aa r X i v : . [ qu a n t - ph ] J u l Maximally coherent states
Zhaofang Bai ∗ and Shuanping Du † School of Mathematical Sciences, Xiamen University, Xiamen 361000, China
The relative entropy measure quantifying coherence, a key property of quantum system, is pro-posed recently. In this note, we firstly investigate structural characterization of maximally coherentstates with respect to the relative entropy measure. It is shown that mixed maximally coherent statesdo not exist and every pure maximally coherent state has the form U | ψ ih ψ | U † , | ψ i = √ d P dk =1 | k i ,U is diagonal unitary. Based on the characterization of pure maximally coherent states, for a bipar-tite maximally coherent state with d A = d B , we obtain that the super-additivity equality of relativeentropy measure holds if and only if the state is a product state of its reduced states. From theviewpoint of resource in quantum information, we find there exists a maximally coherent state withmaximal entanglement. Originated from the behaviour of quantum correlation under the influenceof quantum operations, we further classify the incoherent operations which send maximally coherentstates to themselves. PACS numbers: 03.65.Ud, 03.67.-a, 03.65.TaKeywords: Maximally coherent state, Relative entropy measure, Incoherent operation
I. INTRODUCTION
Being at the heart of interference phenomena, quan-tum coherence plays a central role in physics as it enablesapplications that are impossible within classical mechan-ics or ray optics. It provides an important resource forquantum information processing, for example, Deutsch’salgorithm, Shor’s algorithm, teleportation, superdensecoding and quantum cryptography [1]. Maximally co-herent states are especially important for such quantuminformation processing tasks.Recently, it has attracted much attention to quan-tify the amount of quantum coherence. In [2], the re-searchers establish a quantitative theory of coherence asa resource following the approach that has been estab-lished for entanglement in [3]. They introduce a rigorousframework for quantification of coherence by determiningdefining conditions for measures of coherence and identi-fying classes of functionals that satisfy these conditions.The relative entropy measure and l -norm measure areproposed. Other potential candidates such as the mea-sures induced by the fidelity , l -norm and trace normare also discussed. It is shown that the coherence mea-sure induced by l -norm is not good. Since then, a lotof further considerations about quantum coherence arestimulated [4–15].It has been shown that a good definition of coherencedoes not only depend on the state of the system, but alsodepends on a fixed basis for the quantum system [2]. Theparticular basis (of dimension d ) we choose throughoutthis manuscript is denoted by {| k i} dk =1 . In [2], Baum-gratz etc. identify the pure state | ψ i := √ d P dk =1 | k i asa maximally coherent state (MCS) with respect to any ∗ Electronic address: [email protected] † Corresponding author; Electronic address:[email protected] measure of coherence because every state can be preparedfrom | ψ i by a suitable incoherent operation. Two natu-ral questions arise immediately. Under a given coherencemeasurement, whether it is the unique pure state whosecoherence is maximal and whether there exists a mixedmaximally coherent state? Given a coherence measure C ,we call a state ρ to be a maximally coherent state (MCS)with respect to C if C ( ρ ) attends the maximal value of C .The relative entropy measure is able to not only quan-tify coherence but also quantify superposition and frame-ness [16–21]. In [22], the regularized relative entropymeasure of a resource can be used to describe the opti-mal rate of converting (by asymptotically resource non-generating operations) n copies of a resource state ρ into m copies of another resource state σ . On considering theimportance of the relative entropy measure, we are aimedto characterize the structure of the maximally coherentstates under the relative entropy coherence measure. Weobtain that mixed maximally coherent states do not ex-ist and each pure maximally coherent state has the form U | ψ i , where | ψ i = √ d P dk =1 | k i , U is diagonal unitary.While it does not mean maximally coherent states withrespect to any coherence measure have the form U | ψ i .Indeed there exists a coherence measure such that maxi-mally coherent states with respect to this measure do nothave the form U | ψ i (see the example after Result 1).Quantum correlation includes quantum entanglementand quantum discord. Both entanglement and discordhave a common necessary condition—quantum coherence[23]. In [4], Z. Xi etc. study the relative entropy coher-ence for a bipartite system in a composite Hilbert space H AB = H A ⊗ H B . They obtain an interesting propertyfor the relative entropy of coherence, that is, the super-additivity, C RE ( ρ ) ≥ C RE ( ρ A ) + C RE ( ρ B ) . At the same time, they leave an open question thatwhether the equality holds if and only if ρ = ρ A ⊗ ρ B . Us-ing characterization of MCS with respect to relative en-tropy coherence measure, we will show that this questionholds true if the two subsystems have the same dimen-sion and ρ is a MCS. A counterexample is also given totell us that the answer is negative if the two subsystemshave different dimension. Furthermore, we obtain thatthere is a state with maximal coherence and maximalentanglement.Coherence, as a kind of resource, enables applicationsthat are impossible within classical information. If anincoherent operation sends the MCSs to MCSs, we say itpreserves MCSs. Naturally, does this kind of operationreduce the resource or is it a without noise process? Wewill show that an incoherent operation preserves MCSsif and only if it has the form U · U † , U is a permutationof some diagonal unitary.The structure of this paper is as follows. Section II re-calls the axiomatic postulates for measures of coherence,the concepts of the relative entropy measure and inco-herent operations in [2]. In section III, we focus on thestructural characterization of maximally coherent states.We apply this characterization to bipartite system to an-swer the question on super-additivity equality in sectionIV. The section V is devoted to the incoherent opera-tions preserving maximally coherent states. The paperis ended with the conclusion in section VI. II. PRELIMINARY
Let H be a finite dimensional Hilbert space with d =dim( H ). Fixing a basis {| k i} dk =1 , we call all density op-erators (quantum states) that are diagonal in this basisincoherent, and this set of quantum states will be labelledby I , all density operators ρ ∈ I are of the form ρ = d X k =1 λ k | k ih k | . Quantum operations are specified by a finite set of Krausoperators { K n } satisfying P n K † n K n = I , I is the iden-tity operator on H . From [2], quantum operations areincoherent if they fulfil K n ρK † n /T r ( K n ρK † n ) ∈ I for all ρ ∈ I and for all n . This definition guarantees that inan overall quantum operation ρ P n K n ρK † n , even ifone does not have access to individual outcomes n , noobserver would conclude that coherence has been gener-ated from an incoherent state. Incoherent operations areof particular importance for the decoherence mechanismsof single qubit [24, 25]. As a special case, the unitary in-coherent operation has the form ρ U ρU † , here U is apermutation of a diagonal unitary.Based on Baumgratz et al.’s suggestion [2], any propermeasure of coherence C must satisfy the following ax-iomatic postulates.(i) The coherence vanishes on the set of incoherentstates (faithful criterion), C ( ρ ) = 0 for all ρ ∈ I ;(ii) Monotonicity under incoherent operation Φ, C (Φ( ρ )) ≤ C ( ρ ); (iii) Non-increasing under mixing of quantum states(convexity), C ( X n p n ρ n ) ≤ X n p n C ( ρ n )for any ensemble { p n , ρ n } .For any quantum state ρ on the Hilbert space H , themeasure of relative entropy coherence is defined as C RE ( ρ ) := min σ ∈I S ( ρ || σ ) , where S ( ρ || σ ) = T r ( ρ log ρ − ρ log σ ) is relative en-tropy. In particular, there is a closed form solution thatmakes it easy to evaluate analytical expressions [2]. ForHilbert space H with the fixed basis {| k i} dk =1 , we write ρ = P k,k ′ p k,k ′ | k ih k ′ | and denote ρ diag = P k p kk | k ih k | .By the properties of relative entropy, it is easy to obtain C RE ( ρ ) = S ( ρ diag ) − S ( ρ ) , here S ( · ) is von Neumann entropy. Some basic propertiesof relative entropy coherence have been given in [2].Throughout the paper, if not specified, ρ is a max-imally coherent state (MCS) means that it is with re-spect to C RE . As we mention in introduction, | ψ i := √ d P dk =1 | k i is a maximally coherent state. That is, C RE ( | ψ ih ψ | ) = log d is the maximal value of C RE . Thestructural characterization of MCS plays a key role insection IV and V. An incoherent operation Φ preservesMCSs means that Φ( ρ ) is a MCS if ρ is a MCS. III. MAXIMALLY COHERENT STATES ON H Result 1. ρ is a MCS if and only if ρ = U | ψ ih ψ | U † ,where | ψ i = √ d P dk =1 | k i and U is a diagonal unitary. Proof.
It is easy to see, for every diagonal unitaryelement U , ρ U ρU † is an incoherent operation. Fromthe monotonicity under incoherent operations, it followsthat U | ψ ih ψ | U † is a MCS.For if part, we firstly prove that ρ is pure. Note thatthe maximal value of C RE is log d . For every pure stateensemble ρ = P i p i ρ i . If C RE ( ρ ) = log d , thenlog d = C RE ( ρ ) ≤ X i p i C RE ( ρ i ) ≤ log d. Thus C RE ( ρ ) = P i p i C RE ( ρ i ) and C RE ( ρ i ) = log d . Let C RE ( ρ i ) = S ( ρ i k σ i ) and σ = P i p i σ i . By the jointlyconvex of relative entropy,log d ≤ S ( ρ || σ ) ≤ X i p i S ( ρ i || σ i ) = log d. This implies S ( ρ || σ ) = P i p i S ( ρ i || σ i ). From [26, Theo-rem 10], it follows that ρ i = ρ j and so ρ is a pure state.Now, we write ρ = | φ ih φ | and | φ i = P dk =1 α k | k i . Bythe property of relative entropy, C RE ( ρ ) = S ( ρ diag ) = − d X k =1 | α k | log ( | α k | ) . A direct computation shows that C RE ( ρ ) = log ( d ) im-plies that | α k | = 1 /d . One can write α k = √ d e iθ k , then | φ i = P dk =1 1 √ d e iθ k | k i . Let U = diag( e iθ , e iθ , ..., e iθ d ),so | φ i = U | ψ i . (cid:3) In [2], it is mentioned that if D is distance measuresatisfying contracting under CPTP maps and jointly con-vex, (i.e., satisfying D ( ρ, σ ) ≥ D (Φ CP T P ( ρ ) , Φ CP T P ( σ ))and D ( P n p n ρ n , P n p n σ n ) ≤ P n p n D ( ρ n , σ n )), then onemay define a coherence measure by C D ( ρ ) = min σ ∈I D ( ρ, σ ) . From the proof of Result 1, it is easy to see that if D possesses the property that the equality of jointly convexholds true implies ρ n = ρ m , then the MCSs with respectto the coherence measure induced by D are pure. It isknown that l -norm [2] and quantum skew divergence[27] are with such property.Here we remark that Result 1 does not hold true forany coherence measure. The following is a counter ex-ample. Example.
Let d = 4 and Ω = { x = ( x , x , x , x ) t | P i =1 x i = 1 and x i ≥ } , here ( x , x , x , x ) t denotesthe transpose of row vector ( x , x , x , x ). Assume f ( x ) = (cid:26) − P i =1 x i log x i , x ↓ = 0log , x ↓ = 0 , here x ↓ is the least element in ( x , x , x , x ) t . By [15,Theorem 1], it is easy to check that the nonnegative func-tion f can derive a coherence measure C f . It is clear thatboth | ψ i = P k =1 √ x k | k i , x ↓ = 0 and | φ i = P k =1 q | k i are maximally coherent under C f . IV. MAXIMALLY COHERENT STATES ON H A ⊗ H B Consider a bipartite system in a composite Hilbertspace H AB = H A ⊗ H B of d = d A × d B dimension, here d A = dim( H A ) and d B = dim( H B ). Let {| k i A } d A k =1 and {| j i B } d B j =1 be the orthogonal basis for the Hilbert space H A and H B , respectively. Given a quantum state ρ AB which could be shared between two parties, Alice andBob, and let ρ A and ρ B be the reduced density operatorfor each party.In [4], Xi etc. show the supper-additivity of the relativeentropy coherence: C RE ( ρ AB ) ≥ C RE ( ρ A ) + C RE ( ρ B ) . (1) They leave an question that whether the equality holdsif and only if ρ = ρ A ⊗ ρ B . In the following, we willshow that the answer is affirmative if d A = d B and ρ AB is a MSC. If d A = d B , then the answer is negative. Thisimplies that, in the case of d A = d B , there is a correlationbetween the two subsystems, this leads to the increase ofthe coherence on the bipartite system. Result 2. If d A = d B and ρ AB is a MCS, then theequality in (1) holds if and only if ρ AB = ρ A ⊗ ρ B . Proof.
Let {| i i A } d A i =1 and {| j i B } d B j =1 be the orthogonalbasis for the Hilbert space H A and H B , respectively. Let ρ AB = | φ ih φ | with | φ i = √ d P d A ,d B i,j =1 e iθ ij | i A j B i . Then ρ = 1 d X i,j,s,t e i ( θ ij − θ st ) ) | i A ih s A | ⊗ | j B ih t B | , (2) ρ A = 1 d X i,s ( X j e i ( θ ij − θ sj ) ) | i A ih s A | and ρ B = 1 d X j,t ( X i e i ( θ ij − θ it ) ) | j B ih t B | . Note that C RE ( ρ AB ) = C RE ( ρ A )+ C RE ( ρ B ) ⇔ ρ A , ρ B areMCSs ⇔ | P i e i ( θ ij − θ it ) | = d A and | P j e i ( θ ij − θ sj ) | = d B .The latter equivalence follows from Result 1. By a directcomputation, we have θ ij − θ it = θ i ′ j − θ i ′ t and θ ij − θ sj = θ ij ′ − θ sj ′ . (3)On the other hand, ρ A ⊗ ρ B = 1 d X i,j,s,t α ijst | i A ih s A | ⊗ | j B ih t B | , (4)here α ijst = d ( P j e i ( θ ij − θ sj ) )( P i e i ( θ ij − θ it ) ) . FromEquations (2),(3) and (4), we finish the proof. (cid:3)
What will happen if d A = d B ? The following coun-terexample shows the answer is negative in this case.Assume d A = 2 and d B = 3. Let | φ i = 1 √ | i + e iθ | i + e iθ | i + e iθ | i + e iθ | i + e iθ | i ) ,θ ∈ (0 , π ). Clearly, ρ = | φ ih φ | is a MCS. By an elemen-tary computation, ρ = 16 e − iθ e − iθ e − iθ e − iθ e − iθ e iθ e − iθ e − iθ e − iθ e − iθ e iθ e iθ e − iθ e − iθ e − iθ e iθ e iθ e iθ e − iθ e − iθ e iθ e iθ e iθ e iθ e − iθ e iθ e iθ e iθ e iθ e iθ .ρ A = 12 (cid:18) e − iθ e iθ (cid:19) , ρ B = 13 e − iθ e − iθ e iθ e − iθ e iθ e iθ . It is evident that both ρ A and ρ B are MCSs and C RE ( ρ AB ) = C RE ( ρ A ) + C RE ( ρ B ), however ρ = ρ A ⊗ ρ B .It is wellknown that both coherence and entangle-ment are considered as resource in quantum information.Whether is there a state which is not only maximally co-herent but also maximally entangle? We will discuss thisimportant question at the end of this section. Result 3.
There is a MCS ρ which is maximal entan-glement. Proof.
Let ρ = | φ ih φ | with | φ i = 1 √ d d A ,d B X i,j =1 e iθ ij | i A j B i . Then ρ A = d P i,s ( P j e i ( θ ij − θ sj ) ) | i A ih s A | . Recall that ρ is maximally entangled if and only if ρ A = Id A . Therefore X j e i ( θ ij − θ sj ) = 0 for every pair i = s (5)implies that ρ is a maximally entangle state. Note thatThe equation (5) has a solution. In order to understandthe solution, we list an example in the case of d A = d B =3. θ = θ = θ = 0, θ = 0, θ = − π , θ = − π , θ = 0, θ = − π , and θ = − π . (cid:3) V. INCOHERENT OPERATIONS PRESERVINGMCS
It is an interesting area to study the behavior of quan-tum correlation under the influence of quantum opera- tions [28–43]. For example, local operations that cannotcreate QD is investigated in [34, 37, 39], local operationsthat preserve the state with vanished MIN is character-ized in [38] and local operations that preserve the max-imally entangled states is explored in [40]. The goal ofthis chapter is to discuss when an incoherent operationpreserves MCSs.Here is our main result in this section.
Result 4.
An incoherent operation Φ preserves MCSsif and only if Φ( ρ ) = U ρU † for every quantum state ρ ,here U is a permutation of a diagonal unitary.From Result 4, every incoherent operation preserv-ing MCSs does not reduce the resource and is noiseless.Although this result is not surprising, the proof is nottrivial. Let Φ be specified by a set of Kraus operators { K n } , the main step of our proof is to show that each K n = a n Π n after some reduction, a n is a complex num-ber with P n | a n | = 1 and Π n is a permutation of I . Thereduction process is not trivial because we need to proveΦ is unital which is based on an interesting property thatidentity operator can be described as a sum of d MCSs.
Proof.
The if part can be obtained directly from theResult 1.Now we check the only if part. We firstly claim that I can be written as P dk =1 | φ k ih φ k | with all of | φ k i areMCSs. Choose | φ j i = √ d P dk =1 e iα j,k | k i , all of α j,k arereal numbers. Denote M = P dj =1 | φ j ih φ j | , then M hasthe matrix form d P dj =1 e i ( α j, − α j, ) · · · d P dj =1 e i ( α j, − α j,d )1 d P dj =1 e i ( α j, − α j, ) · · · d P dj =1 e i ( α j, − α j,d ) · · · · · · · · · · · · d P dj =1 e i ( α j,d − α j, ) 1 d P dj =1 e i ( α j,d − α j, ) · · · . If α j,k satisfy α j +1 ,k − α j +1 ,l = α j,k − α j,l + 2( k − l ) d π, then P dj =1 e i ( α j,k − α j,l ) = 0 ( j, k, l = 1 , . . . , d , k = l ). So M = I . There exist solutions of these equations, forexample α j,k = d ( k − j − π .In the following, we show that Φ preserving MCS isunital, that is Φ( I ) = I . Note that Φ is incoherent, we have Φ( I ) is diagonal. From the Result 1 in sectionIII, Φ( | φ k ih φ k | ) = U k | ψ ih ψ | U † k , U k is diagonal unitary.Then (Φ( | φ k ih φ k | )) diag = Id , here (Φ( | φ k ih φ k | )) diag de-notes the state obtained from Φ( | φ k ih φ k | ) by deleting alloff-diagonal elements. This implies thatΦ( I ) = Φ( I ) diag = d X k =1 (Φ( | φ k ih φ k | )) diag = I. Let K n be the Kraus operators of Φ, we obtain P n K n K † n = P n K † n K n = I . From Φ is incoherent,we also have that every column of K n is with at most 1nonzero entry. From Result 1, for every diagonal unitary U , there is a diagonal unitary V U depending on U suchthat Φ( U | ψ ih ψ | U † ) = V U | ψ ih ψ | V † U . That is | ψ ih ψ | is afixed point of V † U Φ( U · U † ) V U . This implies V † U K n U | ψ ih ψ | = | ψ ih ψ | V † U K n U. So V † U K n U | ψ i = λ n,U | ψ i for some scalar λ n,U dependingon U and n . We assert that λ n,I = 0. Otherwise, K n issingular and so there is a row of K n in which all entriesare zero. Note that | ψ i = √ d P dk =1 | k i , therefore all λ n,U equal zero and so K n U | ψ | = 0. Since I can be writtenas a sum of MCSs, we have K n = 0. From λ n,I = 0,there exists a nonzero element of each row of K n . Com-bining this and each column of K n is with at most onenonzero element, we get that there is one and only one nonzero entry in every row and column of K n . Notethat V † I Φ V I possesses the same properties as Φ, withoutloss of generality, we may assume Φ( | ψ ih ψ | ) = | ψ ih ψ | . So K n | ψ i = λ n,I | ψ i . This implies the entries of K n areequal. Therefore K n = a n Π n , a n is a complex numberwith P n | a n | = 1 and Π n is a permutation of I .From Result 1, for arbitrary d real numbers θ , · · · , θ d , | φ i = P k √ d e iθ k | k i is a MCS. By a direct com-putation, K n | φ i = a n √ d P k e iα kn | k i , ( α n , · · · , α dn ) =Π n ( θ , · · · , θ d ). Furthermore, K n | φ ih φ | K † n is the matrix | a n | d e i ( α n − α n ) · · · e i ( α n − α dn ) e i ( α n − α n ) · · · e i ( α n − α dn ) · · · · · · · · · · · · e i ( α dn − α n ) e i ( α dn − α n ) · · · . And Φ( | φ ih φ | ) equals1 d P n | a n | P n | a n | e i ( α n − α n ) · · · P n | a n | e i ( α n − α dn ) P n | a n | e i ( α n − α n ) P n | a n | · · · P n | a n | e i ( α n − α dn ) · · · · · · · · · · · · P n | a n | e i ( α dn − α n ) P n | a n | e i ( α dn − α n ) · · · P n | a n | . By our assumption, it is a MCS. So | X n | a n | e i ( α jn − α kn ) | = 1for j, k = 1 , , · · · , d . The arbitrariness of α jn and α kn implies n = 1. Therefore Φ has the desired form. (cid:3) VI. CONCLUSION
In this paper, we firstly investigate the maximally co-herent states with respect to the relative entropy mea-sure of coherence. We find that there does not exista mixed maximally coherent state and each pure max-imally coherent states have the form U | ψ i , where U is adiagonal unitary and | ψ i := √ d P dk =1 | k i . Applying thisstructural characterization of maximally coherent statesto bipartite system, we answer the question left in [4]whether C RE ( ρ AB ) = C RE ( ρ A ) + C RE ( ρ B ) if and only ρ = ρ A ⊗ ρ B . It is shown that the answer is affirmativeif d A = d B and ρ AB is a MSC. If d A = d B , then theanswer is negative. From the viewpoint of resource of quantum information, we show that there exists a statewhich is not only maximally coherent but also maximallyentangled. By using the form of pure maximally coher-ent states, we obtain the structural characterization ofincoherent operations sending maximally coherent statesinto maximally coherent states. That is, an incoherentoperation Φ preserves MCSs if and only if Φ( ρ ) = U ρU † for every quantum state ρ , here U is a permutation of adiagonal unitary. VII. ACKNOWLEDGEMENT
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