aa r X i v : . [ qu a n t - ph ] A ug Deterministic Coherence Distillation
C. L. Liu and D. L. Zhou ∗ Institute of Physics, Beijing National Laboratory for Condensed Matter Physics, Chinese Academy of Sciences, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China CAS Central of Excellence in Topological Quantum Computation, Beijing 100190, China Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China (Dated: August 15, 2019)Coherence distillation is one of the central problems in the resource theory of coherence. In this Letter, wecomplete the deterministic distillation of quantum coherence for a finite number of coherent states under strictlyincoherent operations. Specifically, we find the necessary and su ffi cient condition for the transformation from amixed coherent state into a pure state via strictly incoherent operations, which recovers a connection between theresource theory of coherence and the algebraic theory of majorization lattice. With the help of this condition, wepresent the deterministic coherence distillation scheme and derive the maximum number of maximally coherentstates obtained via this scheme. Introduction. – Quantum coherence is a valuable resourcein performing quantum information processing tasks [1]. Itcan implement various information processing tasks that can-not be accomplished classically, such as quantum computing[2, 3], quantum cryptography [4], quantum metrology [5, 6],and quantum biology [7]. Recently, the resource theory ofcoherence has attracted a growing interest due to the develop-ment of quantum information science [8–18].All quantum resource theories have two fundamental in-gredients: free states and free operations [19, 20]. For theresource theory of coherence, the free states are the quantumstates that are diagonal in a prefixed reference basis. How-ever, there is no general consensus on the set of free opera-tions. Based on di ff erent physical and mathematical consider-ations, a number of free operations were proposed [8, 9, 11–14]. Here, we focus our discussion on the strictly incoherentoperations. This type of free operation was first given in Ref.[11] and was shown that it can neither create nor use coher-ence and has a physical interpretation in terms of interferome-try in Ref. [12]. Thus, the strictly incoherent operations are aphysically well-motivated set of free operations for coherenceand a strong candidate for free operations.One of the central problems in the resource theory of co-herence is the coherence distillation [9, 11, 19, 21–30], whichis the process that extracts pure coherent states from generalstates via free operations. This problem was approached intwo di ff erent settings: the asymptotic regime [11, 19, 21, 25–28] and the one-shot regime [23, 24, 29, 30]. Althoughmany interesting results have been obtained, however, thereare still some open fundamental questions remaining to besolved. One of which is the deterministic coherence distil-lation, whose aim is to find the condition of conversion froma general mixed state to the maximally coherent state withcertainty [18, 30, 31]. Investigations on this topic have beenstarted in Ref. [30], where the deterministic coherence dis-tillation of pure coherent states under several classes of inco-herent operations was introduced. However, the deterministiccoherence distillation of general mixed states has been left asan open question.In this Letter, we address the above question by completing the framework for deterministic coherence distillation understrictly incoherent operations. We first recall some notions ofthe resource theory of coherence and the notions of majoriza-tion lattice which are related to our topic. Then, we presentthe necessary and su ffi cient condition for the transformationfrom a general state into a pure state via strictly incoherentoperations, which recovers a connection between the resourcetheory of coherence and the algebraic theory of majorizationlattice. With the help of this condition, we present the deter-ministic coherence distillation scheme. Then, we derive themaximum number of maximally coherent states that can beobtained in this deterministic coherence distillation scheme. Resource theory of coherence. –Let H represent the Hilbertspace of a d -dimensional quantum system. A particular basisof H is denoted as {| i i , i = , , · · · , d − } , which is chosenaccording to the physical problem under discussion. Specif-ically, a state is said to be incoherent if it is diagonal in thebasis. We represent the set of incoherent states as I . Anystate that cannot be written as a diagonal matrix is definedas a coherent state. Note that the term coherent state here isdi ff erent from the canonical coherent state or the spin coher-ent state [1]. For a pure state | ϕ i , we will denote | ϕ ih ϕ | as ϕ ,i.e., ϕ : = | ϕ ih ϕ | and we will denote | ϕ dm i = √ d P d − i = | i i as a d -dimensional maximally coherent state.A strictly incoherent operation is a completely positivetrace-preserving map, expressed as Λ ( ρ ) = P n K n ρ K † n , wherethe Kraus operators K n satisfy not only P n K † n K n = I butalso K n I K † n ⊂ I and K † n I K n ⊂ I for K n , i.e., each K n aswell as K † n maps an incoherent state to an incoherent state.With this definition, it is elementary to show that a projec-tor is an incoherent operator if and only if it has the form P I = P i ∈ I | i ih i | with I ⊂ { , , ..., d − } . In what follows, wewill denote P I as strictly incoherent projective operators. Thethe dephasing map, which we will denote as ∆ ( · ), is definedas ∆ ρ = P d − i = | i ih i | ρ | i ih i | . Majorization and majorization lattice. – Majorization [32]is a mathematical tool widely used in quantum informa-tion theory [33–35]. For the n -dimensional probability dis-tributions P n , we say that a probability distribution p = ( p , p , ..., p n ) is majorized by q = ( q , q , ..., q n ), in symbols p ≺ q , if there are P li = p ↓ i ≤ P li = q ↓ i , for all 1 ≤ l ≤ n , where ↓ indicates that the elements are to be taken in descending order.The majorization lattice [36–38] is a quadruple ( P n , ≺ , ∨ , ∧ ).Here ≺ is the relation introduced above. For every pair of p , q ∈ P n , p ∧ q is the unique greatest lower bound of p , q upto a permutation transformation which is defined as a proba-bility distribution, for every s ∈ P n with s ≺ p , s ≺ q , thenthere is s ≺ p ∧ q ; and p ∨ q is the unique least upper boundof p , q which is defined as a probability distribution for every t ∈ P n with p ≺ t and q ≺ t , then there is p ∨ q ≺ t . Sim-ilarly, we write V S as the unique greatest lower bound of S and W S as the unique least upper bound of S , where S is asubset of P n . Hereafter, we will apply majorization to densityoperators and write ρ ≺ ρ if and only if the correspondingmajorization relation holds for the eigenvalues of ρ and ρ .And W S ≺ ρ means that the least upper bound (up to a uni-tary transformation) of S is majorized by ρ . Determined state transformation. –In the following, we willgive the necessary and su ffi cient condition for a state ρ can betransformed into a pure coherent state | ϕ i via strictly incoher-ent operations. Theorem 1.
We can transform a mixed state ρ into a purecoherent state ϕ via strictly incoherent operations if and onlyif there exists an orthogonal and complete set of incoherentprojectors { P α } such that, for all α , there are P α ρ P α Tr( P α ρ P α ) = ψ α and ∆ ψ α ≺ ∆ ϕ, (1)where ψ α are all pure coherent states. In other words, thereexists { P α } such that _ S ≺ ∆ ϕ, (2)where S is the set of { ∆ ψ α } . Proof.
First, we show that ρ can be transformed into ϕ viaa strictly incoherent operation if and only if P ρ P t (superscript t means transpose) can be transformed into ϕ via a strictlyincoherent operation with P being a permutation matrix.For any two strictly incoherent operations Λ with Krausoperators { K n } and Λ with Kraus operators { K m } , the opera-tion Λ = Λ ◦ Λ is also a strictly incoherent operation withKraus operators { K l = K n K m } , since we can easily verify itby examining K l I K † l ⊆ I and K † l I K l ⊆ I . It is straightfor-ward to verify that, for any permutation matrix, both P and itsinverse are strictly incoherent operations. With these knowl-edge, it is easy to show that ρ can be transformed into ϕ via astrictly incoherent operation if and only if P ρ P t can be trans-formed into ϕ via a strictly incoherent operation. Hence, with-out loss of generality, we let ρ = M µ p µ ρ µ , (3)corresponding to the Hilbert space H = L µ H µ with each ρ µ being irreducible. Here, an irreducible matrix ρ µ means that itcannot be transformed into a block diagonal matrix by usinga permutation matrix. Second, we show the if part of the theorem, i.e., if the state ρ satisfies the condition in the theorem above, then we cantransform a mixed state ρ into a pure state ϕ via a strictly in-coherent operation.Let ρ be a state satisfying the condition in the theoremabove. Then, according to the result in Ref. [13, 33, 34] whichsays that a pure coherent state | ψ i can be transformed into an-other pure coherent state | ϕ i via strictly incoherent operationsif and only if there is ∆ ψ ≺ ∆ ϕ , we can always find strictlyincoherent operations Λ α ( · ), which act on the support of P α ,with Λ α ( · ) = P n K n α ( · ) K n α † , such that Λ α ( ψ α ) = ϕ, for all α . With this result, we transform ρ into | ϕ i by using theoperation Λ ( · ) = M α Λ α ( · ) , where the corresponding Kraus operators are K α, n = K n α ⊕ . Here, represents a square matrix with all its elements be-ing zero. It is straightforward to show that Λ ( · ) is a strictlyincoherent operation.Third, we show the only if part of the theorem, i.e., if ϕ canbe obtained from a state ρ via a strictly incoherent operation,then the state ρ should satisfy the condition in the theoremabove.Let us assume that we can obtain a pure coherent state ϕ from a mixed state ρ by using a strictly incoherent operation Λ ( · ). Then, there is Λ ( ρ ) = X n K n ρ K † n = ϕ. (4)Substituting Eq. (3) into (4), we can obtain that Λ ( ρ ) = X n ,µ p µ K n ρ µ K † n = ϕ. (5)Since pure states are extreme points of the set of states, theremust be K n ρ µ K † n = q n ,µ ϕ, for all n and µ , where q n ,µ = Tr( K n ρ µ K † n ).According to the definition of the strictly incoherent oper-ations, there is at most one nonzero element in each column(row) of a strictly incoherent Kraus operator. Thus, any K n can always be decomposed into K n = P π K Dn P n , (6)where the operator P π is a permutation matrix, K Dn = diag( a , ..., a n , , , ... ) is a diagonal matrix with a i beingnonzero complex numbers, and P n is a projective operatorcorresponding to K Dn , i.e., P n = diag(1 , ..., , , , ... ). Let { p µ, i , | ψ µ, i i} be an arbitrary ensemble decomposition of ρ µ .Then, there is K n ρ µ K † n = X µ, i p µ, i P π K Dn P n ψ µ, i P n K Dn † P † π . (7)From Eqs. (5) and (7), we obtain that Λ ( ρ ) = X n ,µ, i p µ p µ, i P π K Dn P n ψ µ, i P n K Dn † P † π = ϕ. Again, by using the fact that pure states are extreme points ofthe set of states, we immediately obtain that P π K Dn P n ψ µ, i P n K Dn † P † π Tr( P π K Dn P n ψ µ, i P n K Dn † P † π ) = ϕ or , (8)for all µ , i , and n . Clearly, | ψ µ, i i are states of the subspace H µ .Thus, we only need to consider the projective operator P n inEq. (6) corresponding to the subspace H µ and we denoted itas P n ,µ . Since Λ is a trace preserving map, we can get that P n K † n K n = I and, furthermore, P n P n ,µ K † n K n P n ,µ = I µ with I µ being the identity matrix of the subspace H µ . Here, sinceevery ρ µ is irreducible, P π K Dn P n | ψ µ, i i cannot be a zero vectorat the same time.From Eq. (8) and P n P n ,µ K † n K n P n ,µ = I µ , we get that P n ,µ ψ µ, i P n ,µ = P n ,µ ψ µ, j P n ,µ or , (9)for all i and j . Both these two cases mean that P n ,µ ρ µ P n ,µ Tr( P n ,µ ρ µ P n ,µ ) isa pure coherent state and we denoted it as ψ n ,µ for the sakeof simplicity. By using the condition that Λ ( ρ ) = ϕ and thecondition in Eq. (9), we immediately derive that Λ ( ψ n ,µ ) = ϕ, for every n and µ . Since the state ψ n ,µ can be transformed into ϕ via a strictly incoherent operation if and only if ∆ ψ n ,µ ≺ ∆ ϕ ,we immediately obtain the conclusion in our theorem. Thiscompletes the proof of the only if part. (cid:3) From Theorem 1, we infer the following corollary:
Corollary . We can transform ρ into a pure coherent state ψ via strictly incoherent operations if and only if ψ α are allcoherent states for some { P α } . Proof.
The only if part follows directly from Theorem 1.To prove the if part, without loss of generality, let us assumethat | ψ α i = P d α i = c α i | i i with the number of c α i > d α ≥ c α ≥ · · · ≥ c α d α . From the definition of the majorizationlattice, we can immediately obtain that W S ≺ W S ′ , where S ′ = { ∆ ψ ′ α } with | ψ ′ α i = c α | i + P d α i = c α i | i i . Noting that the set S ′ is an ordered set [32] and c α <
1, we then obtain that W S ′ equals to one of ∆ ψ ′ α and this corresponds to a coherent state | ψ i where | ψ i = c | i + c | i with 0 < c < (cid:3) Deterministic coherence distillation. –Next, let us move tothe deterministic coherence distillation of a finite number ofcoherent states.Suppose that we have n coherent states ρ , ρ , ..., ρ n , where ρ , ρ , ..., ρ n are not necessarily identical and n is a fi-nite number. The deterministic coherence distillation processis the process that extracts pure coherent states from them withcertainty. Here, we concentrate our discussion on the task thatextracts as more 2-dimensional maximally coherent state | ϕ m i as possible from ρ ⊗ ρ ⊗ · · · ⊗ ρ n via strictly incoherent op-erations.Based on the result above, we take the distillation procedureas the following three steps (See Fig.1). ρ L µ p µ ρ µ ψ ψ ψ n ψ ( S ) ϕ P Π Π Π n ˜ Λ ˜ Λ ˜ Λ n ¯ Λ FIG. 1. Schematic picture of the deterministic coherence transfor-mation via strictly incoherent operations. Here, Π α = P α · P α forincoherent projective operator P α , ψ ( S ) is the pure coherent statedetermined by W S , ˜ Λ α are the strictly incoherent operations suchthat ˜ Λ α ( ψ α ) = ψ , ¯ Λ is the strictly incoherent operation such that¯ Λ ( ψ ) = ϕ , and all the others are the same as in the main text. First, for the given ρ = ρ ⊗ ρ ⊗ · · · ⊗ ρ n , we should transform ρ into a block diagonal matrix.To this end, one should calculate out the permutation matrix P that can transform ρ into a block diagonal matrix, i.e., thepermutation matrix P such that P ( ρ ) = P ρ P t = L M µ = p µ ρ µ M , (10)where each ρ µ = P i , j ρ µ i j | i ih j | ( µ = , , · · · , n ) is an irre-ducible density operator defined on the d µ -dimensional sub-space H µ , p µ > P L µ = p µ =
1, and represents asquare matrix of dimension d = d − P L µ = d µ with all its ele-ments being zero.Second, we should calculate out an incoherent projectiveoperators set { P α } in Theorem 1.To this end, let us first introduce the following three matri-ces, which are useful to obtain the corresponding { P α } . For ρ = P i j ρ i j | i ih j | , we can define two matrices | ρ | and ( ∆ ρ ) − ,where | ρ | reads | ρ | = P i j | ρ i j || i ih j | and ( ∆ ρ ) − is a diagonalmatrix with elements( ∆ ρ ) − ii = ρ − ii , if ρ ii , , if ρ ii = . Next, we recall the following matrix with the help of | ρ | and( ∆ ρ ) − A = ( ∆ ρ ) − | ρ | ( ∆ ρ ) − . (11)A useful property of A is that all the elements of A are 1 ifand only if ρ is a pure coherent state [24]. By substituting theexpression in Eq. (10) into Eq. (11), we obtain that A = ( ∆ ρ ) − | ρ | ( ∆ ρ ) − = L M µ = A µ M , where A µ = ( ∆ ρ µ ) − | ρ µ | ( ∆ ρ µ ) − are also irreducible nonneg-ative matrices. Next, we should find out all the maximallydimensional principal submatrices A n µ of A µ with all its ele-ments being 1, where the maximal dimension means that thedimension of A n µ cannot be enlarged. Let the correspondingHilbert subspaces of principal submatrices A n µ be H n µ spannedby {| i µ i , | i µ i , · · · , | i d n µ i} ⊂ {| i , | i , · · · , | d − i} . Then, the cor-responding incoherent projective operators are P α = | i µ ih i µ | + | i µ ih i µ | + · · · + | i d n µ ih i d n µ | . Performing { P α } on the state ρ , we obtain { ψ α } , i.e., P α ρ P α Tr( P α ρ P α ) = ψ α . By the way, we note that the set of { P α } in Theorem 1 isnot necessarily unique, and we denote the set of { ∆ ψ α } corre-sponding to the maximally dimensional principal submatrices A n µ as S m .Third, we should calculate out the least upper bound of theset S m = { ∆ ψ α } , i.e., W S m .Without loss of generality, suppose that | ψ α i = P d n i = c i α | i i and the corresponding probability distributions of | ψ α i are p ↓ α = ( | c α | , | c α | , ..., | c d n α | , , , ... S m , i.e., W S m . To this end,we first define a probability distribution a = ( a , a , ..., a d ),where a i = max { i X j = | c j | , i X j = | c j | , ..., i X j = | c jL | } − i − X j = a j . We note that the elements of a = ( a , a , ..., a d ) might not be innonincreasing order, i.e., it is not true in general that a j ≥ a j + .Apart from a , we also need the following lemma, which wasproved in Ref. [36]. Lemma . Let a = ( a , a , ... a d ) be a given probability dis-tribution, and let j be the smallest integer in { , ..., n } suchthat a j > a j − . Moreover, let i be the greatest integer in { , , ..., j − } such that a i − ≥ P jr = i a r j − i + = a . Let the probabilitydistribution q = ( q , q , ..., q d ) be defined as q r = ( a , for r = i , i + , ..., j ; a r , otherwise . Then for the probability distribution q , we have that q r − ≥ q r , for all r = , ..., j , and P ks = q s ≥ P ks = a s , k = , ..., d . Moreover, for all t = ( t , t , ..., t d ) such that P ks = t s ≥ P ks = a s , k = , ..., n , we also have P ks = t s ≥ P ks = q s , k = , ..., n . By using the definition of a and the iterate application ofthe above Lemma, we can obtain the least upper bound of S m = { ∆ ψ α } , i.e., W S m and we denoted it as ∆ ψ .Without loss of generality, let the maximum number of ϕ m we can distill from ρ ⊗ ρ ⊗· · ·⊗ ρ n be N . The generalization to d > ∆ ψ ≺ diag(2 − N , ..., − N , ..., . (12)The above relation can be fulfilled if and only if k ψ k ∞ ≤ − N , (13)where k · k ∞ is the max norm on the matrix space. This can beexamined directly since if the first inequality of majorizationrelation in Eq. (12) holds, then the other inequalities for Eq.(12) are automatically satisfied.Thus, the inequality in Eq. (13) gives the maximum num-ber of 2-dimensional maximally coherent state that can be dis-tilled from ρ ⊗ ρ ⊗ · · · ⊗ ρ n and the maximum number is N max = ⌊ log k ψ k − ∞ ⌋ , where ⌊ x ⌋ represents the largest integer equal to or less than x .We can then summarize the above results as Theorem 2. Theorem 2.
The maximum number of 2-dimensional max-imally coherent state that can distill from a set of states, suchas ρ , ρ , ..., ρ n , is N max = ⌊ log k ψ k − ∞ ⌋ . (14)In particular, if the states we chose are all pure coherentstates {| ϕ γ i} with γ = , ..., n , then the maximum number of 2-dimensional maximally coherent state that we can be distilledis N max = ⌊ log ⊗ n γ = k ϕ γ k − ∞ ⌋ , which corresponds to the resultin [30]. This is reminiscent of the case of entanglement [35,39, 40], where the deterministic entanglement distillation ofpure entangled states was studied.We should note that there is a class of states that cannotbe distilled into any pure coherent state via strictly incoherentoperations. If we can transform ρ = P i j ρ i j | i ih j | with the num-ber of ρ ii , m into a pure coherent state | ϕ i = P i c i | i i with the number of c i , n via a strictly incoherentoperation, then the rank of ρ is at most mn . To see this, supposethat we can distill a pure coherent state ϕ from ρ , according toTheorem 1, there must be an orthogonal and complete set ofincoherent projectors { P α } fulfilling the condition in Eq. (1).Let the corresponding decomposition of the Hilbert space of { P α } be H = L α H α , where the dimension of H α is d α , theprojections { P α } of ρ onto each H α are { ψ α } , respectively, and ρ = P li = λ i | λ i ih λ i | is a spectral decomposition for ρ . Then,there are P α | λ i ih λ i | P α Tr( P α | λ i ih λ i | P α ) = ψ α , for all i = , ..., l , with | ψ α i = P i c i α | i i . This means that thenumber, D ρ , of the linear independent vectors of the set {| λ i i} must satisfy D ρ = l − P α ( d α − ≤ m − P α d α + P α = P α ∆ ψ α ≺ ∆ ϕ , we canobtain that the number of c i α , c i ,
0. Thus, there is D ρ = P α ≤ mn .In passing, we would like to point that the phenomenon ofbound coherence under strictly incoherent operations was un-covered in Refs. [23, 27, 28] recently, i.e., there are coherentstates from which no coherence can be distilled via strictly in-coherent operations in the asymptotic regime. The necessaryand su ffi cient condition for a state being bound state was pre-sented in Refs. [27, 28]. Their result shows that a state is abound state if and only if it cannot contain any rank-one sub-matrix. Comparing this result with the Corollary , we obtainthat, for any mixed state ρ , if we can transform it into a purecoherent state | ϕ i , then it cannot be a bound state. However,in general, the converse is not true. Thus, the set of states thatcan be transformed into a pure coherent state | ϕ i is a strictlysmaller set of the set of distillable states. Conclusions. –We have completed the operational task ofdeterministic coherence distillation for a finite number of co-herent states under strictly incoherent operations. Specifically,we have presented the necessary and su ffi cient condition forthe transformation from a mixed coherent state into a pure co-herent state via strictly incoherent operations, which recoversa connection between the resource theory of coherence andthe algebraic theory of majorization lattice. With the help ofthis condition, we have presented the deterministic coherencedistillation scheme and we have derived the maximum num-ber of maximally coherent states that can be obtained via thisscheme.We would like to thank Dian-Min Tong, Xiao-Dong Yu,and Qi-Ming Ding for thoroughly reading the manuscript,and for many suggestions, corrections, and comments, whichhave certainly helped to improve this paper. This work issupported by NSF of China (Grant No.11775300), the Na-tional Key Research and Development Program of China(2016YFA0300603), and the Strategic Priority Research Pro-gram of Chinese Academy of Sciences No. XDB28000000. ∗ [email protected][1] M. A. Nielsen and I. L. Chuang, Quantum Computationand Quantum Information, Cambridge University Press, Cam-bridge, 2000.[2] P. W. Shor, SIAM J. Comput. , 1484 (1997).[3] L. K. Grover, Phys. Rev. Lett. , 325 (1997).[4] C. H. Bennett and G. Brassard, in Proceedings of the IEEE In-ternational Conference on Computers, Systems and Signal Pro-cessing, Bangalore, India , 175 (1984).[5] V. Giovannetti, S. Lloyd, and L. Maccone,
Science , 1330(2004).[6] V. Giovannetti, S. Lloyd, and L. Maccone,
Nat. Photonics ,222 (2011).[7] N. Lambert, Y.-N. Chen, Y.-C. Cheng, C.-M. Li, G.-Y. Chen,and F. Nori, Nat. Phys. , 10 (2013).[8] J. Åberg, arXiv:quant-ph / Phys. Rev. Lett. , 140401 (2014).[10] F. Levi and F. Mintert,
New J. Phys. , 033007 (2014).[11] A. Winter and D. Yang, Phys. Rev. Lett. , 120404 (2016).[12] B. Yadin, J. Ma, D. Girolami, M. Gu, and V. Vedral,
Phys. Rev.X , 041028 (2016).[13] E. Chitambar and G. Gour, Phys. Rev. Lett. , 030401 (2016).[14] E. Chitambar and G. Gour,
Phys. Rev. A , 052336 (2016).[15] J. I. de Vicente and A. Streltsov, J. Phys. A: Math. Theor. ,045301 (2017).[16] X.-D. Yu, D.-J. Zhang, G. F. Xu, and D. M. Tong, Phys. Rev. A , 060302(R) (2016).[17] M.-L. Hu, X. Hu, J.-C. Wang, Y. Peng, Y.-R. Zhang, and H.Fan, Phys. Rep. , 1 (2018).[18] A. Streltsov, G. Adesso, and M. B. Plenio,
Rev. Mod. Phys. ,041003 (2017).[19] F. G. S. L. Brand˜ a o and G. Gour, Phys. Rev. Lett. , 070503(2015).[20] C. L. Liu, X.-D. Yu, and D. M. Tong,
Phys. Rev. A , 042322(2019).[21] X. Yuan, H. Zhou, Z. Cao, and X. Ma, Phys. Rev. A , 022124(2015).[22] K. Fang, X. Wang, L. Lami, B. Regula, and G. Adesso, Phys.Rev. Lett. , 070404 (2018).[23] Q. Zhao, Y. Liu, X. Yuan, E. Chitambar, and A. Winter, arXiv:1808.01885.[24] C. L. Liu, Y. Q. Guo, and D. M. Tong,
Phys. Rev. A , 062325(2017).[25] E. Chitambar, Phys. Rev. A , 050301(R) (2018).[26] K. Bu, U. Singh, S.-M. Fei, A. K. Pati, and J. Wu, Phys. Rev.Lett. , 150405 (2017).[27] L. Lami, arXiv: 1902.02427.[28] L. Lami, B. Regula, and G. Adesso,
Phys. Rev. Lett. ,150402 (2019).[29] B. Regula, L. Lami, and A. Streltsov,
Phys. Rev. A , 052329(2018).[30] B. Regula, K. Fang, X. Wang, and G. Adesso, Phys. Rev. Lett. , 010401 (2018).[31] Q. Zhao, Y. Liu, X. Yuan, E. Chitambar, and X. Ma,
Phys. Rev.Lett. , 070403 (2018).[32] R. Bhatia, Matrix Analysis, Springer-Verlag, New York, 1997.[33] S. Du, Z. Bai and Y. Guo,
Phys. Rev. A , 052120 (2015).[34] H. Zhu, Z. Ma, Z. Cao, S. M. Fei, and V. Vedral, Phys. Rev. A Phys. Rev. Lett. , 436 (1999).[36] F. Cicalese and U. Vaccaro, IEEE Trans. Inf. Theory , 933(2002).[37] B. A. Davey and H. A. Prisetly, Introduction to Lattices andOrder, Cambridge University Press, Cambridge, 1990.[38] X.-D. Yu and O. G¨uhne, Phys. Rev. A , 062310 (2019).[39] F. Morikoshi and M. Koashi, Phys. Rev. A , 022316 (2001).[40] M. Hayashi, M. Koashi, K. Matsumoto, F. Morikoshi, and A.Winter, J. Phys. A: Math. Theor.36