bc^\leftarrow+S_bc,italic_δ start_POSTSUBSCRIPT italic_b italic_c | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_b italic_c | italic_d end_POSTSUBSCRIPT ,

Ecd|bsubscript𝐸conditional𝑐𝑑𝑏\displaystyle E_cditalic_E start_POSTSUBSCRIPT italic_c italic_d | italic_b end_POSTSUBSCRIPT =\displaystyle== δcd|a←+Scd|a,superscriptsubscript𝛿conditional𝑐𝑑𝑎←subscript𝑆conditional𝑐𝑑𝑎\displaystyle\delta_a^\leftarrow+S_a,italic_δ start_POSTSUBSCRIPT italic_c italic_d | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_c italic_d | italic_a end_POSTSUBSCRIPT ,

Eda|csubscript𝐸conditional𝑑𝑎𝑐\displaystyle E_citalic_E start_POSTSUBSCRIPT italic_d italic_a | italic_c end_POSTSUBSCRIPT =\displaystyle== δda|b←+Sda|b,superscriptsubscript𝛿conditional𝑑𝑎𝑏←subscript𝑆conditional𝑑𝑎𝑏\displaystyle\delta_da^\leftarrow+S_da,italic_δ start_POSTSUBSCRIPT italic_d italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_d italic_a | italic_b end_POSTSUBSCRIPT ,

Eab|dsubscript𝐸conditional𝑎𝑏𝑑\displaystyle E_abitalic_E start_POSTSUBSCRIPT italic_a italic_b | italic_d end_POSTSUBSCRIPT =\displaystyle== δab|c←+Sab|c,superscriptsubscript𝛿conditional𝑎𝑏𝑐←subscript𝑆conditional𝑎𝑏𝑐\displaystyle\delta_ab^\leftarrow+S_ab,italic_δ start_POSTSUBSCRIPT italic_a italic_b | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_S start_POSTSUBSCRIPT italic_a italic_b | italic_c end_POSTSUBSCRIPT , (45) where the measurement in the discord is made only in one part. Summing up Eqs. (45), the conditional entropies cancel and we get

Ebc|a+Ecd|b+Eda|c+Eab|d=δbc|d←+δcd|a←+δda|b←+δab|c←.subscript𝐸conditional𝑏𝑐𝑎subscript𝐸conditional𝑐𝑑𝑏subscript𝐸conditional𝑑𝑎𝑐subscript𝐸conditional𝑎𝑏𝑑superscriptsubscript𝛿conditional𝑏𝑐𝑑←superscriptsubscript𝛿conditional𝑐𝑑𝑎←superscriptsubscript𝛿conditional𝑑𝑎𝑏←superscriptsubscript𝛿conditional𝑎𝑏𝑐←E_a+E_b+E_da+E_d=\\ \delta_d^\leftarrow+\delta_a^\leftarrow+\delta_da^% \leftarrow+\delta_c^\leftarrow.start_ROW start_CELL italic_E start_POSTSUBSCRIPT italic_b italic_c | italic_a end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_c italic_d | italic_b end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_d italic_a | italic_c end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT italic_a italic_b | italic_d end_POSTSUBSCRIPT = end_CELL end_ROW start_ROW start_CELL italic_δ start_POSTSUBSCRIPT italic_b italic_c | italic_d end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_c italic_d | italic_a end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_d italic_a | italic_b end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT + italic_δ start_POSTSUBSCRIPT italic_a italic_b | italic_c end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ← end_POSTSUPERSCRIPT . end_CELL end_ROW (46) Therefore, Eq. (46) is a conservation law with the property that all measurements in the discords are made only in one subsystem.

The Eq. (46) can be easily generalized for an arbitrary number of parts. For that, let us consider a system of N𝑁Nitalic_N parts and label them with number from 1 to N𝑁Nitalic_N, instead of letters. We can write the following set of equations:

E2:N-1|1subscript𝐸:2𝑁conditional11\displaystyle\hskip 42.67912ptE_1italic_E start_POSTSUBSCRIPT 2 : italic_N - 1 | 1 end_POSTSUBSCRIPT =δ2:N-1|Nabsentsubscript𝛿:2𝑁conditional1𝑁\displaystyle=\delta_2:N-1= italic_δ start_POSTSUBSCRIPT 2 : italic_N - 1 | italic_N end_POSTSUBSCRIPT +S2:N-1|N,subscript𝑆:2𝑁conditional1𝑁\displaystyle+S_N,\hskip 85.35826pt+ italic_S start_POSTSUBSCRIPT 2 : italic_N - 1 | italic_N end_POSTSUBSCRIPT ,

E3:N|2subscript𝐸:3conditional𝑁2\displaystyle E_2italic_E start_POSTSUBSCRIPT 3 : italic_N | 2 end_POSTSUBSCRIPT =δ3:N|1absentsubscript𝛿:3conditional𝑁1\displaystyle=\delta_3:N= italic_δ start_POSTSUBSCRIPT 3 : italic_N | 1 end_POSTSUBSCRIPT +S3:N|1,subscript𝑆:3conditional𝑁1\displaystyle+S_3:N,+ italic_S start_POSTSUBSCRIPT 3 : italic_N | 1 end_POSTSUBSCRIPT ,

E4:1|3subscript𝐸:4conditional13\displaystyle E_3italic_E start_POSTSUBSCRIPT 4 : 1 | 3 end_POSTSUBSCRIPT =δ4:1|2absentsubscript𝛿:4conditional12\displaystyle=\delta_2= italic_δ start_POSTSUBSCRIPT 4 : 1 | 2 end_POSTSUBSCRIPT +S4:1|2,subscript𝑆:4conditional12\displaystyle+S_4:1,+ italic_S start_POSTSUBSCRIPT 4 : 1 | 2 end_POSTSUBSCRIPT , (47)

⋮⋮\displaystyle\vdots\hskip 8.5359pt⋮ =⋮absent⋮\displaystyle=\hskip 8.5359pt\vdots= ⋮ +⋮⋮\displaystyle+\hskip 8.5359pt\vdots+ ⋮

E1:N-2|Nsubscript𝐸:1𝑁conditional2𝑁\displaystyle E_Nitalic_E start_POSTSUBSCRIPT 1 : italic_N - 2 | italic_N end_POSTSUBSCRIPT =δ1:N-2|N-1absentsubscript𝛿:1𝑁conditional2𝑁1\displaystyle=\delta_N-1= italic_δ start_POSTSUBSCRIPT 1 : italic_N - 2 | italic_N - 1 end_POSTSUBSCRIPT +S1:N-2|N-1,subscript𝑆:1𝑁conditional2𝑁1\displaystyle+S_N-1,+ italic_S start_POSTSUBSCRIPT 1 : italic_N - 2 | italic_N - 1 end_POSTSUBSCRIPT , where the notation X:Y:𝑋𝑌X:Yitalic_X : italic_Y means all the subsystems between numbers X𝑋Xitalic_X and Y𝑌Yitalic_Y when X<y𝑋𝑌xitalic_x>







<italic_y. when x>



Y𝑋𝑌X>Yitalic_X >italic_Y, it meas all subsystems from Y𝑌Yitalic_Y to N𝑁Nitalic_N and 1111 to X𝑋Xitalic_X. Summing up Eqs. (IV.3), all the entropies cancel out and the result is the following conservation law between entanglement and discord:







</italic_y.>







</y𝑋𝑌xitalic_x>

E2:N-1|1+E3:N|2+E4:1|3+⋯+E1:N-2|N=δ2:N-1|N+δ3:N|1+δ4:1|2+⋯+δ1:N-2|N-1.subscript𝐸:2𝑁conditional11subscript𝐸:3conditional𝑁2subscript𝐸:4conditional13⋯subscript𝐸:1𝑁conditional2𝑁subscript𝛿:2𝑁conditional1𝑁subscript𝛿:3conditional𝑁1subscript𝛿:4conditional12⋯subscript𝛿:1𝑁conditional2𝑁1E_1+E_3:N+E_3+\cdots+E_1:N-2\\ =\delta_2:N-1+\delta_1+\delta_4:1+\cdots+\delta_N-1.start_ROW start_CELL italic_E start_POSTSUBSCRIPT 2 : italic_N - 1 | 1 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 3 : italic_N | 2 end_POSTSUBSCRIPT + italic_E start_POSTSUBSCRIPT 4 : 1 | 3 end_POSTSUBSCRIPT + ⋯ + italic_E start_POSTSUBSCRIPT 1 : italic_N - 2 | italic_N end_POSTSUBSCRIPT end_CELL end_ROW start_ROW start_CELL = italic_δ start_POSTSUBSCRIPT 2 : italic_N - 1 | italic_N end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 3 : italic_N | 1 end_POSTSUBSCRIPT + italic_δ start_POSTSUBSCRIPT 4 : 1 | 2 end_POSTSUBSCRIPT + ⋯ + italic_δ start_POSTSUBSCRIPT 1 : italic_N - 2 | italic_N - 1 end_POSTSUBSCRIPT . end_CELL end_ROW (48) This generalized conservation law shows a relation between EF and QD when just one part is measured, as we can note by the right side of Eq. (48). Although the bipartitions in Eq. (48) overlaps, it also shows when the sum of quantum communications needed to form the correlations in the respective bipartitions are equal to the sum of locally inaccessible information.

The way quantum correlations are distributed in a multipartite quantum systems is an aspect of great interest. It is well known that the distribution of correlations can not be made freely and understanding how this mechanism works has implications in the study of the monogamy of quantum correlations as well as in the understanding of protocols and other fundamental aspects of quantum information. Here we present a set of monogamy-like conservative laws that govern how the EF and the QD are distributed in multipartite systems. These equalities links the constrains in the distributed entanglement with the distributed discord and vice-versa, showing that the monogamous properties of these two measures is deeply connected.

We initially focus on four and five-partite systems and after we extend our results to multipartite systems. We show not only a general form of how the EF and the QD are distributed, but also a closed expression that rules how QD is distributed in multipartite systems. These results elucidate important aspects in the distribution of quantum correlation in systems of many parts and may, in the near future, bring several implications and understandings to the quantum information theory.