Towards resource theory of coherence in distributed scenarios
Alexander Streltsov, Swapan Rana, Manabendra Nath Bera, Maciej Lewenstein
TTowards resource theory of coherence in distributed scenarios
Alexander Streltsov,
1, 2, ∗ Swapan Rana, Manabendra Nath Bera, and Maciej Lewenstein
2, 3 Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, D-14195 Berlin, Germany ICFO – Institut de Ciències Fotòniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain ICREA – Institució Catalana de Recerca i Estudis Avançats, Lluis Companys 23, 08010 Barcelona, Spain
The search for a simple description of fundamental physical processes is an important part of quantum theory.One example for such an abstraction can be found in the distance lab paradigm: if two separated parties are con-nected via a classical channel, it is notoriously di ffi cult to characterize all possible operations these parties canperform. This class of operations is widely known as local operations and classical communication (LOCC).Surprisingly, the situation becomes comparably simple if the more general class of separable operations is con-sidered, a finding which has been extensively used in quantum information theory for many years. Here, wepropose a related approach for the resource theory of quantum coherence, where two distant parties can only per-form measurements which do not create coherence and can communicate their outcomes via a classical channel.We call this class local incoherent operations and classical communication (LICC). While the characterizationof this class is also di ffi cult in general, we show that the larger class of separable incoherent operations (SI)has a simple mathematical form, yet still preserving the main features of LICC. We demonstrate the relevanceof our approach by applying it to three di ff erent tasks: assisted coherence distillation, quantum teleportation,and single-shot quantum state merging. We expect that the results obtained in this work also transfer to otherconcepts of coherence which are discussed in recent literature. The approach presented here opens new ways tostudy the resource theory of coherence in distributed scenarios. I. INTRODUCTION
The resource theory of quantum coherence is a vivid re-search topic, and various approaches in this direction havebeen presented over the past few years [1–6]. The formal-ism proposed recently by Baumgratz et al. [5] has triggeredthe interest of several authors, and a variety of results havebeen obtained since then. One line of research is the formula-tion and interpretation of new coherence quantifiers [7–11], inparticular those arising from quantum correlations such as en-tanglement [12, 13]. The study of coherence dynamics undernoisy evolution is another promising research direction [14–18]. The role of coherence in biological systems [19–21],thermodynamics [22–26], spin models [27–29], and other re-lated tasks in quantum theory [30–38] has also been investi-gated.In the framework introduced by Baumgratz et al. [5],quantum states which are diagonal in some fixed basis {| i (cid:105)} are called incoherent: these are all states of the form ρ = (cid:80) k p k | k (cid:105)(cid:104) k | . A quantum operation is called incoherent if it canbe written in the form Λ ( ρ ) = (cid:80) l K l ρ K † l with incoherent Krausoperators K l , i.e., K l | m (cid:105) ∼ | n (cid:105) , where | m (cid:105) and | n (cid:105) are elementsof the incoherent basis. Significant progress within this re-source theory has been achieved by Winter and Yang [39]. Inparticular, they introduced the distillable coherence and pre-sented a closed formula for it for all quantum states. Similarto the distillable entanglement [40, 41], the distillable coher-ence is defined as the maximal rate for extracting maximallycoherent single-qubit states | Ψ (cid:105) = √ | (cid:105) + | (cid:105) ) (1) ∗ [email protected] from a given mixed state ρ via incoherent operations. Anotherclosely related quantity is the relative entropy of coherence,initially defined as [5] C r ( ρ ) = min σ ∈I S ( ρ || σ ) , (2)where S ( ρ || σ ) = Tr[ ρ log ρ ] − Tr[ ρ log σ ] is the relativeentropy, and the minimum is taken over the set of incoher-ent states I . Crucially, the relative entropy of coherence isequal to the distillable coherence and can be evaluated ex-actly [5, 39]: C d ( ρ ) = C r ( ρ ) = S ( ∆ ( ρ )) − S ( ρ ) , (3)where S ( ρ ) = − Tr[ ρ log ρ ] is the von Neumann entropy, and ∆ ( ρ ) = (cid:80) k (cid:104) k | ρ | k (cid:105) | k (cid:105)(cid:104) k | denotes dephasing of ρ in the incoher-ent basis.Recently, various alternative concepts of coherence havebeen presented in the literature. We will briefly reviewthe most important approaches in the following, and referto [6, 42–44] and references therein for more details. Whileall these notions agree on the definition of incoherent statesas states which are diagonal in a fixed reference basis, theydi ff er significantly in the definition of the corresponding freeoperations. A notable approach in this context is the no-tion of translationally invariant operations, these are opera-tions which commute with unitary translations e − iHt for someHamiltonian H [2, 3]. As was shown by Marvian et al . [45],for nondegenerate Hamiltonians translationally invariant op-erations are a proper subset of incoherent operations. More-over, translationally invariant operations have several desir-able properties, such as a free dilation: they can be imple-mented by introducing an incoherent ancilla, performing aglobal incoherent unitary followed by an incoherent measure-ment on the ancilla, and postselection on the outcomes [43].As was also shown in [43], incoherent operations introducedby Baumgratz et al . [5] in general do not have such a free dila-tion. While the existence of a free dilation is clearly appealing a r X i v : . [ qu a n t - ph ] J a n from the resource-theoretic perspective, the question if everyreasonable resource theory should have a free dilation is stillnot fully settled.By a similar motivation, Chitambar and Gour [42] intro-duced the concept of physical incoherent operations. Theseoperations have a free dilation if one allows incoherent pro-jective measurements on the ancilla followed by classical pro-cessing of the outcomes. Interestingly, the resource theory ob-tained in this way does not have a maximally coherent state,i.e., there is no unique state from which all other states canbe obtained via physically incoherent operations. This is alsotrue for genuinely incoherent [46] and fully incoherent oper-ations [44]. Genuinely incoherent operations are defined asoperations which preserve all incoherent states, they capturethe framework of coherence under additional constrains suchas energy preservation [46]. Moreover, all genuinely incoher-ent operations are incoherent regardless of a particular exper-imental realization. Fully incoherent operations is the mostgeneral set having this property [44].An alternative approach to coherence was made by Yadin etal . [47], who studied quantum processes which do not use co-herence. Such a process cannot be used to detect coherence ina quantum state, i.e., an experimenter who has access to thoseoperations and incoherent von Neumann measurements willnot be able to decide if a quantum state has coherence or not.These operations coincide with strictly incoherent operations,which were introduced earlier by Winter and Yang [39].As has been shown in several recent works, quantum co-herence plays an important role in various tasks which arebased on the laws of quantum mechanics. One such task isquantum state merging, which was first introduced and stud-ied in [48, 49]. The interplay between entanglement andlocal coherence in this task was investigated very recentlyin [50]. An important concept in this context is the notionof local quantum-incoherent operations and classical commu-nication [51]. This class of operations is similar to the classof local operations and classical communication where Aliceand Bob can apply local measurements and share their out-comes via a classical channel, with the only di ff erence thatBob’s measurements have to be incoherent [51].In this paper we will consider the situation where both par-ties, Alice and Bob, can perform only incoherent measure-ment on their parts. The corresponding class will be called lo-cal incoherent operations and classical communication. More-over, we will also generalize these notions to separable op-erations known from entanglement theory [41, 52, 53], thusintroducing separable incoherent operations, and separablequantum-incoherent operations. We will study the relation ofall these classes among each other, and apply them to the taskof assisted coherence distillation, which was first introducedin [51]. We also introduce and discuss the task of incoher-ent teleportation, and study the relation between our classeson single-shot quantum state merging. As is discussed in theconclusions of this paper, we expect that the ideas presented inthis work will find applications beyond quantum informationtheory, most prominently in quantum thermodynamics and re-lated research areas. II. CLASSES OF INCOHERENT OPERATIONSIN DISTRIBUTED SCENARIOS
The framework of local operations and classical communi-cation (LOCC) is one of the most important concepts in en-tanglement theory, as it describes all transformations whichtwo separated parties (Alice and Bob) can perform if they ap-ply local quantum measurements and have access to a clas-sical channel [41, 54]. These operations are di ffi cult to cap-ture mathematically, since a general LOCC operation can in-volve an arbitrary number of rounds of classical communica-tion [41, 54]. However, in many relevant cases it is enoughto consider the informal definition given above. In a simi-lar fashion, we define the class of L ocal I ncoherent opera-tions and C lassical C ommunication (LICC): these are LOCCoperations with the additional constraint that the local mea-surements of Alice and Bob have to be incoherent. We willalso consider the case where Alice can perform arbitrary quan-tum measurements, while Bob is restricted to incoherent mea-surements only. The corresponding class of operations iscalled L ocal Q uantum- I ncoherent operations and C lassical C ommunication (LQICC) [51].Another important framework in entanglement theoryare separable operations. While this class is larger thanLOCC [55], it has a simple mathematical description, and stillpreserves the main features of LOCC [41, 54]. Separable op-erations were initially introduced in [52, 53] as follows: Λ S [ ρ AB ] = (cid:88) i A i ⊗ B i ρ AB A † i ⊗ B † i . (4)The product operators A i ⊗ B i are Kraus operators, i.e., theyfulfill the completeness condition (cid:88) i A † i A i ⊗ B † i B i = . (5)The set of all separable operations will be called S. If all theoperators A i and B i are incoherent, i.e., if they satisfy the con-ditions A i | k (cid:105) A ∼ | l (cid:105) A , (6) B i | m (cid:105) B ∼ | n (cid:105) B , (7)we will call the total operation S eparable I ncoherent (SI) [56]. In the more general case where only Bob’s opera-tors B i are incoherent – and thus only Eq. (7) is satisfied – thecorresponding operation will be called S eparable Q uantum- I ncoherent (SQI).Having introduced the notion of LICC, LQICC, SI, and SQIoperations, we will now study the action of these operationson the initial incoherent state | (cid:105) A | (cid:105) B . It is easy to see thatthe set of states created from | (cid:105) A | (cid:105) B via LICC and via SIoperations is the same, and given by states of the form ρ i = (cid:88) k , l p kl | k (cid:105) (cid:104) k | A ⊗ | l (cid:105) (cid:104) l | B . (8)States of this form are known as fully incoherent states [12,14], and the set of all such states will be denoted by I . Sim-ilarly, the set of states created from | (cid:105) A | (cid:105) B via LQICC andvia SQI operations is the set of quantum-incoherent states QI .These are states of the form [51] ρ qi = (cid:88) j p j σ Aj ⊗ | j (cid:105) (cid:104) j | B . (9)From the above results we immediately see that SI operationsmap the set of fully incoherent states I onto itself, and thesame is true for LICC operations. Similarly, SQI operationsand LQICC operations map the set of fully incoherent states I onto the larger set of quantum-incoherent states QI . Thesestatements are summarized in the following equalities: Λ SI [ I ] = Λ LICC [ I ] = I , (10) Λ SQI [ I ] = Λ LQICC [ I ] = QI . (11)In general, LICC is the weakest set of operations, and theset of separable operations S is the most powerful set of oper-ations considered here. Thus, we get the following inclusions:LICC ⊂ SI ⊂ SQI ⊂ S , (12a)LICC ⊂ LQICC ⊂ SQI ⊂ S , (12b)LICC ⊂ LQICC ⊂ LOCC ⊂ S . (12c)For all of the above inclusions it is straightforward to see theweaker form X ⊆ Y , where X and Y is the corresponding set ofoperations. For most of these inclusions, X ⊂ Y can be thenproven by applying the corresponding sets of operations tothe set of fully incoherent states I . As an example, SI ⊂ SQIfollows from the fact that Λ SI [ I ] = I while Λ SQI [ I ] = QI .The same arguments apply to all the above inclusions apartfrom LICC ⊂ SI , (13)LQICC ⊂ SQI , (14)and LOCC ⊂ S. As already noted above Eq. (4), the inclusionLOCC ⊂ S was proven by Bennett et al. [55], and the re-maining two can be proven using very similar arguments. Inparticular, Bennett et al. [55] presented a separable operationwhich cannot be implemented via LOCC. The correspondingproduct operators of this operation have the following form(see Eq. (4) in [55]): A i ⊗ B i = | i (cid:105) (cid:104) α i | A ⊗ | i (cid:105) (cid:104) β i | B . (15)The particular expressions for | α i (cid:105) and | β i (cid:105) were given in [55](see also the Appendix), but are not important for the rest ofour proof. However, it is important to note that the states | i (cid:105) A and | i (cid:105) B are incoherent states of Alice and Bob respectively. Itis straightforward to see that this separable operation is also anSI operation. Moreover, since this operation cannot be imple-mented via LOCC it also cannot be implemented via LICC.This completes the proof of Eq. (13). The proof of Eq. (14)follows by the same reasoning.The hierarchy of the sets LICC, SI, LOCC, and S is shownin Fig. 1. Note that the above reasoning also implies that thesets LOCC and SI have an overlap, but one is not a subsetof other. Moreover, the figure also depicts a region of oper-ations (crossed area) which are simultaneously contained in S LICCLOCC SI ? Figure 1. Hierarchy of LICC, SI, LOCC, and separable operationsS. The set of LICC operations is the weakest set, and S is the mostpowerful set. The crossed region displays operations which are inLOCC and SI, but not in LICC. It remains open if such operationsexist. For simplicity, we do not display the sets LQICC and SQI.
LOCC and SI, but not in LICC. Such operations would havethe property that they cannot create any bipartite coherence.On the other hand, they can be implemented via local oper-ations and classical communication, but for that require localcoherent operations on at least one of the parties. It remains aninteresting open question if such operations exist at all. If theanswer to this question is negative, the intersection of LOCCand SI is equal to LICC. We also mention that similar ques-tions arise if we consider the sets LQICC and SQI. These setswere not included in Fig. 1 for simplicity. Their inclusionsare shown in Eq. (12). In the following section we will ap-ply the tools presented here to the task of assisted coherencedistillation, initially presented in [51].
III. ASSISTED COHERENCE DISTILLATIONA. General setting
The task of assisted coherence distillation via bipartiteLQICC operations was introduced and studied in [51]. Inthis task, Alice and Bob share many copies of a given state ρ = ρ AB . The aim of the process is to asymptotically distillmaximally coherent single qubit states on Bob’s side. In par-ticular, we are interested in the maximal possible rate for thisprocedure.In the following, we will extend this notion beyond LQICCoperations. For this we will consider the maximal amountof coherence that can be distilled on Bob’s side via the set ofoperations X , where X is either LICC, LQICC, SI or SQI [57].The corresponding distillable coherence on Bob’s side will bedenoted by C X and can in general be given as follows: C A | BX ( ρ ) = sup (cid:26) c : lim n →∞ (cid:18) inf Λ ∈ X (cid:13)(cid:13)(cid:13)(cid:13) Tr A (cid:104) Λ [ ρ ⊗ n ] (cid:105) − | c (cid:105) (cid:104) c | ⊗ n (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) = (cid:27) , (16)where | c (cid:105) = | c (cid:105) B is a state on Bob’s subsystem with distillablecoherence C d ( | c (cid:105) ) = c and || M || = Tr[ √ M † M ] is the tracenorm. For more details regarding this definition of C X and forequivalent expressions we refer the reader to section III D. Thequantity C A | B LQICC was introduced in [51], where it was called distillable coherence of collaboration .From Eqs. (12a) and (12b) we immediately see that allquantities C X considered here are between C LICC and C SQI .In the following we will also consider the QI relative entropywhich was defined in [51] as follows: C A | Br ( ρ ) = min σ ∈QI S ( ρ || σ ) . (17)As was also shown in [51], the QI relative entropy can bewritten in closed form: C A | Br ( ρ ) = S ( ∆ B ( ρ )) − S ( ρ ) . (18)Note that the QI relative entropy is additive and does not in-crease under SQI operations. Thus, it does not increase underany set of operations X considered here.It is interesting to compare the QI relative entropy to thebasis-dependent quantum discord, which was initially intro-duced in [58] and can be written as δ A | B ( ρ ) = I A : B ( ρ ) − I A : B ( ∆ B [ ρ ]) (19)with the mutual information I A : B ( ρ ) = S ( ρ A ) + S ( ρ B ) − S ( ρ AB ).Recently, Yadin et al. [47] have studied the role of this quan-tity within the resource theory of coherence. Contrary to theresults presented in [58], the basis-dependent discord vanisheson a larger set of states than the QI relative entropy. While thelatter is zero if and only if the corresponding state is quantum-incoherent, the basis-dependent discord vanishes for all statesof the form ρ = (cid:80) i p i ρ Ai ⊗ ρ Bi , where the states ρ Bi are perfectlydistinguishable by measurements in the incoherent basis [47].This is in particular the case if ρ is quantum-incoherent or aproduct state, and other examples have also been presentedin [47].Quite remarkably, we will see below that C SI is equalto C SQI for all states ρ , and moreover all quantities C X arebounded above by the QI relative entropy. The following in-equality summarizes these results: C A | B LICC ≤ C A | B LQICC ≤ C A | B SI = C A | B SQI ≤ C A | Br . (20)The equality C A | B SI = C A | B SQI will be proven in the followingproposition, and the bound C A | B SQI ≤ C A | Br will be proven be-low in Theorem 2. Proposition 1.
For an arbitrary bipartite state ρ = ρ AB thefollowing equality holds:C A | B SI ( ρ ) = C A | B SQI ( ρ ) . (21) Proof.
In the first step of the proof we will show that for anystate ρ AB and an arbitrary SQI operation Λ SQI there exists anSI operation Λ SI leading to the same reduced state of Bob:Tr A (cid:104) Λ SI [ ρ AB ] (cid:105) = Tr A (cid:104) Λ SQI [ ρ AB ] (cid:105) . (22)The desired SI operation is given by Λ SI [ ρ AB ] = (cid:88) i Π Ai Λ SQI [ ρ AB ] Π Ai , (23)where Π Ai = | i (cid:105) (cid:104) i | A is a complete set of orthogonal projectorsonto the incoherent basis of Alice. It is straightforward tosee that the above operation satisfies Eq. (22) and is indeedseparable and incoherent.Now, given a state ρ = ρ AB with C A | B SQI ( ρ ) = c , for any ε > n ≥ Λ SQI actingon n copies of ρ such that (cid:13)(cid:13)(cid:13)(cid:13) Tr A (cid:104) Λ SQI [ ρ ⊗ n ] (cid:105) − | c (cid:105) (cid:104) c | ⊗ n (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε, (24)where | c (cid:105) is a pure state with C d ( | c (cid:105) ) = c . By using Eq. (22) itfollows that for any ε > n ≥ Λ SI with the same property: (cid:13)(cid:13)(cid:13)(cid:13) Tr A (cid:104) Λ SI [ ρ ⊗ n ] (cid:105) − | c (cid:105) (cid:104) c | ⊗ n (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε. (25)These arguments show that C SI is bounded below by C SQI .The proof of the theorem is complete by noting that C SI isalso bounded above by C SQI since SI ⊂ SQI. (cid:3)
This proposition shows that SQI operations do not providean advantage when compared to SI operations in the consid-ered task: both sets of operations lead to the same maximalperformance. This result is remarkable since the sets SI andSQI are not equal. It is now tempting to assume that the samemethod can also be used to prove that C A | B LICC is equal to C A | B LQICC ,i.e., that quantum operations on Alice’s side do not provideany advantage for assisted coherence distillation. Note how-ever that the above proof does not apply to this scenario, andthus the question remains open for general mixed states. How-ever, as we will see in the next section, for pure states C A | B LICC is indeed equal to C A | B LQICC .In the following theorem we will prove that C SQI is boundedabove by the QI relative entropy. This will complete the proofof the inequality (20).
Theorem 2.
For any bipartite state ρ = ρ AB holds:C A | B SQI ( ρ ) ≤ C A | Br ( ρ ) . (26) Proof.
The proof goes similar lines of reasoning as the proofof Theorem 3 in Ref. [51]. From the definition of C SQI inEq. (16) it follows that for any ε > | φ (cid:105) ,an integer n >
1, and an SQI protocol Λ SQI acting on n copiesof ρ = ρ AB such that C A | B SQI ( ρ ) − C r ( | φ (cid:105) ) ≤ ε, (27) (cid:13)(cid:13)(cid:13)(cid:13) Λ SQI [ ρ ⊗ n ] − ρ ⊗ nf (cid:13)(cid:13)(cid:13)(cid:13) ≤ ε (28)with the final state ρ f = | (cid:105) (cid:104) | A ⊗ | φ (cid:105) (cid:104) φ | B .Eq. (28) together with the continuity of QI relative en-tropy [59] implies that for any 0 < ε ≤ / n ≥ Λ SQI acting on n copies of ρ such that C A | Br ( Λ SQI [ ρ ⊗ n ]) ≥ C A | Br ( ρ ⊗ nf ) − n ε log d − h ( ε ) , (29)where h ( x ) = − x log x − (1 − x ) log (1 − x ) is the binary entropyand d is the dimension of AB . Now we use the fact that the QIrelative entropy is additive [51] and does not increase underSQI operations. This means that for any 0 < ε ≤ / n ≥ C A | Br ( ρ ) ≥ C A | Br ( ρ f ) − ε log d − n h ( ε ) . (30)The above inequality together with the fact C A | Br ( ρ f ) = C r ( | φ (cid:105) )and Eq. (27) implies that for any 0 < ε ≤ / n ≥ C A | Br ( ρ ) ≥ C A | B SQI ( ρ ) − ε − ε log d − n h ( ε ) . (31)This completes the proof of the theorem. (cid:3) Proposition 1 and Theorem 2 in combination implyEq. (20). It remains an open question if the inequalities inEq. (20) are strict. As we will see in the next section, this isnot the case for pure state: in this case all quantities are equalto the von Neumann entropy of the fully decohered state ofBob ∆ ( ρ B ).Before we turn our attention to this question, we will firstcharacterize all quantum states which are useful for assistedcoherence distillation via the sets of operations X consideredabove. Note that a quantum-incoherent state cannot be usedfor extraction of coherence on Bob’s side via any set of op-erations X . On the other hand, as is shown in the followingtheorem, any state which is not quantum-incoherent can beused for extracting coherence via LICC. Theorem 3.
A state ρ = ρ AB has C A | B LICC ( ρ ) > if and only if itis not quantum-incoherent.Proof. As C A | B LICC ( ρ AB ) = ρ AB , the claim follows if we show that ρ AB is not QIimplies C A | B LICC ( ρ AB ) >
0. Without loss of generality, let thenon-QI state be given as ρ AB = (cid:88) i , j | e i (cid:105) (cid:104) e j | A ⊗ N Bi j , (32)where {| e i (cid:105) A } is an orthonormal basis for Alice’s Hilbertspace [60] and N Bi j are some operators on Bob’s space. By thenon-QI assumption, at least one of the { N i j } has o ff -diagonalelements.We note that N ii ≥
0, so N ii (cid:44) ⇔ Tr[ N ii ] >
0. If an N ii has o ff -diagonal elements, then the state of Bob after Alice’sincoherent measurement with Kraus operator K Ai = | i (cid:105) (cid:104) e i | A is ρ Bi ∼ N ii , with non-zero probability Tr[ N ii ] >
0. Hence C A | B LICC ( ρ AB ) >
0. Let us now assume that all N ii are diagonal. In this case – bythe non-QI assumption – some of the operators N kl must haveo ff -diagonal elements for some k (cid:44) l . If one of the operators N kl (by Hermiticity of ρ , we can assume k < l without lossof generality) has some o ff -diagonal elements, then at leastone of the operators P : = N kl + N † kl , Q : = i ( N kl − N † kl ) willalso have o ff -diagonal elements. Depending on whatever thecase, Alice performs an incoherent measurement containingthe Kraus operator K P : = | (cid:105) (cid:104) e P | or K Q : = | (cid:105) (cid:104) e Q | , where wedefine | e P (cid:105) : = cos θ | e k (cid:105) + sin θ | e l (cid:105) , | e Q (cid:105) : = cos θ | e k (cid:105) + i sin θ | e l (cid:105) ,the unknown parameter θ will be determined soon. In the firstcase, the post-measurement state of Bob is given by ρ B θ ∼ cos θ N kk + sin θ N ll + cos θ sin θ ( N kl + N † kl ) , (33)which, by assumption, has o ff -diagonal elements. Note thatsince sin θ, cos θ, sin θ cos θ are linearly independent func-tions, there is always some 0 < θ < π/ ρ B θ is coherent. Similarly, in the other case, where i ( N kl − N † kl ) isassumed to have o ff -diagonal elements, the post measurementstate of Bob is coherent with non-zero probability.Thus, whenever ρ AB is not QI, with non-zero probability Al-ice can steer Bob’s state to a coherent one, which Bob can dis-till by using the methods presented by Winter and Yang [39],so C A | B LICC ( ρ AB ) > (cid:3) Since LICC is the weakest set of operations consideredhere, this theorem also means that a state which is notquantum-incoherent can be used for coherence distillation onBob’s side via any set of operations presented above.
B. Pure states
In the following we will study the scenario where the stateshared by Alice and Bob is pure, and the aim is to distill coher-ence at maximal rate on Bob’s side via the sets of operationspresented above. Before we study this task, we recall the def-inition of coherence of assistance given in [51]: C a ( ρ ) = max (cid:88) i p i C r ( | ψ i (cid:105) ) , (34)where the maximum is performed over all pure state decom-positions of ρ . We will now prove the following lemma. Lemma 4.
For any pure state | Ψ (cid:105) AB there exists an incoherentmeasurement on Alice’s side such that: (cid:88) i p i C r ( | ψ i (cid:105) B ) = C a ( ρ B ) , (35) where | ψ i (cid:105) B are Bob’s post-measurement states with corre-sponding probability p i .Proof. Note that any pure state can be written as | Ψ (cid:105) AB = (cid:88) i √ p i | e i (cid:105) A | ψ i (cid:105) B , (36)where the states | e i (cid:105) A are mutually orthogonal (but not neces-sarily incoherent), and the states | ψ i (cid:105) B together with probabil-ities p i fulfill Eq. (35). The desired incoherent measurementon Alice’s side now consists of the following incoherent Krausoperators: K Ai = | i (cid:105) (cid:104) e i | A . It can be verified by inspection thatBob’s post-measurement states indeed fulfill Eq. (35). Thiscompletes the proof of the lemma. (cid:3) Lemma 4 implies that for any pure state C A | B LICC is boundedbelow by the regularized coherence of assistance of Bob’s re-duced state: C ∞ a ( ρ B ) ≤ C A | B LICC ( | Ψ (cid:105) AB ) , (37)where the regularized coherence of assistance is definedas [51] C ∞ a ( ρ ) = lim n →∞ C a ( ρ ⊗ n ) / n . To prove this statement,consider the situation where Alice and Bob share n · m copiesof the pure state | Ψ (cid:105) = | Ψ (cid:105) AB . Using Lemma 4, it follows thatin the limit of large m Alice and Bob can use n copies of | Ψ (cid:105) to extract coherence at rate C a ( ρ ⊗ nB ) on Bob’s side, and thus C a ( ρ ⊗ nB ) ≤ C A | B LICC ( | Ψ (cid:105) ⊗ n ) . (38)The proof of Eq. (37) is complete by dividing this inequalityby n and taking the limit n → ∞ . Equipped with these resultswe are now in position to prove the following theorem. Theorem 5.
For any bipartite pure state | Ψ (cid:105) = | Ψ (cid:105) AB the fol-lowing equality holds:C A | B LICC ( | Ψ (cid:105) ) = C A | B LQICC ( | Ψ (cid:105) ) = C A | B SI ( | Ψ (cid:105) ) (39) = C A | B SQI ( | Ψ (cid:105) ) = C A | Br ( | Ψ (cid:105) ) = S ( ∆ ( ρ B )) . Proof.
Combining Eqs. (20) and (37) we arrive at the inequal-ity C ∞ a ( ρ B ) ≤ C A | BX ( | Ψ (cid:105) ) ≤ C A | Br ( | Ψ (cid:105) ) , (40)where X is one of the sets considered above. The proof iscomplete by using the following equality which holds for allpure states [51]: C A | Br ( | Ψ (cid:105) ) = C ∞ a ( ρ B ) = S ( ∆ ( ρ B )) . (41) (cid:3) This result is surprising: it implies that the performance ofthe protocol does not depend on the particular set of opera-tions performed by Alice and Bob. In particular, the optimalperformance can already be reaches by the weakest set of op-erations LICC, which restricts both Alice and Bob to localincoherent operations and classical communication. A betterperformance cannot be achieved if Alice is allowed to performarbitrary quantum operations on her side (LQICC), and evenif Alice and Bob have access to the most general set of oper-ations considered here (SQI). This statement is true wheneverAlice and Bob share a pure state.
C. Maximally correlated states
Here we will consider assisted coherence distillations forstates of the form ρ AB = (cid:88) i , j ρ i j | ii (cid:105) (cid:104) j j | AB . (42)States of this form are known as maximally correlatedstates [61]. However, note that the family of states givenin Eq. (42) does not contain all maximally correlated states,since | i (cid:105) A and | j (cid:105) B are incoherent states of Alice and Bob re-spectively. We will call these states maximally correlated inthe incoherent basis . As we show in the following proposi-tion, also for this family of states the inequalities (20) becomeequalities. Proposition 6.
For any state ρ = ρ AB which is maximally cor-related in the incoherent basis the following equality holds:C A | B LICC ( ρ ) = C A | B LQICC ( ρ ) = C A | B SI ( ρ ) = C A | B SQI ( ρ ) = C A | Br ( ρ ) = S ( ∆ B ( ρ )) − S ( ρ ) . (43) Proof.
For proving this statement it is enough to prove theequality C A | B LICC ( ρ ) = S ( ∆ B ( ρ )) − S ( ρ ) . (44)For this, we will present an LICC protocol achieving theabove rate. In particular, we will show that there exist an in-coherent measurement on Alice’s side such that every post-measurement state of Bob has coherence equal to S ( ∆ B ( ρ )) − S ( ρ ). The corresponding incoherent Kraus operators of Aliceare given by K Aj = | j (cid:105) (cid:104) ψ j | A , where the states | ψ j (cid:105) are mutu-ally orthogonal, maximally coherent, and form a complete ba-sis [62]. Since the states | ψ j (cid:105) are all maximally coherent, theycan be written as | ψ j (cid:105) = / √ d A (cid:80) k e i φ jk | k (cid:105) with some phases φ jk , and d A is the dimension of A . The corresponding post-measurement states of Bob are then given by ρ Bj = (cid:88) k , l e i ( φ jl − φ jk ) ρ kl | k (cid:105) (cid:104) l | B . (45)If we now introduce the incoherent unitary U j = (cid:80) k e i φ jk | k (cid:105) (cid:104) k | ,we see that this unitary transforms the state ρ Bj to the state U j ρ Bj U † j = (cid:88) k , l ρ kl | k (cid:105) (cid:104) l | B . (46)Since the relative entropy of coherence is invariant under in-coherent unitaries, it follows that all states ρ Bj have the samerelative entropy of coherence: C r ( ρ Bj ) = C r (cid:16) U j ρ Bj U † j (cid:17) = C r (cid:88) k , l ρ kl | k (cid:105) (cid:104) l | B . (47)It is straightforward to verify that the right-hand side of thisexpression is equal to S ( ∆ B ( ρ )) − S ( ρ ), which completes theproof of the proposition. (cid:3) The above proposition can also be generalized to states ofthe form ρ AB = (cid:88) i , j ρ i j U | i (cid:105) (cid:104) j | A U † ⊗ | i (cid:105) (cid:104) j | B , (48)where the unitary U acts on the subsystem of Alice. In thiscase, the proposition can be proven in the same way, by ap-plying the incoherent Kraus operators K Aj = | j (cid:105) (cid:104) ψ j | A U † onAlice’s side.These results show that the inequality (20) reduces to equal-ity in a large number of scenarios, including all pure states,states which are maximally correlated in the incoherent basis,and even all states which are obtained from the latter classby applying local unitaries on Alice’s subsystem. However, itremains open if Eq. (20) is a strict inequality for any mixedstate. D. Remarks on the definition of C X In the following we will provide some remarks on the def-inition of C X , where the set of operations X is either LICC,LQICC, SI, or SQI. First, we note that the definition of C X given in Eq. (16) is equivalent to the following: C A | BX ( ρ ) = sup (cid:26) R : lim n →∞ (cid:18) inf Λ ∈ X (cid:13)(cid:13)(cid:13)(cid:13) Tr A (cid:104) Λ (cid:104) ρ ⊗ n (cid:105)(cid:105) − Ψ ⊗(cid:98) Rn (cid:99) (cid:13)(cid:13)(cid:13)(cid:13)(cid:19) = (cid:27) , (49)where (cid:98) x (cid:99) is the largest integer below or equal to x and Ψ = | Ψ (cid:105) (cid:104) Ψ | B is a maximally coherent single-qubit state on Bob’sside. To see that the expressions (16) and (49) are indeedequivalent it is enough to note that every set of operations X includes all incoherent operations on Bob’s side, and thatthe theory of quantum coherence is asymptotically reversiblefor pure states, i.e., a state | ψ (cid:105) with distillable coherence c can be asymptotically converted into any other state | ψ (cid:105) withdistillable coherence c at rate c / c [39].In the above discussion we implicitly assumed that theHilbert space of Alice and Bob has a fixed finite dimension,and that the incoherent operations performed by the partiespreserve their dimension. One might wonder if the perfor-mance of any of the assisted distillation protocols X discussedabove changes if this assumption is relaxed, i.e., if Alice andBob have access to additional local incoherent ancillas. Thisamounts to considering operations on the total state of theform ρ AB ⊗ σ A (cid:48) ⊗ σ B (cid:48) , where σ A (cid:48) = | (cid:105) (cid:104) | A (cid:48) and σ B (cid:48) = | (cid:105) (cid:104) | B (cid:48) (50)are additional incoherent states of Alice and Bob respectively.As we will see below, local incoherent ancillas cannot im-prove the performance of the procedure as long as SI and SQIoperations are considered. For this, we will first prove thefollowing lemma. Lemma 7.
For any SI operation ˜ Λ SI acting on the state ρ AB ⊗ σ A (cid:48) ⊗ σ B (cid:48) there exists another SI operation Λ SI acting on ρ AB such that the resulting state of AB is the same: Λ SI [ ρ AB ] = Tr A (cid:48) B (cid:48) (cid:104) ˜ Λ SI [ ρ AB ⊗ σ A (cid:48) ⊗ σ B (cid:48) ] (cid:105) . (51) Proof.
This can be seen explicitly, by considering the form ofa general SI operation ˜ Λ SI acting on ρ AB ⊗ σ A (cid:48) ⊗ σ B (cid:48) :˜ Λ SI [ ρ AB ⊗ σ A (cid:48) ⊗ σ B (cid:48) ] = (cid:88) i ˜ A i ⊗ ˜ B i (cid:16) ρ AB ⊗ σ A (cid:48) ⊗ σ B (cid:48) (cid:17) ˜ A † i ⊗ ˜ B † i , (52)where the operators ˜ A i and ˜ B i act on AA (cid:48) and BB (cid:48) respectively.The corresponding SI operation Λ SI satisfying Eq. (51) is thengiven by Λ SI [ ρ AB ] = (cid:88) k , l , m A klm ⊗ B klm (cid:16) ρ AB (cid:17) A † klm ⊗ B † klm . (53)The incoherent operators A klm and B klm depend on the opera-tors ˜ A i and ˜ B i and have the following explicit form: A klm = Tr A (cid:48) (cid:104) ˜ A k (cid:16) A ⊗ | (cid:105) (cid:104) l | A (cid:48) (cid:17)(cid:105) , (54) B klm = Tr B (cid:48) (cid:104) ˜ B k (cid:16) B ⊗ | (cid:105) (cid:104) m | B (cid:48) (cid:17)(cid:105) , (55)where {| l (cid:105) A (cid:48) } is a complete set of incoherent states on A (cid:48) , and {| m (cid:105) B (cid:48) } is a complete set of incoherent states on B (cid:48) . Usingthe fact that ˜ A k and ˜ B k are incoherent, it is straightforwardto verify that the operators A klm and B klm are also incoherent.Eq. (51) can also be verified by inspection. (cid:3) The above lemma implies that local incoherent ancillas onAlice’s or Bob’s side cannot improve the performance of theprotocol if SI operations are considered. This can be seenby contradiction, assuming that by using a state ρ AB and lo-cal incoherent ancillas Bob can extract maximally coherentsingle-qubit states at rate R > C A | B SI . Applying Theorem 2and noting that the QI relative entropy does not change underattaching local incoherent ancillas, it follows that the rate R is bounded above by C A | Br , which is again bounded above bylog d B , where d B is the dimension of Bob’s subsystem:log d B ≥ C A | Br ( ρ ) ≥ R > C A | B SI ( ρ ) . (56)The inequality log d B ≥ R means that if additional incoher-ent ancillas would improve the procedure at all, they are notneeded at the end of the protocol and can be discarded [63].These results together with Lemma 7 imply that the rate R isalso reachable without additional ancillas as long as SI opera-tions are considered.Very similar arguments can also be applied to the caseof SQI operations. In this case, one can prove an equiva-lent statement to Lemma 7: for any SQI operation acting on ρ AB ⊗ σ A (cid:48) ⊗ σ B (cid:48) there exists an SQI operation acting on ρ AB such that the final state of AB is the same. This implies thatlocal incoherent ancillas cannot improve the performance inthis case as well. It remains unclear if incoherent ancillas canprovide advantage for LICC or LQICC operations. However,if Alice and Bob share a pure state, incoherent ancillas cannotprovide any advantage also in this case due to Theorem 5. IV. INCOHERENT TELEPORTATION
In standard quantum teleportation introduced by Bennett etal . [64], Alice aims to transfer her single-qubit state to Bobby using LOCC together with one singlet. We will now con-sider the task of incoherent teleportation , which is the sameas standard teleportation up to the fact that LOCC is replacedby LICC. This means that Alice and Bob are allowed to applyonly incoherent operations locally, and share their outcomesvia a classical channel.It seems that the restriction to local incoherent operationsprovides a severe constraint, and it is tempting to assume thatAlice and Bob will not be able to achieve perfect teleportationin this way, at least if they have no access to additional coher-ent resource states. As we will show in the following theorem,this intuition is not correct.
Theorem 8.
Perfect incoherent teleportation of an unknownstate of one qubit is possible with one singlet and two bits ofclassical communication.Proof.
To prove this statement, recall that in the standard tele-portation protocol Alice applies a Bell measurement on herqubits A and A (cid:48) of the total initial state | Ψ (cid:105) = | ψ (cid:105) A (cid:48) ⊗ | φ + (cid:105) AB , (57)where | φ + (cid:105) AB = ( | (cid:105) + | (cid:105) ) / √ | ψ (cid:105) is the desired state subject to teleportation. Alicethen communicates the outcome of her measurement to Bob.Depending on the outcome of Alice’s measurement, Bob ei-ther finds his particle B in the desired state | ψ (cid:105) , or he has toadditionally apply one of the Pauli matrices σ , i σ , or σ .It is now crucial to note that all Pauli matrices are incoher-ent: σ i | m (cid:105) ∼ | n (cid:105) . This means that Bob can perform his con-ditional rotation in an incoherent way. We will now show thatalso Alice’s Bell measurement can be performed in a locallyincoherent way. For this, let | φ i (cid:105) denote the four Bell states andconsider the Kraus operators defined as K i = | (cid:105) (cid:104) φ i | AA (cid:48) . Notethat these operators are local in Alice’s lab, and moreover theyare incoherent with respect to the bipartite incoherent basis ofAlice. Finally, note that these Kraus operators lead to the samepost-measurement states of Bob as the projectors | φ i (cid:105) (cid:104) φ i | AA (cid:48) .This completes the proof of the theorem. (cid:3) The above theorem shows that LICC operations are indeedpowerful enough to allow for perfect teleportation. SinceLICC is the weakest set of operations considered here, thesame is true also for all the other sets LQICC, SI, and SQI: allthese sets allow for perfect teleportation of an unknown qubitwith one additional singlet. These results can be immediatelyextended to any system of n qubits, in which case n additionalsinglets are required. V. SUPERIORITY OF SQI OPERATIONS INSINGLE-SHOT QUANTUM STATE MERGING
In the discussion so far, and in particular in Eqs. (12), wehave seen that the set SQI is strictly larger than LICC, LQICC, and SI. At this point it is important to note that a larger set ofoperations is not automatically more useful for real physicalapplications. Nevertheless, the results presented above indeedimply the existence of such physical tasks where the set SQIis more useful, when compared to the other sets individually.For completeness, we will review these results in the follow-ing. • SQI is superior to SI and LICC in the task of quan-tum state preparation . In particular, by starting froman initial state | (cid:105) AB , SQI operations can prepare allquantum-incoherent states, while only fully incoherentstates can be prepared by SI and LICC operations, seeEqs. (10) and (11). • SQI is superior to LICC and LQICC in the task of quan-tum state discrimination . In particular, there exists a setof bipartite states which can be discriminated via SQI(and also via SI), but not via LICC and LQICC. Thiswas discussed in detail in Section II, see Eq. (15).It is now interesting to note that these two arguments areunrelated, and each of the arguments does not automaticallyimply the other one. In particular, the first argument for thesuperiority of SQI in comparison to SI and LICC cannot beused to show superiority in comparison to LQICC, since theset of states that can be prepared via SQI and LQICC is thesame. On the other hand, the second argument showing su-periority of SQI in comparison to LICC and LQICC cannotbe used to show superiority in comparison to SI, since bothSQI and SI are equally well suited for the considered task, seealso Section II for more details. It is thus natural to ask forthe existence of a quantum technological task which showssuperiority of SQI operations with respect to all the other setssimultaneously.In the following, we will present such an application, whichwill be based on the well-known task of quantum state merg-ing. The latter task was introduced and studied in [48, 49],and extended to the framework of coherence in [50]. In thistask, three parties, Alice, Bob, and a referee share a tripartitestate ρ RAB . The aim of the process is to send Bob’s systemto Alice [65] while keeping the overall state intact. In con-trast to [48–50], we do not allow Alice and Bob to share anysinglets, and moreover restrict them to the sets of operationsconsidered in this paper, i.e., LICC, LQICC, SI, or SQI.We will consider merging of the following tripartite state: ρ RAB = (cid:88) i | i (cid:105)(cid:104) i | R ⊗ | ψ i (cid:105)(cid:104) ψ i | AB , (58)where | ψ i (cid:105) = | α i (cid:105) ⊗ | β i (cid:105) are nine 3 × A (cid:48) of dimension 3 in anincoherent initial state | (cid:105) A (cid:48) . Alice will use this register tostore Bob’s system. The total final state is given by ρ RAA (cid:48) Bf = Λ X (cid:104) ρ RAB ⊗ | (cid:105)(cid:104) | A (cid:48) (cid:105) , (59)where X denotes one of the four sets of operations consideredhere. The process is successful if ρ RAA (cid:48) f is the same as ρ RAB up to relabeling B and A (cid:48) . Here, we consider the single-shotscenario, i.e., the corresponding operation is applied to onecopy of the state only.As we will now show, this task can be performed via SQI,but not via any of the other three sets. For proving that thetask can be performed via SQI, it is enough to recall that SQIoperations can be used to distinguish the states | ψ i (cid:105) . For eachoutcome i Alice can then locally prepare her system AA (cid:48) inthe state | ψ i (cid:105) AA (cid:48) , which again can be achieved via SQI oper-ations [66]. In the next step we will show that this task can-not be performed via LQICC operations. This can be seen bycontradiction, assuming that LQICC operations can performmerging in this scenario, i.e., thatTr B (cid:104) Λ LQICC [ ρ RAB ⊗ | (cid:105)(cid:104) | A (cid:48) ] (cid:105) = (cid:88) i | i (cid:105)(cid:104) i | R ⊗ | ψ i (cid:105)(cid:104) ψ i | AA (cid:48) , (60)for some LQICC protocol Λ LQICC . By linearity, it must be that | ψ i (cid:105)(cid:104) ψ i | AA (cid:48) = Tr B (cid:104) Λ LQICC [ | ψ i (cid:105)(cid:104) ψ i | AB ⊗ | (cid:105)(cid:104) | A (cid:48) ] (cid:105) , (61)i.e., Alice and Bob could use the protocol to transfer Bob’spart of | ψ i (cid:105) AB to Alice. Since the states | ψ i (cid:105) AB form an or-thonormal basis, this would imply that Alice and Bob coulddistinguish the states | ψ i (cid:105) AB via LQICC (and thus also viaLOCC), which is a contradiction to the main result of [55].This also proves that the task cannot be performed via LICCoperations.It now remains to show that the task cannot be performedvia SI operations. We will prove this statement by using gen-eral properties of QI relative entropy given in Eq. (17) and itsclosed expression in Eq. (18). In particular, recall that the QIrelative entropy cannot increase under SI operations, whichimplies that C RB | AA (cid:48) r (cid:16) ρ RAA (cid:48) Bf (cid:17) ≤ C RB | AA (cid:48) r (cid:16) ρ RAB ⊗ | (cid:105)(cid:104) | A (cid:48) (cid:17) , (62)where ρ f is the final state given in Eq. (59). In the next stepwe will use the following relations: C R | AA (cid:48) r (cid:16) ρ RAA (cid:48) f (cid:17) ≤ C RB | AA (cid:48) r (cid:16) ρ RAA (cid:48) Bf (cid:17) , (63) C RB | Ar (cid:16) ρ RAB (cid:17) = C RB | AA (cid:48) r (cid:16) ρ RAB ⊗ | (cid:105)(cid:104) | A (cid:48) (cid:17) , (64)which can be proven directly from the properties of QI relativeentropy. Combining these results we arrive at the followinginequality: C R | AA (cid:48) r (cid:16) ρ RAA (cid:48) f (cid:17) ≤ C RB | Ar (cid:16) ρ RAB (cid:17) . (65)In the final step of the proof, assume that SI operations al-low to perform the aforementioned task. We will now showthat this assumption leads to a contradiction. In particular, byour assumption the final state ρ RAA (cid:48) f must be the same as ρ RAB ,up to relabeling B and A (cid:48) . Thus, Eq. (65) is equivalent to C R | ABr (cid:16) ρ RAB (cid:17) ≤ C RB | Ar (cid:16) ρ RAB (cid:17) . (66)By applying Eq. (18) together with the expression for the state ρ RAB in Eq. (58) and using the states | ψ i (cid:105) in Eq. (3) of [55] (see also the Appendix) we can now evaluate both sides ofthis inequality: C R | ABr (cid:16) ρ RAB (cid:17) = , (67) C RB | Ar (cid:16) ρ RAB (cid:17) = , (68)which is the desired contradiction. This finishes the proof thatthe task considered here can be performed with SQI opera-tions, but not with any other set LICC, LQICC, or SI.Thus, we presented the first example for a quantum tech-nological application which can be performed via SQI oper-ations, but cannot be performed with any of the other sets ofoperations considered in this paper. VI. CONCLUSIONS
In this paper we studied the resource theory of coherence indistributed scenarios. In particular, we focused on the follow-ing four classes of operations: local incoherent operations andclassical communication (LICC), local quantum-incoherentoperations and classical communication (LQICC), separableincoherent operations (SI), and separable quantum-incoherentoperations (SQI). We showed that these classes obey inclusionrelations very similar to those between LOCC and separableoperations known from entanglement theory.We further studied the role of these classes for the task ofassisted coherence distillation, first introduced in [51]. Re-gardless of the particular class of operations we proved that abipartite state can be used for coherence extraction on Bob’sside if and only if the state is not quantum-incoherent. Wealso showed that the relative entropy distance to the set ofquantum-incoherent states provides an upper bound for co-herence distillation on Bob’s side, a result which again doesnot depend on the class of operations under scrutiny. Remark-ably, both the SI and the SQI operations lead to the same per-formance in this task for all mixed states. For pure states aneven stronger result has been proven: in this case all classesof operations considered here are equivalent for assisted co-herence distillation. We also studied the task of incoherentteleportation, which arises from standard quantum teleporta-tion by restricting the parties to LICC operations only. Weshowed that in this situation LICC operations do not provideany restriction: perfect teleportation of an unknown qubit canbe achieved with LICC and one additional singlet. Finally,we compared these classes on the task of single-shot quantumstate merging. In this task, SQI operations provide an advan-tage with respect to all the other classes considered here.The tools presented here can be regarded as a first step to-wards a full resource theory of coherence in distributed sce-narios. Indeed, while in the course of this work we focusedon the coherence framework of Baumgratz et al. [5], it is im-portant to note that the presented ideas are significantly moregeneral. As an example, our tools can be directly applied tothe situation where local incoherent operations are replaced byanother well justified set, such as strictly incoherent [39, 47],translationally invariant [2, 3, 43, 45], or physical incoherent0operations [42]. These alternative frameworks have been ex-tensively studied in recent literature, and each of them cap-tures the concept of coherence in a di ff erent scenario [6]. Dueto the close connection between all these frameworks it isclear that the ideas presented in this work also carry over tothese concepts. In particular, the results presented here will ingeneral serve as bounds for other concepts of coherence. Wealso expect that these bounds are tight in several relevant sit-uations, e.g., for pure states. While the investigation of thesequestions is beyond the scope of this work, it has the potentialto provide a unified view for all frameworks of coherence, andultimately put the resource theory of coherence on equal foot-ing with other quantum resource theories, most prominentlythe theory of entanglement [41].Future research in this direction is also important in thelight of the recent progress towards understanding the role ofcoherence in quantum thermodynamics [24, 25]. Here, theframework of thermal operations turned out to be very use-ful [67]. These operations arise from the first and second lawof thermodynamics, and are known to be translationally in-variant [24, 25]. Because of this, the tools developed in ourwork can also be applied to quantum thermodynamics. Thisresearch direction has the potential to reveal new surprising ef-fects, similar to well-known phenomena such as bound entan-glement [68, 69] in quantum information theory, or the work-locking phenomenon [22, 24, 26, 70] in quantum thermody-namics.After the appearance of this article on arXiv, quantum co-herence in multipartite systems has also been discussed byother authors [47, 71, 72]. While these works also study therole of coherence for quantum-state manipulation, their mo-tivation is significantly di ff erent from the concept presentedhere. In particular, the framework of coherence in multipartitesystems is naturally suited for studying general nonclassicalcorrelations such as quantum discord, as discussed in [47, 71].Another important research direction pursued in [71, 72] is therole of coherence in the DQC1 protocol [73]. This quantumprotocol allows for e ffi cient evaluation of the trace of a uni-tary, provided that the unitary has an e ffi cient description interms of two-qubit gates. Remarkably, this protocols does notrequire entanglement, while showing an exponential speedupover the best known classical procedure [74]. The authorsof [71] present a figure of merit for this task, which is relatedto the consumption of coherence in this protocol. Interest-ingly, while their figure of merit vanishes for unitaries of theform U = e i φ
1, it is unclear if a classical algorithm can eval-uate the trace of this unitary e ffi ciently [71]. In the light ofthese results, we expect that the tools presented in our workcan also find applications for the DQC1 protocol and quan-tum computation in general. This is beyond of the scope ofthe current work, and we leave it open for future research. We also note that the class of LICC operations was intro-duced independently by Chitambar and Hsieh [75]. The au-thors study the tasks of asymptotic state creation and distilla-tion of entanglement and coherence via LICC operations, i.e.,local coherence is considered as an additional resource. Theyalso independently obtain the results of our Theorems 3 and 5for LICC operations. ACKNOWLEDGMENTS
We thank Gerardo Adesso, Eric Chitambar, Jens Eisert,Gilad Gour, Iman Marvian, Martin B. Plenio, and AndreasWinter for discussion. We acknowledge financial supportsfrom the Alexander von Humboldt-Foundation, the John Tem-pleton Foundation, the EU grants OSYRIS (ERC-2013-AdGGrant No. 339106), QUIC (H2020-FETPROACT-2014 No.641122), and SIQS (FP7-ICT-2011-9 No. 600645), the Span-ish MINECO grants (FIS2013-46768, FIS2008-01236, andFIS2013-40627-P) with the support of FEDER funds and“Severo Ochoa” Programme (SEV-2015-0522), and the Gen-eralitat de Catalunya grant (2014-SGR-874 and 2014-SGR-966), and Fundació Privada Cellex.
APPENDIX
Here, we list the states | ψ i (cid:105) = | α i (cid:105)⊗| β i (cid:105) from Eq. (3) of [55]: | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = | (cid:105) ⊗ | (cid:105) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = | (cid:105) ⊗ √ | (cid:105) + | (cid:105) ) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = | (cid:105) ⊗ √ | (cid:105) − | (cid:105) ) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = | (cid:105) ⊗ √ | (cid:105) + | (cid:105) ) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = | (cid:105) ⊗ √ | (cid:105) − | (cid:105) ) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = √ | (cid:105) + | (cid:105) ) ⊗ | (cid:105) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = √ | (cid:105) − | (cid:105) ) ⊗ | (cid:105) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = √ | (cid:105) + | (cid:105) ) ⊗ | (cid:105) , | ψ (cid:105) = | α (cid:105) ⊗ | β (cid:105) = √ | (cid:105) − | (cid:105) ) ⊗ | (cid:105) . [1] J. Aberg, (2006), arXiv:quant-ph / , 033023 (2008).[3] G. Gour, I. 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