Quantum Correlations and Global Coherence in Distributed Quantum Computing
QQuantum correlations and global coherence in distributed quantum computing
Farid Shahandeh,
1, 2, ∗ Austin P. Lund, and Timothy C. Ralph Centre for Quantum Computation and Communication Technology,School of Mathematics and Physics, University of Queensland, St Lucia, Queensland 4072, Australia Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
Deviations from classical physics when distant quantum systems become correlated are interesting both fun-damentally and operationally. There exist situations where the correlations enable collaborative tasks that areimpossible within the classical formalism. Here, we consider the efficiency of quantum computation protocolscompared to classical ones as a benchmark for separating quantum and classical resources and argue that thecomputational advantage of collaborative quantum protocols in the discrete variable domain implies the non-classicality of correlations. By analysing a toy model, it turns out that this argument implies the existence ofquantum correlations distinct from entanglement and discord. We characterize such quantum correlations interms of the net global coherence resources inherent within quantum states and show that entanglement anddiscord can be understood as special cases of our general framework. Finally, we provide an operational inter-pretation of such correlations as those allowing two distant parties to increase their respective local quantumcomputational resources only using locally incoherent operations and classical communication.
The complete characterization of correlations between con-stituent elements of quantum systems is important both fun-damentally and operationally. Two well-known examples ofsuch attempts are quantum entanglement and discord. En-tanglement is a resource for many nonlocal tasks [1, 2] thatcannot be created between spatially separated subsystems us-ing local operations and classical communication (LOCC) [3].However, in many other tasks entanglement is known to playno or very minor role [4–6], putting forward quantum dis-cord [7–10] as a necessary resource [11–13], although thereare ongoing controversies [9, 14, 15]. The latter arises fromthe discrepancy between the entropic measures of correlationsin classical and quantum physics showing that not all the in-formation encoded via LOCC into spatially separated systemscan be extracted using the same type of operations [11, 16].Here we offer a new viewpoint on the quantumness ofcorrelations. Our ultimate objective is three-fold: first, tounderstand the fundamental border (if any) between clas-sical and quantum correlations in light of the nonclassicalpower of quantum computers; second, to put forward a novelunified and consistent framework for characterizing quan-tum correlations in both continuous and discrete variable do-mains [15, 17]; third, to obtain a deeper understanding of theresources that might be responsible for the nonclassical powerof quantum computation models. The present manuscript fo-cuses on the first aim and provides a proposal for the secondone, in complement to our recent investigation of the contin-uous variable protocols [17]. In view of our results, we alsoobtain a new perspective on the third goal. To this end, we firstexamine nonclassicality from two viewpoints, namely, thatof computational science and the resource theory of coher-ence [18–20] and provide two nonclassicality criteria basedon them. We establish a fundamental correspondence betweenclassical computation protocols and the formalism of quan-tum coherence. This close affinity benchmarks computationalefficiency for quantum-classical separation and gives rise to ∗ Electronic address: [email protected] an equivalence between our nonclassicality criteria. We thenintroduce a toy model called nonlocal deterministic quantumcomputing with two qubits (NDQC2) which performs a non-local collaborative computation exponentially faster than anyclassical algorithm via correlation measurements without us-ing any entanglement or discord. Making use of our computa-tional efficiency benchmark, this protocol shows quantumnessof correlations that are not captured by the standard classifi-cation in quantum information theory. This is complemen-tary to a similar conclusion for the continuous variable do-main where we showed that the nonlocal B
OSON S AMPLING protocol contains correlations that cannot be efficiently simu-lated on a classical computer while the input and output mixedstates contain no entanglement or discord [17]. In contrast tothis, for the specific case of pure state quantum computations,entanglement is known to be required for a computational ad-vantage over classical algorithms [21]. Inspired by this featureof NDQC2 and the role of coherence as a primitive property ofquantum systems [22–28] in quantum computation [27, 29],we show that the nonclassical advantage of the correlationswithin NDQC2 can be quantitatively explained in terms of thenet global coherence inherent in the input and output statesto the protocol. We thus argue that the net global quantum-coherence should be understood as the more general conceptof quantum correlations. We show the relevance of our defi-nition by proving that the current standard hierarchy of quan-tum correlations defines special classes of globally-coherentstates and further providing an operational interpretation forsuch correlations. To be specific, quantum correlations as pre-sented here are manifested in the ability of two distant partiesto increase their local quantum computational resources byapplying classical operations locally and exploiting classicalcommunication. a r X i v : . [ qu a n t - ph ] J u l I. PRELIMINARIESA. Standard Correlations in Quantum Information
From the viewpoint of quantum information theory, not allthe global information can be encoded within or decoded froma bipartite (or multipartite) physical system via local opera-tions and classical communication. This leads to the follow-ing hierarchy of quantum-correlated states [3, 11, 30]:(i) entangled states that cannot be written in the separable form ˆ (cid:37) AB = (cid:80) i p i ˆ (cid:37) A; i ⊗ ˆ (cid:37) B; i ;(ii) two-way quantum correlated (discordant) states whichcannot be represented via a set of locally orthogonalstates on either side;(iii) one-way nondiscordant (or one-way quantum-classical correlated) states that can be written as ˆ (cid:37) AB = (cid:80) j p j ˆ (cid:37) A; j ⊗| j (cid:105) B (cid:104) j | , or ˆ (cid:37) AB = (cid:80) i p i | i (cid:105) A (cid:104) i |⊗ ˆ (cid:37) B; i using at most one set of locally orthonormal states;(iv) two-way or fully nondiscordant (or strictlyclassical-classical) states admitting the form ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i |⊗| j (cid:105) B (cid:104) j | .The latter simply encode the joint probability distributions { p ij } using locally orthonormal states and have previouslybeen assumed to possess no quantum advantage in a nonlocalinformation processing task. The correlations within each ofthe classes above are usually measured using an entropic func-tion as the discrepancy between their total-correlation con-tents and the amount accessible via LOCC, most commonlycalled quantum discord [7, 8]. For a bipartite quantum state,quantum discord is asymmetric and a quantum state has zerodiscord both from Alice to Bob and vice versa if and only ifit is of the form (iv). For this reason, speaking of classicalcorrelations is assumed to be synonymous with nondiscordantstates. In addition, every entangled state is necessarily discor-dant. Quantum discord is thus considered as the most generalmeasure of quantum correlations in quantum information the-ory [11, 30]. In a recent work, however, we considered thecomputational advantage obtained in the nonlocal B OSON -S AMPLING quantum computation protocol, that exploits par-ticular types of mixed states, to show that there exists quantumcorrelations that are not captured by this picture, namely theglobal P-function nonclassicality [17].
B. Quantum Coherence
The resource theory of coherence comprises (i) a set of purequantum states as extreme points E = {| i (cid:105)} which generatesthe set of (cost) free states of the theory as its convex hull S inc =conv {| i (cid:105)(cid:104) i | : | i (cid:105)∈ E } ; (ii) the set of free transformations O inc which leave the set of free states invariant. The extremepoints, usually termed the computational basis , in general,does not need to satisfy any orthogonalization or complete-ness conditions. This is, for example, the case for the non-classicality theory of continuous variable bosonic systems in which the set of bosonic coherent states {| α (cid:105) : α ∈ C } , as ex-treme points, are nonorthogonal and overcomplete [31]. It isalso clear that having infinite freedom, our choice of the com-putational basis depends on the physical system of interest andfundamental or operational restrictions. For instance, in pho-tonics the computational basis can be chosen to be the verticaland horizontal, or the diagonal and antidiagonal componentsof the radiation field. In contrast, in an atomic realization ofqubits, the preferred computational basis could be the energyeigenstates to which the systems decohere.We assume that the computational basis is a finite completeorthonormal basis. In particular, we are interested in col-laborative global computations consisting of two local com-putations. In such scenarios, when the local computationalbases for the two parties, Alice and Bob, are E A = {| i (cid:105) A } and E B = {| j (cid:105) B } , respectively, the global computational basis isgiven by E AB = E A ⊗ E B = {| i (cid:105) A ⊗ | j (cid:105) B } if Alice and Bob areconfined to separable operations (S).Incoherent states are invariant under various classes of freeoperations. Several of such operations for the resource the-ory of coherence have been studied so far, e.g., general,strict [20], and genuine incoherent operations [32]. A re-view of these operations and their operational meaning canbe found in Refs. [33–35]. The first class of interest here,is called the general incoherent operations as the most gen-eral incoherent operations possible and possess Kraus decom-position Λ( · ) = (cid:80) i ˆ F i ( · ) ˆ F † i such that (cid:80) i ˆ F † i ˆ F i = ˆ and ˆ F i S inc ˆ F † i ⊂ S inc for all i [20]. The latter condition ensuresthat even by subselection of the operation output one cannotgenerate coherence from incoherent states. Every Kraus oper-ator then must be of the form ˆ F = (cid:88) i c i | i (cid:105)(cid:104) ψ i | , (1)in which c i ∈ C and | ψ i (cid:105) ∈ span {| j (cid:105) ∈ E i } so that E i s aredisjoint subsets of E .The second class of operations are called strict incoher-ent operations. They are simply incoherent operations ofthe above given form with the extra restriction that for ev-ery Kraus operator ˆ F i it holds true that ˆ F † i is also incoherentso that the adjoint map of Λ given by Λ ‡ ( · ) = (cid:80) i ˆ F † i ( · ) ˆ F i is also incoherent [20]. Using Eq. (1) this implies that theKraus operators of strict incoherent operations has the form ˆ F = (cid:80) i c i | i (cid:105)(cid:104) j ( i ) | , with both | i (cid:105) , | j (cid:105) ∈ E and j ( i ) beinga one-to-one function. As shown by Yadin et al [36], theseoperations correspond to those also not consuming quantumcoherence within the given computational basis. A necessaryand sufficient for strictness of incoherent operations is givenbelow. Lemma 1. [33, 36, 37] An incoherent operation Λ is strictincoherent if and only if it possesses a set of Kraus operators { ˆ F i } such that for all quantum states ˆ (cid:37) holds ∀ i : ∆[ ˆ F i ˆ (cid:37) ˆ F † i ] = ˆ F i ∆[ˆ (cid:37) ] ˆ F † i . (2)Here, ∆[ · ] = (cid:80) i (cid:104) i | · | i (cid:105)| i (cid:105)(cid:104) i | is the fully depolarizing map.Equation (2) is sometimes notationally compressed into acommutation relation as [∆ , ˆ F i ] = 0 , where the implicit mul-tiplication must be understood as a concatenation of superop-erators.In nonlocal scenarios, the relevant class of operations to ourstudy is local incoherent operations and classical communica-tion (LICC) where Alice and Bob perform only incoherentoperations and share their possible outcomes via a classicalchannel. We also note that LICC ⊂ S [34, 38].
II. NONCLASSICALITY IN QUANTUM COHERENCEAND QUANTUM COMPUTATIONA. A Computational Perspective on Nonclassicality
Let us begin with the definition of “a nonclassical physicalprocess” from a computational perspective, highlighting therole of computational efficiency in our physical picture. Avalid empirical theory is one which is plausibly testable andfalsifiable as per below. Let T be a physical theory and P aphysical process consisting of preparations, transformations,and measurements of some physical system. Then, testing thetheory T in process P consists of three steps:1. Write down the equations provided by T that are as-sumed to govern P .2. Efficiently compute the predictions of T regarding theoutcomes of measurements on the outputs of P . Thiscan be deterministic or probabilistic, e.g., quantum the-ory is intrinsically nondeterministic. Here, by efficientwe mean probabilistic in polynomial time.3. Compare the predictions of T against the experimentalresults obtained in P and check the validity of the the-ory.First note that, we only speak of the properties of processes rather than systems with respect to given theories. Second,for the above procedure to be consistent it is crucial that thecomputation used in step 2 be itself efficiently described by T .To clarify the reason, denote the specific computational pro-cess leading to predictions of T for P by P (cid:63) . Both P (cid:63) and P are physical processes irrespective of the presumed underly-ing theory, therefore, the fact that the particular computation P (cid:63) resembles P also means that P replicates P (cid:63) . It immedi-ately follows that if a theory other than T , say T (cid:63) , is necessaryfor efficiently describing P (cid:63) , then it must be necessary for anefficient description of P too. Equivalently, if we assume that T provides a sufficient explanation for P , then it must also re-count P (cid:63) . All theories including classical ones, quantum me-chanics, general relativity, etc., are physical theories that havebeen subject to such tests. A non- T -process can now definedas follows. Definition 1.
A process P is said not to be a T -process (or,said to be a non- T -process) if and only if T fails the three-stepvalidity test in P . For instance, spectroscopic measurement of a black-body ra-diation is a nonclassical process because a classical theoryfails to give an account for it in terms of the above test.The important point that is commonly missed in assigningthe adjective “ T ” to a process, however, is the role of the com-putational efficiency in step 2. Suppose that we have a theory T (cid:63) for which we cannot efficiently compute (at least approx-imately up to some error ε ) the result of its equations for agiven physical process P (cid:63) on a computer efficiently describedby T (cid:63) . Then, it would be practically implausible for us to fig-ure out if T (cid:63) passes the validity test in P (cid:63) . In other words,we do not have a way to determine within a reasonable timeif T (cid:63) is the suitable theory for describing P (cid:63) without runninginto contradictions. Therefore, it is meaningless to consider P (cid:63) a T (cid:63) -process. Similarly, if there exists a process for whichwe cannot efficiently compute the predictions of the classicaltheory on a classical computer, it cannot carry the prefix “clas-sical”. Hence, in information science and from an operationalperspective, all classical physical processes are premised tobe efficiently simulatable on a (probabilistic or deterministic)classical Turing machine, corresponding to the BPP class ofcomputational complexity.
Criterion 1.
A physical process that cannot be efficiently sim-ulated on a classical computer is nonclassical.
We emphasize here that, Criterion 1 only provides a sufficientcondition, meaning that, not every nonclassical process is notefficiently simulatable on classical computers. For instance,many quantum processes can be efficiently classically simu-lated.
B. Nonclassicality in Resource Theory of Coherence
Superpositions of generic states of a physical system arenot allowed in classical theories. Hence, coherence is consid-ered to be a unique feature of post-classical theories. Thus,one can also investigate nonclassicality within the frameworkof quantum theory of coherence. It is sometimes stated thatstrict incoherent operations are classical ones in the resourcetheory of coherence because they are represented by stochas-tic transformations with respect to the computational basis E ,resembling a classical process (see e.g., Refs. [36, 37]). Thisconclusion, however, is debatable in view of our discussion inSec. II A as follows. Consider the map Υ D = ∆ ◦ Υ (cid:63) ◦ ∆ where Υ (cid:63) is a quantum computation, where we assume thatit is fixed by a given set of input parameters up to the inputquantum state. ∆[ · ]= (cid:80) (cid:104) i | ·| i (cid:105)| i (cid:105)(cid:104) i | represents the fully depo-larizing map with respect to the computational basis E = {| i (cid:105)} and ◦ is the composition operation between superoperators.Now, assuming any Kraus representation of the computationprocess, say Υ (cid:63) ( · ) = (cid:80) i ˆ F i ( · ) ˆ F † i , we have Υ D = ∆ ◦ Υ (cid:63) ◦ ∆ = (cid:88) ijk | k (cid:105)(cid:104) k | ˆ F i | j (cid:105)(cid:104) j | · | j (cid:105)(cid:104) j | ˆ F † i | k (cid:105)(cid:104) k | . (3)Due to the fact that the set of input states of a fixed size to thecomputation are finite, the Hilbert space, and consequently thespace of operators acting on it, are considered to be finite di-mensional. Hence, we can define new indices r = ( i, j, k ) sothat Υ D = (cid:80) r c r c ∗ r | s ( r ) (cid:105)(cid:104) r | · | r (cid:105)(cid:104) s ( r ) | , with | s ( r ) (cid:105) ∈ S inc and c r = (cid:104) k | ˆ F i | j (cid:105) . This gives the set of Kraus operators forthe map Υ D as { ˆ G r = c r | s ( r ) (cid:105)(cid:104) r |} . It can readily be seenthat, from Lemma 1, Υ D is strict incoherent. Notice that, be-cause inputs and outputs of a quantum computer can alwaysbe considered to be computational basis states (i.e., incoher-ent states), the two depolarizing maps in Υ D leave them un-changed and do not affect the computation. Therefore, for aclassical user of the quantum computer, Υ D is computation-ally as powerful as Υ (cid:63) , implying that it cannot be classicaleven though it is strict incoherent. The catch is that, there existincoherent input states for which the map Υ D is not efficiently decomposable into a polynomial number of operations from afinite set of universal stochastic operations. Such inputs cor-respond to the cases in which the map performs a quantumcomputation.The important class of incoherent operations to our discus-sion in this section is thus a subset of strict incoherent opera-tions that we name universal strict incoherent (USI) denotedby O USI . Elements of O USI are those generating the symmet-ric group (i.e., the group of permutations) on E . In mathe-matical terms sym E = (cid:104)(cid:104) O USI (cid:105)(cid:105) , where (cid:104)(cid:104)·(cid:105)(cid:105) is the group gen-eration operation via group composition, that is, the group isformed by repeatedly composing the elements of the generat-ing set. It is also straightforward to show that sym E is iso-morphic to sym { , . . . , d } , where d is the dimensionality of E . The universality of O USI must be understood over the setof incoherent states, that is, USI operations are necessary andsufficient to transform any incoherent state ˆ σ ∈ S inc to anyother incoherent state ˆ σ (cid:48) ∈ S inc via their composition andmixing, and subselection of outcomes. Note also that, startingfrom a pure state the subselection can be disregarded. Impor-tant to this construction is that any strict incoherent operationcan be obtained from a (not necessarily efficient) compositionof the elements in O USI . That is, ∀ Γ ∈ O inc : ∃{ Λ i } ⊆ O USI , (4)such that Γ[ · ] = Λ ◦ Λ ◦ Λ ◦ · · · [ · ] . The final remark is that, Υ D does not belong to O USI . This is because, Υ D ∈ O USI may hold only if we know the output of the process for allpure incoherent inputs. However, given the fact that all theother input computation parameters are fixed, the latter im-plies that Υ D , and thus Υ (cid:63) , would not be a computation sincewe already know the output of the process for all relevant in-put states.Speaking of the separation between classical and quantumcorrelations we should first be clear about what we mean by“nonclassicality”. We thus first propose the following notionof “classicality” within the context of coherence theory. Definition 2.
Within the context of coherence theory, a clas-sical observer is one who is restricted to universal strict inco-herent operations O USI , their probabilistic mixture, and sub-selection.
Here, the universality of strict incoherent operations impliesthat every incoherent state can be obtained via a successive
FIG. 1.
Geometrical illustration of nonclassicality within the con-text of coherence theory.
A classical process (the zigzag solid blackline) is represented as an efficient composition of strictly incoherentoperations evolving inside the set of incoherent states at all times.A quantum process from a classical observer’s point of view (thegreen curve), on the other hand, is equivalent to a strict incoherentoperation. However, it may involve generation and consumption ofcoherence at some stages and thus, it may partially be traversing out-side the incoherent set. Such maps may or may not be efficientlyrepresentable as a composition of USI operations. operation of O USI elements on another incoherent state, theirconvex combination and subselection of the outcomes; seeFig. 1.Definition 2 naturally gives rise to the following sufficientcondition for nonclassicality within the context of coherencetheory.
Criterion 2.
For a classical observer equipped with a set ofUSI operations O USI , a process that cannot be efficiently rep-resented as a compositions of O USI elements and their convexcombinations for at least one input state is nonclassical. Here,by efficiency we mean a polynomial number of USI maps in thesize of the input state.
We also emphasize here that, not every nonclassical process isnot efficiently decomposable into universal strictly incoherentoperations.
C. The Equivalence Theorem
We now state the first result of the present manuscript whichestablishes a fundamental link between classical computationand quantum coherence formalism.
Theorem 1.
There exists an isomorphism between classicalcomputations and the formalism of coherence theory equippedwith a set of USI operations.Proof.
The claimed isomorphism can be constructed as fol-lows.1. The set of all possible states of a deterministic classicalcomputer (bits) can be represented as elements of a fi-nite, but sufficiently large, set S p = { s i } i ∈ I with theindex set I = 1 , . . . , N for some N < ∞ . They areperfectly distinguishable and thus, they can be mappedonto an orthonormal basis set of vectors within a Hilbertspace as M ps : S p → {| i (cid:105)(cid:104) i |} i ∈ I . These vectorsform the computational basis E = {| i (cid:105)(cid:104) i |} i ∈ I . In aprobabilistic classical computer, the input as well as thereadout state of the computation could be a probabilis-tic mixture of the pure state elements as s = (cid:80) i p i s i for s i ∈ S p such that p = ( p , . . . , p N ) is a vec-tor of probabilities with (cid:80) i p i = 1 . Hence, the statespace of such a computer is S cl = conv S p . Clearlythere is a bijection between elements of S cl and S inc as M ms : S cl → S inc with M ms [ s ] = (cid:80) i p i | i (cid:105)(cid:104) i | = ˆ σ .The converse is also true. Given a computational basis E = {| i (cid:105)(cid:104) i |} i ∈ I with the index set I = 1 , . . . , N forsome N < ∞ , one can define the map M − : E → S p , where S p = { s i } i ∈ I is a set of distinguishablestates identifying different preparations of pure inputsto a classical computer. Similarly, given a mixed inco-herent state ˆ σ ∈ S inc , one can define a vector of proba-bilities p = ( p , . . . , p N ) for which ˆ σ = (cid:80) i p i | i (cid:105)(cid:104) i | andthen map it onto a probabilistic state of a classical com-puter via M − : S inc → S cl where S cl = conv S p and M − [ˆ σ ] = (cid:80) i p i s i .2. Every classical algorithm running on a classical com-puter can be decomposed into a sequence of successiveoperations of universal classical logic gates from a finiteset G UCL . Each logic gate is represented by a stochasticmap acting on the state s of the computer [39]. Impor-tantly, such gates do not create or consume superposi-tions of computational states and thus, are representedby strictly incoherent transformations with respect tothe defined computational basis. As a result, the classof universal classical gates is mapped onto a subset ofstrictly incoherent operations as M ops : G UCL → O USI where M ops is bijective and can be implicitly defined asfollows. For every classical gate G ∈ G UCL and everyincoherent state ˆ σ ∈ S inc , ˆ σ (cid:48) = Λ[ˆ σ ] = M ops [ G ][ˆ σ ] = M ms ◦ G ◦ M − [ˆ σ ] . The invertibility of M ms sim-ply implies the invertibility of M ops : for every USIquantum gate Λ ∈ O USI and every computational state s ∈ S cl , s (cid:48) = Gs = M − [Λ][ s ] = M − ◦ Λ ◦ M ms [ s ] .The universality of the classical logic gates then imme-diately implies the universality of the strictly incoherentmaps Λ defined above over the set of incoherent states.The two steps above can be summarized as S cl M ms ←−−−→ M − S inc , G UCL M ops ←−−−→ M − O USI , (5)establishing an isomorphism between classical computationand the structure of incoherent states equipped with a USI setof operations. (cid:4) Now, we use the fact that any efficient classical compu-tation can be implemented in a polynomial number of stepsin combination with the above isomorphism to conclude that the efficiency of classical algorithms implies application of apolynomial number of USI operations. The converse is alsoobvious. Consequently, every physical process that cannot beefficiently simulated on a classical computer, cannot be repre-sented as an efficient composition of USI maps within somecoherence theory and vice versa. We formalize this in a the-orem highlighting the connection between the two nonclassi-cality criteria above as our second result.
Theorem 2.
The nonclassicality Criteria 1 and 2 are equiva-lent.
Our third result, which is an immediate consequence ofTheorems 1 and 2, and proved within Appendix A, tells uswhen it is not possible to do quantum computation.
Theorem 3.
Production or consumption of quantum coher-ence provides the necessary resource for the exponentialspeed up of quantum computations versus classical ones.
Having these results at hand, we can now answer the question“What do we learn about quantum correlations from collabo-rative quantum computing?”
III. A TOY PROTOCOL
Our aim is to use Criterion 1, i.e., the power of quantumcomputation models, to show the quantumness of correlations.To this end, we need to consider quantum computation proto-cols that are nonlocal and use mixed quantum states, notingthat any pure correlated quantum state is necessarily entan-gled. We thus first consider a toy protocol for which we al-ready have available the minimal tools for a characterizationof the resources used.In classical computations, to run large computational taskson multiple supercomputers in parallel and then combine theiroutputs to get a final result is a common protocol; a modelcalled distributed computing . It is thus intriguing to considera situation in which each of the servers is equipped with aquantum computer to run quantum computations. In suchscenarios, the input to each server is possibly classical infor-mation accompanied with quantum states. We consider twoversions of a distributed computing task in which a client,Charlie, exploits nondiscordant states to run quantum com-putations on two servers, Alice and Bob. We assume that thefollowing rules apply: (i) servers are forbidden to commu-nicate; (ii) they do not have access to any sources of quan-tum states —they can only perform unitary transformationsand make destructive measurements on their outputs; (iii) theclient, on the other hand, does not possess any quantum pro-cessors —he may only have limited capability of preparingquantum states. We also assume that there are no losses, inef-ficiencies, or errors, as they are not essential to our argumentsand conclusions about the quantumness of the correlations.
Task 1.—
Charlie has classical descriptions of two n X -qubit unitary matrices ˆ U X ( X=A , B ) in terms of polynomialsized network of universal gates. His task is to estimate thequantity ι = Tr ˆ U A · Tr ˆ U B n A + n B . (6)He can also prepare up to two pure qubit states—any otherstates are maximally mixed.There is strong evidence to suggest that estimating the nor-malised trace of a n -qubit unitary matrix Tr ˆ
U / n , generatedfrom a polynomial sized network of universal gates, is hardfor a classical computer [12, 40–44]. These hardness argu-ments imply that even with a classical description of the poly-nomial sized network forming the unitary, Charlie (as wellas Alice and Bob) cannot efficiently estimate the normalisedtrace ι using his classical resources. The latter follows fromthe fact that, assuming an exponential growth in the classicalresources required to estimate the normalized trace of one uni-tary Tr ˆ U X / n X ( X = A , B ) with the number of input qubits n X , classically estimating their normalized tensor product willalso require at least an exponential effort in the total numberof input qubits n A + n B . Thus, he is encountering a classicallychallenging task. He can, however, conquer the difficulty withthe help of the two servers using a duplicated DQC1 proto-col [4], termed here as nonlocal deterministic quantum com-puting with two qubits (NDQC2). FIG. 2.
The schematic of a nonlocal deterministic quantum com-putation with two qubits (NDQC2).
A client, Charlie, who doesnot possess any quantum processors aims to estimate the quantity ι = Tr ˆ U A · Tr ˆ U B / n A + n B . This is believed to be hard to perform ona classical computer. Therefore, he asks two servers, Alice and Bob,who are capable of performing unitary transformations and makingdestructive measurements to realize the controlled unitaries ˆ U contX ( X=A , B ). The servers are forbidden to communicate and do nothave access to any sources of quantum states. Charlie then sendsstrictly classical states to the servers and receives the results of themeasurements of the Pauli operators as per Eq. (7). By manipulat-ing the received data, he is able to efficiently estimate ι . Depend-ing on his choice of state, Charlie is also able to hide the local esti-mates from Alice and Bob without reducing his global computationalpower. NDQC2 (see Fig. 2).—
We assume it is always possiblefor Charlie to ask Alice and Bob to realize the controlledunitaries ˆ U contX = | (cid:105) X (cid:104) |⊗ ˆ I X + | (cid:105) X (cid:104) |⊗ ˆ U X ( X=A , B ), re-spectively [45]. He then prepares the two-qubit control system in either of the two pure product (nondiscordant)states ˆ (cid:37) contAB;1 = |±(cid:105) A (cid:104)±| ⊗ |±(cid:105) B (cid:104)±| , where |±(cid:105) X =( | (cid:105) X ±| (cid:105) X ) / √ , and the ancillary qubits in the maximallymixed states, ˆ τ X =ˆ I ⊗ n X / n X , and sends them to theservers. Alice and Bob operate on their respective ancil-lae and control inputs, locally and independently, to ob-tain ˆ (cid:37) outX;1 = ˆ U contX ( |±(cid:105) X (cid:104)±|⊗ ˆ τ X ) ˆ U cont † X , and make measure-ments of the Pauli operators on the output control states, Trˆ (cid:37) outX;1 [( ˆ σ x ;X + i ˆ σ y ;X ) ⊗ ˆ I ⊗ n X ]= ± Tr ˆ U X / n X . Finally, theservers send their statistics to Charlie, who will combine themto obtain an estimate of ι = (cid:89) X=A , B (cid:104) ˆ σ x ;X + i ˆ σ y ;X (cid:105) = (cid:42) (cid:79) X=A , B ( ˆ σ x ;X + i ˆ σ y ;X ) (cid:43) , (7)where (cid:104)·(cid:105) denotes the quantum expectation value of the outputcontrol state. Task 2.—
Consider Task 1 where Charlie also wants tohide the local estimates
Tr ˆ U X / n X from Alice and Bob at alltimes, given the constraints (i)-(iii) on the protocol.Clearly Task 2 is classically, if not impossible, as hard asTask 1, because estimating the normalized trace of the globalunitary ˆ U A ⊗ ˆ U B / n A + n B is also classically hard. However,this can be done efficiently using NDQC2 protocol above ifCharlie prepares the control in the nondiscordant state ˆ (cid:37) contAB;2 = 12 (cid:88) x = ± | x (cid:105) A (cid:104) x |⊗| x (cid:105) B (cid:104) x | . (8)Following the same procedure as in Task 1 and making thesame measurements, the correlations within the measure-ment outcomes are processed by Charlie as per the r.h.s ofEq. (7), which results in ι = (cid:104) ( ˆ σ x ;A + i ˆ σ y ;A ) ( ˆ σ x ;B + i ˆ σ y ;B ) (cid:105) .The marginals of the control state (8) are maximally mixedstates, so that independent measurements do not result in anyinformation about ι . This hides the local estimates from Aliceand Bob. Hence, NDQC2 enables a classically hard collabo-rative task only using correlated inputs and correlation mea-surements. A. Quantum Correlations in NDQC2.
As we discussed earlier, any process that cannot be effi-ciently simulated on a classical computer is nonclassical . Wehave shown that NDQC2 cannot be efficiently simulated usingclassical resources, i.e., classical communication and classi-cal computers held by Alice and Bob, implying that it is anexample of a nonlocal nonclassical process.We are now able to give an operational meaning to the term“quantum correlations”, inspired by the NDQC2 toy model.First, from Criterion 1, we infer the quantumness of the re-sources used in collaborative quantum computations from thenonclassical advantages obtained in them. Second, wheneverthe locally accessible quantum resources are inadequate tofully account for such advantages, they are necessarily the re-sult of correlations. A closer look at the NDQC2 protocolshows that in Tasks 2, not only the input, but also the outputstate is nondiscordant with respect to the Alice-Bob partition-ing in which the correlations are measured, because the globaloperation ˆ U contAB = ˆ U contA ⊗ ˆ U contB preserves local orthonormal-ity of bases. In addition, from Eq. (8), it is clear that there isno local entanglement or discord within each server in Task 2in contrast to the DQC1 protocol [12, 40]. In fact, in thiscase, Alice and Bob have no local quantum computational re-sources as they only receive locally maximally-mixed states.Hence, from the common perspective of quantum informationtheory, the input and output states in this scenario are consid-ered to possess no quantum correlations between Alice andBob. One thus should wonder if there is nothing quantumgoing on locally, and there is nothing quantum about the cor-relations between Alice and Bob as characterized by the stan-dard measures of quantum information, then where does thequantum power of the joint Alice-Bob party in NDQC2 comefrom? And why is that obtained only through correlation mea-surements? Our answer is that the computational power ofNDQC2 in this case is indeed a manifestation of quantumnessof correlations distributed between Alice and Bob through theinput quantum state. Importantly, the standard classificationof quantum correlations does not account for these sort of cor-relations.One might object to calling these correlations quantum byconsidering the following scenarios: (i) suppose that the lo-cal servers in our toy model are granted the ability to preparequantum states. Then, Charlie can send classical encryptedmessages instructing the servers to prepare either the super-position state | + (cid:105) X or |−(cid:105) X with equal probabilities to Al-ice and Bob, where they would have created the state ˆ (cid:37) contAB;2 in Eq. (8) locally without accessing the content of the mes-sage, and thus, simulating the protocol locally. Regardless ofthe complexity of such a semi-classical protocol compared toours, we emphasize that the possibility to prepare the statesvia LOCC does not imply the classicality of its inherent cor-relations. Similarly, any separable discordant state can be pre-pared using LOCC, and yet it is believed that quantum discordimplies quantum correlations; (ii) if Alice and Bob have ac-cess to local quantum resources, i.e., perfect qubits, they mayextract the correlations encoded within the input state and ac-cess the local estimates of Tr ˆ U X / n X . Equivalently, if theyare allowed to communicate during the protocol, then they canobtain the same information from the correlations as Charliedoes. In this case, the resolution is that the extractibility ofthe encoded probability distribution and the final result of thecomputation also does not imply the classicality of the cor-relations within quantum states. The counterexample is, forinstance, a one-way discordant state. If the two-way classicalcommunication is allowed between parties, then they can ex-tract the probability distribution encoded within such states.However, one-way discordant states are also quantum corre-lated. The lesson we learn is thus that the quantumness ofcorrelations can be revealed only if appropriate restrictions are imposed on particular tasks. In our case, the required re-strictions are exactly (i)-(iii) given for the NDQC2. B. Global Coherence in NDQC2
In order to characterize quantum correlations present inNDQC2, we should identify the resources empowering it. Weobserved in Theorem 3 that quantum coherence is necessaryfor the exponential speed-up of quantum computers. Recently,it has been shown that coherence also provides the sufficientresource for the particular case of DQC1 protocol in the sensethat the precision of the quantity estimated in DQC1 is a func-tion of the amount of quantum coherence inherent within theinput state [29]. Since NDQC2 enjoys a construction similarto DQC1, we anticipate that the power of NDQC2 in Tasks 1and 2 is also due to the coherence of the input and outputstates. Here we show this fact quantitatively.We start by choosing the relative entropy of coherence(REC) [19, 20] as our measure of coherence. For any den-sity operator ˆ (cid:37) , the REC is given by C r (ˆ (cid:37) ):= S (∆[ˆ (cid:37) ]) − S (ˆ (cid:37) ) , (9)in which S (ˆ (cid:37) )= − Trˆ (cid:37) log ˆ (cid:37) is the von Neumann entropy. Weomit the dependence of C r is the basis from the function’sargument for brevity. REC satisfies the three main require-ments for any faithful measure of quantum coherence: (i) aquantum state is incoherent if and only if C r (ˆ (cid:37) )=0 ; (ii) itis nonincreasing on average under all incoherent transforma-tions, i.e., C r (ˆ (cid:37) ) (cid:62) (cid:80) i p i C r ( ˆ (cid:37) i ) where p i is the probabil-ity of obtaining ˆ (cid:37) i upon measurement; (iii) it is convex, i.e., C r ( (cid:80) i p i ˆ (cid:37) i ) (cid:54) (cid:80) i p i C r ( ˆ (cid:37) i ) for any set of states { ˆ (cid:37) i } andprobability distribution { p i } . Operationally, REC is equiva-lent to the distillable coherence and quantifies the optimal rateat which maximally coherent states can be prepared from in-finitely many copies of a given mixed state using incoherentoperations [20].First, we choose the local computational bases in ourprotocol to be E X = {| (cid:105) X ⊗ | ξ i (cid:105) X , | (cid:105) X ⊗ | ξ i (cid:105) X } n X i =1 , where {| ξ i (cid:105) X } n X i =1 are eigenvectors of ˆ U X for X = A , B . Now,we see that, in both Tasks 1 and 2, the input states ˆ (cid:37) inAB;1 =ˆ (cid:37) contAB;1 ⊗ ˆ τ A ⊗ ˆ τ B and ˆ (cid:37) inAB;2 =ˆ (cid:37) contAB;2 ⊗ ˆ τ A ⊗ ˆ τ B are glob-ally coherent with respect to the global computational basis E AB = E A ⊗ E B with C r (ˆ (cid:37) inAB;1 )=2 log 2 and C r (ˆ (cid:37) inAB;2 )= log 2 ,respectively.Second, we consider the local coherences of the marginalstates in Tasks 1 and 2 to obtain C r ( |±(cid:105) X (cid:104)±|⊗ ˆ τ X )= log 2 and C r (ˆ I X / ⊗ ˆ τ X )=0 , respectively in each task. From Refs. [27,29] we know that the less the input coherence to a DQC1 pro-tocol is, the worse the estimation of the normalized trace of theunitary will be. Therefore, these values justify the fact that inTask 1 local traces are accessible to Alice and Bob, due to the local computational powers provided by locally coherent re-sources, while in Task 2 they remain hidden to them becauseno local computational power is available to parties. We em-phasize here that ˆ U contAB = ˆ U contA ⊗ ˆ U contB neither increases nordecreases the amount of global and local REC, as E AB is aneigenbasis of ˆ U contAB [46].Now, we show that only global coherence plays a role inthe nonclassical performance of NDQC2 protocol. Lemma 2.
The precision of the estimated quantity ι in Eq. (6) is given by the amount of global coherence inherent within theinput quantum state as quantified by REC. Please see Appendix B for the proof. We now make use thefact that local coherence of marginal states implies the globalcoherence of the joint state of the system, in combination withTheorem 3 and the above lemma, to conclude the fourth mainresult of the present paper.
Theorem 4.
Global coherence is necessary and sufficient forthe nonclassical performance of the NDQC2 protocol.
Theorem 4, clearly shows the role of global coherence in ourprotocol. There is, however, a difference between NDQC2 ofTask 1 and Task 2 to be discussed shortly.
IV. NET GLOBAL COHERENCE AS QUANTUMCORRELATIONS
In the Task 2, we conclude the quantumness of the corre-lations from nonclassical performance of the protocol, since,(i) the input state to the protocol is indeed correlated; our con-siderations are merely regarding whether they are quantum orclassical, and, (ii) other than observing correlations betweenAlice and Bob outcomes, Charlie would not be able to obtainthe result of the task. In Task 1, on the other hand, such aconclusion is not valid. The obvious reason is that, the prod-uct state used in Task 1 represents independent preparationprocedures and hence, by postulates of quantum mechanics,uncorrelated states.To address this difference, suppose that a measure of coher-ence C has been chosen to characterize quantum correlationsin some computational basis. We then define C net (ˆ (cid:37) AB ) = C (ˆ (cid:37) AB ) − C (ˆ (cid:37) A ) − C (ˆ (cid:37) B ) , (10)to determine the net-global quantum computational-power ofthe quantum states with the interpretation that we subtractthe local quantum powers from the overall one. The require-ment that product states show no quantum correlations, andhence no global computational power except those due to lo-cal resources, imposes the condition “if ˆ (cid:37) AB = ˆ (cid:37) A ⊗ ˆ (cid:37) B then C net (ˆ (cid:37) AB )=0 ”. This holds true if and only if the coherencemeasure C is additive, i.e., C (ˆ (cid:37) A ⊗ ˆ (cid:37) B )= C (ˆ (cid:37) A )+ C (ˆ (cid:37) B ) . Im-portantly, the relative entropy of coherence [19, 20] is anadditive measure, while, for instance, the (cid:96) -norm of co-herence [19] is not. It immediately follows that in Task 1 C netr (ˆ (cid:37) inAB;1 )=0 , that is, all the global quantum computationalpower is due to the local resources. In sharp contrast, in Task 2 C netr (ˆ (cid:37) inAB;2 )= log 2 , interpreted as the amount of global quan-tum computational power purely due to quantum correlations.We thus notice that in Task 2 all the computational advantagecan be associated with the net global coherence. We draw in-spiration from this fact and, with a little foresight, define thequantum correlated states as per below. Definition 3. (Quantum-Correlated States) A bipartite quan-tum state ˆ (cid:37) AB is said to contain quantum correlations with respect to a global computational basis E AB if and only if C netr (ˆ (cid:37) AB ) > within E AB . In what follows, we show that this definition is indeed welljustified, demonstrating that previously known classes ofquantum correlations are emergent from our extended notion,and that it allows for a proper operational interpretation.
A. Some Properties of C netr Using relative entropy of coherence, as shown in Ap-pendix C, we can rewrite Eq. (10) as C netr (ˆ (cid:37) AB ) = I (ˆ (cid:37) AB ) − I (∆ AB [ˆ (cid:37) AB ]) , (11)in which I (ˆ (cid:37) AB ):= S (ˆ (cid:37) A )+ S (ˆ (cid:37) B ) − S (ˆ (cid:37) AB ) is the mutual in-formation and ∆ AB is the bipartite fully dephasing channelwithin the global computational basis. C netr is a further gener-alization of the basis-dependent discord in which only one ofthe parties undergoes the dephasing [36]. According to Def-inition 2 of a classical observer in coherence theory, the per-ception of two classical observers from the quantum world islimited to the globally incoherent state ˆ σ AB = ∆ AB [ˆ (cid:37) AB ] .Hence, the quantity I (∆ AB [ˆ (cid:37) AB ]) can be considered as themutual information between these two classical observerswith particular local bases E A and E B . It then follows that C netr (ˆ (cid:37) AB ) represents the net quantum information shared be-tween the two in the global basis E AB . It is also worth pointingout that a quantity called global quantum discord was previ-ously introduced by Rulli and Sarandy in Ref. [47]. The im-portant difference between this quantity and net global coher-ence, however, is that a minimization over all computationalbases is involved in the definition of the former, in tradition ofdiscord quantities. Here, in contrast, we showed, within theframework of coherence theory, that the computational basisplays a significant role in the characterisation of quantum cor-relations. In this regard, we obtain the following properties ofthe net global coherence. Theorem 5.
For every bipartite quantum state ˆ (cid:37) AB it holdsthat C netr (ˆ (cid:37) AB ) (cid:62) . The equality holds if and only if thequantum state is a product state or has the form ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i | ⊗ | j (cid:105) B (cid:104) j | with respect to the global compu-tational basis E AB = E A ⊗ E B = {| i (cid:105) A ⊗ | j (cid:105) B } , with { p ij } beinga probability distribution. Please see Appendix D for a detailed proof. It is necessarythat every extension of the standard classes of quantum corre-lated states includes the hierarchy of entangled and discordantstates as special cases. To show that indeed this holds true forour approach in a well-defined way, we first state a feature of C netr , the proof of which is given in Appendix E. Theorem 6.
Any multipartite pure state has a nonzero netglobal coherence if and only if it is entangled.
Second, we give some more general results on the global-coherence properties of the standard classification. As it hasbeen shown in Appendix F, the following holds.
Theorem 7.
A bipartite quantum state ˆ (cid:37) AB is nondiscordantif and only if C netr (ˆ (cid:37) AB )=0 within some appropriate globalcomputational basis E (cid:63) AB . The following is then an immediate conclusion.
Corollary 1. [48] For every quantum state that is charac-terized as quantum correlated in quantum information theory(entangled and discordant states), C netr (ˆ (cid:37) AB ) > independentof the chosen (pure product) global-basis. From Corollary 1 we see that the standard picture of quantumcorrelations follows from our formalism if a net global coher-ence is required to exist with respect to every computationalbasis. The latter is indeed a very strong condition. In par-ticular, because classical observers are restricted to a specificcomputational basis, having global coherence with respect tothat basis is sufficient for obtaining possible quantum advan-tage of the correlations, provided that appropriate fine-grainedoperations at a quantum level are accessible.Arguably, one of the desirable properties of a theory ofquantum correlations as a resource, which is not met by quan-tum discord, is convexity [29]. Important to our constructionis that there exist convex combinations of two product statesthat possess quantum correlations with a positive C netr (ˆ (cid:37) AB ) .This shows that convex combinations of uncorrelated statesare not necessarily uncorrelated. In other words, the two de-sirable properties of “a convex resource theory of quantumcorrelations” and “the preparation independence of the prod-uct states” seem incompatible, unless considering a theory ofnonclassicality which identifies products of coherent states asnonclassical [49].As a final word, quantum-correlated states provide an inter-esting nonlocal feature which we call coherence localization .,namely, the process of providing local computational powerfor one party with the aid of another. The trivial case is thatthere already exists some local coherence available to parties.However, there exists a nontrivial scenario that is given belowand proved in Appendix G. Theorem 8. [34] Given the local computational bases E A = {| i (cid:105) A } and E B = {| j (cid:105) B } , and Alice and Bob sharing a bi-partite quantum state ˆ (cid:37) AB , they can distil quantum coherenceon Bob’s side using LICC if and only if ˆ (cid:37) AB cannot be writtenas ˆ (cid:37) AB = (cid:80) j p j ˆ (cid:37) A; j ⊗| j (cid:105) B (cid:104) j | . Corollary 2.
Given the local computational bases E A = {| i (cid:105) A } and E B = {| j (cid:105) B } , Alice and Bob sharing abipartite quantum state ˆ (cid:37) AB , and C r (ˆ (cid:37) A )= C r (ˆ (cid:37) B )=0 ,they cannot distil quantum coherence on neither sides us-ing LICC if and only if ˆ (cid:37) AB is not quantum correlated, i.e. C netr (ˆ (cid:37) AB )=0 , with respect to the global bases E AB = E A ⊗ E B . The above corollary gives us a way to see quantum correla-tions from a different perspective. Suppose that a bipartitestate is shared between two classical agents Alice and Bob.If we ask under what conditions one of the parties can pro-vide quantum coherence (and hence, computational power)for the other party using local classical operations and clas-sical communication, i.e., LICC, then the answer is given by Corollary 2: if C netr (ˆ (cid:37) AB ) (cid:54) =0 with respect to the global bases E AB = E A ⊗ E B . This is simply the operational interpretation ofour extended notion of quantum correlations. DISCUSSION
We have built a rigorous framework for benchmarking theefficiency of computational models as a separation criterionof classical and quantum resources. We then introduced theNDQC2 model of quantum computation, a collaborative non-local algorithm for estimating the product of the normal-ized traces of two unitary matrices that shows an exponen-tial speedup compared to the best known classical algorithms.We demonstrated that this task can be done using a separa-ble and nondiscordant input state. In essence, based on ourquantum-classical separation criterion we argued that the ex-ponential speedup of NDQC2 over classical algorithms is amanifestation of the quantum correlations beyond entangle-ment and discord inherent within the input quantum state toour toy model.Our main observation, however, was that the standard clas-sification of quantum correlations in quantum informationtheory does not capture the quantumness of such correlations,and thus require a revision. It is noteworthy that, similar ar-guments exists within the quantum optics community, wherenonclassical phase-space quasiprobability distributions are thesignatures of quantumness [15, 50]. This viewpoint has alsobeen extended to composite discrete-continuous variables sys-tems [51]. Very recently, we have shown that phase-spacenonclassicality provides a resource for nonlocal B
OSON S AM - PLING in absence of the standard quantum correlations ofquantum information [17], in favour of the quantum opticalviewpoint.The approach we presented here, extends the standardquantum information theoretic classification of quantum cor-relations to include quantum advantages obtained in dis-tributed quantum computation protocols. We quantitativelyshowed that the net global coherence emerging from correla-tions between subsystems can be considered as equivalent toquantum correlations. We showed that our generalized defini-tion of quantum correlations characterizes the necessary andsufficient quantum resources in NDQC2 and properly containsthe standard classification as a particular case.It is worth noting that within the rapidly developing fieldof quantum coherence, there has recently been interests in es-tablishing an appropriate framework for the paradigm of localincoherent operations and classical communication (LICC),in which parties are restricted to locally incoherent opera-tions [34, 38]. One can think of LICC as the class of classicaloperations on distant multipartite systems. Quantum correla-tions, as we have defined, then are the weakest in the sensethat they allow a party to remotely provide quantum computa-tional resources for a distant party using LICC. Thus, we seethat it is possible to obtain local resource states for efficientquantum computation, even if no local resource states are ini-tially available and the parties are locally restricted to classicaloperations which do not generate resource states, if and only0if they share quantum correlations of the type introduced here.The relation between multipartite quantum coherence andquantum correlations has also been studied recently by otherresearchers [24, 26–28]. However, their approaches are fun-damentally different from the perspective presented in thismanuscript. Specifically, rather than to investigate the con-version of local quantum coherence into standard types ofquantum correlations, we have considered net global quan-tum coherence as a primitive notion of quantum correlation,which can be distributed and might reveal its unique quantumsignatures only within distributed quantum protocols. As aconsequence, our results opens up the possibility for explor-ing new protocols that use such correlations, which are gener- ically cheaper than entanglement and discord, to perform col-laborative tasks more efficiently than any classical algorithm.
ACKNOWLEDGEMENTS
The authors gratefully acknowledge Werner Vogel, FabioCosta, and Eric Chitambar for valuable discussions. Thisproject was supported by the Australian Research CouncilCentre of Excellence for Quantum Computation and Commu-nication Technology (CE110001027).
Appendix A: Proof of Theorem 3
Theorem 3.
Production or consumption of quantum coherence provides the necessary resource for the exponential speed up ofquantum computations versus classical ones.
Proof.
First, notice that a quantum computer runs algorithms in a polynomial time (number of steps or gates), otherwise it wouldnot be efficient. Then, the contrary of the above statement would be that,• there exist a quantum computer in which all steps are strict incoherent operations, i.e., neither they produce nor theyconsume quantum coherence.We now use the fact that one is able to produce any incoherent state ˆ σ ∈ S inc at the output of a quantum computer, which isequivalent to the universality of the computer’s gates over the set of incoherent states as given by Eq. (4). Hence, for one such acomputer, the set of gates used is, in fact, a USI set of operations. As a result, our counter assumption further translates into that,• there exists a quantum computer which merely performs a polynomial number of USI operations.Using the isomorphism between classical computers and the formalism of coherence theory with a set of USI operations givenin Eq. (5), the latter just means that whatever such a quantum computer does can be equally performed on a classical computer.This contradicts the assumption that we have a quantum computer running algorithms faster than any classical computer, hencethe result. (cid:4) Appendix B: Proof of Lemma 2: Estimate Precision in NDQC2
Following Ref. [29], it is easy to show that the precision in the estimate of the quantity ι =Tr ˆ U A · Tr ˆ U B / n A + n B is given bythe relative entropy of coherence of the control qubit C r (ˆ (cid:37) contAB ) . The latter is defined as the optimal rate at which maximallycoherent states can be distilled from infinitely many copies of the state ˆ (cid:37) contAB using incoherent operations [20]. Hence, given M copies of the state ˆ (cid:37) contAB a total number N ≈ M C r (ˆ (cid:37) contAB ) pairs of maximally coherent states can be distilled in Alice’s and Bob’slocal laboratories. Assuming M being large, as shown in [29], the standard error in the estimation of each local normalizedtrace ι X =Tr ˆ U X / n X is given by SE ( ι X ) ≈ (cid:113) −| ι X | N = (cid:113) −| ι X | M C r (ˆ (cid:37) contAB ) for X = A , B . The standard error of the quantity ι is thusgiven by SE ( ι ) = (cid:112) SE ( ι A ) + SE ( ι B ) ≈ (cid:115) − | ι A | − | ι B | M C r (ˆ (cid:37) contAB ) . (B1)The binary precision in the estimation of a number µ with | µ | (cid:54) goes like BP( µ ) ≈ − log [SE( µ )] . Noting that the nominatorin Eq. (B1) is bounded by (cid:54) − | ι A | − | ι B | (cid:54) , we find BP( ι ) ≈
12 log C r (ˆ (cid:37) contAB ) . (B2)1 Appendix C: Derivation of Eq. (11) of the main text
To show the truth of the Eq. (11) of the main text, again, we recall from Eq. (10) of the main text that C netr (ˆ (cid:37) AB ) = C r (ˆ (cid:37) AB ) − C r (ˆ (cid:37) A ) − C r (ˆ (cid:37) B ) , (C1)in which C r refers to the REC. We also restate that C r (ˆ (cid:37) ):= S (∆[ˆ (cid:37) ]) − S (ˆ (cid:37) ) , in which S (ˆ (cid:37) )= − Trˆ (cid:37) log ˆ (cid:37) and ∆[ˆ (cid:37) ]:= (cid:80) (cid:104) i | ˆ (cid:37) | i (cid:105)| i (cid:105)(cid:104) i | are the von Neumann entropy and the fully dephasing channel with respect to the computational bases E = {| i (cid:105)} . Consequently,we have C r (ˆ (cid:37) AB )= S (∆ AB [ˆ (cid:37) AB ]) − S (ˆ (cid:37) AB ) , C r (ˆ (cid:37) A )= S (∆ A [ˆ (cid:37) A ]) − S (ˆ (cid:37) A ) , C r (ˆ (cid:37) B )= S (∆ B [ˆ (cid:37) B ]) − S (ˆ (cid:37) B ) . (C2)Substituting the relations of Eq. (C2) into Eq. (C1), after a simple rearrangement and using I (ˆ (cid:37) AB ):= S (ˆ (cid:37) A )+ S (ˆ (cid:37) B ) − S (ˆ (cid:37) AB ) ,we get C netr (ˆ (cid:37) AB ) = { S (ˆ (cid:37) A ) + S (ˆ (cid:37) B ) − S (ˆ (cid:37) AB ) } − { S (∆ A [ˆ (cid:37) A ]) + S (∆ B [ˆ (cid:37) B ]) − S (∆ AB [ˆ (cid:37) AB ]) } , = I (ˆ (cid:37) AB ) − I (∆ AB [ˆ (cid:37) AB ]) . (C3)We note that, for the second equality to be true, we need that Tr B ∆ AB [ˆ (cid:37) AB ] = ∆ A [Tr B ˆ (cid:37) AB ] = ∆ A [ˆ (cid:37) A ] , and similarly Tr A ∆ AB [ˆ (cid:37) AB ] = ∆ B [Tr A ˆ (cid:37) AB ] = ∆ B [ˆ (cid:37) B ] . This can be easily verified, as Tr B ∆ AB [ˆ (cid:37) AB ] = (cid:88) ij (cid:104) i, j | ˆ (cid:37) AB | i, j (cid:105)| i (cid:105)(cid:104) i | (Tr | j (cid:105)(cid:104) j | )= (cid:88) i | i (cid:105)(cid:104) i | ⊗ (cid:104) i | (cid:88) j (cid:104) j | ˆ (cid:37) AB | j (cid:105) | i (cid:105) = (cid:88) i | i (cid:105)(cid:104) i | ⊗ (cid:104) i | (Tr B ˆ (cid:37) AB ) | i (cid:105) = ∆ A [ˆ (cid:37) A ] . (C4) (cid:4) Appendix D: Proof of Theorem 5
Theorem 5.
For every bipartite quantum state ˆ (cid:37) AB it holds that C netr (ˆ (cid:37) AB ) (cid:62) . The equality holds if and only if the quan-tum state is a product state or has the form ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i | ⊗ | j (cid:105) B (cid:104) j | with respect to the global computational basis E AB = E A ⊗ E B = {| i (cid:105) A ⊗ | j (cid:105) B } , with { p ij } being a probability distribution. Proof.
Recall that the basis-dependent discord [36] is defined as D A → B (ˆ (cid:37) AB ) := I (ˆ (cid:37) AB ) − I (∆ A [ˆ (cid:37) AB ]) , (D1)in which ∆ A [ˆ (cid:37) AB ] = (cid:80) i | i (cid:105)(cid:104) i | ⊗ (cid:104) i | ˆ (cid:37) AB | i (cid:105) = (cid:80) i p i | i (cid:105)(cid:104) i | ⊗ ˆ (cid:37) B; i , with p i = Tr B (cid:104) i | ˆ (cid:37) AB | i (cid:105) , is the one-sided dephasing channelfor Alice’s subsystem. It has been shown that D A → B (ˆ (cid:37) AB ) (cid:62) [8]. Similarly, D B → A (ˆ (cid:37) AB ) (cid:62) ⇔ I (ˆ (cid:37) AB ) (cid:62) I (∆ B [ˆ (cid:37) AB ]) . (D2)We can easily verify that ∆ AB [ˆ (cid:37) AB ] = (cid:88) ij (cid:104) i, j | ˆ (cid:37) AB | i, j (cid:105)| i (cid:105)(cid:104) i | ⊗ | j (cid:105)(cid:104) j | = (cid:88) i | i (cid:105)(cid:104) i | ⊗ (cid:104) i | (cid:88) j | j (cid:105)(cid:104) j | ⊗ (cid:104) j | ˆ (cid:37) AB | j (cid:105) | i (cid:105) = ∆ A [∆ B [ˆ (cid:37) AB ]] = ∆ B [∆ A [ˆ (cid:37) AB ]] . (D3)2Combining this with Eq. (D1) and (D2), we have D A → B (∆ B [ˆ (cid:37) AB ]) = I (∆ B [ˆ (cid:37) AB ]) − I (∆ A [∆ B [ˆ (cid:37) AB ]])= I (∆ B [ˆ (cid:37) AB ]) − I (∆ AB [ˆ (cid:37) AB ]) (cid:62) ⇔ I (∆ B [ˆ (cid:37) AB ]) (cid:62) I (∆ AB [ˆ (cid:37) AB ]) ⇒ I (ˆ (cid:37) AB ) (cid:62) I (∆ B [ˆ (cid:37) AB ]) (cid:62) I (∆ AB [ˆ (cid:37) AB ]) ⇒ C netr (ˆ (cid:37) AB ) = I (ˆ (cid:37) AB ) − I (∆ AB [ˆ (cid:37) AB ]) (cid:62) . (D4)For the equality to hold, either both I (ˆ (cid:37) AB ) and I (∆ AB [ˆ (cid:37) AB ]) must be zero, which implies that the state is a product. Or, onemust have D A → B (∆ B [ˆ (cid:37) AB ]) = 0 in Eq. (D4), which implies the form of ∆ B [ˆ (cid:37) AB ] to be ∆ B [ˆ (cid:37) AB ] = (cid:80) ij p ij | i (cid:105) A (cid:104) i | ⊗ | j (cid:105) B (cid:104) j | .This in turn means that ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i | ⊗ ˆ (cid:37) B; j . Also, from symmetry of A and B, and by similar arguments, it must betrue that ˆ (cid:37) AB = (cid:80) ij p ij ˆ (cid:37) A; i ⊗ | j (cid:105) B (cid:104) j | . Together, we must have ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i | ⊗ | j (cid:105) B (cid:104) j | . (cid:4) Appendix E: Proof of Theorem 6
Theorem 6.
Any multipartite pure state has a nonzero net global coherence if and only if it is entangled.
Proof.
Using Theorem 5 it is clear that any entangled state has a nonzero net global coherence. The converse easily followsfrom the fact that any pure multipartite state is either a product state or entangled so that the net global coherence of nonentangledstates becomes zero by the additivity condition for the coherence measures. (cid:4)
Appendix F: Proof of Theorem 7
Theorem 7.
A bipartite quantum state ˆ (cid:37) AB is a CC state if and only if C netr (ˆ (cid:37) AB )=0 within some appropriate global computa-tional basis E (cid:63) AB .To prove Theorem 7, let us first prove the following useful Lemma. Lemma.
A bipartite quantum state ˆ (cid:37) AB is a CC state if and only if it is incoherent within some appropriate global computationalbasis E (cid:63) AB . Proof.
If: Assuming that there exists a computational basis E (cid:63) AB = {| i (cid:105) A ⊗| j (cid:105) B } in which the state ˆ (cid:37) AB is incoherent impliesthat the state can be written as ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i |⊗| j (cid:105) B (cid:104) j | . Due to the orthogonality of the local vectors in the basis set, ˆ (cid:37) AB is CC. Only if: A CC state, by definition, admits the form ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i |⊗| j (cid:105) B (cid:104) j | , which is clearly incoherent with thechoice of computational basis E (cid:63) AB = {| i (cid:105) A ⊗| j (cid:105) B } . (cid:4) Now we can prove Theorem 7.
Proof.
Using Lemma 1 above, it is sufficient to prove that a bipartite quantum state ˆ (cid:37) AB is incoherent within some appropri-ate global computational basis E (cid:63) AB if and only if C netr (ˆ (cid:37) AB )=0 .Only if: A CC state, by definition, admits the form ˆ (cid:37) AB = (cid:80) ij p ij | i (cid:105) A (cid:104) i |⊗| j (cid:105) B (cid:104) j | , which is clearly incoherent with the choiceof the computational basis E (cid:63) AB = {| i (cid:105) A ⊗| j (cid:105) B } . Also, global incoherence implies marginal incoherence, evidently. Consequently,one has C r (ˆ (cid:37) AB )= C r (ˆ (cid:37) A )= C r (ˆ (cid:37) B )=0 , which results C netr (ˆ (cid:37) AB )=0 .If: Assuming that C netr (ˆ (cid:37) AB )=0 , from Eq. (C3) we have I (ˆ (cid:37) AB ) = I (∆ AB [ˆ (cid:37) AB ]) . Combining this with the facts I (ˆ (cid:37) AB ) (cid:62) I (∆ A [ˆ (cid:37) AB ]) (cid:62) I (∆ AB [ˆ (cid:37) AB ]) and I (ˆ (cid:37) AB ) (cid:62) I (∆ B [ˆ (cid:37) AB ]) (cid:62) I (∆ AB [ˆ (cid:37) AB ]) (see Eq. (D4)), we conclude that I (ˆ (cid:37) AB ) = I (∆ B [ˆ (cid:37) AB ]) and I (ˆ (cid:37) AB ) = I (∆ B [ˆ (cid:37) AB ]) . On one hand, using Eq. (D1), these equalities imply that the state must possesses avanishing basis-dependent discord [36] both from Alice to Bob and from Bob to Alice. On the other hand, the standard discordis the minimum of the basis-dependent discord over all measurements on either sides [36]. As a result, the state must have avanishing (standard) discord both from Alice to Bob and from Bob to Alice. It is already known that this is possible only if thestate is CC. (cid:4) Appendix G: Proof of Corollary 2
Corollary 2.
Given the local computational bases E A = {| i (cid:105) A } and E B = {| j (cid:105) B } , Alice and Bob sharing a bipartite quantumstate ˆ (cid:37) AB , and C r (ˆ (cid:37) A )= C r (ˆ (cid:37) B )=0 , they cannot distil quantum coherence on neither sides using LICC if and only if ˆ (cid:37) AB is notquantum correlated, i.e. C netr (ˆ (cid:37) AB )=0 , with respect to the global bases E AB = E A ⊗ E B .3 Proof.
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