Assisted coherence distillation of certain mixed states
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Assisted coherence distillation of certain mixedstates
Xiao-Li Wang · Qiu-Ling Yue · Su-JuanQin † Received: date / Accepted: date
Abstract
In the task of assisted coherence distillation via the set of opera-tions X , where X is either local incoherent operations and classical commu-nication (LICC), local quantum-incoherent operations and classical communi-cation (LQICC), separable incoherent operations (SI), or separable quantum-incoherent operations (SQI), two parties, namely Alice and Bob, share manycopies of a bipartite joint state. The aim of the process is to generate themaximal possible coherence on the subsystem of Bob. In this paper, we in-vestigate the assisted coherence distillation of some special mixed states, thestates with vanished basis-dependent discord and Werner states. We show thatall the four sets of operations are equivalent for assisted coherence distillation,whenever Alice and Bob share one of those mixed quantum states. Moreover,we prove that the assisted coherence distillation of the former can reach theupper bound, namely QI relative entropy, while that of the latter can not.Meanwhile, we also present a sufficient condition such that the assistance ofAlice via the set of operations X can not help Bob improve his distillablecoherence, and this condition is that the state shared by Alice and Bob hasvanished basis-dependent discord. Keywords quantum coherence · coherence distillation · QI relative entropy
X.-L. Wang · Q.-L. Yue · S.-J. Qin † State Key Laboratory of Networking and Switching Technology, Beijing University of Postsand Telecommunications, Beijing 100876, China † E-mail: [email protected]. WangSchool of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo,454000, China Xiao-Li Wang et al.
Quantum coherence, another embodiment of the superposition principle ofstates, is essential for many distinctive and captivating characteristics of quan-tum systems [1-3]. The quantification of of quantum coherence was recentlyproposed by Baumgratz et al. [4]. Under this framework, many new coherencequantifiers have been presented since then [5-8]. Meanwhile, various propertiesof quantum coherence have been investigated such as the connections betweenquantum coherence and quantum correlations [9-14], the distillation of coher-ence [15-17], the dynamics under noisy evolution of quantum coherence [18,19],among others. The role of coherence in biological system [20], and thermody-namics [21] has also been explored.In the framework of coherence given by Baumgratz et al. [4], let {| i i} besome fixed reference basis (incoherent basis) in the finite dimensional Hilbertspace, and a state is said to be incoherent if it is diagonal in this basis, be-ing of the form P i p i | i ih i | . A quantum operation is identified by a set ofKraus operators { K l } satisfying P l K † l K l = I . Specially, a quantum oper-ation is called incoherent if it can be written in the form Λ ( ρ ) = K l ρK † l ,with incoherent Kraus operators K l , i.e., K l | m i ∼ | n i , where | m i and | n i are elements of the fixed reference basis. As a quantifier of quantum co-herence, we will use the relative entropy of coherence, initially defined as C re ( ρ ) = min σ ∈I S ( ρ k σ ) , where S ( ρ k σ ) = T r ( ρ log ρ ) − T r ( ρ log σ ) is thequantum relative entropy [22] and the minimization is taken over the set ofall incoherent states I . Crucially, the relative entropy of coherence can beevaluate exactly: C re ( ρ ) = S (Π( ρ )) − S ( ρ ) , where S ( ρ ) = − T r ( ρ log ( ρ )) isthe von Neumann entropy [22] and Π( ρ ) = P i | i ih i | ρ | i ih i | denotes the vonNeumann measurement (dephasing operation) of ρ with respect to the fixedreference basis. An important progress within the resource theory of coher-ence is the operational theory of coherence, which is introduced by Winterand Yang [15]. Particularly, they presented the distillable coherence for anystate ρ , i.e., C d ( ρ ) = C re ( ρ ), where the distillable coherence C d ( ρ ) is definedas the maximal rate for extracting maximally coherent single-qubit states, | Ψ i = √ ( | i + | i ), from a given state ρ via incoherence operations.For a bipartite system AB shared by Alice and Bob, its reference ba-sis is usually assumed to be the tenser product basis of local bases [6], i.e., {| ij i AB } , where {| i i A } and {| j i B } are the fixed reference bases of subsystems A and B , respectively. In the task of assisted coherence distillation [16, 17],two parties, namely Alice and Bob, share many copies of some bipartite state ρ AB in system AB and their goal is to maximize coherence on Bob’s sub-system via special operations. The task of assisted coherence distillation vialocal quantum-incoherent operations and classical communication (LQICC)was firstly introduced in Ref. [16]. Then, Streltsov et al. [17] further studiedthe assisted coherence distillation via the set of operations X , where X is ei-ther local incoherent operations and classical communication (LICC), LQICC,separable incoherent operations (SI), or separable quantum-incoherent oper- ssisted coherence distillation of certain mixed states 3 ations (SQI). In these sets of operations, Bob is restricted to just incoherentoperations on his subsystem, since they do not allow for local creation of co-herence on Bob’s side. The corresponding distillable coherence on Bob’s sideis denoted by C A | BX and can be given as follows [17]: C A | BX ( ρ AB ) = sup { R : lim n →∞ ( inf Λ ∈ X k T r A [ Λ [ ρ ⊗ n ]] − Ψ ⊗⌊ Rn ⌋ k ) = 0 } , where ⌊ x ⌋ is the largest integer below or equal to x and Ψ = | Ψ ih Ψ | B is amaximally coherent single-qubit state on Bob’s side. For any bipartite state ρ AB , the distillable coherence C A | BX ( ρ AB ) is upper bounded by the QI relativeentropy C A | Bre ( ρ AB ) = min σ AB ∈QI S ( ρ AB k σ AB ), where the minimization istaken over the set of all quantum-incoherent states QI [17]. Quite remarkably,both sets of operations SQI and SI lead to the same maximal performance forall states in the task of assisted coherence distillation. Besides, for all purestates and maximally correlated states in the incoherent basis, all the sets ofoperations we consider are always equivalent in this task and the distillablecoherence can reach the upper bound, namely QI relative entropy. However,we do not know whether the sets of operations such as LICC, LQICC andSI (SQI) are equivalent in this task and whether the distillable coherence canreach the upper bound for general mixed states. Therefore, in this paper,we will investigate the assisted coherence distillation of two special classesof mixed states. The results suggest that there exist states, namely Wernerstates, whose distillable coherence can not reach the upper bound via the setof operations X we consider here. In the following we discuss the scenario where the state shared by Alice andBob has vanished basis-dependent discord [23]. In this section, we will fre-quently refer to a bipartite system AB , and without other stated, we use {| i i A } and {| j i B } as the fixed reference bases of subsystems A and B , re-spectively. Before we study this work, we recall the condition of states withvanished basis-dependent discord as follows. Lemma 1 . ( Yadin et al. [23]) Let Π B be the von Neumann measurementwith respect to the fixed reference basis {| j i B } of subsystem B . For any givenstate ρ AB in system AB , let ρ A and ρ B be the reduced states of ρ AB on sub-system A and B , respectively. Then, its basis-dependent discord D A | B Π B ( ρ AB )is zero, i.e. D A | B Π B ( ρ AB ) = S ( ρ AB k ρ A ⊗ ρ B ) − S (Π B ( ρ AB ) k ρ A ⊗ Π B ( ρ B )) = 0,if and only if there exists a decomposition of ρ AB , ρ AB = X α p α ρ Aα O ρ Bα , (1)such that ρ Bα are perfectly distinguishable by projective measurements in thefixed reference basis {| j i B } . Xiao-Li Wang et al.
It is well known that coherence is a basis-dependent measure of quantum-ness in single systems, and similarly, D A | B Π B can be seen as a basis-dependentmeasure of quantumness of correlation [23]. From the Lemma 1, we know thatthe states with vanished basis-dependent discord are special separable stateswhose ρ Bα have disjoint coherence support. Another important result we willrefer to is the following Lemma 2 which is about the distillable coherence C A | BX via the set of operations X we consider here. Lemma 2 . (Streltsov et al. [17]) For an arbitrary bipartite state ρ = ρ AB in system AB , the following inequality holds: C A | BLICC ( ρ ) ≤ C A | BLQICC ( ρ ) ≤ C A | BSI ( ρ ) = C A | BSQI ( ρ ) ≤ C A | Bre ( ρ ) , (2)Lemma 2 presents the power of all the sets of operations such as LICC,LQICC, SI and SQI in the task of assisted coherence distillation for an arbi-trary bipartite state. Equipped with these results, we are now in position toprove the following theorem. Theorem 3 . For any bipartite state ρ = ρ AB given in Eq.(1), the followingequality holds: C A | BLICC ( ρ ) = C A | BLQICC ( ρ ) = C A | BSI ( ρ ) = C A | BSQI ( ρ ) = C A | Bre ( ρ ) = C re ( ρ B ) , (3)where ρ B is the reduced state of ρ AB on subsystem B . Proof . Note that the QI relative entropy of any bipartite state ρ can beevaluated exactly [16]: C A | Bre ( ρ ) = S (Π B ( ρ )) − S ( ρ ) = D A | B Π B ( ρ ) + C re ( ρ B ) . (4)Where Π B is the von Neumann measurement in the fixed reference basis {| j i B } and D A | B Π B ( ρ ) is the basis-dependent discord as given in Lemma 1. CombiningEq. (4) and Eq. (2) in Lemma 2, we get the inequality C A | BLICC ( ρ ) ≤ C A | BLQICC ( ρ ) ≤ C A | BSI ( ρ ) = C A | BSQI ( ρ ) ≤ D A | B Π B ( ρ ) + C re ( ρ B ) . (5)Besides, it is straightforward that C A | BX ( ρ AB ) ≥ C re ( ρ B ) , where X is eitherLICC, LQICC, SI or SQI. According to Lemma 1, for any bipartite state ρ = ρ AB given in Eq.(1), its basis-dependent discord D A | B Π B ( ρ ) is equal to zero.Applying Eq. (5) and combing the aforementioned results we arrive at thedesired equality given in Eq. (3). ⊓⊔ From Theorem 3, we obtain that for all states with vanished (incoherent)basis-dependent discord, there does not exist basis-dependent quantum corre-lation between Alice and Bob to help Bob improve his distillable coherence.However, we do not know whether the necessity of Theorem 3 is true. In otherwords, if the distillable coherence C A | BX ( ρ AB ) via the set of operations X , isequal to the relative entropy of coherence C re ( ρ B ), does the state ρ AB has theform given in Eq. (1)? However, as will see in the next section, the distillablecoherence of a Werner state can not reach the upper bound and it means thatthe basis-dependent discord D A | B Π B ( ρ AB ) of a Werner state can not be com-pletely transformed into the coherence on Bob’s side via the set of operations X we consider here. ssisted coherence distillation of certain mixed states 5 Here, we consider assisted coherence of distillation for the Werner states of theform ρ AB = p | Φ + ih Φ + | + 1 − p I, (6)where p ∈ (0 , | Φ + i is the Bell state denoted as | Φ + i = √ ( | i + | i (or other Bell states [22]). Here I is the identity operator on the compositesystem or on the marginal systems, depending on the context. For the two-qubit system, we choose the computable basis as the fixed reference basis ofeach subsystem. In this case, the reduced state of Bob ρ B is incoherent and C re ( ρ B ) = 0. Then, the distillable coherence of Bob via the set of operations X comes from the basis-dependent quantum correlation between Alice and Bob.Moreover, in the following theorem we show that the distillable coherence for aWerner state can not reach the upper bound, namely QI relative entropy andin this case all sets of operations we consider here are equivalent for assistedcoherence distillation. Theorem 4 . For any Werner state ρ = ρ AB given in Eq. (6), the followinginequality holds: C A | BLICC ( ρ ) = C A | BLQICC ( ρ ) = C A | BSI ( ρ ) = C A | BSQI ( ρ ) < C A | Bre ( ρ ) , (7) Proof . In the first step of the proof we show that for any Werner state ρ AB , its distillable coherence C A | BLQICC ( ρ AB ) is strictly less than its QI relativeentropy C A | Bre ( ρ AB ). By direct calculations, the QI relative entropy of theWerner state ρ AB is C A | Bre ( ρ AB ) = D A | B Π B ( ρ AB )= 1 − p (1 − p ) − p (1 + p ) + 1 + 3 p (1 + 3 p ) . (8)where p ∈ (0 , ρ B + = p | + ih + | + (1 − p ) I , (9)where | + i = | Ψ i = √ ( | i + | i ) is a maximally coherent single-qubit state,and in this case Bob does not need to do anything. If the result of Alice’s mea-surement is −
1, Alice informs Bob her result and Bob performs the incoherentZ gate | ih | − | ih | on his subsystem to obtain the state with the form givenin Eq. (9). Repeat this process, and Bob obtains many copies of ρ B + . Then, thedistillable coherence of Bob via this LQICC protocol is C re ( ρ B + ) = 1 + p (1 + p ) + 1 − p (1 − p ) . (10)Consequently, we show the fact that C re ( ρ B + ) is the largest distillable co-herence of Bob assisted by Alice through all LQICC protocols. Suppose that Xiao-Li Wang et al.
Alice performs a local quantum measurement E on her subsystem, and informsBob her results. According to the results of Alice, Bob performs the incoherentunitary operations on his subsystem, and then the state on Bob’s side can bedenoted as ρ B E = p ˜ ρ B E + (1 − p ) I B . With the convexity of the relative entropyof coherence [4], we only need to consider the case that ˜ ρ B E is a pure state,which can be written in the Bloch representation,˜ ρ B E = 12 (cid:18) z x − iyx + iy − z (cid:19) , where x, y and z are real and satisfy the equation x + y + z = 1 . (11)In this case, direct calculations show that C re ( ρ B E ) = − p log (1 − p ) + p log (1 + p ) − pz log (1 + pz ) − − pz log (1 − pz ) . Examine the maximum value of C re ( ρ B E ) under the constraint condition Eq.(11), and we obtain the maximum value C re ( ρ B + ) at x = 1 and y = z = 0.Then, it follows that C re ( ρ B E ) ≤ C re ( ρ B + ). Therefore, for the Werner state ρ AB ,it holds that C A | BLQICC ( ρ AB ) = C re ( ρ B + ).Now, we show that C re ( ρ B + ) is strictly less than the upper bound C A | Bre ( ρ AB ).Let Eq. (8) subtract Eq. (10) and the difference is denoted as f ( p ) = C A | Bre ( ρ AB ) − C re ( ρ B + )= 1 + 3 p (1 + 3 p ) − − p (1 − p ) − (1 + p ) log (1 + p ) , (12)with p ∈ (0 , f (0) = f (1) = 0 , f ( ) > f ′′ ( p ) = − p (1+3 p )(1 − p ) ln 2 .Then, f ( p ) is strictly convex for all p ∈ (0 , ) and strictly concave for all p ∈ ( , f ′ ( p ) = 0, we get the minimum value at p = 0.Therefore, we obtain that f ( p ) > p ∈ (0 ,
1) and plot the function f ( p )in Fig. 1 and Fig. 2 , which highlights the distillable coherence C re ( ρ B + ) via theset of operations X is strictly less than the upper bound, namely QI relativeentropy for any Werner state given in Eq. (6).Finally, we show that for the Werner state ρ AB , the distillable coherenceof Bob is also equal to C re ( ρ B + ) via other sets of operations such as SQI, SI andLICC. Since LQICC is a subset of SQI and SI, the similar approach as aboveleads to the equality: C A | BSI ( ρ ) = C A | BSQI ( ρ ) = C re ( ρ B + ) . In order to prove that C A | BLICC ( ρ ) = C re ( ρ B + ), we present a LICC protocol achieving the rate C re ( ρ B + ).Let Alice perform an erasing (incoherent) measurement on her subsystem [24],whose Kraus operators are K A = (cid:18) √ i √ (cid:19) , K A = (cid:18) − √ i √ (cid:19) , ssisted coherence distillation of certain mixed states 7 −0.2 0 0.2 0.4 0.6 0.8 1 1.200.010.020.030.040.050.060.070.080.090.1 ← f(p) p Fig. 1
Graph of function f ( p ) which is the difference between the QI relative entropy andthe distillable coherence via LQICC for any Werner state given in Eq. (6). reA|B ( ρ AB ) → p ← C re ( ρ +B ) ← f(p) Fig. 2
Distillable coherence of Bob is strictly less than the upper bound for any Wernerstate ρ AB given in Eq. (6) in the task of assisted coherence distillation. We plot the distillablecoherence C re ( ρ B + ) (the black line), the upper bound C A | Bre ( ρ AB ) (the dashed line), and thefunction f ( p ) given in Eq. (12) (the red line). with ( K A ) † K A +( K A ) † K A = I . If the result of Alice’s measurement is 1, Bob’spost-measurement state is ρ B = p ( i | i + | i√ )( − i h | + h |√ ) + (1 − p ) I . In this case,Bob can get the state ρ B + by applying first an X gate | ih | + | ih | and thenan incoherent unitary gate | ih | − i | ih | to ρ B . If the result of Alice’s mea-surement is 2, Bob’s post-measurement state is ρ B = p ( − i | i + | i√ )( i h | + h |√ ) +(1 − p ) I , and then Bob can get ρ B + by applying first an X gate | ih | + | ih | and then a phase gate | ih | + i | ih | to ρ B . Therefore, via LICC, the distill-able coherence of Bob is C re ( ρ B + ) with the similar arguments as above. Theseresults imply the inequality given in Eq. (7). ⊓⊔ Note that for standard coherence distillation the relative entropy coherenceis in fact equal to the optimal distillation rate [15]. In the task of assistedcoherence distillation via the set of operations X , the QI relative entropy C A | Bre is an upper bound on C A | BX [16, 17]. However, by Theorem 4, we knowthat the distillation coherence C A | BX is not always able to reach the upperbound, namely QI relative entropy. It is surprising that for a Werner state Xiao-Li Wang et al. ρ AB given in Eq. (6) the distillable coherence C A | BX ( ρ AB ) does not depend onthe particular set of operations performed by Alice and Bob. In particular, inthis case the optimal distillation rate can already be obtained by the weakestset of operations LICC. The other sets of operations, such as LQICC, SQI,and SI, can not provide any advantage to improve the distillable coherence onBob’s side. Through the above results, we strongly conjecture that all the setsof operations we consider here are equivalent in the task of assisted coherencedistillation for any mixed state and we leave this question open for furtherstudy. In this paper, we have discussed the assisted coherence distillation of somemixed states, such as the states with vanished (incoherent) basis-dependentdiscord and Werner states, via the set of operations X , where X is eitherLICC, LQICC, SI or SQI. For these mixed states, all sets of operations weconsider here are equivalent for assisted coherence distillation. In particular,we have provided a sufficient condition such that Alice’s assistance via theset of operations X can not help Bob improve his distillable coherence, andthe condition is that the state shared by Alice and Bob has vanished basis-dependent discord. Moreover, we have proved that the distillable coherence ofa Werner state via the set of operations X can not reach the upper bound,namely QI relative entropy. This result suggest that there exist a state forexample the Werner state whose distillable coherence is not its QI relativeentropy, even through the largest set of operations SQI. We hope that ourwork helps to well understand the resource theory of coherence in distributedscenarios. References
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