General Bounds For Quantum Discord And Discord Distance
For any bipartite state, how strongly can one subsystem be quantum correlated with another? Using the Koashi-Winter relation, we study the upper bound of purified quantum discord, which is given by the sum of the von Neumann entropy of the unmeasured subsystem and the entanglement of formation shared between the unmeasured subsystem with the environment. In particular, we find that the Luo et al.’s conjecture on the quantum correlations and the Lindblad conjecture are all ture, when the entanglement of formation vanishes. Let the difference between the left discord and the right discord be captured by the discord distance. If the Lindblad conjecture is true, we show that the joint entropy is a tight upper bound for the discord distance. Further, we obtain a necessary and sufficient condition for saturating upper bounds of purified quantum discord and discord distance separately with the equality conditions for the Araki-Lieb inequality and the Lindblad conjecture. Furthermore, we show that the subadditive relation holds for any bipartite quantum discord.
03.65.Ud, 03.65.Ta, 03.67.Mn
pacs: Valid PACS appear here ††preprint: APS/123-QED
Entanglement has always been identified as a key ingredient in quantum information processing, and it can be used to perform certain tasks more efficiently than with classical correlation only. However, entanglement does not account for all the nonclassical properties of quantum correlations. This is because 1) Knill and Laflamme [1] showed that quantum computation, in which a collection of qubits in a completely mixed state couple to a single control qubit that has nonzero purity, can achieve an exponential improvement in efficiency over classical computers for a limited set of tasks; 2) within the context of the quantum measurement problem, Zurek and Vedral [2] concluded that even separable states usually contain correlations which are not entirely classical. These correlations are aptly named quantum discord (QD𝑄𝐷QDitalic_Q italic_D). As a measure of quantum correlations, QD𝑄𝐷QDitalic_Q italic_D encapsulates entanglement but goes further, as it is also present even in separable states. Intriguingly, this could be of practical significance because QD𝑄𝐷QDitalic_Q italic_D is more easily produced and maintained than entanglement [3]. Over the past decade, QD𝑄𝐷QDitalic_Q italic_D has been the focus of several theoretical and experimental studies addressing its formal characterization [4], its connection with entanglement distribution [5], remote state preparation [6] and unambiguous quantum state discrimination [7].
Despite the significance, the value of QD𝑄𝐷QDitalic_Q italic_D is notoriously difficult to calculate due to the optimization procedure involved. Analytical results are known only for certain special classes of states [8]. Particularly, it has been proved that it is impossible to obtain a closed expression for QD𝑄𝐷QDitalic_Q italic_D, even for general states of two qubits [9]. This fact makes it desirable to obtain some computable bounds for QD𝑄𝐷QDitalic_Q italic_D, and several attempts have been devoted to this issue in the past few years [10-12].
In 2010, Luo et al. [10] conjectured that both the classical and quantum correlations are (like the classical mutual information) bounded above by each subsystem’s entropy. The conjecture seems intuitively reasonable since the marginal entropies quantify the effective sizes of the two subsystems in view of the Schumacher noiseless coding theorem [13], and some further supporting evidences for the conjecture are given [10]. On the conjecture of classical correlations in bipartite states, it has been rigorous proved that it is upper bounded by the von Neumann entropies of each subsystem in [14-16]. In any bipartite state, the quantum mutual information can be separated into two parts: classical correlation and QD𝑄𝐷QDitalic_Q italic_D, and it is considered as the total amount of correlations in the bipartite state. Under these conditions, it has been proved [14,16-17] that QD𝑄𝐷QDitalic_Q italic_D is upper bounded by the von Neumann entropy of the measured subsystem. Recently, Xi et al. [11] revisited the upper bound of QD𝑄𝐷QDitalic_Q italic_D given by the von Neumann entropy of the measured subsystem using a tradeoff between the amount of classical correlation and QD𝑄𝐷QDitalic_Q italic_D in a tripartite pure state.
However, it has remained an open question as to what is the size relationship of QD𝑄𝐷QDitalic_Q italic_D and the von Neumann entropy of the unmeasured subsystem in any bipartite state (or the corresponding tripartite pure state). For this question, some class of states have been given in Refs. [11,16], which conform Luo et al.’s conjecture [10]. (From now on, when we refer to the Luo et al.’s conjecture we mean the quantum correlation is bounded by each subsystem’s entropy.) Recent investigations [14,18] give compelling evidence that there exist some bipartite states to disprove the conjecture. These facts motivate us to systematically investigate the upper bound of QD𝑄𝐷QDitalic_Q italic_D. And, it is interesting to give a necessary and sufficient condition for saturating the upper bound of QD𝑄𝐷QDitalic_Q italic_D. On the other hand, QD𝑄𝐷QDitalic_Q italic_D is generally not symmetric, i.e. the left discord and the right discord are unequal, which may be expected because conditional entropy is not symmetric [19]. The difference between the left discord and the right discord is captured by the discord distance. Then, how close is the discord distance?
By purifying the bipartite quantum systems and using the Koashi-Winter relation [20], we prove that, for every bipartite systems, QD𝑄𝐷QDitalic_Q italic_D cannot exceed the sum of the entropy of the unmeasured subsystem and entanglement of formation shared between the unmeasured subsystem with the environment. We further prove that the Luo et al.’s conjecture on the quantum correlations [10] and the Lindblad conjecture [21] are all true, when entanglement of formation vanishes. When the Lindblad conjecture is true, we show that the discord distance is upper bounded by the joint entropy. Intriguingly, we find that the subadditivity is a new important property of tripartite quantum discord.
This paper is organized as follows. In Sec. 2, we give a brief review on QD𝑄𝐷QDitalic_Q italic_D and Luo et al.’s conjecture [10]. In Sec. 3, we show that a subsystem being quantum correlated with another one limits its possible entanglement of formation with the environment, and then we prove Luo et al.’s conjecture when the entanglement of formation vanishes. In Sec. 4, we show that the Lindblad conjecture can be always true for a class of states, and then we prove the discord distance is upper bounded by the joint entropy. In Sec. 5, we show that for any tripartite state the subadditivity holds. Section 6 is the conclusion.
II Review of quantum discord
Two systems A𝐴Aitalic_A and B𝐵Bitalic_B are correlated if together they contain more information than taken separately. If A𝐴Aitalic_A and B𝐵Bitalic_B are classical systems and we measure the lack of information by entropy, correlations between two random variables of them are in information theory quantified by the mutual information
ℐ(A:B)=ℋ(A)+ℋ(B)-ℋ(A,B)fragmentsIfragments(A:B)Hfragments(A)Hfragments(B)Hfragments(A,B)\mathcalI(A:B)=\mathcalH(A)+\mathcalH(B)-\mathcalH(A,B)caligraphic_I ( italic_A : italic_B ) = caligraphic_H ( italic_A ) + caligraphic_H ( italic_B ) - caligraphic_H ( italic_A , italic_B ) (1) where ℋ(⋅)ℋ⋅\mathcalH(\cdot)caligraphic_H ( ⋅ ) stands for the Shannon entropy. For quantum systems A𝐴Aitalic_A and B𝐵Bitalic_B, i.e., Hilbert space ℋAB=ℋA⊗ℋBsubscriptℋ𝐴𝐵tensor-productsubscriptℋ𝐴subscriptℋ𝐵\mathcalH_AB=\mathcalH_A\otimes\mathcalH_Bcaligraphic_H start_POSTSUBSCRIPT italic_A italic_B end_POSTSUBSCRIPT = caligraphic_H start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ⊗ caligraphic_H start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT is a bipartite quantum composite system, and a bipartite quantum state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT (density matrix) of the composite system, the total amount of correlations is quantified by quantum mutual information [22] between the two subsystems A𝐴Aitalic_A and B𝐵Bitalic_B
ℐ(ρAB)=S(ρA)+S(ρB)-S(ρAB)ℐsuperscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝐵\mathcalI(\rho^AB)=S(\rho^A)+S(\rho^B)-S(\rho^AB)caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 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 ) (2) where S(⋅)𝑆⋅S(\cdot)italic_S ( ⋅ ) is the von Neumann entropy and ρA(B)=TrB(A)(ρAB)superscript𝜌𝐴𝐵𝑇subscript𝑟𝐵𝐴superscript𝜌𝐴𝐵\rho^A(B)=Tr_B(A)(\rho^AB)italic_ρ start_POSTSUPERSCRIPT italic_A ( italic_B ) end_POSTSUPERSCRIPT = italic_T italic_r start_POSTSUBSCRIPT italic_B ( italic_A ) end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) are reduced density matrices.
In contradistinction to the classical case, in the quantum analog there are many different measurements that can be performed on a subsystem, and measurements generally disturb the quantum state [19]. A measurement on subsystem A𝐴Aitalic_A is described by a positive-operator-valued measure (POVM) with elements EkAsuperscriptsubscript𝐸𝑘𝐴E_k^Aitalic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT, where k𝑘kitalic_k is the classical outcome. In analogy with Eq. (2) and to quantify the classical correlations (on B𝐵Bitalic_B) of the state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT independently of a measurement on A𝐴Aitalic_A, Vedral et al. [19] define an alternative version of the quantum mutual information
𝒥A(ρAB)=S(ρB)-minEkAS(B|EkA)subscript𝒥𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵𝑚𝑖subscript𝑛superscriptsubscript𝐸𝑘𝐴𝑆conditional𝐵superscriptsubscript𝐸𝑘𝐴\mathcalJ_A(\rho^AB)=S(\rho^B)-min_\E_k^A\S(B|\E_k^A\)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_m italic_i italic_n start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S ( italic_B | italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) (3) where the minimum is taken over all POVM measurements EkAsuperscriptsubscript𝐸𝑘𝐴\E_k^A\ italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT on A𝐴Aitalic_A, and S(B|EkA)=∑kpkS(ρB|k)𝑆conditional𝐵superscriptsubscript𝐸𝑘𝐴subscript𝑘subscript𝑝𝑘𝑆superscript𝜌conditional𝐵𝑘S(B|\E_k^A\)=\sum_kp_kS(\rho^B)italic_S ( italic_B | italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = ∑ start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B | italic_k end_POSTSUPERSCRIPT ) is the averaged conditional von Neumann entropy of the nonselective postmeasurement state ρB|k=TrA(EkAρAB)/pksuperscript𝜌conditional𝐵𝑘𝑇subscript𝑟𝐴superscriptsubscript𝐸𝑘𝐴superscript𝜌𝐴𝐵subscript𝑝𝑘\rho^k=Tr_A(E_k^A\rho^AB)/p_kitalic_ρ start_POSTSUPERSCRIPT italic_B | italic_k end_POSTSUPERSCRIPT = italic_T italic_r start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) / italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT with pk=Tr(EkAρAB)subscript𝑝𝑘𝑇𝑟superscriptsubscript𝐸𝑘𝐴superscript𝜌𝐴𝐵p_k=Tr(E_k^A\rho^AB)italic_p start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT = italic_T italic_r ( italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ). Then, QD𝑄𝐷QDitalic_Q italic_D (on B𝐵Bitalic_B) of a state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT under a measurement EkAsuperscriptsubscript𝐸𝑘𝐴\E_k^A\ italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT is defined as a difference between the total correlation, as given by the quantum mutual information in Eq. (2), and the classical correlation Eq. (3) [2,19]:
𝒟A(ρAB)=ℐ(ρAB)-𝒥A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵ℐsuperscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)=\mathcalI(\rho^AB)-\mathcalJ_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT )
=minEkAS(B|EkA)-S(B|A)absent𝑚𝑖subscript𝑛superscriptsubscript𝐸𝑘𝐴𝑆conditional𝐵superscriptsubscript𝐸𝑘𝐴𝑆conditional𝐵𝐴=min_\E_k^A\S(B|\E_k^A\)-S(B|A)= italic_m italic_i italic_n start_POSTSUBSCRIPT italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_POSTSUBSCRIPT italic_S ( italic_B | italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_B | italic_A ) (4) where S(B|A)=S(ρAB)-S(ρA)𝑆conditional𝐵𝐴𝑆superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴S(B|A)=S(\rho^AB)-S(\rho^A)italic_S ( italic_B | italic_A ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) denotes the conditional von Neumann entropy of ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT [22], and the minimum is taken over all POVM measurements EkAsuperscriptsubscript𝐸𝑘𝐴\E_k^A\ italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT . QD𝑄𝐷QDitalic_Q italic_D is not symmetric, i.e., in general 𝒟A(ρAB)≠𝒟B(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB) eq\mathcalD_B(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≠ caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), which can be interpreted in terms of the probability of confusing certain quantum states [19]. Here, 𝒟A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) refers to the left discord, while 𝒟B(ρAB)subscript𝒟𝐵superscript𝜌𝐴𝐵\mathcalD_B(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) refers to the right discord. From now on, when we refer to the discord we mean the left discord 𝒟A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ).
There are interesting points as follows
ℐ(A:B)≤minℋ(A),ℋ(B),fragmentsIfragments(A:B)minfragmentsHfragments(A),Hfragments(B),\mathcalI(A:B)\leq min\\mathcalH(A),\mathcalH(B)\,caligraphic_I ( italic_A : italic_B ) ≤ italic_m italic_i italic_n caligraphic_H ( italic_A ) , caligraphic_H ( italic_B ) ,
ℐ(ρAB)≤2⋅minS(ρA),S(ρB).ℐsuperscript𝜌𝐴𝐵⋅2𝑚𝑖𝑛𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵\mathcalI(\rho^AB)\leq 2\cdot min\S(\rho^A),S(\rho^B)\.caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ 2 ⋅ italic_m italic_i italic_n italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) , italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) . (5) In particular, if ρAB=|ψ⟩AB⟨ψ|superscript𝜌𝐴𝐵superscriptket𝜓𝐴𝐵bra𝜓\rho^AB=|\psi\rangle^AB\langle\psi|italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = | italic_ψ ⟩ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ⟨ italic_ψ | is a pure state, then ℐ(ρAB)=2S(ρA)=2S(ρB)ℐsuperscript𝜌𝐴𝐵2𝑆superscript𝜌𝐴2𝑆superscript𝜌𝐵\mathcalI(\rho^AB)=2S(\rho^A)=2S(\rho^B)caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 2 italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = 2 italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ), which saturates Eq. (5). Now the total correlation ℐ(ρAB)ℐsuperscript𝜌𝐴𝐵\mathcalI(\rho^AB)caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) is separated into classical correlation 𝒥A(ρAB)subscript𝒥𝐴superscript𝜌𝐴𝐵\mathcalJ_A(\rho^AB)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) and quantum correlation 𝒟A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ). Then, Eq. (5) can be recast into 𝒥A(ρAB)+𝒟A(ρAB)≤2⋅minS(ρA),S(ρB)subscript𝒥𝐴superscript𝜌𝐴𝐵subscript𝒟𝐴superscript𝜌𝐴𝐵⋅2𝑚𝑖𝑛𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵\mathcalJ_A(\rho^AB)+\mathcalD_A(\rho^AB)\leq 2\cdot min\S(\rho^% A),S(\rho^B)\caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ 2 ⋅ italic_m italic_i italic_n italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) , italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) . From the perspective of correlative capacities, Luo et al. naturally proposed the following conjectures on the quantum correlation in a bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, which splits the preceding inequalities [10,14]:
𝒟A(ρAB)≤S(ρA)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴\mathcalD_A(\rho^AB)\leq S(\rho^A)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) (I)
𝒟A(ρAB)≤S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)\leq S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) (II)
To illuminate the progress of the two conjectures, we first introduce two lemmas as follows.
Lemma 1 [16]. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, the QD𝑄𝐷QDitalic_Q italic_D satisfies (i) 𝒟A(ρAB)≤S(ρA)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴\mathcalD_A(\rho^AB)\leq S(\rho^A)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ); (ii) 𝒟A(ρAB)≤minS(ρA),S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑚𝑖𝑛𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)\leq min\S(\rho^A),S(\rho^B)\caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_m italic_i italic_n italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) , italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) , whenever S(ρA)≤S(ρB)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵S(\rho^A)\leq S(\rho^B)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) or ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is separable.
It is clear that Zhang and Wu [16] have proved Conjecture (I) after taking over all von Neumann measurements on the subsystem A𝐴Aitalic_A. They also found that 𝒟A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) is upper bounded by the von Neumann entropy of the subsystem B𝐵Bitalic_B for a class of states. However, the constraints on the class of states in (ii) are so strong that all entangled pure states satisfying Conjecture (II) are excluded. It is necessary to consider relaxing the constraints in Lemma 1. Using the method of purification and a tradeoff relation between QD𝑄𝐷QDitalic_Q italic_D and classical correlation, Xi et al. [11] got the following result.
Lemma 2 [11]. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, there is always a tripartite pure state ρABC=|ψ⟩ABC⟨ψ|superscript𝜌𝐴𝐵𝐶superscriptket𝜓𝐴𝐵𝐶bra𝜓\rho^ABC=|\psi\rangle^ABC\langle\psi|italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = | italic_ψ ⟩ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ⟨ italic_ψ | such that ρAB=TrC(ρABC)superscript𝜌𝐴𝐵𝑇subscript𝑟𝐶superscript𝜌𝐴𝐵𝐶\rho^AB=Tr_C(\rho^ABC)italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_T italic_r start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ), and the QD𝑄𝐷QDitalic_Q italic_D satisfies (i) 𝒟A(ρAB)≤S(ρA)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴\mathcalD_A(\rho^AB)\leq S(\rho^A)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ), with equality if and only if S(ρB)-S(ρA)=S(ρC)𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝑆superscript𝜌𝐶S(\rho^B)-S(\rho^A)=S(\rho^C)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ); (ii) 𝒟A(ρAB)=S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)=S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) if the equality S(ρA)-S(ρB)=S(ρC)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐶S(\rho^A)-S(\rho^B)=S(\rho^C)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_C end_POSTSUPERSCRIPT ) is satisfied.
Taking over all POVM measurements on the subsystem A𝐴Aitalic_A, Conjecture (I) is proved. And, the above sufficient condition (ii) for the situation of 𝒟A(ρAB)=S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)=S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) is also true for a class of states, including all entangled pure states as shown by Example 2 in Sec. 3.
The conclusions above appear to indicate that conjecture (II) is true. However, Li and Luo [14] pointed out that the inequality (II) is not valid in general and provided a counter-example [18]:
Example 1. Let ρACsuperscript𝜌𝐴𝐶\rho^ACitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT be a Werner state ρAC=d-xd3-dℐAC+dx-1d3-d∑i,j=1d|ij⟩AC⟨ij|superscript𝜌𝐴𝐶𝑑𝑥superscript𝑑3𝑑superscriptℐ𝐴𝐶𝑑𝑥1superscript𝑑3𝑑superscriptsubscript𝑖𝑗1𝑑superscriptket𝑖𝑗𝐴𝐶bra𝑖𝑗\rho^AC=\fracd-xd^3-d\mathcalI^AC+\fracdx-1d^3-d\sum_i,j=1% ^d|ij\rangle^AC\langle ij|italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT = divide start_ARG italic_d - italic_x end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_d end_ARG caligraphic_I start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT + divide start_ARG italic_d italic_x - 1 end_ARG start_ARG italic_d start_POSTSUPERSCRIPT 3 end_POSTSUPERSCRIPT - italic_d end_ARG ∑ start_POSTSUBSCRIPT italic_i , italic_j = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT | italic_i italic_j ⟩ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ⟨ italic_i italic_j | acting on 𝒞d⊗𝒞dtensor-productsuperscript𝒞𝑑superscript𝒞𝑑\mathcalC^d\otimes\mathcalC^dcaligraphic_C start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT ⊗ caligraphic_C start_POSTSUPERSCRIPT italic_d end_POSTSUPERSCRIPT, where x∈(-1,0)𝑥10x\in(-1,0)italic_x ∈ ( - 1 , 0 ), d≥6𝑑6d\geq 6italic_d ≥ 6, ket𝑖𝑗\ij\rangle\ is an orthogonal basis of product states for the composite system. Gaming News If ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT is a purification of ρACsuperscript𝜌𝐴𝐶\rho^ACitalic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT, then 𝒟A(ρAB)>S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)>S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) >italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ), which is quite counterintuitive.
III The upper bound of purified discord
For any general bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, we study bipartite QD𝑄𝐷QDitalic_Q italic_D in the tripartite purified system ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT such that ρAB=TrC(ρABC)superscript𝜌𝐴𝐵𝑇subscript𝑟𝐶superscript𝜌𝐴𝐵𝐶\rho^AB=Tr_C(\rho^ABC)italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_T italic_r start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ). There is a Koashi-Winter relation [20]
ℰF(ρBC)+𝒥A(ρAB)=S(ρB),subscriptℰ𝐹superscript𝜌𝐵𝐶subscript𝒥𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalE_F(\rho^BC)+\mathcalJ_A(\rho^AB)=S(\rho^B),caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) + caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) , (6) which is a tradeoff between entanglement of formation and classical correlation. Here ℰF(ρBC)subscriptℰ𝐹superscript𝜌𝐵𝐶\mathcalE_F(\rho^BC)caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) is entanglement of formation, defined as ℰF(ρBC)=minψi⟩ΣipiS[TrC(|ψi⟩⟨ψi|)]subscriptℰ𝐹superscript𝜌𝐵𝐶𝑚𝑖subscript𝑛subscript𝑝𝑖ketsubscript𝜓𝑖subscriptΣ𝑖subscript𝑝𝑖𝑆delimited-[]𝑇subscript𝑟𝐶ketsubscript𝜓𝑖brasubscript𝜓𝑖\mathcalE_F(\rho^BC)=min_\p_i,\Sigma_ip_iS[Tr% _C(|\psi_i\rangle\langle\psi_i|)]caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = italic_m italic_i italic_n start_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT , end_POSTSUBSCRIPT roman_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_S [ italic_T italic_r start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( | italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ ⟨ italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | ) ], and the minimum is taken over all pure ensembles pi,subscript𝑝𝑖ketsubscript𝜓𝑖\p_i, italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ satisfying ρBC=Σipi|ψi⟩⟨ψi|superscript𝜌𝐵𝐶subscriptΣ𝑖subscript𝑝𝑖ketsubscript𝜓𝑖brasubscript𝜓𝑖\rho^BC=\Sigma_ip_i|\psi_i\rangle\langle\psi_i|italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT = roman_Σ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT italic_p start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT | italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ⟩ ⟨ italic_ψ start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT |. Using the interplay between mutual information and QD𝑄𝐷QDitalic_Q italic_D, Xi et al. [11] obtained the monogamic relation between QD𝑄𝐷QDitalic_Q italic_D and the classical correlation as follows
𝒟A(ρAB)+𝒥A(ρAC)=S(ρA)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐶𝑆superscript𝜌𝐴\mathcalD_A(\rho^AB)+\mathcalJ_A(\rho^AC)=S(\rho^A)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) (7) which is universal for any tripartite pure states ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT. The equation tells us that the amount of quantum correlation between A𝐴Aitalic_A and B𝐵Bitalic_B, plus the amount of classical correlation between A𝐴Aitalic_A and the environment C𝐶Citalic_C, must be equal to the entropy of the measured subsystem A𝐴Aitalic_A. In particular, the monogamic relation (7) directly supplies a general upper bound for QD𝑄𝐷QDitalic_Q italic_D, i.e., the Conjecture (I), which has been proved in Refs. [11,14,16]. Analogously, substituting 𝒥A(ρAB)=ℐ(ρAB)-𝒟A(ρAB)subscript𝒥𝐴superscript𝜌𝐴𝐵ℐsuperscript𝜌𝐴𝐵subscript𝒟𝐴superscript𝜌𝐴𝐵\mathcalJ_A(\rho^AB)=\mathcalI(\rho^AB)-\mathcalD_A(\rho^AB)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) and -S(B|A)=ℐ(ρAB)-S(B)𝑆conditional𝐵𝐴ℐsuperscript𝜌𝐴𝐵𝑆𝐵-S(B|A)=\mathcalI(\rho^AB)-S(B)- italic_S ( italic_B | italic_A ) = caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_B ) into Eq. (6), we can also obtain the tradeoff [24] between QD𝑄𝐷QDitalic_Q italic_D and the entanglement of formation as follows
𝒟A(ρAB)-ℰF(ρBC)=-S(B|A)subscript𝒟𝐴superscript𝜌𝐴𝐵subscriptℰ𝐹superscript𝜌𝐵𝐶𝑆conditional𝐵𝐴\mathcalD_A(\rho^AB)-\mathcalE_F(\rho^BC)=-S(B|A)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = - italic_S ( italic_B | italic_A ) (8) which gives an operational interpretation of the negative conditional entropy as the difference between the two non-classical correlations.
By applying the Araki-Lieb inequality to Eq. (8), we get a general upper bound for QD𝑄𝐷QDitalic_Q italic_D as the following result, and determine which states saturate this bound.
Theorem 1. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT is a purification of it, then we have
𝒟A(ρAB)≤S(ρB)+ℰF(ρBC)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵subscriptℰ𝐹superscript𝜌𝐵𝐶\mathcalD_A(\rho^AB)\leq S(\rho^B)+\mathcalE_F(\rho^BC)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) + caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) (9) with equality if and only if S(ρA)-S(ρB)=S(ρC)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐶S(\rho^A)-S(\rho^B)=S(\rho^C)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_C end_POSTSUPERSCRIPT ).
This result shows that the necessary and sufficient conditions for saturating the upper bound of QD𝑄𝐷QDitalic_Q italic_D in Theorem 1 and the result (ii) in Lemma 2 are consistent results. The causes of consistent results is attributed to S(ρA)-S(ρB)=S(ρC)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐶S(\rho^A)-S(\rho^B)=S(\rho^C)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_C end_POSTSUPERSCRIPT ) and ρBC=ρB⊗ρCsuperscript𝜌𝐵𝐶tensor-productsuperscript𝜌𝐵superscript𝜌𝐶\rho^BC=\rho^B\otimes\rho^Citalic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT = italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT are equivalent [11].
As an illustration of the necessary and sufficient condition, let us consider the following two examples.
Example 2. Let ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT is a pure state. For any bipartite pure state, S(ρA)=S(ρB)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵S(\rho^A)=S(\rho^B)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) and QD𝑄𝐷QDitalic_Q italic_D is just the entropy of the reduced states of the subsystems. Using the fact that a entropy is zero if and only if the state is pure, we get the equality condition of the Araki-Lieb inequality S(ρA)-S(ρB)=S(ρAB)=0𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝐵0S(\rho^A)-S(\rho^B)=S(\rho^AB)=0italic_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 ) = 0. On the base of the Theorem 1 and Lemma 2, 𝒟A(ρAB)=𝒟B(ρAB)=S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)=\mathcalD_B(\rho^AB)=S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) hold.
If ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT is a purification of ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, we now apply Eq. (5) to the tripartite pure state ρABC=|ψ⟩ABC⟨ψ|superscript𝜌𝐴𝐵𝐶superscriptket𝜓𝐴𝐵𝐶bra𝜓\rho^ABC=|\psi\rangle^ABC\langle\psi|italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = | italic_ψ ⟩ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ⟨ italic_ψ |, and get ℐ(ρABC)≤2⋅S(ρAB)=0ℐsuperscript𝜌𝐴𝐵𝐶⋅2𝑆superscript𝜌𝐴𝐵0\mathcalI(\rho^ABC)\leq 2\cdot S(\rho^AB)=0caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ≤ 2 ⋅ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = 0. Then, ρABC=ρAB⊗ρCsuperscript𝜌𝐴𝐵𝐶tensor-productsuperscript𝜌𝐴𝐵superscript𝜌𝐶\rho^ABC=\rho^AB\otimes\rho^Citalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ⊗ italic_ρ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT. So we have 𝒥B(ρBC)=0subscript𝒥𝐵superscript𝜌𝐵𝐶0\mathcalJ_B(\rho^BC)=0caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, and obviously ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0. These results show that the condition of 𝒟A(ρAB)=S(ρA)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴\mathcalD_A(\rho^AB)=S(\rho^A)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) follows from Theorem 1 in Ref. [11], and the condition of 𝒟A(ρAB)=S(ρB)+EF(ρBC)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵subscript𝐸𝐹superscript𝜌𝐵𝐶\mathcalD_A(\rho^AB)=S(\rho^B)+E_F(\rho^BC)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) + italic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) follows from our Theorem 1.
Example 3. As given in Ref. [11], let
ρAB=14[(|00⟩A⟨00|+|01⟩A⟨01|)⊗|0⟩B⟨0|+(|00⟩A⟨10|+|01⟩A⟨11|)⊗|0⟩B⟨1|+fragmentssuperscript𝜌𝐴𝐵14superscriptfragments[superscriptfragments(|00⟩𝐴superscriptfragments⟨00||01⟩𝐴fragments⟨01|)tensor-product|0⟩𝐵superscriptfragments⟨0|superscriptfragments(|00⟩𝐴superscriptfragments⟨10||01⟩𝐴fragments⟨11|)tensor-product|0⟩𝐵fragments⟨1|\rho^AB=\frac14[(|00\rangle^A\langle 00|+|01\rangle^A\langle 01|)% \otimes|0\rangle^B\langle 0|+(|00\rangle^A\langle 10|+|01\rangle^A% \langle 11|)\otimes|0\rangle^B\langle 1|+italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG 4 end_ARG [ ( | 00 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 00 | + | 01 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 01 | ) ⊗ | 0 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⟨ 0 | + ( | 00 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 10 | + | 01 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 11 | ) ⊗ | 0 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⟨ 1 | +
(|10⟩A⟨00|+|11⟩A⟨01|)⊗|1⟩B⟨0|+(|10⟩A⟨10|+|11⟩A⟨11|)⊗|1⟩B⟨1|]fragmentssuperscriptfragments(|10⟩𝐴superscriptfragments⟨00||11⟩𝐴fragments⟨01|)tensor-product|1⟩𝐵⟨0|superscriptfragments(|10⟩𝐴superscriptfragments⟨10||11⟩𝐴fragments⟨11|)tensor-product|1⟩𝐵⟨1|](|10\rangle^A\langle 00|+|11\rangle^A\langle 01|)\otimes|1\rangle^B% \langle 0|+(|10\rangle^A\langle 10|+|11\rangle^A\langle 11|)\otimes|1% \rangle^B\langle 1|]( | 10 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 00 | + | 11 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 01 | ) ⊗ | 1 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⟨ 0 | + ( | 10 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 10 | + | 11 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ⟨ 11 | ) ⊗ | 1 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ⟨ 1 | ] where ℋA=𝒞2⊗𝒞2superscriptℋ𝐴tensor-productsuperscript𝒞2superscript𝒞2\mathcalH^A=\mathcalC^2\otimes\mathcalC^2caligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = caligraphic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ⊗ caligraphic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT, ℋB=𝒞2superscriptℋ𝐵superscript𝒞2\mathcalH^B=\mathcalC^2caligraphic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = caligraphic_C start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT. Notice that this state is a mixed state, since Tr[(ρAB)2]=12<1𝑇𝑟delimited-[]superscriptsuperscript𝜌𝐴𝐵2121Tr[(\rho^AB)^2]=\frac12<1italic_T italic_r [ ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) start_POSTSUPERSCRIPT 2 end_POSTSUPERSCRIPT ] = divide start_ARG 1 end_ARG start_ARG 2 end_ARG <1. We can always find a tripartite pure state ρABC=|Ψ⟩ABC⟨Ψ|superscript𝜌𝐴𝐵𝐶superscriptketΨ𝐴𝐵𝐶braΨ\rho^ABC=|\Psi\rangle^ABC\langle\Psi|italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = | roman_Ψ ⟩ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ⟨ roman_Ψ | such that ρAB=TrC(ρABC)superscript𝜌𝐴𝐵𝑇subscript𝑟𝐶superscript𝜌𝐴𝐵𝐶\rho^AB=Tr_C(\rho^ABC)italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = italic_T italic_r start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) and |Ψ⟩ABC=12(|ψ0⟩AB|0⟩C+|ψ1⟩AB|1⟩C)superscriptketΨ𝐴𝐵𝐶12superscriptketsubscript𝜓0𝐴𝐵superscriptket0𝐶superscriptketsubscript𝜓1𝐴𝐵superscriptket1𝐶|\Psi\rangle^ABC=\frac1\sqrt2(|\psi_0\rangle^AB|0\rangle^C+|\psi% _1\rangle^AB|1\rangle^C)| roman_Ψ ⟩ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟩ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT | 0 ⟩ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT + | italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT | 1 ⟩ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ), where |ψ0⟩=12(|00⟩A|0⟩B+|10⟩A|1⟩B)ketsubscript𝜓012superscriptket00𝐴superscriptket0𝐵superscriptket10𝐴superscriptket1𝐵|\psi_0\rangle=\frac1\sqrt2(|00\rangle^A|0\rangle^B+|10\rangle^A% |1\rangle^B)| italic_ψ start_POSTSUBSCRIPT 0 end_POSTSUBSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 00 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT | 0 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT + | 10 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT | 1 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) and |ψ1⟩=12(|01⟩A|0⟩B+|11⟩A|1⟩B)ketsubscript𝜓112superscriptket01𝐴superscriptket0𝐵superscriptket11𝐴superscriptket1𝐵|\psi_1\rangle=\frac1\sqrt2(|01\rangle^A|0\rangle^B+|11\rangle^A% |1\rangle^B)| italic_ψ start_POSTSUBSCRIPT 1 end_POSTSUBSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 01 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT | 0 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT + | 11 ⟩ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT | 1 ⟩ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ). The reduced states can be obtained ρA=IA4superscript𝜌𝐴superscript𝐼𝐴4\rho^A=\fracI^A4italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT = divide start_ARG italic_I start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT end_ARG start_ARG 4 end_ARG and ρB=IB2superscript𝜌𝐵superscript𝐼𝐵2\rho^B=\fracI^B2italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT = divide start_ARG italic_I start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT end_ARG start_ARG 2 end_ARG, where IAsuperscript𝐼𝐴I^Aitalic_I start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and IBsuperscript𝐼𝐵I^Bitalic_I start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT are respectively identity operators on ℋAsuperscriptℋ𝐴\mathcalH^Acaligraphic_H start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and ℋBsuperscriptℋ𝐵\mathcalH^Bcaligraphic_H start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT. After some calculations one obtains S(ρA)=2𝑆superscript𝜌𝐴2S(\rho^A)=2italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) = 2, S(ρAB)=S(ρB)=1𝑆superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵1S(\rho^AB)=S(\rho^B)=1italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) = 1.
Using the equality 𝒟A(ρAB)=ℰF(ρBC)+S(ρA)-S(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscriptℰ𝐹superscript𝜌𝐵𝐶𝑆superscript𝜌𝐴𝑆superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)=\mathcalE_F(\rho^BC)+S(\rho^A)-S(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), we know the upper bound in Eq. (9) is achievable if and only if S(ρA)-S(ρAB)=S(ρB)=1𝑆superscript𝜌𝐴𝑆superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵1S(\rho^A)-S(\rho^AB)=S(\rho^B)=1italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) = 1.
From the above results, we know Theorem 1 with equality if and only if the equality in the Araki-Lieb inequality holds. And, the unmeasured subsystem cannot correlate with the environment if QD𝑄𝐷QDitalic_Q italic_D between A𝐴Aitalic_A and B𝐵Bitalic_B is equal to the entropy of the unmeasured subsystem. What is astonishing is that the result and Theorem 2 in Ref. [11] are almost entirely the same with equality. However, the upper bound of QD𝑄𝐷QDitalic_Q italic_D is obtained in a quite different form from Conjecture (II). To this end, we have the following sufficient condition for the situation of Conjecture (II), i.e., 𝒟A(ρAB)≤S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)\leq S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ).
Corollary 1. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT is a purification of it and ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, then we have
𝒟A(ρAB)≤S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)\leq S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) (10) with equality if and only if S(ρA)-S(ρB)=S(ρC)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐶S(\rho^A)-S(\rho^B)=S(\rho^C)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_C end_POSTSUPERSCRIPT ).
The result shows that Conjecture (II) is true if the equality ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0 is satisfied. In other words, QD𝑄𝐷QDitalic_Q italic_D between A𝐴Aitalic_A and B𝐵Bitalic_B is upper bounded by the entropy of the unmeasured subsystem if the unmeasured subsystem B𝐵Bitalic_B cannot entangled with the environment C𝐶Citalic_C. Further, the maximal QD𝑄𝐷QDitalic_Q italic_D between A𝐴Aitalic_A and B𝐵Bitalic_B will even forbid B𝐵Bitalic_B from being correlated to other systems outside this composite system. One thing to be noted is 𝒟A(ρAB)=-S(B|A)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆conditional𝐵𝐴\mathcalD_A(\rho^AB)=-S(B|A)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = - italic_S ( italic_B | italic_A ) by the Koashi-Winter equality when ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, so we are just trying to formally establish the relation between discord and the unmeasured subsystem.
IV The upper bound of discord distance
The discord distance of a quantum state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT under POVM measurements EkAsuperscriptsubscript𝐸𝑘𝐴\E_k^A\ italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT and EkBsuperscriptsubscript𝐸𝑘𝐵\E_k^B\ italic_E start_POSTSUBSCRIPT italic_k end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT is defined as a difference between the two one-side discords (left discord and right disocrd):
|𝒟A(ρAB)-𝒟B(ρAB)|.subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵|\mathcalD_A(\rho^AB)-\mathcalD_B(\rho^AB)|.| caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) | . (11) To be clearer, we substitute 𝒟i(ρAB)=ℐ(ρAB)-𝒥i(ρAB)subscript𝒟𝑖superscript𝜌𝐴𝐵ℐsuperscript𝜌𝐴𝐵subscript𝒥𝑖superscript𝜌𝐴𝐵\mathcalD_i(\rho^AB)=\mathcalI(\rho^AB)-\mathcalJ_i(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_J start_POSTSUBSCRIPT italic_i end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), where i=A,B𝑖𝐴𝐵i=A,Bitalic_i = italic_A , italic_B, and obtain the classical correlation distance |𝒥A(ρAB)-𝒥B(ρAB)|subscript𝒥𝐴superscript𝜌𝐴𝐵subscript𝒥𝐵superscript𝜌𝐴𝐵|\mathcalJ_A(\rho^AB)-\mathcalJ_B(\rho^AB)|| caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) |. A suitable physical interpretation of the discord distance is how close are two classical correlations for A𝐴Aitalic_A versus B𝐵Bitalic_B in the quantum state. The Eq. (11) shows that smaller difference implies more fair communication in an ideal quantum network. From Lemma 2 in Sec. 2 and Corollary 1 in Sec. 3, we have a sufficient condition for the situation of 𝒟A(ρAB)=𝒟B(ρAB)=S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)=\mathcalD_B(\rho^AB)=S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) if the equality S(ρA)-S(ρB)=S(ρAB)𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵𝑆superscript𝜌𝐴𝐵S(\rho^A)-S(\rho^B)=S(\rho^AB)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 ) is satisfied. Meantime, 𝒥A(ρAB)=𝒥B(ρAB)=-S(B|A)subscript𝒥𝐴superscript𝜌𝐴𝐵subscript𝒥𝐵superscript𝜌𝐴𝐵𝑆conditional𝐵𝐴\mathcalJ_A(\rho^AB)=\mathcalJ_B(\rho^AB)=-S(B|A)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = - italic_S ( italic_B | italic_A ) can also be deduced from the above sufficient condition. Though, in its present form, a necessary and sufficient condition cannot be obtained. We leave it as an open question whether one could consider variations of the proof procedure that would render it suitable for fair quantum communication tasks.
Combining the monogamic relation in Eq. (6) with the equality condition for the strong subadditivity inequality
S(B|A)+S(B|C)=0,𝑆conditional𝐵𝐴𝑆conditional𝐵𝐶0S(B|A)+S(B|C)=0,italic_S ( italic_B | italic_A ) + italic_S ( italic_B | italic_C ) = 0 , (12) we have the following result.
Theorem 2. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT is a purification of it and ℰF(ρAC)=ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐴𝐶subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^AC)=\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, then we have
|𝒟A(ρAB)-𝒟B(ρAB)|≤S(ρAB).subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝐵|\mathcalD_A(\rho^AB)-\mathcalD_B(\rho^AB)|\leq S(\rho^AB).| caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) | ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) . (13)
To prove this theorem, we first introduce the Lindblad conjecture [21], which states that the classical correlation account for at least half of the total correlation, or equivalently, correlations are more classical than quantum.
Lindblad conjecture. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, Lindblad proposed the following conjecture 𝒟A(ρAB)≤𝒥A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)\leq\mathcalJ_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), which is based on several intuitive observations [10,21].
Luo and Zhang [21] disproved the intuitive conjecture of Lindblad by evaluating an observable correlations for generic two-qubit states and obtain analytical expressions in some particular cases. And, they provided a counter-example:
Example 4. Let ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT be a Werner state ρAB=IAB6+13|ψ-⟩AB⟨ψ-|superscript𝜌𝐴𝐵superscript𝐼𝐴𝐵613superscriptketsuperscript𝜓𝐴𝐵brasuperscript𝜓\rho^AB=\fracI^AB6+\frac13|\psi^-\rangle^AB\langle\psi^-|italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT = divide start_ARG italic_I start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT end_ARG start_ARG 6 end_ARG + divide start_ARG 1 end_ARG start_ARG 3 end_ARG | italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ⟩ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ⟨ italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT |, where |ψ-⟩=12(|01⟩-|10⟩)ketsuperscript𝜓12ket01ket10|\psi^-\rangle=\frac1\sqrt2(|01\rangle-|10\rangle)| italic_ψ start_POSTSUPERSCRIPT - end_POSTSUPERSCRIPT ⟩ = divide start_ARG 1 end_ARG start_ARG square-root start_ARG 2 end_ARG end_ARG ( | 01 ⟩ - | 10 ⟩ ). In such a situation,
𝒟A(ρAB)>𝒥A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)>\mathcalJ_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) >caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ), where 𝒟A(ρAB)≃0.126similar-to-or-equalssubscript𝒟𝐴superscript𝜌𝐴𝐵0.126\mathcalD_A(\rho^AB)\simeq 0.126caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≃ 0.126, 𝒥A(ρAB)≃0.082similar-to-or-equalssubscript𝒥𝐴superscript𝜌𝐴𝐵0.082\mathcalJ_A(\rho^AB)\simeq 0.082caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≃ 0.082.
From Corollary 1, we find that Lindblad conjecture can be true, and introduce a lemma as follows.
Lemma 3. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT is a purification of it and ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, then we have 𝒟A(ρAB)≤𝒥A(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)\leq\mathcalJ_A(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ).
Proof. From the Corollary 1 in Sec. 3, we get 𝒟A(ρAB)≤S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)\leq S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) when ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0. While under the same condition, we have 𝒥A(ρAB)=S(ρB)subscript𝒥𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalJ_A(\rho^AB)=S(\rho^B)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) due to the Koashi-Winter relation 𝒥A(ρAB)+ℰF(ρBC)=S(ρB)subscript𝒥𝐴superscript𝜌𝐴𝐵subscriptℰ𝐹superscript𝜌𝐵𝐶𝑆superscript𝜌𝐵\mathcalJ_A(\rho^AB)+\mathcalE_F(\rho^BC)=S(\rho^B)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ), i.e. Eq. (6). It turns out that Lemma 3 is true.
The result shows that QD𝑄𝐷QDitalic_Q italic_D is always bounded from above by the classical correlation when the entanglement of formation, between the unmeasured subsystem B𝐵Bitalic_B and the environment C𝐶Citalic_C, vanishes. Further, the maximal QD𝑄𝐷QDitalic_Q italic_D between A𝐴Aitalic_A and B𝐵Bitalic_B will forbid system B𝐵Bitalic_B from being correlated to other systems outside this composite system when Lindblad conjecture is true. The phenomenon is very real significance in quantum information theory. The accessible information is a measure of how well the receiver can do at inferring the information being included in the other subsystem [22]. And, the difference between the accessible information achieves the maximum of the classical correlation 𝒥B(ρBC)subscript𝒥𝐵superscript𝜌𝐵𝐶\mathcalJ_B(\rho^BC)caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) if and only if Lindblad conjecture is true and ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0.
Using Lemma 3, we will complete the proof of Theorem 2.
Proof of Theorem 2. Consider that 𝒟B(ρBC)≤𝒥B(ρBC)subscript𝒟𝐵superscript𝜌𝐵𝐶subscript𝒥𝐵superscript𝜌𝐵𝐶\mathcalD_B(\rho^BC)\leq\mathcalJ_B(\rho^BC)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) ≤ caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ), i.e., 2⋅𝒥B(ρBC)≥ℐ(ρBC)⋅2subscript𝒥𝐵superscript𝜌𝐵𝐶ℐsuperscript𝜌𝐵𝐶2\cdot\mathcalJ_B(\rho^BC)\geq\mathcalI(\rho^BC)2 ⋅ caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) ≥ caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ), the inequality follows from the result of Lemma 3 with ℰF(ρAC)=0subscriptℰ𝐹superscript𝜌𝐴𝐶0\mathcalE_F(\rho^AC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = 0. We have
2𝒥B(ρBC)+S(B|C)≥ℐ(ρBC)+S(B|C)2subscript𝒥𝐵superscript𝜌𝐵𝐶𝑆conditional𝐵𝐶ℐsuperscript𝜌𝐵𝐶𝑆conditional𝐵𝐶2\mathcalJ_B(\rho^BC)+S(B|C)\geq\mathcalI(\rho^BC)+S(B|C)2 caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) + italic_S ( italic_B | italic_C ) ≥ caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) + italic_S ( italic_B | italic_C )
=S(ρB)≥𝒥A(ρAB).absent𝑆superscript𝜌𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵=S(\rho^B)\geq\mathcalJ_A(\rho^AB).= italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) ≥ caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) . (14) The first inequality follows by adding the item S(B|C)𝑆conditional𝐵𝐶S(B|C)italic_S ( italic_B | italic_C ) to the left and right hand sides of 2⋅𝒥B(ρBC)≥ℐ(ρBC)⋅2subscript𝒥𝐵superscript𝜌𝐵𝐶ℐsuperscript𝜌𝐵𝐶2\cdot\mathcalJ_B(\rho^BC)\geq\mathcalI(\rho^BC)2 ⋅ caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) ≥ caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ). The first equality follows because the quantum mutual information has an equivalent form for the classical mutual information ℐ(ρBC)=S(ρB)-S(B|C)ℐsuperscript𝜌𝐵𝐶𝑆superscript𝜌𝐵𝑆conditional𝐵𝐶\mathcalI(\rho^BC)=S(\rho^B)-S(B|C)caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_B | italic_C ). The second inequality follows from the Eq. (6), which is the Koashi-Winter relation.
Using Eqs. (12) and (14), the classical correlation 𝒥A(ρAB)subscript𝒥𝐴superscript𝜌𝐴𝐵\mathcalJ_A(\rho^AB)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) admits the following representation:
𝒥A(ρAB)≤2⋅𝒥B(ρBC)-S(B|A)subscript𝒥𝐴superscript𝜌𝐴𝐵⋅2subscript𝒥𝐵superscript𝜌𝐵𝐶𝑆conditional𝐵𝐴\mathcalJ_A(\rho^AB)\leq 2\cdot\mathcalJ_B(\rho^BC)-S(B|A)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ 2 ⋅ caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_B | italic_A )
=2⋅𝒥B(ρBC)+ℐ(ρAB)-S(ρB).absent⋅2subscript𝒥𝐵superscript𝜌𝐵𝐶ℐsuperscript𝜌𝐴𝐵𝑆superscript𝜌𝐵=2\cdot\mathcalJ_B(\rho^BC)+\mathcalI(\rho^AB)-S(\rho^B).= 2 ⋅ caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) + caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) . (15) In terms of QD𝑄𝐷QDitalic_Q italic_D and the von Neumann entropy of the corresponding unmeasured subsystem, Eq. (15) can be rephrased as follows.
S(ρB)≤2𝒥B(ρBC)+𝒟A(ρAB).𝑆superscript𝜌𝐵2subscript𝒥𝐵superscript𝜌𝐵𝐶subscript𝒟𝐴superscript𝜌𝐴𝐵S(\rho^B)\leq 2\mathcalJ_B(\rho^BC)+\mathcalD_A(\rho^AB).italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) ≤ 2 caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) + caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) . (16)
In addition, we have
𝒟A(ρAB)≤S(ρB)subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)\leq S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) (17) from the Corollary 1 and ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0.
Using Eq. (7) and Eqs. (16-17), we have
|𝒟A(ρAB)-𝒟B(ρAB)|=|𝒟A(ρAB)+𝒥B(ρBC)-S(ρB)|≤𝒥B(ρBC).subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒥𝐵superscript𝜌𝐵𝐶𝑆superscript𝜌𝐵subscript𝒥𝐵superscript𝜌𝐵𝐶|\mathcalD_A(\rho^AB)-\mathcalD_B(\rho^AB)|=|\mathcalD_A(\rho^% AB)+\mathcalJ_B(\rho^BC)-S(\rho^B)|\leq\mathcalJ_B(\rho^BC).| caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) | = | caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) | ≤ caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) .
Under the assumption ℰF(ρAC)=ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐴𝐶subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^AC)=\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, it can be shown that JA(ρAC)=JB(ρBC)=S(ρAB)subscript𝐽𝐴superscript𝜌𝐴𝐶subscript𝐽𝐵superscript𝜌𝐵𝐶𝑆superscript𝜌𝐴𝐵J_A(\rho^AC)=J_B(\rho^BC)=S(\rho^AB)italic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = italic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) by using the Koashi-Winter relation for other permutations of A𝐴Aitalic_A, B𝐵Bitalic_B, and C𝐶Citalic_C.
This completes the proof of Theorem 2.
Note: In fact, this is a direct result of the Koashi-Winter equality (8) and the Araki-Lied inequality, see the following:
𝒟A(ρAB)=ℰF(ρBC)-S(B|A),subscript𝒟𝐴superscript𝜌𝐴𝐵subscriptℰ𝐹superscript𝜌𝐵𝐶𝑆conditional𝐵𝐴\mathcalD_A(\rho^AB)=\mathcalE_F(\rho^BC)-S(B|A),caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_B | italic_A ) , 𝒟B(ρAB)=ℰF(ρAC)-S(A|B).subscript𝒟𝐵superscript𝜌𝐴𝐵subscriptℰ𝐹superscript𝜌𝐴𝐶𝑆conditional𝐴𝐵\mathcalD_B(\rho^AB)=\mathcalE_F(\rho^AC)-S(A|B).caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) - italic_S ( italic_A | italic_B ) . From the above two Koashi-Winter equalities one can obtain
𝒟A(ρAB)-𝒟B(ρAB)=S(ρA)-S(ρB),subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝑆superscript𝜌𝐵\mathcalD_A(\rho^AB)-\mathcalD_B(\rho^AB)=S(\rho^A)-S(\rho^B),caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) , Using ℰF(ρAC)=ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐴𝐶subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^AC)=\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, we have |𝒟A(ρAB)-𝒟B(ρAB)|≤𝒥A(ρAB).subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵|\mathcalD_A(\rho^AB)-\mathcalD_B(\rho^AB)|\leq\mathcalJ_A(% \rho^AB).| caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) | ≤ caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) . This completes the proof of Theorem 2. Using the above detailed proof process, it is mainly because we want to show the Lindblad conjecture’s value.
Further, we give the explicit characterization of the quantum states saturating the upper bound of discord distance as follows.
Corollary 2. For any bipartite state ρABsuperscript𝜌𝐴𝐵\rho^ABitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT, ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT is a purification of it and ℰF(ρAC)=ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐴𝐶subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^AC)=\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) = caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0, then we have
𝒟B(ρAB)-𝒟A(ρAB)≤S(ρAB)subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝐵\mathcalD_B(\rho^AB)-\mathcalD_A(\rho^AB)\leq S(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) (18) with equality if and only if ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0 and 𝒟B(ρAB)=JB(ρAB)subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝐽𝐵superscript𝜌𝐴𝐵\mathcalD_B(\rho^AB)=J_B(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ).
Proof. Eq. (18) is true which can easily be deduced from Theorem 2.
Based on Lemma 3 and the Koashi-Winter relation, we can give a necessary and sufficient condition for the situation of 𝒟B(ρAB)-𝒟A(ρAB)=S(ρAB)subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝐵\mathcalD_B(\rho^AB)-\mathcalD_A(\rho^AB)=S(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ). The equality can be rephrased as follows.
0=𝒟B(ρAB)-𝒟A(ρAB)-S(ρAB)=S(ρB)-2𝒥B(ρBC)-𝒟A(ρAB)0subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝒟𝐴superscript𝜌𝐴𝐵𝑆superscript𝜌𝐴𝐵𝑆superscript𝜌𝐵2subscript𝒥𝐵superscript𝜌𝐵𝐶subscript𝒟𝐴superscript𝜌𝐴𝐵0=\mathcalD_B(\rho^AB)-\mathcalD_A(\rho^AB)-S(\rho^AB)=S(\rho^B% )-2\mathcalJ_B(\rho^BC)-\mathcalD_A(\rho^AB)0 = caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - 2 caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) - caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT )
=S(ρB)-2𝒥B(ρBC)-(ℐ(ρAB)-𝒥A(ρAB))absent𝑆superscript𝜌𝐵2subscript𝒥𝐵superscript𝜌𝐵𝐶ℐsuperscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵=S(\rho^B)-2\mathcalJ_B(\rho^BC)-(\mathcalI(\rho^AB)-\mathcalJ_% A(\rho^AB))= italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - 2 caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) - ( caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) )
=𝒥A(ρAB)-S(B|C)-2𝒥B(ρBC)absentsubscript𝒥𝐴superscript𝜌𝐴𝐵𝑆conditional𝐵𝐶2subscript𝒥𝐵superscript𝜌𝐵𝐶=\mathcalJ_A(\rho^AB)-S(B|C)-2\mathcalJ_B(\rho^BC)= caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_B | italic_C ) - 2 caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ). For the second equality, by the Koashi-Winter relation 𝒟B(ρAB)+𝒥B(ρBC)=S(ρB)subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝒥𝐵superscript𝜌𝐵𝐶𝑆superscript𝜌𝐵\mathcalD_B(\rho^AB)+\mathcalJ_B(\rho^BC)=S(\rho^B)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ); for the second equality, by 𝒟A(ρAB)+𝒥A(ρAB)=ℐ(ρAB)subscript𝒟𝐴superscript𝜌𝐴𝐵subscript𝒥𝐴superscript𝜌𝐴𝐵ℐsuperscript𝜌𝐴𝐵\mathcalD_A(\rho^AB)+\mathcalJ_A(\rho^AB)=\mathcalI(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ); for the last equality, by ℐ(ρAB)=S(ρB)-S(B|A)ℐsuperscript𝜌𝐴𝐵𝑆superscript𝜌𝐵𝑆conditional𝐵𝐴\mathcalI(\rho^AB)=S(\rho^B)-S(B|A)caligraphic_I ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) - italic_S ( italic_B | italic_A ) and Eq. (12).
From the result in Eq. (15), we know that 𝒥A(ρAB)=S(B|C)+2𝒥B(ρBC)subscript𝒥𝐴superscript𝜌𝐴𝐵𝑆conditional𝐵𝐶2subscript𝒥𝐵superscript𝜌𝐵𝐶\mathcalJ_A(\rho^AB)=S(B|C)+2\mathcalJ_B(\rho^BC)caligraphic_J start_POSTSUBSCRIPT italic_A end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_S ( italic_B | italic_C ) + 2 caligraphic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) if and only if ℰF(ρBC)=0subscriptℰ𝐹superscript𝜌𝐵𝐶0\mathcalE_F(\rho^BC)=0caligraphic_E start_POSTSUBSCRIPT italic_F end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) = 0 and 𝒟B(ρAB)=JB(ρAB)subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝐽𝐵superscript𝜌𝐴𝐵\mathcalD_B(\rho^AB)=J_B(\rho^AB)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) = italic_J start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ). This completes the proof of Corollary 2.
V Subadditivity of discord
Quantum discord has the following properties [19]: asymmetric, nonnegative, invariant under local-unitary transformations, and vanishes if and only if the related state is classical quantum. Furthermore, discord is bounded from above by the von Neumann entropy of the measured subsystem [11,12], and the sum of the von Neumann entropy of the unmeasured subsystem and the entanglement of formation shared between the unmeasured subsystem with the environment (Theorem 1 in Sec. 3). In 2012, Prabhu et al. [25] investigated the monogamy relationship for quantum discord. They showed that for any tripartite state ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT, the inequality 𝒟B(ρAB)+𝒟C(ρAC)≤𝒟BC(ρABC)subscript𝒟𝐵superscript𝜌𝐴𝐵subscript𝒟𝐶superscript𝜌𝐴𝐶subscript𝒟𝐵𝐶superscript𝜌𝐴𝐵𝐶\mathcalD_B(\rho^AB)+\mathcalD_C(\rho^AC)\leq\mathcalD_BC(\rho% ^ABC)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B end_POSTSUPERSCRIPT ) + caligraphic_D start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_C end_POSTSUPERSCRIPT ) ≤ caligraphic_D start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) holds if and only if I(A:B:C)≥JBC(ρABC)fragmentsIfragments(A:B:C)subscript𝐽𝐵𝐶fragments(superscript𝜌𝐴𝐵𝐶)I(A:B:C)\geq J_BC(\rho^ABC)italic_I ( italic_A : italic_B : italic_C ) ≥ italic_J start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ), where I(A:B:C)=I(A:B)-I(A:B|C)fragmentsIfragments(A:B:C)Ifragments(A:B)Ifragments(A:B|C)I(A:B:C)=I(A:B)-I(A:B|C)italic_I ( italic_A : italic_B : italic_C ) = italic_I ( italic_A : italic_B ) - italic_I ( italic_A : italic_B | italic_C ).
Using the definition of discord, we prove a very important property relating discord to the subadditivity as follows.
Theorem 3. For any tripartite state ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT, we have
𝒟BC(ρABC)≤𝒟B(ρABC)+𝒟C(ρABC).subscript𝒟𝐵𝐶superscript𝜌𝐴𝐵𝐶subscript𝒟𝐵superscript𝜌𝐴𝐵𝐶subscript𝒟𝐶superscript𝜌𝐴𝐵𝐶\mathcalD_BC(\rho^ABC)\leq\mathcalD_B(\rho^ABC)+\mathcalD_C(% \rho^ABC).caligraphic_D start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ≤ caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) + caligraphic_D start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) . (19)
Proof. Let ΠB*superscriptsubscriptΠ𝐵\Pi_B^*roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT and ΠC*superscriptsubscriptΠ𝐶\Pi_C^*roman_Π start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT be the optimal complete projective measurements over B𝐵Bitalic_B and C𝐶Citalic_C for the sake of 𝒟B(ρABC)subscript𝒟𝐵superscript𝜌𝐴𝐵𝐶\mathcalD_B(\rho^ABC)caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) and 𝒟C(ΠB*(ρABC))subscript𝒟𝐶superscriptsubscriptΠ𝐵superscript𝜌𝐴𝐵𝐶\mathcalD_C(\Pi_B^*(\rho^ABC))caligraphic_D start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) separately. Then the inequality (19) is obtained as follows:
𝒟BC(ρABC)≤S(ρABC||ΠB*⊗ΠC*(ρABC))fragmentssubscript𝒟𝐵𝐶fragments(superscript𝜌𝐴𝐵𝐶)Sfragments(superscript𝜌𝐴𝐵𝐶||superscriptsubscriptΠ𝐵tensor-productsuperscriptsubscriptΠ𝐶fragments(superscript𝜌𝐴𝐵𝐶))\mathcalD_BC(\rho^ABC)\leq S(\rho^ABC||\Pi_B^*\otimes\Pi_C^*(% \rho^ABC))caligraphic_D start_POSTSUBSCRIPT italic_B italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT | | roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) )
=S(ΠB*⊗ΠC*(ρABC))-S(ρABC)absent𝑆tensor-productsuperscriptsubscriptΠ𝐵superscriptsubscriptΠ𝐶superscript𝜌𝐴𝐵𝐶𝑆superscript𝜌𝐴𝐵𝐶=S(\Pi_B^*\otimes\Pi_C^*(\rho^ABC))-S(\rho^ABC)= italic_S ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT )
=[S(ΠB*(ρABC))-S(ρABC)]+[S(ΠB*⊗ΠC*(ρABC))-S(ΠB*(ρABC))]absentdelimited-[]𝑆superscriptsubscriptΠ𝐵superscript𝜌𝐴𝐵𝐶𝑆superscript𝜌𝐴𝐵𝐶delimited-[]𝑆tensor-productsuperscriptsubscriptΠ𝐵superscriptsubscriptΠ𝐶superscript𝜌𝐴𝐵𝐶𝑆superscriptsubscriptΠ𝐵superscript𝜌𝐴𝐵𝐶=[S(\Pi_B^*(\rho^ABC))-S(\rho^ABC)]+[S(\Pi_B^*\otimes\Pi_C^*(% \rho^ABC))-S(\Pi_B^*(\rho^ABC))]= [ italic_S ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) - italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ] + [ italic_S ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ⊗ roman_Π start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) - italic_S ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) ]
=𝒟B(ρABC)+S[ΠC*(ΠB*(ρABC))]-S(ΠB*(ρABC))]fragmentssubscript𝒟𝐵fragments(superscript𝜌𝐴𝐵𝐶)Sfragments[superscriptsubscriptΠ𝐶fragments(superscriptsubscriptΠ𝐵fragments(superscript𝜌𝐴𝐵𝐶))]Sfragments(superscriptsubscriptΠ𝐵fragments(superscript𝜌𝐴𝐵𝐶))]=\mathcalD_B(\rho^ABC)+S[\Pi_C^*(\Pi_B^*(\rho^ABC))]-S(\Pi_B% ^*(\rho^ABC))]= caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) + italic_S [ roman_Π start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) ] - italic_S ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) ]
=𝒟B(ρABC)+𝒟C(ΠB*(ρABC))≤𝒟B(ρABC)+𝒟C(ρABC)absentsubscript𝒟𝐵superscript𝜌𝐴𝐵𝐶subscript𝒟𝐶superscriptsubscriptΠ𝐵superscript𝜌𝐴𝐵𝐶subscript𝒟𝐵superscript𝜌𝐴𝐵𝐶subscript𝒟𝐶superscript𝜌𝐴𝐵𝐶=\mathcalD_B(\rho^ABC)+\mathcalD_C(\Pi_B^*(\rho^ABC))\leq% \mathcalD_B(\rho^ABC)+\mathcalD_C(\rho^ABC)= caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) + caligraphic_D start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( roman_Π start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT start_POSTSUPERSCRIPT * end_POSTSUPERSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) ) ≤ caligraphic_D start_POSTSUBSCRIPT italic_B end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ) + caligraphic_D start_POSTSUBSCRIPT italic_C end_POSTSUBSCRIPT ( italic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT ).
This completes the proof of Theorem 3.
We now apply Eq. (19) to a tripartite pure state ρABCsuperscript𝜌𝐴𝐵𝐶\rho^ABCitalic_ρ start_POSTSUPERSCRIPT italic_A italic_B italic_C end_POSTSUPERSCRIPT. For any bipartite pure state, the relative entropy of entanglement and the relative entropy of discord coincide with the entropy of the reduced states of the parts [23]. Thus, Eq. (19) becomes
S(ρBC)≤S(ρB)+S(ρC)𝑆superscript𝜌𝐵𝐶𝑆superscript𝜌𝐵𝑆superscript𝜌𝐶S(\rho^BC)\leq S(\rho^B)+S(\rho^C)italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B italic_C end_POSTSUPERSCRIPT ) ≤ italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_B end_POSTSUPERSCRIPT ) + italic_S ( italic_ρ start_POSTSUPERSCRIPT italic_C end_POSTSUPERSCRIPT ) which is the subadditivity [22] of entropy for subsystems BC𝐵𝐶BCitalic_B italic_C. Accordingly, Eq. (19) can also be seen as a generalization of the subadditivity of entropy valid for tripartite mixed states.
VI Conclusion
In this work, we prove that the Luo et al.’s conjecture [10] and the Lindblad conjecture [21] are all true, when the entanglement of formation between the unmeasured subsystem and the environment vanishes. Further, the maximal quantum discord will even forbid the unmeasured subsystem from being correlated to the environment. We have also shown that the discord distance is always bounded from above by the amount of the joint entropy when the Lindblad conjecture is true. Though, there in no exemplary state which can show the meaningfulness of Theorem 2, we leave it as an open question whether one could give an operational interpretation of it. Finally, we find that the subadditivity is a new important property of tripartite quantum discord. Acknowledgments We thank Professor Shunlong Luo and Nan Li for their helpful discussions. This work is supported by NSFC (Grant Nos. 61300181, 61272057, 61202434, 61170270, 61100203, 61121061), Beijing Natural Science Foundation (Grant No. 4122054), Beijing Higher Education Young Elite Teacher Project (Grant Nos. YETP0475, YETP0477), and BUPT Excellent Ph.D. Students Foundation (Grant No. CX201434).
