Measuring Quantum Discord Using The Most Distinguishable Steered States

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Any two-qubit state can be represented, geometrically, as an ellipsoid with a certain size and center located within the Bloch sphere of one of the qubits. Points of this ellipsoid represent the post-measurement states when the other qubit is measured. Based on the most demolition concept in the definition of quantum discord, we study the amount of demolition when the two post-measurement states, represented as two points on the steering ellipsoid, have the most distinguishability. We use trace distance as a measure of distinguishability and obtain the maximum distinguishability for some classes of states, analytically. Using the optimum measurement that gives the most distinguishable steered states, we extract quantum correlation of the state and compare the result with the quantum discord. It is shown that there are some important classes of states for which the most demolition happens exactly at the most distinguished steered points. Correlations gathered from the most distinguished post-measurement states provide a faithful and tight upper bound touching the quantum discord in most of the cases.



Quantum steering ellipsoid, Quantum correlations, Quantum discord, Demolition ††preprint: APS/123-QED



In the light of the specific geometry of composite two-qubit systems at the Bloch sphere of one of the parts, Alice’s part for example, it is useful to study some non-classical features such as entanglement, separability, negativity, fully entangled fraction, quantum discord, Bell non-locality, monogamy, EPR steering and even the dynamics of a quantum system MilneNJP2014-1 ; MilneNJP2014-2 ; MilneNJP2015 ; MilnePRA2014 ; HuPRA2015 ; ShiNJP2011 ; WangPRA2014 ; CloskeyPRA2017 ; Schrödinger1935 ; WisemanPRA2007 ; WisemanPRL2007 ; CavalcantiPRL2014 . Each part of a two-qubit system has a specific unique set of post-measurement states, within them the part can be steered after doing measurement on the other part. This set geometrically forms the so-called “Quantum Steering Ellipsoid” (QSE) MilneNJP2014-1 , may be degenerate, within the Bloch sphere with the specific size and center. The measurement performed locally by Bob cannot affect the Alice’s reduced density matrix. Therefore the ensemble average of the Alice’s Bloch vectors of the post-measurement states, produced by a set of POVM on the Bob’s part, must be equal to the coherence vector of the Alice’s reduced state, meaning that the coherence vector lies inside the ellipsoid. In particular, when Bob’s reduced state is totally mixed, the Alice’s coherence vector coincides on her ellipsoid center MilneNJP2014-1 . Such states are called “canonical states”.



Conversely, we can reconstruct a two-qubit state from its ellipsoid, given the coherence vectors of two parts MilneNJP2014-1 . However, not any ellipsoid can belong to a physical state. For example any physical ellipsoid touches the Bloch sphere at most at two points unless it is the whole Bloch sphere BraunJPMT2014 . Physical ellipsoids have been studied in Ref. MilneNJP2014-2 .



By employing local filtering transformation (LFT) VerstraetePRA2001 , we can promote our insight about the QSE. It is shown that the physicality and separability of states are unchanged under LFT MilneNJP2014-2 . Furthermore, the Alice’s ellipsoid is invariant under LFT on Bob’s side, therefore the LFT makes orbits such that states on the same orbit have equal ellipsoids MilneNJP2014-1 . In view of this, the canonical states can be considered as the representatives on the corresponding orbits, therefore physicality and separability of the states on a general orbit can be determined from the ones of the canonical states. Given the ellipsoid center, authors in MilneNJP2014-2 have been studied conditions of physicality and separability of canonical states. Based on the Pres-Horodecki criterion PresPRL1996 ; HorodeckiPL1996 , the authors of MilneNJP2014-1 have shown that the separability of the canonical states depends on the shape of their ellipsoids.



All of the above symmetric features can be observed from the Bob’s ellipsoid which its dimension is the same as the Alice’s one MilneNJP2014-1 . Quantum discord (QD) is an asymmetric measure of quantum correlations that could be obtained by eliminating the classical correlation from the total correlation measured by the mutual information, by means of the most destructive measurement on the one party of the system. The total shared information in a composite quantum state ρ𝜌\rhoitalic_ρ is given by



I(ρ)=S(ρA)+S(ρB)-S(ρAB),𝐼𝜌𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝐵I(\rho)=S(\rho^A)+S(\rho^B)-S(\rho^AB),italic_I ( italic_ρ ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) , (1) where ρA=TrBρsuperscript𝜌𝐴subscriptTr𝐵𝜌\rho^A=\mathrmTr_B\rhoitalic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = roman_Tr start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT italic_ρ is the reduced density matrix of the Alice’s side, and ρBsuperscript𝜌𝐵\rho^Bitalic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT is defined similarly. Moreover, S(ρ)=-Tr[ρlog2ρ]𝑆𝜌Trdelimited-[]𝜌subscript2𝜌S(\rho)=-\mathrmTr[\rho\,\log_2\,\rho]italic_S ( italic_ρ ) = - roman_Tr [ italic_ρ roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT italic_ρ ] is the von Neumann entropy of the state ρ𝜌\rhoitalic_ρ. Quantum discord at Bob’s side reads



QB(ρ)=I(ρ)-CB(ρ),subscript𝑄𝐵𝜌𝐼𝜌subscript𝐶𝐵𝜌Q_B(\rho)=I(\rho)-C_B(\rho),italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = italic_I ( italic_ρ ) - italic_C start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) , (2) where



CB(ρ)=supΠkBΠkB).subscript𝐶𝐵𝜌subscriptsupremumsuperscriptsubscriptΠ𝑘𝐵𝑆superscript𝜌𝐴𝑆conditionalsuperscript𝜌𝐴superscriptsubscriptΠ𝑘𝐵C_B(\rho)=\mathop\sup\limits_\\Pi_k^B\\\,\\Pi_k^B\)\.italic_C start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = roman_sup start_POSTSUBSCRIPT roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT . (3) Here S(ρA|ΠkB)=∑kpkS(ρkA)𝑆conditionalsuperscript𝜌𝐴superscriptsubscriptΠ𝑘𝐵subscript𝑘subscript𝑝𝑘𝑆subscriptsuperscript𝜌𝐴𝑘S\left(\,\\Pi_k^B\\right)=\sum_kp_kS(\rho^A_k)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT | roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) is the Alice’s conditional entropy due to the Bob’s measurement.



Equation (3) shows that in order to calculate quantum discord we shall be concerned about the set ΠkBsuperscriptsubscriptΠ𝑘𝐵\\Pi_k^B\ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT of all measurements on the Bob’s qubit ZurekPRL2002 . This allows one to extract the most information about the Alice’s qubit. In this paper we use the notion of distinguishability of the Alice’s outcomes and look to those measurements on Bob’s qubit that lead to the most distinguishability of the Alice’s steered states. We show that such obtained optimum measurement coincide in some cases with the optimum measurement of Eq. (3). The correlations gathered from the most distinguished measurements give, in general, a tight upper bound for the quantum discord.



The paper is organized as follows. In Section II we present our terminology and provide a brief review for quantum steering ellipsoid. In section III the notion of distinguishability of the Alice’s outcomes is defined and we provide some important classes of states for which the maximum distinguishability can be calculated, analytically. Section IV is devoted to compare our results with quantum discord. The paper is conclude in section V with a brief conclusion.



II Framework: Quantum Steering Ellipsoid



We start from a two-qubit state in the general form as



ρ=14(𝟙⊗𝟙+𝒙⋅𝝈⊗𝟙+𝟙⊗𝒚⋅𝝈+∑i,j=13tijσi⊗σj),𝜌14tensor-productdouble-struck-𝟙double-struck-𝟙tensor-product⋅𝒙𝝈double-struck-𝟙⋅tensor-productdouble-struck-𝟙𝒚𝝈superscriptsubscript𝑖𝑗13tensor-productsubscript𝑡𝑖𝑗subscript𝜎𝑖subscript𝜎𝑗\rho=\frac14\left(\mathbb1\otimes\mathbb1+\boldsymbolx\cdot% \boldsymbol\sigma\otimes\mathbb1+\mathbb1\otimes\boldsymboly\cdot% \boldsymbol\sigma+\sum_i,j=1^3t_ij\sigma_i\otimes\sigma_j% \right),italic_ρ = divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( blackboard_𝟙 ⊗ blackboard_𝟙 + bold_italic_x ⋅ bold_italic_σ ⊗ blackboard_𝟙 + blackboard_𝟙 ⊗ bold_italic_y ⋅ bold_italic_σ + ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) , (4) where 𝒙𝒙\boldsymbolxbold_italic_x and 𝒚𝒚\boldsymbolybold_italic_y are Alice and Bob coherence vectors, respectively, T=[tij]𝑇delimited-[]subscript𝑡𝑖𝑗T=[t_ij]italic_T = [ italic_t start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT ] is the correlation matrix, 𝝈=(σ1,σ2,σ3)𝝈subscript𝜎1subscript𝜎2subscript𝜎3\boldsymbol\sigma=(\sigma_1,\sigma_2,\sigma_3)bold_italic_σ = ( italic_σ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_σ start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) are the Pauli matrices, and 𝟙double-struck-𝟙\mathbb1blackboard_𝟙 denotes the unit 2×2222\times 22 × 2 matrix. If Bob performs a projective measurement



ΠkB=12(𝟙+𝒏^k⋅𝝈),k=0,1,formulae-sequencesuperscriptsubscriptΠ𝑘𝐵12double-struck-𝟙⋅subscript^𝒏𝑘𝝈𝑘01\Pi_k^B=\frac12\left(\mathbb1+\hat\boldsymboln_k\cdot% \boldsymbol\sigma\right),\qquad k=0,1,roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( blackboard_𝟙 + over^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋅ bold_italic_σ ) , italic_k = 0 , 1 , (5) on his qubit, where 𝒏^0=(sinθcosϕ,sinθsinϕ,cosθ)t=-𝒏^1subscript^𝒏0superscript𝜃italic-ϕ𝜃italic-ϕ𝜃tsubscript^𝒏1\hat\boldsymboln_0=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos% \theta)^\mathrmt=-\hat\boldsymboln_1over^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT = ( roman_sin italic_θ roman_cos italic_ϕ , roman_sin italic_θ roman_sin italic_ϕ , roman_cos italic_θ ) start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT = - over^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT and tt\mathrmtroman_t denotes the transposition, the shared bipartite state collapses to



ρ=p0ρ0A⊗Π0B+p1ρ1A⊗Π1B,𝜌tensor-productsubscript𝑝0superscriptsubscript𝜌0𝐴superscriptsubscriptΠ0𝐵tensor-productsubscript𝑝1superscriptsubscript𝜌1𝐴superscriptsubscriptΠ1𝐵\rho=p_0\rho_0^A\otimes\Pi_0^B+p_1\rho_1^A\otimes\Pi_1^% B,italic_ρ = italic_p start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT + italic_p start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT , (6) with



ρkA=12(𝟙+𝒙~k⋅𝝈),superscriptsubscript𝜌𝑘𝐴12double-struck-𝟙⋅subscript~𝒙𝑘𝝈\displaystyle\rho_k^A=\frac12(\mathbb1+\widetilde\boldsymbolx_% k\cdot\boldsymbol\sigma),italic_ρ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( blackboard_𝟙 + over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋅ bold_italic_σ ) , (7) as the post-measurement state of the Alice’s side associated to the outcome k𝑘kitalic_k, with the corresponding probability



pk=12(1+𝒚⋅𝒏^k).subscript𝑝𝑘121⋅𝒚subscript^𝒏𝑘\displaystyle p_k=\frac12(1+\boldsymboly\cdot\hat\boldsymboln_k).italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( 1 + bold_italic_y ⋅ over^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ) . (8) Above, the Alice’s post-measurement coherence vector 𝒙~ksubscript~𝒙𝑘\widetilde\boldsymbolx_kover~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT is defined by



𝒙~k=𝒙+T𝒏^k1+𝒚⋅𝒏^k,subscript~𝒙𝑘𝒙𝑇subscript^𝒏𝑘1⋅𝒚subscript^𝒏𝑘\widetilde\boldsymbolx_k=\frac\boldsymbolx+T\hat\boldsymboln_k% 1+\boldsymboly\cdot\hat\boldsymboln_k,over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG bold_italic_x + italic_T over^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG start_ARG 1 + bold_italic_y ⋅ over^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT end_ARG , (9) for k=0,1𝑘01k=0,1italic_k = 0 , 1.



Canonical states.-It can be easily seen that starting from a generic two-qubit state ρ𝜌\rhoitalic_ρ with nonzero Bob’s coherence vector 𝒚𝒚\boldsymbolybold_italic_y, one can obtain the canonical state ρ(can)superscript𝜌c𝑎𝑛\rho^(\mathrmcan)italic_ρ start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT with 𝒚(can)=0superscript𝒚c𝑎𝑛0\boldsymboly^(\mathrmcan)=0bold_italic_y start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT = 0 as MilneNJP2014-1



ρ(can)superscript𝜌c𝑎𝑛\displaystyle\rho^(\mathrmcan)italic_ρ start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT =\displaystyle== (𝟙⊗𝟙2ρB)ρ(𝟙⊗𝟙2ρB)tensor-productdouble-struck-𝟙double-struck-𝟙2superscript𝜌𝐵𝜌tensor-productdouble-struck-𝟙double-struck-𝟙2superscript𝜌𝐵\displaystyle\left(\mathbb1\otimes\frac\mathbb1\sqrt2\rho^B% \right)\rho\left(\mathbb1\otimes\frac\mathbb1\sqrt2\rho^B\right)( blackboard_𝟙 ⊗ divide start_ARG blackboard_𝟙 end_ARG start_ARG square-root start_ARG 2 italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG end_ARG ) italic_ρ ( blackboard_𝟙 ⊗ divide start_ARG blackboard_𝟙 end_ARG start_ARG square-root start_ARG 2 italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG end_ARG )



=\displaystyle== 14(𝟙⊗𝟙+𝒙(can)⋅𝝈⊗𝟙+∑i,j=13tij(can)σi⊗σj).14tensor-productdouble-struck-𝟙double-struck-𝟙tensor-product⋅superscript𝒙c𝑎𝑛𝝈double-struck-𝟙superscriptsubscript𝑖𝑗13tensor-productsuperscriptsubscript𝑡𝑖𝑗c𝑎𝑛subscript𝜎𝑖subscript𝜎𝑗\displaystyle\frac14\left(\mathbb1\otimes\mathbb1+\boldsymbolx^% (\mathrmcan)\cdot\boldsymbol\sigma\otimes\mathbb1+\sum_i,j=1^3% t_ij^(\mathrmcan)\sigma_i\otimes\sigma_j\right).divide start_ARG 1 end_ARG start_ARG 4 end_ARG ( blackboard_𝟙 ⊗ blackboard_𝟙 + bold_italic_x start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT ⋅ bold_italic_σ ⊗ blackboard_𝟙 + ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT italic_t start_POSTSUBSCRIPT italic_i italic_j end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT italic_σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⊗ italic_σ start_POSTSUBSCRIPT italic_j end_POSTSUBSCRIPT ) .



Regarding that Alice’s ellipsoid is invariant under LFT on the Bob’s side MilneNJP2014-1 , we just need to know the ellipsoid of the associated canonical state of the original state. For canonical states for which 𝒚(can)=0superscript𝒚c𝑎𝑛0\boldsymboly^(\mathrmcan)=0bold_italic_y start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT = 0, it is easy to construct Alice’s ellipsoid from the above formalism. In this particular case, Alice’s post-measurement Bloch vector (9) reduces to



𝒙~k(can)=𝒙(can)+T(can)𝒏^k.subscriptsuperscript~𝒙c𝑎𝑛𝑘superscript𝒙c𝑎𝑛superscript𝑇c𝑎𝑛subscript^𝒏𝑘\widetilde\boldsymbolx^(\mathrmcan)_k=\boldsymbolx^(\mathrmc% an)+T^(\mathrmcan)\hat\boldsymboln_k.over~ start_ARG bold_italic_x end_ARG start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = bold_italic_x start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT + italic_T start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT over^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT . (11) with probability pk(can)=12subscriptsuperscript𝑝c𝑎𝑛𝑘12p^(\mathrmcan)_k=\frac12italic_p start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG for k=0,1𝑘01k=0,1italic_k = 0 , 1. Since the unit vector 𝒏^ksubscript^𝒏𝑘\hat\boldsymboln_kover^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT defines a unit sphere centered at origin, the above equation states that the set of all points Alice’s coherence vector steers to forms an ellipsoid. This canonical ellipsoid, associated to the canonical state ρ(can)superscript𝜌c𝑎𝑛\rho^(\mathrmcan)italic_ρ start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT for which 𝒚(can)=0superscript𝒚𝑐𝑎𝑛0\boldsymboly^(can)=0bold_italic_y start_POSTSUPERSCRIPT ( italic_c italic_a italic_n ) end_POSTSUPERSCRIPT = 0, is obtained by shrinking and rotating the sphere 𝒏^ksubscript^𝒏𝑘\hat\boldsymboln_kover^ start_ARG bold_italic_n end_ARG start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT by matrix T(can)superscript𝑇c𝑎𝑛T^(\mathrmcan)italic_T start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT, and then translating it by vector 𝒙(can)superscript𝒙c𝑎𝑛\boldsymbolx^(\mathrmcan)bold_italic_x start_POSTSUPERSCRIPT ( roman_c italic_a italic_n ) end_POSTSUPERSCRIPT MilneNJP2014-1 .



III Measuring Bob’s qubit with the most disruptive Alice’s qubit



As we mentioned already, in order to calculate quantum discord we shall be concerned about the set ΠkBsuperscriptsubscriptΠ𝑘𝐵\\Pi_k^B\ roman_Π start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT of all measurements on the Bob’s qubit. This allows one to extract the most information about the Alice’s qubit ZurekPRL2002 and that at the same time disturbs least the overall quantum state ρ𝜌\rhoitalic_ρ. This corresponds also to finding measurements that maximize Eq. (3). When the results of the measurement are not recorded, the measurement on Bob’s qubit does not disturb Alice’s state ρAsuperscript𝜌𝐴\rho^Aitalic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT. However, corresponding to the measurement outcomes, Alice’s state steers to some states ρkAsubscriptsuperscript𝜌𝐴𝑘\rho^A_kitalic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT in her ellipsoid following the route of Eqs. (7) and (8). The ability to extract information about Alice’s qubit by measuring Bob’s qubit comes from correlations shared between them and this is, in general, accompanied by disrupting the Alice’s outcome states. It seems that extracting the most information about Alice’s qubit can be associated with the most disturbing her outcomes states. In the following we are looking to those measurements on Bob’s qubit that cause the most disturbance in the Alice’s post-measurements states. To this aim we use trace distance as a measure of quantum distinguishability between two outcomes NielsenBook2000



D(ρ0A,ρ1A)=Tr|ρ0A-ρ1A|.𝐷subscriptsuperscript𝜌𝐴0subscriptsuperscript𝜌𝐴1Trsubscriptsuperscript𝜌𝐴0subscriptsuperscript𝜌𝐴1\displaystyle D(\rho^A_0,\rho^A_1)=\mathrmTr\rho^A_0-\rho^A% _1.italic_D ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = roman_Tr | italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | . (12) This, in turns, reduces simply to the Euclidian distance D(𝒙~0,𝒙~1)𝐷subscript~𝒙0subscript~𝒙1D(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)italic_D ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) between Bloch vectors of the two post-measurement states ρ0Asuperscriptsubscript𝜌0𝐴\rho_0^Aitalic_ρ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and ρ1Asuperscriptsubscript𝜌1𝐴\rho_1^Aitalic_ρ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT. For the squared distance we find



D2(𝒙~0,𝒙~1)=|𝒙~0-𝒙~1|2=4𝒏^tM𝒏^(1-𝒏^tY𝒏^)2,superscript𝐷2subscript~𝒙0subscript~𝒙1superscriptsubscript~𝒙0subscript~𝒙124superscript^𝒏t𝑀^𝒏superscript1superscript^𝒏t𝑌^𝒏2\displaystyle D^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_% 1)=|\widetilde\boldsymbolx_0-\widetilde\boldsymbolx_1|^2=\frac% 4\hat\boldsymboln^\mathrmtM\hat\boldsymboln(1-\hat\boldsymbol% n^\mathrmtY\hat\boldsymboln)^2,italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = | over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT - over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 4 over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_M over^ start_ARG bold_italic_n end_ARG end_ARG start_ARG ( 1 - over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_Y over^ start_ARG bold_italic_n end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , (13) where Y=𝒚t𝒚𝑌superscript𝒚t𝒚Y=\boldsymboly^\mathrmt\boldsymbolyitalic_Y = bold_italic_y start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT bold_italic_y and M=mtm𝑀superscript𝑚t𝑚M=m^\mathrmtmitalic_M = italic_m start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_m with m=(T-𝒙𝒚t)𝑚𝑇𝒙superscript𝒚tm=(T-\boldsymbolx\boldsymboly^\mathrmt)italic_m = ( italic_T - bold_italic_x bold_italic_y start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT ). Maximum distinguishability corresponds therefore to the maximum distance given by



Dmax2(𝒙~0,𝒙~1)=max𝒏^[4𝒏^tM𝒏^(1-𝒏^tY𝒏^)2],subscriptsuperscript𝐷2subscript~𝒙0subscript~𝒙1subscript^𝒏4superscript^𝒏t𝑀^𝒏superscript1superscript^𝒏t𝑌^𝒏2\displaystyle D^2_\max(\widetilde\boldsymbolx_0,\widetilde% \boldsymbolx_1)=\max_\hat\boldsymboln\left[\frac4\hat\boldsymbol% n^\mathrmtM\hat\boldsymboln(1-\hat\boldsymboln^\mathrmtY% \hat\boldsymboln)^2\right],italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = roman_max start_POSTSUBSCRIPT over^ start_ARG bold_italic_n end_ARG end_POSTSUBSCRIPT [ divide start_ARG 4 over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_M over^ start_ARG bold_italic_n end_ARG end_ARG start_ARG ( 1 - over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_Y over^ start_ARG bold_italic_n end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] , (14) where maximum is taken over all unit vectors 𝒏^∈ℝ3^𝒏superscriptℝ3\hat\boldsymboln\in\mathbbR^3over^ start_ARG bold_italic_n end_ARG ∈ roman_ℝ start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT. Before we proceed further to find conditions under which D2(𝒙~0,𝒙~1)superscript𝐷2subscript~𝒙0subscript~𝒙1D^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) is maximize, let us turn our attention on some particular cases for which the maximum is obtained analytically without any need for rigorous optimization. (i) Canonical states 𝐲=0𝐲0\boldsymboly=0bold_italic_y = 0.-For the important class of canonical states for which the Bob’s coherence vector is zero, the optimum measurement leading to the maximum distance between Bloch vectors of the post-measurement states is nothing but the eigenvector of T𝑇Titalic_T corresponding to its largest eigenvalue. Therefore in this case we have Dmax2(𝒙~0,𝒙~1)=4maxt12,t22,t32subscriptsuperscript𝐷2subscript~𝒙0subscript~𝒙14superscriptsubscript𝑡12superscriptsubscript𝑡22superscriptsubscript𝑡32D^2_\max(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)=4% \max\t_1^2,t_2^2,t_3^2\italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 4 roman_max italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (ii) States for which 𝐲𝐲\boldsymbolybold_italic_y is an eigenvector corresponding to the largest eigenvalue of M𝑀Mitalic_M.-In this case maximum of the enumerate happens in the direction of coherence vector of the part B𝐵Bitalic_B, i.e. max𝒏^𝒏^tM𝒏^=𝒚tM𝒚/y2subscript^𝒏superscript^𝒏t𝑀^𝒏superscript𝒚t𝑀𝒚superscript𝑦2\max_\hat\boldsymboln\hat\boldsymboln^\mathrmtM\hat\boldsymbol% n=\boldsymboly^\mathrmtM\boldsymboly/y^2roman_max start_POSTSUBSCRIPT over^ start_ARG bold_italic_n end_ARG end_POSTSUBSCRIPT over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_M over^ start_ARG bold_italic_n end_ARG = bold_italic_y start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_M bold_italic_y / italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. For such states we get Dmax2(𝒙~0,𝒙~1)=[4𝒚tM𝒚y2(1-y2)2]subscriptsuperscript𝐷2subscript~𝒙0subscript~𝒙1delimited-[]4superscript𝒚t𝑀𝒚superscript𝑦2superscript1superscript𝑦22D^2_\max(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)=% \left[\frac4\boldsymboly^\mathrmtM\boldsymbolyy^2(1-y^2)^2\right]italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = [ divide start_ARG 4 bold_italic_y start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_M bold_italic_y end_ARG start_ARG italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( 1 - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ]. Gaming News (iii) X𝑋Xitalic_X states.-The important class of X𝑋Xitalic_X states is defined by 𝒙=(00x)t𝒙superscript00𝑥t\boldsymbolx=\left(\beginarray[]ccc0&0&x\endarray\right)^\mathrmtbold_italic_x = ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_x end_CELL end_ROW end_ARRAY ) start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT, 𝒚=(00y)t𝒚superscript00𝑦t\boldsymboly=\left(\beginarray[]ccc0&0&y\endarray\right)^\mathrmtbold_italic_y = ( start_ARRAY start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_y end_CELL end_ROW end_ARRAY ) start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT, and T=diagt1,t2,t3𝑇diagsubscript𝑡1subscript𝑡2subscript𝑡3T=\mathrmdiag\t_1,t_2,t_3\italic_T = roman_diag italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT . In this case M𝑀Mitalic_M is also a diagonal matrix given by M=diagM1,M2,M3𝑀diagsubscript𝑀1subscript𝑀2subscript𝑀3M=\mathrmdiag\M_1,M_2,M_3\italic_M = roman_diag italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT with M1=t12subscript𝑀1superscriptsubscript𝑡12M_1=t_1^2italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, M2=t22subscript𝑀2superscriptsubscript𝑡22M_2=t_2^2italic_M start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and M3=(t3-xy)2subscript𝑀3superscriptsubscript𝑡3𝑥𝑦2M_3=(t_3-xy)^2italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT = ( italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT - italic_x italic_y ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. In what follows we assume that |t1|≥|t2|subscript𝑡1subscript𝑡2|t_1|\geq|t_2|| italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ | italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | (|t1|≤|t2|subscript𝑡1subscript𝑡2|t_1|\leq|t_2|| italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≤ | italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | can be obtained just by replacing 1→2→121\rightarrow 21 → 2 and x→y→𝑥𝑦x\rightarrow yitalic_x → italic_y). In this case we find the following results.



1. M1≤M3subscript𝑀1subscript𝑀3M_1\leq M_3italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. For such case we get Dmax2(𝒙~0,𝒙~1)=4M3(1-y2)2superscriptsubscript𝐷2subscript~𝒙0subscript~𝒙14subscript𝑀3superscript1superscript𝑦22D_\max^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)=% \frac4M_3(1-y^2)^2italic_D start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = divide start_ARG 4 italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG ( 1 - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG with σzsubscript𝜎𝑧\sigma_zitalic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT as the optimal measurement.



2. M1≥M3subscript𝑀1subscript𝑀3M_1\geq M_3italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. In this case the optimal measurement is defined by (n^1∗)2=1-(n^3∗)2superscriptsubscriptsuperscript^𝑛∗121superscriptsubscriptsuperscript^𝑛∗32(\hatn^\ast_1)^2=1-(\hatn^\ast_3)^2( over^ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = 1 - ( over^ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, n^2∗=0subscriptsuperscript^𝑛∗20\hatn^\ast_2=0over^ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT = 0, and (n^3∗)2=2M1y2-(M1-M3)(M1-M3)y2superscriptsubscriptsuperscript^𝑛∗322subscript𝑀1superscript𝑦2subscript𝑀1subscript𝑀3subscript𝑀1subscript𝑀3superscript𝑦2(\hatn^\ast_3)^2=\frac2M_1y^2-(M_1-M_3)(M_1-M_3)y^2( over^ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT = divide start_ARG 2 italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - ( italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) end_ARG start_ARG ( italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG if



M1-M32M1≤y2≤M1-M3M1+M3,subscript𝑀1subscript𝑀32subscript𝑀1superscript𝑦2subscript𝑀1subscript𝑀3subscript𝑀1subscript𝑀3\fracM_1-M_32M_1\leq y^2\leq\fracM_1-M_3M_1+M_3,divide start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ≤ italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ divide start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT - italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG , (15) or equivalently



(2M3M1+M3)2≤(1-y2)2≤(M1+M32M1)2.superscript2subscript𝑀3subscript𝑀1subscript𝑀32superscript1superscript𝑦22superscriptsubscript𝑀1subscript𝑀32subscript𝑀12\left(\frac2M_3M_1+M_3\right)^2\leq(1-y^2)^2\leq\left(\fracM_% 1+M_32M_1\right)^2.( divide start_ARG 2 italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( 1 - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≤ ( divide start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (16) On the other hand, the optimal measurement is σzsubscript𝜎𝑧\sigma_zitalic_σ start_POSTSUBSCRIPT italic_z end_POSTSUBSCRIPT, i.e. Dmax2(𝒙~0,𝒙~1)=4M3(1-y2)2superscriptsubscript𝐷2subscript~𝒙0subscript~𝒙14subscript𝑀3superscript1superscript𝑦22D_\max^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)=% \frac4M_3(1-y^2)^2italic_D start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = divide start_ARG 4 italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG ( 1 - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG, if



(2M3M1+M3)2≥(1-y2)2,superscript2subscript𝑀3subscript𝑀1subscript𝑀32superscript1superscript𝑦22\left(\frac2M_3M_1+M_3\right)^2\geq(1-y^2)^2,( divide start_ARG 2 italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( 1 - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , (17) and it is σxsubscript𝜎𝑥\sigma_xitalic_σ start_POSTSUBSCRIPT italic_x end_POSTSUBSCRIPT, i.e. Dmax2(𝒙~0,𝒙~1)=4M1superscriptsubscript𝐷2subscript~𝒙0subscript~𝒙14subscript𝑀1D_\max^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)=4M% _1italic_D start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = 4 italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT, if



(1-y2)2≥(M1+M32M1)2.superscript1superscript𝑦22superscriptsubscript𝑀1subscript𝑀32subscript𝑀12(1-y^2)^2\geq\left(\fracM_1+M_32M_1\right)^2.( 1 - italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ ( divide start_ARG italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT + italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT end_ARG start_ARG 2 italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT . (18)



Now, after giving the maximum distance for some particular classes of states without rigorous optimization, we provide in what follows an analytical procedure for optimization of Eq. (14). In order to determine the maximum distance, we have to calculate its derivatives with respect to θ𝜃\thetaitalic_θ and ϕitalic-ϕ\phiitalic_ϕ. For derivative with respect to θ𝜃\thetaitalic_θ we get



∂D2∂θ=8n^,θtℳn^(1-𝒏^tY𝒏^)2,superscript𝐷2𝜃8superscriptsubscript^𝑛fragments,θtℳ^𝑛superscript1superscript^𝒏t𝑌^𝒏2\displaystyle\frac\partial D^2\partial\theta=\frac8\hatn_,\theta^% \mathrmt\mathcalM\hatn(1-\hat\boldsymboln^\mathrmtY\hat% \boldsymboln)^2,divide start_ARG ∂ italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_θ end_ARG = divide start_ARG 8 over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_θ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT caligraphic_M over^ start_ARG italic_n end_ARG end_ARG start_ARG ( 1 - over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_Y over^ start_ARG bold_italic_n end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , (19) where ℳ(n^)ℳ^𝑛\mathcalM(\hatn)caligraphic_M ( over^ start_ARG italic_n end_ARG ) is a n^^𝑛\hatnover^ start_ARG italic_n end_ARG-dependent symmetric matrix given by



ℳ(n^)=M+2n^tMn^(1-𝒏^tY𝒏^)Y,ℳ^𝑛𝑀2superscript^𝑛t𝑀^𝑛1superscript^𝒏t𝑌^𝒏𝑌\mathcalM(\hatn)=M+\frac2\hatn^\mathrmtM\hatn(1-\hat% \boldsymboln^\mathrmtY\hat\boldsymboln)Y,caligraphic_M ( over^ start_ARG italic_n end_ARG ) = italic_M + divide start_ARG 2 over^ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_M over^ start_ARG italic_n end_ARG end_ARG start_ARG ( 1 - over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT italic_Y over^ start_ARG bold_italic_n end_ARG ) end_ARG italic_Y , (20) and the unit vector n^,θsubscript^𝑛fragments,θ\hatn_,\thetaover^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_θ end_POSTSUBSCRIPT is defined by



n^,θ=∂n^∂θ=(cosθcosϕ,cosθsinϕ,-sinθ)t.subscript^𝑛fragments,θ^𝑛𝜃superscript𝜃italic-ϕ𝜃italic-ϕ𝜃t\hatn_,\theta=\frac\partial\hatn\partial\theta=(\cos\theta\cos% \phi,\cos\theta\sin\phi,-\sin\theta)^\mathrmt.over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_θ end_POSTSUBSCRIPT = divide start_ARG ∂ over^ start_ARG italic_n end_ARG end_ARG start_ARG ∂ italic_θ end_ARG = ( roman_cos italic_θ roman_cos italic_ϕ , roman_cos italic_θ roman_sin italic_ϕ , - roman_sin italic_θ ) start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT . (21) Evidently n^,θ⋅n^=0⋅subscript^𝑛fragments,θ^𝑛0\hatn_,\theta\cdot\hatn=0over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_θ end_POSTSUBSCRIPT ⋅ over^ start_ARG italic_n end_ARG = 0. By defining the nonunit vector n~,ϕsubscript~𝑛fragments,ϕ\tilden_,\phiover~ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_ϕ end_POSTSUBSCRIPT by



n~,ϕ=∂n^∂ϕ=(-sinθsinϕ,sinθcosϕ,0)t,subscript~𝑛fragments,ϕ^𝑛italic-ϕsuperscript𝜃italic-ϕ𝜃italic-ϕ0t\tilden_,\phi=\frac\partial\hatn\partial\phi=(-\sin\theta\sin% \phi,\sin\theta\cos\phi,0)^\mathrmt,over~ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_ϕ end_POSTSUBSCRIPT = divide start_ARG ∂ over^ start_ARG italic_n end_ARG end_ARG start_ARG ∂ italic_ϕ end_ARG = ( - roman_sin italic_θ roman_sin italic_ϕ , roman_sin italic_θ roman_cos italic_ϕ , 0 ) start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT , (22) orthogonal to both n^^𝑛\hatnover^ start_ARG italic_n end_ARG and n^,θsubscript^𝑛fragments,θ\hatn_,\thetaover^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_θ end_POSTSUBSCRIPT, we get a similar equation for the derivative of the distance with respect to ϕitalic-ϕ\phiitalic_ϕ, but now n^,θsubscript^𝑛fragments,θ\hatn_,\thetaover^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_θ end_POSTSUBSCRIPT is replaced by n~,ϕsubscript~𝑛fragments,ϕ\tilden_,\phiover~ start_ARG italic_n end_ARG start_POSTSUBSCRIPT , italic_ϕ end_POSTSUBSCRIPT. Excluding the case y=1𝑦1y=1italic_y = 1 which happens if and only if the overall state is pure, we find the following relation for the stationary condition ∂D2∂θ=∂D2∂ϕ=0superscript𝐷2𝜃superscript𝐷2italic-ϕ0\frac\partial D^2\partial\theta=\frac\partial D^2\partial\phi=0divide start_ARG ∂ italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_θ end_ARG = divide start_ARG ∂ italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ∂ italic_ϕ end_ARG = 0,



n^⟂tℳ(n^)n^=0,superscriptsubscript^𝑛perpendicular-totℳ^𝑛^𝑛0\hatn_\perp^\mathrmt\;\mathcalM(\hatn)\;\hatn=0,over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT caligraphic_M ( over^ start_ARG italic_n end_ARG ) over^ start_ARG italic_n end_ARG = 0 , (23) where n^⟂subscript^𝑛perpendicular-to\hatn_\perpover^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT is any vector perpendicular to n^^𝑛\hatnover^ start_ARG italic_n end_ARG, i.e. n^⟂.n^=0formulae-sequencesubscript^𝑛perpendicular-to^𝑛0\hatn_\perp.\hatn=0over^ start_ARG italic_n end_ARG start_POSTSUBSCRIPT ⟂ end_POSTSUBSCRIPT . over^ start_ARG italic_n end_ARG = 0. This implies that the stationary points are achieved if and only if n^^𝑛\hatnover^ start_ARG italic_n end_ARG be an eigenvector of ℳ(n^)ℳ^𝑛\mathcalM(\hatn)caligraphic_M ( over^ start_ARG italic_n end_ARG ). Note that knowing the extremum points of the distance is not enough to establish its maximum, and we are required a further investigation of the distance over all extremum points to get the maximum one. Although condition (23) does not provide an easy solution for the maximum of the distance, due to the dependence of the symmetric matrix ℳ(n^)ℳ^𝑛\mathcalM(\hatn)caligraphic_M ( over^ start_ARG italic_n end_ARG ) on the unknown direction n^^𝑛\hatnover^ start_ARG italic_n end_ARG, it provides still a simple condition to evaluate the stationary points numerically. Not surprisingly, the above stationary condition is fulfilled for the special classes of states for which we have already obtained the maximum distance without rigorous optimization.



From the discussion given at the beginning of this section, two questions are being raised. The first one is that, is there any relation between the optimum measurement associated to the maximum distinguishability of the Alice’s outcomes with the one that allows one to extract the most information about the Alice’s qubit? We demonstrate in the following section that this is, indeed, the case. To do so, we provide some examples for which these two optimum measurements coincide exactly. The second question is that, when the optimum measurement of the maximum distinguished-outcomes process differs from the most information-gathering one, whether the former can be used to find a tight and faithful upper bound on the quantum discord? We will address these questions in the next section.



IV Maximum distinguished-outcomes measurement versus the most information-gathering one



Suppose Bob performs a measurement on his qubit in a direction 𝒏^∗superscript^𝒏∗\hat\boldsymboln^\astover^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT which fulfills the maximum distinguishability condition (14). Using this in the definition of quantum discord we find that



QB(ρ)≤QB∗(ρ),subscript𝑄𝐵𝜌superscriptsubscript𝑄𝐵∗𝜌Q_B(\rho)\leq Q_B^\ast(\rho),italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) ≤ italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) , (24) where QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) is the quantum discord of ρ𝜌\rhoitalic_ρ and QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) is its upper bound given by the measurement Πk∗B=12(𝟙+𝒏^k∗⋅𝝈)subscriptsuperscriptΠ∗absent𝐵𝑘12double-struck-𝟙⋅subscriptsuperscript^𝒏∗𝑘𝝈\Pi^\ast B_k=\frac12\left(\mathbb1+\hat\boldsymboln^\ast_k% \cdot\boldsymbol\sigma\right)roman_Π start_POSTSUPERSCRIPT ∗ italic_B end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( blackboard_𝟙 + over^ start_ARG bold_italic_n end_ARG start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT ⋅ bold_italic_σ ) for k=0,1𝑘01k=0,1italic_k = 0 , 1 that maximizes D2(𝒙~0,𝒙~1)superscript𝐷2subscript~𝒙0subscript~𝒙1D^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) . The following lemma shows that the above upper bound is faithful in a sense that it vanishes if and only if the bounded quantity vanishes.



QB∗(ρ)=0subscriptsuperscript𝑄∗𝐵𝜌0Q^\ast_B(\rho)=0italic_Q start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = 0 if and only if QB(ρ)=0subscript𝑄𝐵𝜌0Q_B(\rho)=0italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = 0.



The sufficient condition is a simple consequence of Eq. (24). To prove the necessary condition, let ρ𝜌\rhoitalic_ρ be a zero-discord on the Bob’s side. A two-qubit state has zero discord on Bob’s side if and only if either T=0𝑇0T=0italic_T = 0, or rank(T)=1r𝑎𝑛𝑘𝑇1\mathrmrank(T)=1roman_r italic_a italic_n italic_k ( italic_T ) = 1 and 𝒚𝒚\boldsymbolybold_italic_y belongs to the range of T𝑇Titalic_T LuPRA2011 ; Saman . For the first case, T=0𝑇0T=0italic_T = 0, from (13) it is easy to show the maximum of D2(𝒙~0,𝒙~1)superscript𝐷2subscript~𝒙0subscript~𝒙1D^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) happens if 𝒏^^𝒏\hat\boldsymbolnover^ start_ARG bold_italic_n end_ARG be parallel to 𝒚𝒚\boldsymbolybold_italic_y. We can easily check that AkhtarshenasJTP2015 such a 𝒏^^𝒏\hat\boldsymbolnover^ start_ARG bold_italic_n end_ARG yields QB(ρ)=0subscript𝑄𝐵𝜌0Q_B(\rho)=0italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = 0 and therefore QB(ρ)=QB∗(ρ)=0subscript𝑄𝐵𝜌superscriptsubscript𝑄𝐵∗𝜌0Q_B(\rho)=Q_B^\ast(\rho)=0italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) = 0. For the second case, without any loose of generality, we assume that 𝒚𝒚\boldsymbolybold_italic_y and T𝑇Titalic_T have the following form 𝒚=y3k^𝒚subscript𝑦3^𝑘\boldsymboly=y_3\,\hatkbold_italic_y = italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT over^ start_ARG italic_k end_ARG and T=diag0,0,t3𝑇diag00subscript𝑡3T=\mathrmdiag\0,0,t_3\italic_T = roman_diag 0 , 0 , italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT , so that the optimum measurement to get zero discord is noting but along the z𝑧zitalic_z-direction. On the other hand, for the distance between coherence vectors of the post-measurement states we get D2(𝒙~0,𝒙~1)=(t32+x2y32-2t3x3y3)(n^tk^k^tn^)/(1-y34n^tk^k^tn^)2,superscript𝐷2subscript~𝒙0subscript~𝒙1superscriptsubscript𝑡32superscript𝑥2superscriptsubscript𝑦322subscript𝑡3subscript𝑥3subscript𝑦3superscript^𝑛𝑡^𝑘superscript^𝑘𝑡^𝑛superscript1superscriptsubscript𝑦34superscript^𝑛𝑡^𝑘superscript^𝑘𝑡^𝑛2D^2(\widetilde\boldsymbolx_0,\widetilde\boldsymbolx_1)=\left(% t_3^2+x^2\kern 1.0pt\kern 1.0pty_3^2-2\kern 1.0pt\kern 1% .0ptt_3\kern 1.0pt\kern 1.0ptx_3\kern 1.0pt\kern 1.0pty_3% \right)(\hatn^t\kern 1.0pt\kern 1.0pt\hatk\hatk^t% \kern 1.0pt\kern 1.0pt\hatn)/(1-y_3^4\kern 1.0pt\kern 1.0% pt\hatn^t\kern 1.0pt\kern 1.0pt\hatk\hatk^t\kern 1.0pt% \kern 1.0pt\hatn)^2,italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ( over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT , over~ start_ARG bold_italic_x end_ARG start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ) = ( italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - 2 italic_t start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_x start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT ) ( over^ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT over^ start_ARG italic_k end_ARG over^ start_ARG italic_k end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT over^ start_ARG italic_n end_ARG ) / ( 1 - italic_y start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 4 end_POSTSUPERSCRIPT over^ start_ARG italic_n end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT over^ start_ARG italic_k end_ARG over^ start_ARG italic_k end_ARG start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT over^ start_ARG italic_n end_ARG ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , which takes its maximum value if n^=k^^𝑛^𝑘\hatn=\hatkover^ start_ARG italic_n end_ARG = over^ start_ARG italic_k end_ARG. We therefore find that for any zero-discord state we have QB∗(ρ)=0superscriptsubscript𝑄𝐵∗𝜌0Q_B^\ast(\rho)=0italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) = 0. ∎



In what follows we show that the above upper bound is tight in a sense that in more situations the equality is saturated. To this aim we consider states that we have considered in the last subsection.



(i) Canonical states 𝐲=0𝐲0\boldsymboly=0bold_italic_y = 0.-There is no complete solution to the quantum discord of the canonical states, although their geometry and optimization formula are simpler than the general states. Without losing generality we assume |t1|≥|t2|subscript𝑡1subscript𝑡2\left|t_1\right|\geq\left|t_2\right|| italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT | ≥ | italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT | and then do measurement along the greater semi-axis between i^^𝑖\hatiover^ start_ARG italic_i end_ARG and k^^𝑘\hatkover^ start_ARG italic_k end_ARG. We do this and plot the results versus the quantum discord in Fig. 1 for more than 20000 random states. There are many points on the bisector line showing that the optimized direction is very near to the direction of our upper bound. Moreover, non-exact results are not too far from quantum discord and distribution of points near the bisector line shows that the upper bound is very near to the quantum discord. Canonical states with Ttx→=0superscript𝑇t→𝑥0T^\mathrmt\vecx=0italic_T start_POSTSUPERSCRIPT roman_t end_POSTSUPERSCRIPT over→ start_ARG italic_x end_ARG = 0 analytically have been solved in AkhtarshenasJTP2015 and it is easy to see that for this subclass we have QB(ρ)=QB∗(ρ)subscript𝑄𝐵𝜌superscriptsubscript𝑄𝐵∗𝜌Q_B(\rho)=Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ).



(ii) States for which 𝐲𝐲\boldsymbolybold_italic_y is an eigenvector corresponding to the largest eigenvalue of M𝑀Mitalic_M.-In this case there are some classes of state for which there exist a good agreement between QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) and QB∗(ρ)subscriptsuperscript𝑄∗𝐵𝜌Q^\ast_B(\rho)italic_Q start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ). Consider states with



𝒙=xk^,𝒚=yi^,T=diagt1,t2,0.formulae-sequence𝒙𝑥^𝑘formulae-sequence𝒚𝑦^𝑖𝑇diagsubscript𝑡1subscript𝑡20\displaystyle\boldsymbolx=x\hatk,\qquad\boldsymboly=y\hati,\qquad T=% \mathrmdiag\t_1,t_2,0\.bold_italic_x = italic_x over^ start_ARG italic_k end_ARG , bold_italic_y = italic_y over^ start_ARG italic_i end_ARG , italic_T = roman_diag italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT , 0 . (25) In this case M=diagt12+x2y2,t22,0𝑀diagsuperscriptsubscript𝑡12superscript𝑥2superscript𝑦2superscriptsubscript𝑡220M=\mathrmdiag\t_1^2+x^2y^2,t_2^2,0\italic_M = roman_diag italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , 0 , and when t12+x2y2≥t22superscriptsubscript𝑡12superscript𝑥2superscript𝑦2superscriptsubscript𝑡22t_1^2+x^2y^2\geqt_2^2italic_t start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_x start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT italic_y start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ≥ italic_t start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT both QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) and QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) are obtained by measurement along 𝒚𝒚\boldsymbolybold_italic_y. In Fig. 2 we have plotted QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) versus QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) for more than 3000 random states of this category.



(iii) X𝑋Xitalic_X states.-For X𝑋Xitalic_X-states we consider the following classes separately.



1. M1≤M3subscript𝑀1subscript𝑀3M_1\leq M_3italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≤ italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. In this case QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) is very near to QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) and the relative error is less than 10-6superscript10610^-610 start_POSTSUPERSCRIPT - 6 end_POSTSUPERSCRIPT. For 10000 random states of this category any point lies on the bisector line (Fig. 3).



2. M1≥M3subscript𝑀1subscript𝑀3M_1\geq M_3italic_M start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ≥ italic_M start_POSTSUBSCRIPT 3 end_POSTSUBSCRIPT. In Figs . 4 and 5 we plot QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) vs. QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) for 20000 random states satisfying one of the Eqs. (17) and (18), and 5000 random states satisfying Eq. (16), respectively.



IV.1 QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) as a tight upper bound



Now we proceed to employ QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) as an upper bound and check if it is a tight one. Here we focus on a two parameters state as AlqasimiPRA2011



ρ=12(a00a01-a-b00001-a+b0a00a),𝜌12𝑎00𝑎missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression01𝑎𝑏00missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression001𝑎𝑏0missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression𝑎00𝑎missing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpressionmissing-subexpression\rho=\frac12\left(\beginarray[]*20ca&0&0&a\\ 0&1-a-b&0&0\\ 0&0&1-a+b&0\\ a&0&0&a\endarray\right)\,,italic_ρ = divide start_ARG 1 end_ARG start_ARG 2 end_ARG ( start_ARRAY start_ROW start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_a 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 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 start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 1 - italic_a - italic_b end_CELL start_CELL 0 end_CELL start_CELL 0 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 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 start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 1 - italic_a + italic_b end_CELL start_CELL 0 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 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 start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW start_ROW start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL italic_a 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 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 start_CELL end_CELL start_CELL end_CELL start_CELL end_CELL end_ROW end_ARRAY ) , (26) where 0≤a≤10𝑎10\leq a\leq 10 ≤ italic_a ≤ 1 and a-1≤b≤1-a𝑎1𝑏1𝑎a-1\leq b\leq 1-aitalic_a - 1 ≤ italic_b ≤ 1 - italic_a. The discord of this state is AlqasimiPRA2011



QB(ρ)=mina,q,subscript𝑄𝐵𝜌𝑎𝑞Q_B(\rho)=\min\a,q\,italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = roman_min italic_a , italic_q , (27) where



q𝑞\displaystyle qitalic_q =\displaystyle== a2log2[4a2(1-a)2-b2]-b2log2[(1+b)(1-a-b)(1-b)(1-a+b)]𝑎2subscript24superscript𝑎2superscript1𝑎2superscript𝑏2𝑏2𝑙𝑜subscript𝑔2delimited-[]1𝑏1𝑎𝑏1𝑏1𝑎𝑏\displaystyle\fraca2\,\log_2[\frac4\,a^2\left(1-a\right)^% 2-b^2]-\fracb2\,log_2[\frac(1+b)(1-a-b)(1-b)(1-a+b)]divide start_ARG italic_a end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ divide start_ARG 4 italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_a ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG ] - divide start_ARG italic_b end_ARG start_ARG 2 end_ARG italic_l italic_o italic_g start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ divide start_ARG ( 1 + italic_b ) ( 1 - italic_a - italic_b ) end_ARG start_ARG ( 1 - italic_b ) ( 1 - italic_a + italic_b ) end_ARG ]



+\displaystyle++ 12log2[4((1-a)2-b2)(1-b2)(1-a2-b2)]-a2+b22log2[1+a2+b21-a2+b2].12subscript24superscript1𝑎2superscript𝑏21superscript𝑏21superscript𝑎2superscript𝑏2superscript𝑎2superscript𝑏22subscript21superscript𝑎2superscript𝑏21superscript𝑎2superscript𝑏2\displaystyle\frac12\,\log_2[\frac4((1-a)^2-b^2)(1-b^2% )(1-a^2-b^2)]-\frac\sqrta^2+b^22\,\log_2[\frac% 1+\sqrta^2+b^21-\sqrta^2+b^2].divide start_ARG 1 end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ divide start_ARG 4 ( ( 1 - italic_a ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG start_ARG ( 1 - italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) ( 1 - italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT - italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) end_ARG ] - divide start_ARG square-root start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG start_ARG 2 end_ARG roman_log start_POSTSUBSCRIPT 2 end_POSTSUBSCRIPT [ divide start_ARG 1 + square-root start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG start_ARG 1 - square-root start_ARG italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG end_ARG ] . Here a𝑎aitalic_a and q𝑞qitalic_q are obtained by measurements n^=i^^𝑛^𝑖\hatn=\hatiover^ start_ARG italic_n end_ARG = over^ start_ARG italic_i end_ARG and n^=k^^𝑛^𝑘\hatn=\hatkover^ start_ARG italic_n end_ARG = over^ start_ARG italic_k end_ARG, respectively. Gaming News Marginal coherence vectors and the correlation matrix of this state are given by



𝒙=-𝒚=(00-b),T=(a000-a0002a-1).formulae-sequence𝒙𝒚00𝑏𝑇𝑎000𝑎0002𝑎1\boldsymbolx=-\boldsymboly=\left(\beginarray[]c0\\ 0\\ -b\endarray\right),\qquad T=\left(\beginarray[]ccca&0&0\\ 0&-a&0\\ 0&0&2a-1\endarray\right).bold_italic_x = - bold_italic_y = ( start_ARRAY start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL end_ROW start_ROW start_CELL - italic_b end_CELL end_ROW end_ARRAY ) , italic_T = ( start_ARRAY start_ROW start_CELL italic_a end_CELL start_CELL 0 end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL - italic_a end_CELL start_CELL 0 end_CELL end_ROW start_ROW start_CELL 0 end_CELL start_CELL 0 end_CELL start_CELL 2 italic_a - 1 end_CELL end_ROW end_ARRAY ) . (29) On the other hand, the maximal distance of this state is



Dmax2=maxa2,((2a-1)+b2)2(1-b2)2,subscriptsuperscript𝐷2superscript𝑎2superscript2𝑎1superscript𝑏22superscript1superscript𝑏22D^2_\max=\max\left\a^2,\frac\left((2a-1)+b^2\right)^2% \left(1-b^2\right)^2\right\,italic_D start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT start_POSTSUBSCRIPT roman_max end_POSTSUBSCRIPT = roman_max italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT , divide start_ARG ( ( 2 italic_a - 1 ) + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG , (30) where a2superscript𝑎2a^2italic_a start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT and ((2a-1)+b2)2(1-b2)2superscript2𝑎1superscript𝑏22superscript1superscript𝑏22\frac\left((2a-1)+b^2\right)^2\left(1-b^2\right)^2% divide start_ARG ( ( 2 italic_a - 1 ) + italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG start_ARG ( 1 - italic_b start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT end_ARG are obtained by measurements n^=i^^𝑛^𝑖\hatn=\hatiover^ start_ARG italic_n end_ARG = over^ start_ARG italic_i end_ARG and n^=k^^𝑛^𝑘\hatn=\hatkover^ start_ARG italic_n end_ARG = over^ start_ARG italic_k end_ARG, respectively.



Evidently, for b=0𝑏0b=0italic_b = 0 we have QB(ρ)=QB∗(ρ)subscript𝑄𝐵𝜌superscriptsubscript𝑄𝐵∗𝜌Q_B(\rho)=Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ). In Fig. 6 we plot QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) and QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) as a function of a𝑎aitalic_a for two cases b=0.3𝑏0.3b=0.3italic_b = 0.3 and b=0.7𝑏0.7b=0.7italic_b = 0.7, respectively. Except at a very small interval, we have QB(ρ)=QB∗(ρ)subscript𝑄𝐵𝜌superscriptsubscript𝑄𝐵∗𝜌Q_B(\rho)=Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) = italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ). A comparison of these figures reveals that as b𝑏bitalic_b increases the amounts of QD decreases and also the interval in which QB(ρ)≠QB∗(ρ)subscript𝑄𝐵𝜌superscriptsubscript𝑄𝐵∗𝜌Q_B(\rho) eq Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) ≠ italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ), grows up. Therefore, we observe that in high discordant states QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) is more precise.



Figure 6: (Color online) QB(ρ)subscript𝑄𝐵𝜌Q_B(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) and QB∗(ρ)superscriptsubscript𝑄𝐵∗𝜌Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) as a function of a𝑎aitalic_a for (a) b=0.3𝑏0.3b=0.3italic_b = 0.3 and (b) b=0.7𝑏0.7b=0.7italic_b = 0.7. The insets show behaviour in the rigion near QB(ρ)≠QB∗(ρ)subscript𝑄𝐵𝜌superscriptsubscript𝑄𝐵∗𝜌Q_B(\rho) eq Q_B^\ast(\rho)italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ ) ≠ italic_Q start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ).



Here we have defined Q∗(ρ)superscript𝑄∗𝜌Q^\ast(\rho)italic_Q start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) as the correlations that Bob can extract about Alice’s qubit by means of the most distinguishable measurements, i.e. measurements that Bob steers Alice to two post-measurement states with the most distinguishability. For some classes of state, we showed that this quantity is equal to the quantum discord Q(ρ)𝑄𝜌Q(\rho)italic_Q ( italic_ρ ). Although Q∗(ρ)superscript𝑄∗𝜌Q^\ast(\rho)italic_Q start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) may contain some classical correlations, the amount of classical correlations are not so much in particular for high discordant states. It provides, in general, a faithful and tight upper bound for the quantum discord. Marginal states at high discordant states have high mixedness and so they are near to Bell-diagonal states, where Q∗(ρ)superscript𝑄∗𝜌Q^\ast(\rho)italic_Q start_POSTSUPERSCRIPT ∗ end_POSTSUPERSCRIPT ( italic_ρ ) exactly equals to Q(ρ)𝑄𝜌Q(\rho)italic_Q ( italic_ρ ).



Acknowledgements.This work was supported by Ferdowsi University of Mashhad under Grant No. 3/38668 (1394/06/31).