Relative quantum coherence, incompatibility, and quantum

PHYSICAL REVIEW A 95, 052106 (2017)

Relative quantum coherence, incompatibility, and quantum correlations of states Ming-Liang Hu1,2,* and Heng Fan1,3,4,† 1

Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China 3 School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100190, China 4 Collaborative Innovation Center of Quantum Matter, Beijing 100190, China (Received 22 January 2017; published 10 May 2017) 2

Quantum coherence, incompatibility, and quantum correlations are fundamental features of quantum physics. A unified view of those features is crucial for revealing quantitatively their intrinsic connections. We define the relative quantum coherence of two states as the coherence of one state in the reference basis spanned by the eigenvectors of another one and establish its quantitative connections with the extent of mutual incompatibility of two states. We also show that the proposed relative quantum coherence, which can take any form of measures such as l1 norm and relative entropy, can be interpreted as or connected to various quantum correlations such as quantum discord, symmetric discord, entanglement of formation, and quantum deficits. Our results reveal conceptual implications and basic connections of quantum coherence, mutual incompatibility, and quantum correlations. DOI: 10.1103/PhysRevA.95.052106 I. INTRODUCTION

Quantum coherence is rooted in the superposition of quantum states, and also remains as a research focus since the early days of quantum mechanics [1]. It plays a key role for nearly all the novel quantum phenomena in the fields of quantum optics [1], quantum thermodynamics [2–6], and quantum biology [7]. However, its characterization and quantification from a mathematically rigorous and physically meaningful perspective has been achieved only very recently [8], when Baumgratz et al. introduced the defining conditions for a bona fide measure of coherence, and proved that the l1 norm and relative entropy satisfy the required conditions. In the past few years, many other coherence measures that satisfy the defining conditions have been proposed from different aspects [9–15]. Moreover, the freezing phenomenon of coherence in open systems [16–21], the coherence-preserving channels [22], and the creation of coherence by local or nonlocal operations [23–26] have been extensively studied. Other topics such as the coherence distillation [27–29], the complementarity relations of coherence [30,31], the connections of coherence with path distinguishability [32,33] and asymmetry [34,35], the coherence averaged over all basis sets [30] or the Haar distributed pure states [36], and the role of coherence in state merging [37] were also studied. For bipartite and multipartite systems, quantum coherence also underpins different forms of quantum correlations [9,24,38–44]. The mutual incompatibility of states, which represents another fundamental feature of the nonclassical systems, is also intimately related to quantum correlations, e.g., for any nondiscordant state there must exist local measurements which commute with it [45]. Intuitively, coherence, incompatibility, and quantum correlations are all closely related concepts. A direct and quantitative connection between them can provide a whole view and a measure in characterizing

* †

[email protected] [email protected]

2469-9926/2017/95(5)/052106(6)

the quantumness of the nonclassical systems. In this paper, by defining the relative quantum coherence (RQC) of two states as the coherence of one state in the basis spanned by the eigenvectors of another one, we find connections between coherence, incompatibility, and quantum correlations (Fig. 1). Our observations include: (i) the quantitative connection between RQC and mutual incompatibility of states, and (ii) the interpretation of quantum discord (QD), symmetric discord, measurement-induced disturbance, quantum deficits, entanglement of formation, and distillable entanglement via the proposed RQC. We expect the connections established in this paper may contribute to a unified view of the resource theory of coherence, incompatibility, and quantum correlations. II. QUANTIFYING THE RQC

Consider two states ρ and σ in the same Hilbert space H. When σ is nondegenerate with the eigenvectors = {|ψi }, i.e., σ = i i |ψi ψi | (i are the eigenvalues of σ ), we define the RQC of ρ with respect to σ as C(ρ,σ ) = C (ρ),

(1)

where C (ρ) denotes any bona fide measure of quantum coherence defined in the reference basis . The rationality for this choice of reference basis lies in that the RQC of a state with respect to itself equals zero. Although this definition is essentially the same as that of the coherence measure introduced by Baumgratz et al. [8], the choice of the eigenvectors of another state as the reference basis allows one to establish quantitatively the connections between quantum coherence, incompatibility, and quantum correlations of states. (For example, the excess of RQC of the total state with respect to the postmeasurement state and the sum of RQC of the reduced states with respect to the local postmeasurement states gives an interpretation of the QD.) Hence this can deepen our understanding about distribution of quantumness in a composite system. Moreover, by choosing σ = ρ0 as the initial state and ρ = ρt as the evolved state,

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©2017 American Physical Society

MING-LIANG HU AND HENG FAN


FIG. 1. (a) Equivalence between RQC and commutativity of states. The vanishing (nonvanishing) RQC implies commutativity (noncommutativity) of states and vice versa. (b) Interpretations of quantum correlations via RQC. The various correlations can be interpreted as or connected to the discrepancy between RQC for the total state (top) and those localized in the reduced states (bottom).

C(ρt ,ρ0 ) also allows one to characterize the decoherence or coherence process of a system relative to its initial state. If one takes the l1 norm or the relative entropy as the coherence measure [8], then |ψi |ρ|ψj |, Cl1 (ρ,σ ) = i=j (2) Cre (ρ,σ ) = S(|ρ) − S(ρ), where S(|ρ) = − i ψi |ρ|ψi log2 ψi |ρ|ψi , and S(ρ) = −tr(ρ log2 ρ) denotes the von Neumann entropy. When σ is degenerate, is not uniquely defined; however, we can take the supremum over all possible eigenvectors of σ and obtain the maximum RQC as ˜ C(ρ,σ ) = sup C (ρ).

(3)

Although there exists an optimization process, for certain cases it can be obtained analytically. For example, for the maximally ˜ mixed state σm , C(ρ,σ m ) is in fact the maximum coherence of ρ over all possible reference basis. By decomposing ρ (with dimension d) via the orthonormal operator bases {Xi } [20], one can obtain (see Appendix A) C˜ l1 (ρ,σm ) = (d 2 − d)/2| x |, (4) C˜ re (ρ,σm ) = log2 d − S(ρ), where | x | is the length of the vector x = (x1 ,x2 , · · · ,xd 2 −1 ), and xi = tr(ρXi ). As | x |2 ≤ 2(d − 1)/d and S(ρ) ≥ 0 [20], we have C˜ l1 (ρ,σm ) ≤ d − 1 and C˜ re (ρ,σm ) ≤ log2 d, and the bounds are achieved for pure states ρ. Equation (4) reveals that the states without quantum coherence with respect to any basis are the maximally mixed states and the states with maximum coherence with respect to any basis are the pure states. This property is similar to the quantumness captured by the noncommutativity of the algebra of observables A, in which a state ρ is defined to be classical if and only if tr(ρ[A,B]) = 0, ∀A,B ∈ A [46]. Here, it is easy to check that only σm is classical in this sense of the definition. Moreover, the maximal coherence of Eq. (4) is also intimately related to the complementarity of coherence under mutually unbiased bases (MUBs). For

example, from Eq. (13) of Ref. [30] one can obtain that the upper bound for the sum of the squared l1 norm of coher 2 2 ence is just dCl21 (ρ,σm ), i.e., d+1 j =1 Cl1 (Aj ,ρ) ≤ dCl1 (ρ,σm ), d+1 with {Aj }j =1 being the MUBs, and for the one-qubit state the equality always holds. Similarly, from Eq. (23) of Ref. [30] we obtain d+1 j =1 Cre (Aj ,ρ) ≤ (d + 1)Cre (ρ,σm ) − 2 Cl1 (ρ,σm ) log2 (d − 1)/[d(d − 2)]. This gives an upper bound for the sum of the relative entropy of coherence under MUBs via the difference of maximal RQCs measured by the relative entropy and the l1 norm. The RQC is unitary invariant in the sense that Cl1 (ρ,σ ) = Cl1 (UρU † ,U σ U † ) for any unitary operation be U . UThis can U † proved directly by noting that U σ U = |ψ ψ | and i i i i UρU † = kl λkl |ψkU ψlU |, with |ψiU = U |ψi . That is, U rotates both the basis of ρ and σ simultaneously. III. LINKING RQC TO INCOMPATIBILITY OF STATES

We establish connections between the RQC and mutual incompatibility of two states in this section. For this purpose, we rewrite ρ in the basis as ρ = kl λkl |ψk ψl |, where λkl = ψk |ρ|ψl . Then by Eq. (2) we obtain |λij |. (5) Cl1 (ρ,σ ) = i=j

On the other hand, the commutator [ρ,σ ] of states ρ and σ can be calculated as λij (j − i )|ψi ψj |, (6) [ρ,σ ] = i=j

then if we quantify the extent of the mutual incompatibility of the two states by l1 norm of the commutator, we have |λij (i − j )|, (7) Ql1 (ρ,σ ) = i=j

where Ql1 (ρ,σ ) is by definition nonnegative, vanishes if and only if the commutator [ρ,σ ] vanishes, and is unitary invariant. Different from the mutual incompatibility QF (ρ,σ ) = 2 [ρ,σ ] 22 measured by square of the Frobenius norm, which takes the value between √ 0 and 1 [46], the maximum possible value of Ql1 (ρ,σ ) is d − 1. But apart from the single-qubit case, this maximum is reached on the pure but not maximally coherent states (see Appendix B). Here, by comparing Eqs. (5) and (7), one can obtain a quantitative connection between Cl1 (ρ,σ ) and Ql1 (ρ,σ ), Cl1 (ρ,σ ) ≥ Ql1 (ρ,σ ),

(8)

due to |i − j | ≤ 1, ∀i,j . For the special case that σ is pure, it has only one nonvanishing eigenvalue; this gives Ql1 (ρ,σ ) = 2 i=1 |λi1 |. Thus when the elements λkl of ρ not in the first row, the first column, and the main diagonal equal to zero, we have Cl1 (ρ,σ ) = Ql1 (ρ,σ ). That is to say, the bound in Eq. (8) is tight. In particular, for pure state σ and arbitrary state ρ of one qubit, the bound is always saturated. From Eq. (8), one can see that the mutual incompatibility of two states provides a lower bound for their RQC. Alternatively, the RQC provides an upper bound for the extent of their mutual incompatibility. It also implies that when there is no RQC between two states, then they must commute with each other.

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RELATIVE QUANTUM COHERENCE, INCOMPATIBILITY, . . .


On the other hand, when two states commute, they either share a common eigenbasis or are orthogonal, for which we always have the vanishing RQC. What is more, by using Eqs. (6), (7), and the definition of QF (ρ,σ ) [46], one can show that Q2l1 (ρ,σ ) ≥ QF (ρ,σ )/2. As QF (ρ,σ ) can be measured via an interferometric setup and without performing the full state tomography [46], it provides an experimentally accessible lower bound for the mutual incompatibility and RQC of two quantum states. In fact, for every basis state |ψi of σ , we can associate with it a pure state σi = |ψi ψi |. Then by summing Ql1 (ρ,σi ) over the set {σi }, one can obtain Ql1 (ρ,σi ) = 2Cl1 (ρ,σ ), (9)

minimal entropy increase of ρ˜PB for tightening the above bound. Moreover, the eigenbasis of ρ˜PQ (ρ˜P ) for obtaining ˜ ⊗ |l} ˜ ({|k}), ˜ the RQC in Eq. (11) are chosen to be {|k which B Ã ˜ corresponds to the optimal measurement { ⊗ k l }, i.e., here we do not perform the optimization of Eq. (3) even if the postmeasurement states are degenerate. The same holds for other discussions of this section. In Eq. (11), Cre (ρAB ,ρ˜PQ ) is the RQC (defined by the relative entropy) between ρAB and ρ˜PQ , while Cre (ρA ,ρ˜P ) is that between ρA and ρ˜P , which can be recognized as the quantum coherence localized in subsystem A. From this point of view, the QD of a state is always smaller than or equal to δ1 (ρAB ), which characterizes the discrepancy between the RQCs for the total system and that for the subsystem to be measured in the definition of QD [see Fig. 1(b)]. When the RQC discrepancy δ1 (ρAB ) vanishes, there will be no QD in the state. For the case of a quantum-classical state χAB = l pl ρA|l ⊗ |ϕl ϕl |, with ρA|l being the density operator in HA , |ϕl the orthonormal basis (also the eigenvectors of χB = trA χAB ) in HB , and {pl } the probability distribution, we have ρ˜PB = ρ˜PQ if we choose Q = {|ϕl ϕl |}. Then the RQC discrepancy δ1 (χAB ) equals exactly the QD of χAB , i.e.,

i

that is to say, the sum of Ql1 (ρ,σi ) over {σi } equals twice the RQC between ρ and σ . This establishes another connection between RQC and the mutual incompatibility of two states. IV. LINKING RQC TO QUANTUM CORRELATIONS

The RQC and incompatibility of states are also intimately related to quantum correlations such as QD [47]. For bipartite state ρAB with the reduced state ρA , if the QD DA (ρAB ) defined with respect to party A equals zero, then ρAB and ρA ⊗ IB commute. Equivalently, if ρAB does not commute with ρA ⊗ IB , then it is quantum discordant [48]. By Eq. (8), we know that the nonvanishing mutual incompatibility implies the nonvanishing RQC of ρAB with respect to ρA ⊗ IB . In fact, a necessary and sufficient condition for ρAB to have zero discord has also been proved, which says that DA (ρAB ) = 0 if and only if all the operators ρA|b1 b2 = b1 |ρAB |b2 commute with each other for any orthonormal basis {|bi } in HB [49]. From the analysis below Eq. (8), we know that the commutativity of two operators corresponds to the vanishing RQC of them. Hence, the RQC can be linked to QD of a state. A By denoting P = {A k } (k = |kk|) the local projective measurements on party A, and likewise for Q = {Bl }, the postmeasurement states after the measurements P ⊗ IB and P ⊗ Q are given, respectively, by ρPB = p k A k ⊗ ρB|k , k

ρPQ =

B pkl A k ⊗ l ,

(10)

kl A where ρB|k = trA (A pk = k|ρA |k, pkl = k ρAB k )/pk , kl|ρAB |kl. Then, by the definition of QD [47], we obtain

DA (ρAB ) = S(ρA ) − S(ρAB ) − S(ρ˜P ) + S(ρ˜PB ) ≤ S(ρA ) − S(ρAB ) − S(ρ˜P ) + S(ρ˜PQ ) = Cre (ρAB ,ρ˜PQ ) − Cre (ρA ,ρ˜P ) ≡ δ1 (ρAB ),

(11)

where S(ρ˜PB ) ≤ S(ρ˜PQ ) as project measurements do not decrease entropy. Moreover, ρ˜PB and ρ˜PQ denote, respectively, Ã Ã ˜B the postmeasurement states of { k } and {k ⊗ l }, where A ˜ k } are the optimal measurements for obtaining DA (ρAB ), { ˜ Bl } can in fact be arbitrary projective measurements, while { and here we fix it to be the measurement that gives the

DA (χAB ) = Cre (χAB ,χ˜ PQ ) − Cre (χA ,χ˜ P ),

(12)

and hence the bound given in Eq. (11) is tight. Besides the QD defined with one-sided measurements [47], one can also demonstrate the role of RQC discrepancy in interpreting the symmetric discord Ds (ρAB ) = I (ρAB ) − I (ρ˜PQ ) Ã ˜B defined via two-sided optimal measurements { k ⊗ l } [50]. For this case, one can prove that Ds (ρAB ) =Cre (ρAB ,ρ˜PQ ) − Cre (ρA ,ρ˜P ) − Cre (ρB ,ρ˜Q ) ≡ δ2 (ρAB ),

(13)

with δ2 (ρAB ) being the RQC discrepancy. It implies that the symmetric discord Ds (ρAB ) is nonzero if and only if there exists RQC not localized in the subsystems [see Fig. 1(b)]. This establishes a direct connection between the RQC discrepancy and the symmetric discord. The RQC can also be linked to other discordlike correlation measures such as measurement-induced disturbance [51], measurement-induced nonlocality [52], and quantum deficits [53,54]. (i) For the measurement-induced disturbance, it is just the RQC of ρAB with respect to ρAB = ij ξij ρξij , i.e., M(ρAB ) = Cre (ρAB ,ρAB ) [55]. Here ξij = |eiA eiA | ⊗ B B A B |ej ej |, and {|ei ,|ej } are local eigenvectors of the reduced states. (ii) For the measurement-induced nonlocality defined as Nv (ρAB ) = maxA S(A [ρ]) − S(ρ) (A are restricted to the locally invariant measurements), from Eq. (9) of Ref. [52] one can obtain that Nv (ρAB ) ≤ Cre (ρAB ,ρ˜PQ ). Here, ρ˜PQ is similar to that in Eq. (11), and the difference is that Ã { k } should be locally invariant. (iii) For the zero-way deficit ∅ = S(ρ˜PQ ) − S(ρAB ) and one-way deficit (equal to the thermal discord [45]) → = S(ρ˜PB ) − S(ρAB ) (where ρ˜PQ and ρ˜PB denote, respectively, the corresponding optimal postmeasurement states) [54], it is direct to see that

052106-3

∅ (ρAB ) = Cre (ρAB ,ρ˜PQ ),

→ (ρAB ) ≤ Cre (ρAB ,ρ˜PQ ),

(14)



where for the quantum-classical states, the inequality becomes equality when Q = {|ϕl ϕl |}. These relations give interpretations of the corresponding correlation measures in terms of RQC, and hence bridge the gap between quantum coherence for a single quantum system and quantum correlations for a system with two parties, which are two fundamentals of quantum physics. For certain cases, the RQC can also be linked to quantum entanglement. For example, by resorting to the Koashi-Winter equality [56], the chain inequality [57], and equality condition of the Araki-Lieb inequality [58], one can obtain Ef (ρAB ) ≤ δ1 (ρAB )

(15)

when the conditional entropy S(B|A) is non-negative or when the equality S(ρAB ) = |S(ρA ) − S(ρB )| is satisfied (see Appendix C). Here, Ef (ρAB ) denotes the entanglement of formation for ρAB [59]. The relation shows that for these cases, the entanglement of formation for a state is always bounded from above by its RQC discrepancy δ1 (ρAB ). When one performs local measurements {A k } on a system ρAB , there will be entanglement created between the measurement apparatus M and the system AB. Streltsov et al. showed that the minimal distillable entanglement min ED (ρ˜M:AB ) = minU ED (ρ˜M:AB ) created between M and AB equals → (ρAB ) [60]. Here, ρ˜MAB = U (|0M 0M | ⊗ ρAB )U † , and U are unitaries acting on MAB which realize a von NeuA mann measurement on A, i.e., trM ρ˜MAB = k A k ρAB k . min (ρ˜M:AB ) ≤ Cre (ρAB ,ρ˜PQ ). Then by Eq. (14) we obtain ED Moreover, the minimal partial entanglement PEmin (ρ˜MAB ) = D minU [ED (ρ˜M:AB ) − ED (ρ˜MA )] created in a von Neumann measurement on A equals DA (ρAB ) [60], and this gives PEmin (ρ˜M:AB ) ≤ δ1 (ρAB ). All these relations show the role of D RQC in interpreting quantum entanglement. V. CONCLUSION

In this paper, we explored the connections among quantum coherence, incompatibility, and quantum correlations. We defined the RQC and proved several of its connections to the extent of mutual incompatibility of states. We also gave interpretations of various quantum correlations (QD, entanglement of formation, symmetric discord, measurementinduced disturbance, measurement-induced nonlocality, and quantum deficits, etc.) via the RQC discrepancy between the total system and those localized in the respective subsystems. We mainly considered the l1 norm and relative entropy of coherence, but we remark that the coherence measure in other forms can also be extended to the case of RQC, and more connections among coherence, mutual incompatibility, and quantum correlations can be expected in future research. Note added. Recently Yao et al. presented a similar study on the problem of maximum coherence under a generic basis [61]. ACKNOWLEDGMENTS

This work was supported by NSFC (Grants No. 11675129 and No. 91536108), MOST (Grants No. 2016YFA0302104 and No. 2016YFA0300600), New Star Project of Science and Technology of Shaanxi Province (Grant No. 2016KJXX-27),

CAS (Grants No. XDB01010000 and No. XDB21030300), and New Star Team of XUPT. APPENDIX A: PROOF OF EQ. (4)

For any d-dimensional stateρ, it can always be decomposed as d −1 1 1 Id + xi Xi , d 2 i=1 2

ρ=

(A1)

where {Xi } constitutes the orthonormal operator bases, and the l1 norm of coherence can be obtained as [31] Cl1 (ρ) =

d0 2 2 1/2 x2r−1 + x2r ,

(A2)

r=1

where d0 = d(d − 1)/2. The maximization of Cl1 (ρ) over all possible reference basis is equivalent to its maximization over all the unitary transformations of ρ which keep | x | unchanged. Then, by denoting x˜i = tr(ρ U Xi ) with ρ U = UρU † , and by using the mean inequality, we obtain d0 2 2 Cl1 (ρ ) = x˜2r−1 + x˜2r U

r=1

2d0 x˜k2 ≤ d0 k=1

≤ d0 | x |, (A3) √ 2 2 x | when U gives rise to x˜2r−1 + x˜2r = and Cl1 (ρ U ) = d0 | 2 2 ˜ ˜ ˜x2r −1 + x2r , ∀r,r ∈ [1,d0 ], and xl = 0, ∀l ≥ 2d0 + 1. Thus we arrive at the first equality of Eq. (4). Second, the maximization of Cre (ρ,σm ) over all the referU ence basis is equivalent to the maximization of S(ρdiag ) over all U . Still, by using the mean inequality, we obtain U )=− S(ρdiag

d

ρiiU log2 ρiiU

i=1

d 2 ≤ d ρiiU log2 ρiiU i=1

≤ log2 d,

(A4)

U U U where S(ρdiag ) = log2 d when ρiiU log2 ρiiU = ρjj log2 ρjj , ∀i,j , and x˜l = 0, ∀l ≥ 2d0 + 1. This completes the proof of the second equality of Eq. (4).

APPENDIX B: MAXIMUM OF THE MUTUAL INCOMPATIBILITY

To obtain the maximum of Ql1 (ρ,σ ), we consider ρ = ||, with | = di=1 ai(d) |ψi and di=1 |ai(d) | = 1, and σ1 = |ψ1 ψ1 |, for which we have (d) (d) a a . (B1) Ql1 (ρ ,σ1 ) = 2 i 1

052106-4

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RELATIVE QUANTUM COHERENCE, INCOMPATIBILITY, . . .

Then by taking a1(2) = cos θ2 and a2(2) = sin θ2 for d = 2, and for j ≥ 2 and d ≥ 3 (the a1(d) = cos θd , aj(d) = sin θd aj(d−1) −1 (d) phases of ai do not affect the incompatibility), one can show √ d − 1, (B2) Qmax l1 (ρ ,σ1 ) = which is obtained with√θ2 = θd = (2n + 1)π/4 (n ∈ Z), and θk = π/2 − arctan (1/ k − 1),√ ∀k ∈ [3,d − 1]. Now, we further prove that d − 1 is also the maximum of Ql1 (ρ,σ ) for general cases. First, for the pure states ρ and general σ , the convexity of the l1 norm gives Ql1 (ρ ,σ ) ≤ i Ql1 (ρ ,σi ) i

≤

i Ql1 (ρ˜ ,σ1 )

i

√ = d − 1,

(B3) Qmax l1 (ρ ,σ1 ),

where ρ˜ is the optimal state for obtaining and the second inequality comes from the fact that ρ˜ is not necessarily the optimal state for Qmax l1 (ρ ,σi ) when i ≥ 2. Second, by denoting ρ = i pi |φi φi | any pure state decomposition of the general state ρ, we have Ql1 (ρ,σ ) ≤ pi Ql1 (|φi φi |,σ ) i

≤

√

PHYSICAL REVIEW A 95, 052106 (2017) APPENDIX C: PROOF OF EQ. (15)

First, S(B|A) ≥ 0 implies S(ρC ) ≥ S(ρA ) for any pure state |ABC . Then, by using the chain inequality [57], one can obtain S(ρC ) + Ef (ρAB ) ≤ S(ρA ) + Ef (ρBC ),

(C1)

which is equivalent to Ef (ρAB ) ≤ Ef (ρBC ) − S(B|A). This, together with Eq. (11) and the Koashi-Winter equality [56] DA (ρAB ) + S(B|A) = Ef (ρBC ),

(C2)

gives Ef (ρAB ) ≤ δ1 (ρAB ). Second, the equality S(ρAB ) = S(ρB ) − S(ρA ) if and only if the Hilbert space HB can be decomposed as HB = HB L ⊗ HB R such that ρAB = |ψAB L ψ| ⊗ ρB R [58]. For this case, it is direct to show that Ef (ρAB ) = DA (ρAB ) = DB (ρAB ) = S(ρA ).

(C3)

This, together with Eq. (11), gives Ef (ρAB ) ≤ δ1 (ρAB ). Similarly, S(ρAB ) = S(ρA ) − S(ρB ) if and only if there exists a decomposition HA = HAL ⊗ HAR such that ρAB = ρAL ⊗ |ψAR B ψ|, and for this kind of state we have

d − 1,

(B4) √ and the second inequality is due to Ql1 (|φi φi |,σ √ )≤ d −1 for any |φi . Therefore, we proved Qmax (ρ,σ ) = d − 1. l1

Hence, we still have Ef (ρAB ) ≤ δ1 (ρAB ).

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Ef (ρAB ) = DA (ρAB ) = DB (ρAB ) = S(ρB ).

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(C4)



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