Quantum steering with positive operator valued measures
QQuantum steering with positive operator valued measures
H. Chau Nguyen,
1, 2, ∗ Antony Milne,
3, 4
Thanh Vu, and Sania Jevtic Max-Planck-Institut f¨ur Physik komplexer Systeme,N¨othnitzer Straße 38, D-01187 Dresden, Germany Naturwissenschaftlich-Technische Fakult¨at, Universit¨at Siegen,Walter-Flex-Straße 3, D-57068 Siegen, Germany Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom Department of Computing, Goldsmiths, University of London,New Cross, London SE14 6NW, United Kingdom Hanoi University of Science and Technology, 1 Dai Co Viet, Hai Ba Trung, Hanoi, Vietnam Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom (Dated: May 12, 2019)We address the problem of quantum nonlocality with positive operator valued measures (POVM)in the context of Einstein-Podolsky-Rosen quantum steering. We show that, given a candidatefor local hidden state (LHS) ensemble, the problem of determining the steerability of a bipartitequantum state of finite dimension with POVMs can be formulated as a nesting problem of twoconvex objects. One consequence of this is the strengthening of the theorem that justifies choosingthe LHS ensemble based on symmetry of the bipartite state. As a more practical application, westudy the classic problem of the steerability of two-qubit Werner states with POVMs. We showstrong numerical evidence that these states are unsteerable with POVMs up to a mixing probabilityof within an accuracy of 10 − . Introduction.
Ever since its first examination by Ein-stein, Podolsky, and Rosen (EPR) in 1935 [1], quantumnonlocality has been a puzzling phenomenon. In the EPRthought experiment, one observer, Alice, can perform ameasurement on her half of an entangled pair to steerthe other half (that belongs to a distant observer, Bob)to ensembles that conflict with the very intuition of clas-sical locality [2]. This conflict was so profound that itprompted EPR to conclude that quantum theory was“incomplete” [1] and caused the longest debate in the his-tory of quantum mechanics [3]. A more “complete” the-ory would be supplemented by hidden variables; however,the seminal work of Bell [4] demonstrated that no suchtheory, when constrained by locality, is capable of ex-plaining all quantum mechanical predictions for bipartitesystems. Nowadays, quantum nonlocality is perceived asone of the hallmarks of quantum theory that sets it apartfrom classical notions and underlies numerous quantuminformation applications [5].Bell’s work defined the first class of quantum nonlo-cality, now known as Bell nonlocality [5]. Some 25 yearslater, Werner realised that Bell nonlocality and entan-glement (nonseparability) were in fact two independentforms of quantum nonlocality [6]. In 2007, Wiseman,Jones and Doherty [7] recognised that the original ideaof the EPR thought experiment is actually best capturedby yet another form of quantum nonlocality – quantumsteerability. Since then, quantum steerability has beensuccessfully demonstrated experimentally in loophole-free tests [8–10]. It has been employed in a range ofpractical quantum information tasks, including quantum ∗ [email protected] cryptography [11], randomness certification [12, 13], andself-testing [14, 15].Among this surge of discoveries, a fundamental ques-tion remains: which bipartite states manifest quantumsteerability? In fact, determining the steerability of a bi-partite state when considering all possible measurements,i.e., positive operator valued measure (POVM) measure-ments, has been such a challenging task that it is unan-swered for even the simplest case of two-qubit Wernerstates [16, Problem 39]. The problem remains open inspite of many significant advances towards understandingquantum steering under particular subsets of POVMs,e.g., with projection valued measure (PVM) measure-ments [7, 17, 18], with finite subsets of POVMs [19, 20],and with highly noisy POVMs or highly noisy states [21].In this Letter, we are concerned with the problem ofquantum steering with POVMs for bipartite systems ofarbitrary (but finite) dimension. We demonstrate thatfor a given choice of local hidden state ensemble, the taskof determining whether a quantum state is steerable canbe considered as a nesting problem of two convex objects.As a consequence, we derive an inequality which allowsa test of steerability for all measurements. Surprisingly,the inequality also reveals a fundamental aspect of quan-tum steering. Namely, in quantum steering, the choiceof local hidden variable is no longer arbitrary as in Bellnonlocality, but can be limited to the set of Bob’s purestates. This in fact makes the study of quantum steer-ing significantly simpler than its partner Bell nonlocality.In particular, one can strengthen the theorem (Lemma 1of Ref. [7]) which limits the choice of local hidden stateensemble based on the symmetry of the state. As thefirst application, we then apply the inequality to studythe steerability of the two-qubit Werner states. Contraryto the fact that general POVMs provide an advantage a r X i v : . [ qu a n t - ph ] J a n over PVMs in many situations [22–24], we provide strongnumerical evidence that POVMs and PVMs are in factequivalent for steering two-qubit Werner states. Quantum steerability.
Suppose Alice and Bob sharea bipartite quantum state ρ over the finite-dimensionalHilbert space H A ⊗ H B . We use A H (or B H ) to de-note the space of Hermitian operators over H A (or H B ).A POVM measurement with n outcomes ( n -POVM) E implemented by Alice is an (ordered) collection of n pos-itive operators, E = { E i } ni =1 with E i ∈ A H , E i ≥ (cid:80) ni =1 E i = I A , where I A is the identity operator on A H . On performing the measurement, Alice steers Bob’ssystem to the steering ensemble { Tr A [ ρ ( E i ⊗ I B )] } ni =1 .However, despite the arbitrary choice of measurements,for certain bipartite states, the steering experiment canbe locally simulated. More specifically, let u be an en-semble (that is, a probability distribution) on the set ofBob’s pure states, denoted by S B . A state ρ is thencalled u - unsteerable (always considered from Alice’s side)with respect to n -POVMs if, for any n -POVM E , Alicecan find n response functions G i (with G i ( P ) ≥ (cid:80) ni =1 G i ( P ) = 1 for all P ∈ S B ) such that the steer-ing ensemble can be simulated via a local hidden statemodel [7],Tr A [ ρ ( E i ⊗ I B )] = (cid:90) d ω ( P ) u ( P ) G i ( P ) P, (1)where the integral is taken over the Haar measure ω onBob’s pure states S B . Equation (1) ensures that Bob,when performing state tomography conditioned on Al-ice’s outcomes, obtains the same result as if Alice weresteering his system [7]. The ensemble u is called a localhidden state (LHS) ensemble. In principle, the domainof the ensemble u can be extended to mixed states. How-ever, as a mixed state can be written as a convex combi-nation of pure states, restricting the domain of u to purestates causes no loss of generality. We say then that ρ isunsteerable with n -POVMs if there exists u such that ρ is u -unsteerable with n -POVMs.We note that this definition of quantum steering isslightly different from the original definition [7]. In thelatter, the LHS ensemble is indexed by a local hiddenvariable. We will prove the two definitions are equivalentas parts of our results. Our seemingly minor simplifica-tion in fact has very important consequences, which willbe discussed below. The set of n -POVMs and its geometry. The key ideain our approach is that an n -POVM E can be thought ofas a point in the real vector space of composite operators( A H ) ⊕ n = ⊕ ni =1 A H . We therefore write E = ⊕ ni =1 E i ,which explicitly indicates that it is a composite oper-ator in ( A H ) ⊕ n with components E i each bounded by0 ≤ E i ≤ I A . The space ( A H ) ⊕ n can be made Euclideanby defining an inner product (cid:104) X, Y (cid:105) = (cid:80) ni =1 (cid:104) X i , Y i (cid:105) forany composite operators X and Y , where (cid:104) X i , Y i (cid:105) denotesthe Hilbert–Schmidt inner product of A H , (cid:104) X i , Y i (cid:105) =Tr( X † i Y i ). The set of n -POVMs is then a convex andcompact subset of this space [25], which we denote by M n . Since (cid:80) ni =1 E i = I A , M n in fact belongs to thelinear manifold P n = { X | (cid:80) ni =1 X i = I A } . FIG. 1. (Colour online) The similarity between a probabilitysimplex and the set of 3-POVMs.
While the set of POVMs M n is perhaps unfamiliar,it is similar to the classical probability simplexes. Fig-ure 1 illustrates this similarity when n = 3. The basis toconstruct a probability simplex is the probability range[0 , R , one aligns3 probability ranges [0 ,
1] along the 3 axes and forms thetriangle with vertices (1 , , , , , , , ≤ X ≤ I A , which for qubitsforms a double cone in A H (illustrated in Figure 1) [26].One then ‘aligns’ these 3 sets 0 ≤ X ≤ I A along thethree orthogonal component spaces of ( A H ) ⊕ . The setof 3-POVMs is formed in between the points ( I A , , , I A , , , I A ). This analogy between a probabil-ity simplex and the set of n -POVMs applies in the sameway for any n . There is, however, a crucial difference be-tween classical probability simplexes and sets of POVMs:while [0 ,
1] is 1-dimensional with 2 extreme points 0 and1, the set 0 ≤ X ≤ I A is generally high-dimensional andwith more extreme points other than 0 and I A . As a re-sult, the set of n -POVMs is also of high dimension andcarries other extreme points apart from the special onesat the ‘corners’, which are of the form ⊕ ni =1 δ ik I A with k = 1 , , . . . n . The steering assemblage of n -POVMs. Now eachPOVM measurement E performed on Alice’s sidegives rise to a steering ensemble on Bob’s side, ⊕ ni =1 Tr A [ ρ ( E i ⊗ I B )]. This is most easily implementedby the concept of a steering function ρ A → B : A H → B H ,which maps X ∈ A H to Tr A [ ρ ( X ⊗ I B )] ∈ B H [26].This induces the map ( ρ A → B ) ⊕ n : ( A H ) ⊕ n → ( B H ) ⊕ n .For X being an element or a subset of A H , we denote X (cid:48) = ρ A → B ( X ). The same notation is used for compos-ite vectors, namely, for X being an element or a subsetof ( A H ) ⊕ n , X (cid:48) = ( ρ A → B ) ⊕ n ( X ).Geometrically, the map ( ρ A → B ) ⊕ n maps a point inthe set of POVMs M n to a point in ( B H ) ⊕ n . The set( M n ) (cid:48) = ( ρ A → B ) ⊕ n ( M n ) is called the steering assem-blage of n -POVMs . Being a linear image of M n , which isconvex and compact [25], ( M n ) (cid:48) is also convex and com-pact. Moreover, since M n belongs to P n , ( M n ) (cid:48) belongsto ( P n ) (cid:48) . The capacity of an ensemble of Bob’s pure states.
Foran ensemble u of Bob’s pure states S B , the n - capacity K n ( u ) is the set of n -component ensembles that it cansimulate. That is to say, the capacity K n ( u ) is a subset of( B H ) ⊕ n consisting of composite operators K = ⊕ ni =1 K i ,each component being given by K i = (cid:90) d ω ( P ) u ( P ) G i ( P ) P, (2)with all possible choices of response functions G i thatsatisfy G i ( P ) ≥ (cid:80) ni =1 G i ( P ) = 1. It is easy toshow that the n -capacity K n ( u ) is also a convex com-pact set, which has n special extreme points of the form ⊕ ni =1 δ ik (cid:82) d ω ( P ) u ( P ) P with k = 1 , , . . . , n . Steerability as a nesting problem.
With the above def-initions, the following lemma is obvious.
Lemma 1.
A state ρ is u -unsteerable with n -POVMs ifand only if ( M n ) (cid:48) ⊆ K n ( u ) . We first consider the special extreme points of M n . Itis easy to show that for their steering images to be in K n ( u ), one has (cid:90) d ω ( P ) u ( P ) P = I (cid:48) A , (3)which is referred to as the minimal requirement for u [27].Once reformulated in terms of a nesting problem ofconvex objects (Lemma 1), one can apply nesting criteriato test steerability. The following lemma is such a nestingcriterion based on a duality representation. Lemma 2 (Nesting criterion by duality) . Let X be aconvex compact subset of a finite-dimensional Euclideanspace. Then a compact subset Y is contained in X if andonly if max X ∈ X (cid:104) Z, X (cid:105) ≥ max Y ∈ Y (cid:104) Z, Y (cid:105) for all vectors Z in the space. The idea behind this lemma is that if X contains Y ,then its projection onto any direction contains that of Y and vice versa (see Figure 2). A full proof is given inAppendix B. Y X
FIG. 2. (Colour online) Nesting by duality.
To apply this lemma with X = K n ( u ), Y =( M n ) (cid:48) , we need to solve two maximisation problems:max K ∈ K n ( u ) (cid:104) Z, K (cid:105) and max E ∈ M n (cid:104) Z, E (cid:48) (cid:105) for a givencomposite operator Z in ( B H ) ⊕ n . While the latter is asemidefinite program, the former is a linear constrained maximisation, which can be solved explicitly (see Ap-pendix C for the details):max K ∈ K n ( u ) (cid:104) Z, K (cid:105) = (cid:90) d ω ( P ) u ( P ) max i (cid:104) Z i , P (cid:105) . (4)From Lemma 1 and Lemma 2, the following theoremimmediately follows. Theorem 1.
A bipartite state ρ is u -unsteerable if andonly if (cid:90) d ω ( P ) u ( P ) max i (cid:104) Z i , P (cid:105) ≥ max E ∈ M n n (cid:88) i =1 (cid:104) Z i , E (cid:48) i (cid:105) (5) for all composite operators Z = ⊕ ni =1 Z i in ( B H ) ⊕ n . Inequality (5) is the main result in this Letter: it isvalid for systems of arbitrary dimension and POVMs ofarbitrary number of outcomes. We now discuss some ofits important consequences; details of proofs and furtherdiscussions are given in Appendix D.Had one started with the original definition of quan-tum steering with an “indexed LHS ensemble”, that is,an LHS ensemble indexed by some hidden variable, onewould arrive at a similar inequality as (5). In that case,the integration is taken over the hidden variable instead(see Appendix D). However, one can rewrite it as an in-tegral over the push-forward measure over Bob’s purestates S B [28]. This implies that it is only the push-forward measure on S B that really determines the capac-ity of an LHS ensemble. Two indexed LHS ensemblesgenerating the same measure on Bob’s pure states wouldhave the same capacity. In other words, our definitionof quantum steering where the local hidden variable isomitted is equivalent to the original definition of quan-tum steering (Corollary 1, Appendix D). Having elim-inated the arbitrary choice of local hidden variable inquantum steering, the symmetry of the state directly hasa stronger implication on the symmetry of LHS ensem-bles (see Theorem 2). In fact, it is this stronger implica-tion of symmetry that actually renders many unsolvableproblems in Bell nonlocality solvable in the context ofquantum steering [7].More specifically, the state ρ is said to have ( G , U, V )-symmetry with G being a group and U and V being itstwo representations on H A and H B , respectively, if ρ = U † ( g ) ⊗ V † ( g ) ρU ( g ) ⊗ V ( g ) for all g ∈ G . The action V of G on Bob’s pure states S B generates an action R V on thespace of distributions on S B defined by [ R V ( g ) u ]( P ) = u [ V † ( g ) P V ( g )]. We then have a strengthened form ofLemma 1 of Ref. [7] on the symmetry of LHS ensemble. Theorem 2 (Symmetry of LHS ensembles) . For a givenstate ρ which is ( G , U, V ) -symmetric with a compactgroup G , if ρ is unsteerable with n -POVMs then it ad-mits an LHS ensemble u ∗ which is ( G , R V ) -invariant,i.e., u ∗ = R V ( g ) u ∗ for all g in G . This theorem is applicable as well when measurementsare restricted to PVMs. To understand the differencewith Lemma 1 of Ref. [7], we consider the example where G acts transitively on Bob’s pure states S B (i.e., a singleorbit covers all of S B ). If ρ is unsteerable, Lemma 1 ofRef. [7] then states the existence of an indexed LHS en-semble, on which G acts covariantly on the indices andthe states. However, due to the arbitrariness in choiceof the local hidden variable, there exist in fact infinitelymany different G -covariant indexed LHS ensembles (seeAppendix D). One then could not single out an uniquechoice of LHS ensemble. On the other hand, under thesame conditions, Theorem 2 implies that the state is un-steerable with the unique uniform distribution on Bob’spure states as an LHS ensemble.Beyond revealing very general aspects of quantumsteering, Theorem 1 can also be used to test steerabilityin practice. The most difficult part is to determine theexistence of a LHS ensemble u . Even when measurementsare limited to PVMs, the question is so far solved only forhighly symmetric states, e.g., the Werner state [7] and thetwo-qubit T -state (mixtures of Bell states) [17, 18]. Theimplication of our approach on this problem will be dis-cussed elsewhere. However, as we mentioned, even when u is known, the problem of determining steerability withPOVMs is still open [16, Problem 39]. It is this latterproblem that we are concerned with in the following. Wewill show that Theorem 1 can provide a strong numericalevidence for steering with POVMs with a given identifiedcandidate for the LHS ensemble. The gap function.
We first note that for a LHS en-semble u satisfying the minimal requirement (3), the in-equality (5) is invariant with respect to the transforma-tion ⊕ ni =1 Z i → √ D ⊕ ni =1 ( Z i − C ), where C = n (cid:80) ni =1 Z i and D = (cid:80) ni =1 (cid:104) Z i − C, Z i − C (cid:105) . We can therefore re-strict Z to the set of those satisfying (cid:80) ni =1 Z i = 0 and (cid:80) ni =1 (cid:104) Z i , Z i (cid:105) = 1, denoted by C n . For clarity, we intro-duce the gap function ∆[( M n ) (cid:48) , K n ( u )], defined to bemin Z ∈ C n (cid:40)(cid:90) d ω ( P ) u ( P ) max i (cid:104) Z i , P (cid:105) − max E ∈ M n n (cid:88) i =1 (cid:104) Z i , E (cid:48) i (cid:105) (cid:41) . (6)The gap function characterises the gap between theboundary of ( M n ) (cid:48) and that of K n ( u ) from inside. Thestate ρ is u -unsteerable with n -POVMs if and only if∆[( M n ) (cid:48) , K n ( u )] ≥ Restriction to rank- POVMs.
As all POVMs can bepost-processed from those of rank-1, to test quantumsteerability we can concentrate on the latter [29]. To thisend, we define (cid:101) M n = {⊕ ni =1 α i P i } where P i are (not nec-essarily independent) rank-1 projections and 0 ≤ α i ≤ (cid:80) ni =1 α i P i = I A . To test the steerability with M n , we therefore only need to calculate ∆[( (cid:101) M n ) (cid:48) , K n ( u )]. Steerability of two-qubit Werner states with POVMs.
Consider the two-qubit Werner state, W p = p (cid:12)(cid:12) ψ − (cid:11) (cid:10) ψ − (cid:12)(cid:12) + (1 − p ) I A ⊗ I B , (7) .
490 0 .
495 0 .
500 0 .
505 0 . p − . − . . . . ∆ with rank-1 4-POVMswith PVMswith PVMs (analytic) FIG. 3. (Colour online) The gap function for the two-qubitWerner state with mixing probability around for rank-14-POVMs and for PVMs. which is a mixing between the singlet Bell state | ψ − (cid:105) = √ ( | (cid:105) − | (cid:105) ) and the maximally mixed state with mix-ing parameter p (0 ≤ p ≤ p ≤ [7]. By adapting Barrett’s model [29]of local hidden variables, it was further possible to showthat Werner states are unsteerable with POVMs when p ≤ [21]. We are to study the conjecture [30]: Conjecture 1.
The Werner state with mixing parameter p ≤ is unsteerable for all n -POVMs. That is to say,POVMs and PVMs are equivalent for steering two-qubitWerner states. Although further analyses restricted to finite subsets ofPOVMs [19, 20] or POVMs with special symmetry [30]support unsteerability of the Werner state for ≤ p ≤ [30, 31], there has not been a concrete evidence whenone considers all POVMs. For n = 1, the conjecture isobvious. For n = 2, it has been proven by demonstratingthe equivalence to steering with PVMs [26]. The prooffor n = 3 is also known [30]. Finally, it is known that itis sufficient to consider the conjecture for n = 4 [30, 32].Here, by computing the corresponding gap function forthe Werner state, we provide strong numerical evidencefor Conjecture 1 for n = 4.It is easy to see that the Werner state has U(2) symme-try as defined above. Moreover, since the action is transi-tive on Bob’s Bloch sphere, by Theorem 2, the candidatefor the LHS ensemble can be limited to the uniform dis-tribution, u = 1. To simplify the notation, from nowon we simply use ∆ to denote the gap function with-out specifying the set of measurements and the capacity,which can be understood from the context. The com-putation of the gap function ∆ generally requires globalminimisation over Z , which is carried out by the stan-dard (non-deterministic) simulated annealing algorithm(see Appendix E for details).In Figure 3, we present values of ∆ found for the mixingprobability p ≈ . For comparison, we also present thenumerical results of ∆ when the measurements are lim-ited to PVMs, which are in very good agreement with theanalytical calculation (see Appendix F). For p > , oneobserves that the gap function ∆ for rank-1 4-POVMs isnegative and coincides with the gap function for PVMs.For all p ≤ − − , the best obtained values for ∆ areless than 10 − but persistently non-negative. The factthat the gap function tends to vanish for p ≤ instead ofattending finite positive values seems to be because of thehigh dimensionality of POVMs [33] The presented resultstherefore support conjecture 1. Unfortunately, withinthis work, the ambiguity region of − − ≤ p ≤ cannot be resolved due to limits of numerical accuracy(see Appendix E). Conclusion.
Our work simplifies the definition of quan-tum steering, where we show that the local hidden vari-able indexing LHS ensemble can be omitted. As a directconsequence, a stronger theorem on the symmetry of theLHS ensemble is derived. We have thereby opened ageneral approach to studying quantum steerability withPOVMs. Further works to strengthen the numerical evi-dence for the unsteerability of the Werner state at p = and testing steerability with POVMs of arbitrary two-qubit states are underway. Moreover, although the cur-rent illustrative applications are based on two qubits, ourapproach is not limited by the dimensionality of the sys-tems. It is hoped that systematic tests for steering withPOVMs (particularly for high dimensional systems) willgive a complete answer to the fundamental question ofthe equivalence between POVMs and PVMs for steer-ability. Beyond quantum steering, we leave open thequestion of whether this approach can be extended tocharacterise Bell nonlocality with POVMs. ACKNOWLEDGMENTS
We thank Jessica Bavaresco, Johannes Berg, Anna C.S. Costa, Otfried G¨uhne, Marcus Huber, Michael Jarret,Brad Lackey, X. Thanh Le, N. Duc Le, H. Viet Nguyen,Matthew Nicol, Ana Bel´en Sainz, Roope Uola, ReinhardWerner, and Howard Wiseman for useful discussions ongeneral and technical aspects of this work. SJ is sup-ported by an Imperial College London Junior ResearchFellowship. CN acknowledges the support from ErwinSchr¨odinger International Institute for Mathematics andPhysics (ESI) during his participation in the programQuantum Physics and Gravity.
Appendix A: The compactness of the capacity
In this appendix, we prove the compactness of the ca-pacity K n ( u ). On the Bloch sphere S B , the distribu- tion u defines a measure µ . We assume that the re-sponse functions G i ( P ) are squared-integrable with re-spect to µ . Consider the space L ( S B , µ ) of squared-integrable functions on S B with respect to µ , which is aHilbert space (as usual, we ignore the difference on a zero-measured set) [34, Chapter V]. We then construct theHilbert space [ L ( S B , µ )] ⊕ n as usual. We define the sub-set of Ω = { G = ⊕ ni =1 G i ∈ [ L ( S B , µ )] ⊕ n : 0 ≤ G i ( P ) ≤ , (cid:80) ni =1 G i ( P ) = 1 } , which is closed and convex, thusweakly closed [35, Chapter V, Corollary 1.5]. Moreover,it is obviously bounded, thus weakly compact [35, Chap-ter V, Theorem 4.2].Now consider the linear operator T : [ L ( S B , µ )] ⊕ n → ( B H ) ⊕ n , defined by T ( G ) = ⊕ ni =1 (cid:82) d µ ( P ) G i ( P ) P . It isobvious that T is bounded, thus continuous and weaklycontinuous [35, Chapter VI, Theorem 1.1]. It then followsdirectly that K n ( u ) = T (Ω) is compact. Appendix B: Nesting criterion by duality
In this Appendix, we provide the proof for the nestingcriterion by duality.
Lemma 2 (Nesting criterion by duality) . Let X be aconvex compact subset of a finite-dimensional Euclideanspace. Then a compact subset Y is contained in X if andonly if max X ∈ X (cid:104) Z, X (cid:105) ≥ max Y ∈ Y (cid:104) Z, Y (cid:105) for all vectors Z in the space.Proof. It is obvious that if Y ⊆ X then max X ∈ X (cid:104) Z, X (cid:105) ≥ max Y ∈ Y (cid:104) Z, Y (cid:105) for all Z . Now suppose max X ∈ X (cid:104) Z, X (cid:105) ≥ max Y ∈ Y (cid:104) Z, Y (cid:105) for all Z and Y (cid:54)⊆ X . Because Y (cid:54)⊆ X ,there exists A ∈ Y , A (cid:54)∈ X . Since X is a convex andcompact set, by the separation theorem, A is separatedfrom X by a hyperplane, i.e., there exists a vector Z such that (cid:104) Z, A (cid:105) > max X ∈ X (cid:104) Z, X (cid:105) [36]. It follows thatmax Y ∈ Y (cid:104) Z, Y (cid:105) ≥ (cid:104)
Z, A (cid:105) > max X ∈ X (cid:104) Z, X (cid:105) , contradict-ing the assumption.
Appendix C: Solving the first optimisation problem
Here we provide details of the solution to the con-strained maximisation problem (4). Using the definitionof K n ( u ), we havemax K ∈ K n ( u ) (cid:104) Z, K (cid:105) = max G (cid:90) d ω ( P ) u ( P ) n (cid:88) i =1 G i ( P ) (cid:104) Z i , P (cid:105) , (C1)subject to the constraints G i ( P ) ≥ (cid:80) ni =1 G i ( P ) =1. This is a linear maximisation problem with linear con-straints, which can be solved easily by Lagrange’s mul-tipliers. For every constraint (cid:80) ni =1 G i ( P ) = 1 for each P ∈ S A , we introduce a Lagrange’s multiplier λ ( P ). Thisleads us to a modified unconstrained maximisation prob-lem I [ λ ( P )] = max G (cid:40)(cid:90) d ω ( P ) u ( P ) n (cid:88) i =1 G i ( P ) (cid:104) Z i , P (cid:105)− (cid:90) d ω ( P ) λ ( P ) (cid:34) n (cid:88) i =1 G i ( P ) − (cid:35)(cid:41) = max G (cid:40)(cid:90) d ω ( P ) n (cid:88) i =1 G i ( P ) [ u ( P ) (cid:104) Z i , P (cid:105) − λ ( P )] (cid:41) + (cid:90) d ω ( P ) λ ( P ) . (C2)Note that the function under maximisation is a linearfunction of G i ( P ), which is bounded by 0 ≤ G i ( P ) ≤ I [ λ ( P )] is saturated by G ∗ i ( P ) = Θ[ u ( P ) (cid:104) Z i , P (cid:105) − λ ( P )] , (C3)where Θ is the Heaviside step function, Θ( x ) = 1 if x ≥ x ) = 0 otherwise. One now needs to choose λ ( P )such that the constraint is satisfied, n (cid:88) i =1 Θ[ u ( P ) (cid:104) Z i , P (cid:105) − λ ( P )] = 1 . (C4)Consider some fixed P . The last equation means that λ ( P ) must be such that out of { u ( P ) (cid:104) Z i , P (cid:105)} ni =1 , onlyone is larger than or equal to λ ( P ). In other words, thesuitable choice for λ ( P ) is λ ( P ) = u ( P ) max i (cid:104) Z i , P (cid:105) . (C5)With this solution for λ ( P ), substituting (C3) to (C1)one then obtains (4).So far we actually ignored the case where for some P ,max i (cid:104) Z i , P (cid:105) is attained by two indices, say, i = i and i = i . In this case, one then has to slightly modify (C3):at such a point P , while G ∗ i ( P ) = 0 for i (cid:54) = i , i , G ∗ i ( P )and G ∗ i ( P ) can take arbitrary values between 0 and 1,provided that G ∗ i ( P )+ G ∗ i ( P ) = 1. Similar modificationis needed if max i (cid:104) Z i , P (cid:105) is attained by more indices. Themaximal value (4) however remains the same. Appendix D: Corollaries of Theorem 11. The equivalence between our definition and theoriginal definition of steering
The original definition of quantum steering [7] goes asfollows. Let (Λ , ν ) be a probability measure space (weignore the symbol which denotes the σ -algebra for themeasure ν ). Let F : Λ → S B be a measurable functionfrom the index space (Λ , ν ) to the set of Bob’s pure states S B . A state ρ is then called (Λ , ν )- unsteerable (from Al-ice’s side) with respect to n -POVMs if, for any n -POVM E , Alice can find n response functions G i (with G i ( λ ) ≥ (cid:80) ni =1 G i ( λ ) = 1 for all λ ∈ Λ) such that the steer-ing ensemble can be simulated via a local hidden statemodel [7],Tr A [ ρ ( E i ⊗ I B )] = (cid:90) Λ d ν ( λ ) G i ( λ ) F ( λ ) , (D1)where the integral is taken over the index space Λ. Inthis case, we say ρ admits an indexed LHS model .If ρ satisfies the definition of steering in the main text,it is cleared that it admits an indexed LHS model, wherethe index space is Bob’s pure states themselves. On theother hand, if ρ admits an indexed LHS model, it is un-clear that the existence of a response function on Bob’spure states is guaranteed. This is particularly importantif the indexing function F is a many-to-one function. Oneof the strengths of our approach is that it provides a proofthat a response function on Bob’s pure states does exist.We know of no other (constructive) proof at the moment. Corollary 1.
Our definition of quantum steering isequivalent to the conventional definition of quantumsteering where the LHS ensemble is indexed by a hiddenvariable.Proof.
Suppose ρ admits an indexed LHS ensemble,which is indexed by (Λ , ν ). Then following the argumentthat leads to Theorem 1 in the main text, we arrive atthe following statement: Lemma.
A bipartite state ρ is (Λ , ν ) -unsteerable if andonly if (cid:90) Λ d ν ( λ ) max i (cid:104) Z i , F ( λ ) (cid:105) ≥ max E ∈ M n n (cid:88) i =1 (cid:104) Z i , E (cid:48) i (cid:105) (D2) for all composite operators Z = ⊕ ni Z i in ( B H ) ⊕ n . Now denote by µ the push-forward measure generatedby F on the set of Bob’s pure states [28]. Changing thevariable in the integral, one has (cid:90) Λ d ν ( λ ) max i (cid:104) Z i , F ( λ ) (cid:105) = (cid:90) d µ ( P ) max i (cid:104) Z i , P (cid:105) , (D3)where the latter integral is taken over Bob’s pure states.Let u be the distribution on Bob’s pure states generatedby µ with respect the Haar measure ω . This means theinequality (5) in the main text is satisfied for distribution u . According to Theorem 1, ρ is u -unsteerable.
2. The symmetry of LHS ensembles
For a given state ρ , we denote by Ω n ( ρ ) the set ofensembles u over Bob’s pure states such that ρ is u -unsteerable with n -POVMs (Ω n ( ρ ) is empty if ρ is steer-able). From inequality (5), it is easy to see that Ω n ( ρ ) isconvex (Corollary 2). Moreover, we show that the sym-metry of ρ implies the symmetry of Ω n ( ρ ). Corollary 2.
For a given state ρ , Ω n ( ρ ) is convex.Proof. Suppose u and u are in Ω n ( ρ ). That is to say, u and u satisfy inequality (5). It is then easy to check thatinequality (5) is also satisfied for all convex combinationsof u and u . In other words, all convex combinations of u and u are in Ω n ( ρ ).Let us recall from the main text that the state ρ issaid to have ( G , U, V )-symmetry with G being a groupand U and V being its two representations on H A and H B , respectively, if ρ = U † ( g ) ⊗ V † ( g ) ρU ( g ) ⊗ V ( g ) forall g ∈ G . The action V of G on Bob’s pure states S B generates an action R V on the space of distributions on S B defined by [ R V ( g ) u ]( P ) = u [ V † ( g ) P V ( g )]. Corollary 3.
For a given state ρ which is ( G , U, V ) -symmetric, then Ω n ( ρ ) is ( G , V ) -symmetric, i.e., Ω n ( ρ ) = R V ( g )Ω n ( ρ ) for all g ∈ G .Proof. We need to show that if ρ is u -unsteerable for some u , that is, if inequality (5) holds for u , then it also holdsfor R V ( g ) u . Due to the symmetry of ρ , we havemax E ∈ M n n (cid:88) i =1 Tr[ ρ ( E i ⊗ Z i )] = max E ∈ M n n (cid:88) i =1 Tr[ U † ( g ) ⊗ V † ( g ) ρU ( g ) ⊗ V ( g )( E i ⊗ Z i )]= max E ∈ M n n (cid:88) i =1 Tr[ ρ U ( g ) E i U † ( g ) ⊗ V ( g ) Z i V † ( g )]= max E ∈ M n n (cid:88) i =1 Tr[ ρ E i ⊗ V ( g ) Z i V † ( g )] , (D4)where the last equality is because M n is symmetric underthe action U of G . Then inequality (5) is equivalent to (cid:90) d ω ( P ) u ( P ) max i (cid:104) Z i , P (cid:105) ≥ max E ∈ M n n (cid:88) i =1 (cid:10) V ( g ) Z i V † ( g ) , E (cid:48) i (cid:11) . (D5) That this inequality holds for all Z is then equivalent to (cid:90) d ω ( P ) u ( P ) max i (cid:10) V † ( g ) Z i V ( g ) , P (cid:11) ≥ max E ∈ M n n (cid:88) i =1 (cid:104) Z i , E (cid:48) i (cid:105) (D6)for all Z . Now we manipulate the left-hand side, (cid:90) d ω ( P ) u ( P ) max i (cid:10) V † ( g ) Z i V ( g ) , P (cid:11) = (cid:90) d ω ( P ) u ( P ) max i (cid:10) Z i , V ( g ) P V † ( g ) (cid:11) = (cid:90) d ω ( P ) u [ V † ( g ) P V ( g )] max i (cid:104) Z i , P (cid:105) , (D7)where the last inequality is a change of integration vari-able. By definition, R V ( g ) u ( P ) = u ( V † ( g ) P V ( g )).Therefore, inequality (5) indeed holds for R V ( g ) u . Theorem 2 (Symmetry of LHS ensemble) . For a givenstate ρ which is ( G , U, V ) -symmetric with a compactgroup G , if ρ is unsteerable with n -POVMs then it ad-mits an LHS ensemble u ∗ which is ( G , R V ) -invariant,i.e., u ∗ = R V ( g ) u ∗ for all g in G .Proof 1. Since ρ is unsteerable, there exists u ∈ Ω n ( ρ ).By Corollary 3, R V ( g ) u ∈ Ω n ( ρ ) for all g ∈ G . SinceΩ n ( ρ ) is convex (Corollary 2), the average over the Haarmeasure of G , i.e., u ∗ = (cid:82) G d µ ( g ) R V ( g ) u ∈ Ω n ( ρ ), alsobelongs to Ω n ( ρ ). This averaged distribution u ∗ is obvi-ously invariant under the action of G , i.e., u ∗ = R V ( g ) u ∗ for all g in G .For completeness, we also provide an alternative proofof this theorem without the use of inequality (5). Thisproof is more similar to the original proof in Ref. [7]; toget the stronger statement, one has to apply the so-calledmean value theorem for integrals [28], though. Proof 2.
Suppose ρ is u -unsteerable with n -POVMs, thenfor a POVM E , there exists response function G such that E (cid:48) i = (cid:90) d ω ( P ) u ( P ) G Ei ( P ) P. (D8)Here, to track the dependence of the response function G on the measurement, we introduce the superscript E for G . Now due to the symmetry of the state, we also have E (cid:48) i = (cid:90) d ω ( P ) R V ( g ) u ( P ) G U † ( g ) EU ( g ) i [ V † ( g ) P V ( g )] P. (D9)Since the right-hand-side is independent of g , we can takethe average over g with respect to the Haar measure µ of G , E (cid:48) i = (cid:90) d ω ( P ) P (cid:90) dµ ( g ) R V ( g ) u ( P ) G U † ( g ) EU ( g ) i [ V † ( g ) P V ( g )] . (D10)According to the mean value theorem [28], there exists afunction ¯ G i ( P ) with 0 ≤ ¯ G i ( P ) ≤ (cid:90) d µ ( g ) R V ( g ) u ( P ) G U † ( g ) EU ( g ) i [ V † ( g ) P V ( g )] =¯ G i ( P ) (cid:90) d µ ( g ) u [ V † ( g ) P V ( g )] . (D11)Let u ∗ ( P ) = (cid:82) d µ ( g ) u [ V † ( g ) P V ( g )], which is obviously R V − covariant. Then E (cid:48) i = (cid:90) d ω ( P ) u ∗ ( P ) ¯ G i ( P ) . (D12)To see that ¯ G satisfies the normalisation, we sum (D11)over i : (cid:34) n (cid:88) i =1 ¯ G i ( P ) (cid:35) u ∗ ( P ) = u ∗ ( P ) , (D13)which means (cid:80) ni =1 ¯ G i ( P ) = 1 almost everywhere withmeasure generated by u ∗ ( P ). Therefore ¯ G is a properresponse function for measurement E with LHS ensemble u ∗ . Thus ρ is also u ∗ -unsteerable.One can easily check that both proofs work equally wellwhen the measurements are restricted to PVMs. To bet-ter understand the relation of Theorem 2 with Lemma 1of Ref. [7], we come back to the example of the two-qubitWerner state W p in the main text. According to Lemma1 of Ref. [7], if W p is unsteerable, there exists a covariantindexed LHS ensemble for which W p is unsteerable, andthen it is deduced that this singles out the uniform dis-tribution over the hidden variables as the “optimal” LHSensemble u ∗ . In fact, there exist infinitely many indexedensembles that are covariant under the U(2) action. Forexample, consider and index space Λ = Z K × S B with Z K = { , , , ..., K − } , i.e., we have a composite hid-den variable λ = ( α, P ) with α = 0 , , , ..., K − P ∈ S B . The measure ν on Λ is generated by the distri-bution u on Λ, defined by u ( α, P ) = c α independent of P with c α ≥ (cid:80) K − α =0 c α = 1.The action of g ∈ U(2) on Λ can be defined as gλ = g ( α, P ) = ( α, gP g † ). This action is not transitive on Λ;it has K orbits indexed by α . As a result, despite thefact that u is G -invariant, u ( λ ) = u ( gλ ), it is not uniformover Λ if c α are distinct numbers. Now consider the indexed LHS ensemble given by theindexing function F : Λ → S B , ( α, P ) (cid:55)→ F ( α, P ) = P (which is many-to-one). The LHS ensemble is apparentlycovariant, that is, F ( λ ) = g † F ( gλ ) g or even u ( λ ) F ( λ ) = u ( gλ ) g † F ( gλ ) g .Being unsteerable with respect to this indexed LHSensemble implies that for any n -POVM E , there existresponse functions G i ( α, P ) such that E (cid:48) i = K − (cid:88) α =0 π (cid:90) d S ( P ) G i ( α, P ) u ( α, P ) P = K − (cid:88) α =0 π (cid:90) d S ( P ) G i ( α, P ) c α P. (D14)where S is surface measure on Bob’s Bloch sphere. Onthe face of it, this does not imply that one can choose theuniform distribution on Bob’s Bloch sphere to be the LHSensemble. The latter requires that there exists responsefunction ¯ G i ( P ) such that E (cid:48) i = 14 π (cid:90) d S ( P ) ¯ G i ( P ) P. (D15)The existence of ¯ G i ( P ) only follows upon applying themean value theorem as in Proof 2, which states that thereexist ¯ G ( P ) such that¯ G ( P ) = K − (cid:88) α =0 c α G ( α, P ) . (D16)More complicated examples can be easily constructedby replacing Z K with a more complicated measurablespace. Corollary 1 then implies that all these differentconstructions are actually equivalent when one concernswith simulating steering assemblages, since they generatethe same uniform distribution on the Bloch sphere. Thusone sees that Lemma 1 of Ref. [7], when augmented withCorollary 1, can also identify the uniform distribution asthe optimal choice for LHS ensemble as stated directlyin Theorem 2. Appendix E: Simulated annealing and computationof the gap function
Simulated annealing is a standard heuristic algorithmto solve a generic global optimisation problem [37]. Inour case, we wish to compute∆ = min Z ∈ C ,E ∈ N F ( Z, E ) (E1)with F ( Z, E ) = 14 π (cid:90) d S ( P ) max i (cid:104) Z i , P (cid:105) − (cid:88) i =1 Tr[ ρ ( Z i ⊗ E i )] , (E2) − − − − T − − − − − ∆ FIG. 4. Typical trajectories of the energy of the system duringthe cooling procedure. The trajectories of 64 different replicasare plotted in different colours. The data is for a Werner statewith p = 0 . where S is the sureface measure of the Bloch sphere(which is different from the Haar measure by a factor π ).The simulated annealing algorithm goes as follows. Onefirst regards F ( Z, E ) as an energy function of a system inthe state space (
Z, E ). Simulated annealing couples thissystem to an effective heat bath, whose temperature isthen lowered slowly, so that configurations with decreas-ing energy are explored. At each temperature the systemfollows stochastic dynamics leading to equilibrium withthe heat bath. The system is cooled down slowly to suf-ficiently small temperature T f . It is known that if thetemperature schedule is sufficiently slow then the systemconverges to a global minimum of F ( Z, E ) [38]. How-ever, the required cooling schedule is too slow that it isnot useful in practice and an alternative cooling scheduleis used. Here we use an exponential cooling scheme, i.e.,in each step the temperature is cooled down by a factor f . The system can in principle become stuck in a localminimum at T f . It is then necessary to repeat the coolingprocedure multiple times. Coordinisation of variables.
Note that for two qubitsystems, A = B = M ( C , M ( C ,
2) is the alge-bra of 2 × { σ i } i =0 = { I , σ x , σ y , σ z } to coordinate the real subspace M H ( C , X ∈ M H ( C ,
2) is thereforecharacterised by 4 (real) coordinates x i , X = 12 (cid:88) i =0 x i σ i . (E3)The boundary of the positive cone of M H ( C ,
2) is givenby x − x − x − x = 0 with x ≥
0, consisting of vectorsof the form α (cid:18) nnn (cid:19) with α ≥ Z = ⊕ i =1 Z i and E = ⊕ i =1 E i are thought of as 4 × Z i and E i respectively. From now .
490 0 .
495 0 .
500 0 .
505 0 . p − . − . . . . ∆ with rank-1 4-POVMswith PVMswith PVMs (analytic) FIG. 5. The gap function for the Werner state with PVMsand rank-1 4-POVMs from 512 replicas. Note that 512 tri-als strongly concentrate at the minimum values, which givesconfidence to the algorithm. on, we will use Z and E to denote these matrices, and Z i , E i to denote the i th column.To implement the constraint C to Z , we write Z = XR , where R = 12 − − − − − − , (E4)and X satisfies Tr( X T X ) = 2, X = (0 , , , T .To implement the constraint N on E , we note thatevery component E i is on the boundary of the positivecone, E i = α i (cid:18) nnn i (cid:19) , where α i ≥ nnn i is a unit vec-tor. The constraint (cid:80) i =1 E i = I can be considered as aconstraint for { α i } i =1 . In fact, if (cid:18) nnn i (cid:19) are independent, { α i } i =1 are uniquely determined. Note that the set of E where (cid:18) nnn i (cid:19) are dependent are zero-measured in N .In practice, we therefore do not need to worry about thecase that (cid:18) nnn i (cid:19) are dependent if the linear solver is rela-tively stable. Here we use the Householder linear solver,provided by Eigen 3 [39]. Further, we use a commontechnique [37] to take care of the constraint 0 ≤ α i ≤ α i are outside [0 , N by 4 vectors { nnn i } i =1 . Annealing.
At temperature T , the stochastic dynam-ics of the system is simulated by the Metropolis algo-rithm [37]: at every time step, the system tries an el-ementary step, which will be accepted with probabilitymin { , e − ∆ F/T } , where ∆ F is the change in energy due0 . . . . . p − . − . . . ∆ with rank-1 4-POVMswith PVMswith PVMs (analytic) FIG. 6. The gap function for the Werner state with PVMsand ePOVMs at high resolution around the transition point p = . to the trial step. In each elementary trial step, either Z or E is updated with equal probability. If Z is updated, wechoose randomly two elements of X , say X ij , X kl , where j, l <
4, and perform a rotation Q ( θ ) ∈ SO (2) on the vec-tor ( X ij , X kl ) T by a random angle θ normally distributedwith mean 0 and standard deviation 2 π √ T . The compo-nents of the vector Q ( θ )( X ij , X kl ) T replace the ij and kl elements of X . Note that the constraints Tr( X T X ) = 2and X = (0 , , , T are respected in the new X . Then Z is updated as Z = XR . If E is updated, we chooseone of the vectors nnn i randomly, and rotate it around oneof the 3 axes x , y , z by a random angle normally dis-tributed with mean 0 and standard deviation 2 π √ T . Ateach temperature, the number of the simulated steps areat least 100 times the degree of freedom. Cooling schedule.
After annealing the system at tem-perature T , the temperature is decreased by a factor f ;here f = 0 .
95. This is known as exponential temperaturescheduling [37]. The initial temperature T i is chosen asthe maximal value minus the minimal value of the en-ergy function sampled at 1000 times the degree of free-dom points. The algorithm is stopped at temperature T f = 10 − . Lowering the final temperature does notsignificantly improve the results. Replicas.
We repeat the cooling procedure M = 512times. 64 such typical cooling trajectories are presentedin Figure 4. As shown in Figure 5, all 512 replicas pro-duce a very similar minimum energy, suggesting thatthere is no major local minimum in the energy land-scape. This provides confidence that the system indeedconverges close to a global minimum. Numerical accuracy.
As seen in Figure 4, the gap canbe overestimated by some order of 10 − . This makes itdifficult to study very small gap values when p is around ± − . Other cooling schedules [37], parallel temper-ing [40] or more subtle global optimisation techniques [41] can be considered to increase the accuracy. However, thenumerical accuracy is limited by another critical factor:the accuracy of the spherical integral in (E2). Here, weused Lebedev’s quadrature with 5810 points to computespherical integrals. This effectively replaces the optimaluniform LHS ensemble by a suboptimal discrete distri-bution at 5810 quadrature points. Accordingly, the ex-pected transition probability p is shifted by some valueof 2 × − to the left of , smearing out the accuracyof the simulated annealing optimisation as seen in Fig-ure 6. Similar problems occur for T -states at a resolutionof | (cid:15) | ≈ − around the surface of unsteerable states. Appendix F: The gap function for steering theWerner states with PVMs
The numerical calculation of the gap function for theWerner states with PVMs is carried out similarly to Ap-pendix E. The analytical calculation of the gap functionfor steering a Werner state with PVMs is rather straight-forward. We are to calculate∆ = min ( Z ,Z ) ∈ C (cid:26) π (cid:90) d S ( P ) max {(cid:104) Z , P (cid:105) , (cid:104) Z , P (cid:105)}− max ( P ,P ) Tr[ W p ( P ⊗ Z + P ⊗ Z )] (cid:27) , (F1)where ( P , P ) forms a projective measurement, i.e., P , P are orthogonal projections such that P + P = I .Since ( Z , Z ) ∈ C implies that Z + Z = 0, we canset Z = X and Z = − X . Moreover, because of theU(2) symmetry of the problem, we can suppose X = λ | (cid:105) (cid:104) | + λ | (cid:105) (cid:104) | with λ ≥ λ . Because (cid:104) Z , Z (cid:105) + (cid:104) Z , Z (cid:105) = 1, we have (cid:104) X, X (cid:105) = , or λ + λ = .If we write the projections in Pauli coordinates as P = (cid:18) nnn (cid:19) , then (cid:104) Z , P (cid:105) ≥ (cid:104) Z , P (cid:105) is equivalent n z ≥ − ab , with a = λ + λ and b = λ − λ . Therefore14 π (cid:90) d S ( P ) max {(cid:104) Z , P (cid:105) , (cid:104) Z , P (cid:105)} =14 π (cid:90) n z ≥− ab d S ( P ) (cid:104) X, P (cid:105) − (cid:90) n z ≤− ab d S ( P ) (cid:104) X, P (cid:105) , (F2)which evaluates to b .On the other handmax ( P ,P ) Tr[ W p ( P ⊗ Z + P ⊗ Z )] =max P Tr[ W p ( P ⊗ X )] − Tr[ W p ( I ⊗ X )] , (F3)which evaluates to pb .1Thus we have ∆ = min b (cid:110) b − pb (cid:111) . Note that λ + λ = implies that a + b = 1, and so b ≤
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