The resource theory of steering

The resource theory of steering

Rodrigo Gallego Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany    Leandro Aolita Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany Instituto de Física, Universidade Federal do Rio de Janeiro, P. O. Box 68528, Rio de Janeiro, RJ 21941-972, Brazil
Abstract

We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. For arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device, we show that local operations assisted by one-way classical communication (1W-LOCCs) from the quantum part to the black box cannot create steering. Based on this, we build a resource theory of steering with 1W-LOCCs as the free operations. We introduce the notion of convex steering monotones as the fundamental axiomatic quantifiers of steering. As a convenient example thereof, we present the relative entropy of steering. In addition, we prove that two previously proposed quantifiers, the steerable weight and the robustness of steering, are also convex steering monotones. To end up with, for minimal-dimensional systems, we establish, on the one hand, necessary and sufficient conditions for pure-state steering conversions under stochastic 1W-LOCCs and prove, on the other hand, the non-existence of steering bits, i.e., measure-independent maximally steerable states from which all states can be obtained by means of the free operations. Our findings reveal unexpected aspects of steering and lay foundations for further research, with potential implications in Bell non-locality.

I Introduction

Steering, as Schrödinger named it Schrodinger35 (), is an exotic quantum effect by which ensembles of quantum states can be remotely prepared by performing local measurements at a distant lab. It allows Wiseman07 (); Reid09 () to certify the presence of entanglement between a user with an untrusted measurement apparatus, Alice, and another with a trusted quantum-measurement device, Bob. Thus, it constitutes a fundamental notion between quantum entanglement Horodecki09 (), whose certification requires quantum measurements on both sides, and Bell non-locality Brunner13 (), where both users possess untrusted black-box devices. Steering can be detected through simple tests analogous to Bell inequalities Cavalcanti09 (), and has been verified in a variety of remarkable experiments Experiments (), including steering without Bell non-locality Saunders10 () and a fully loop-hole free steering demonstration loopholefree (). Apart from its fundamental relevance, steering has been identified as a resource for one-sided (1S) device-independent (DI) quantum key-distribution (QKD), where only one of the parts has an untrusted apparatus while the other ones possess trusted devices Branciard12 (); He13 (). There, the experimental requirements for unconditionally secure keys are less stringent than in fully (both-sided) DI-QKD Deviceindependent ().

The formal treatment of a physical property as a resource is given by a resource theory. The basic component of this is a restricted class of operations, called the free operations. These are typically the operations at hand in physical scenarios where the property in question acts as a useful resource. They fulfil the essential requirement of mapping every free state, i.e., every one without the property, into a free state. Furthermore, they provide a formal recipe for the quantification of the resource: The fundamental necessary condition for a function to be a measure of the resource is that it is monotonous –non-increasing– under the free operations. That is, the operations that do not increase the resource on the free states do not increase it on all other states either.

Entanglement theory is the most popular and best understood VedralPlenio (); Brandao08 () resource theory. There, local operations assisted by classical communication (LOCCs) are usually the natural free operations Bennett96 (). However, other sets of operations that do not create entanglement either have also been considered Jonathan99 (); Vedral97 (). On the other hand, resource theories have been formulated also for states out of thermal equilibrium Brandao13 (), asymmetry Ahmadi13 (), reference frames Gour08 (), and non-locality Gallego12 (); deVicente14 (), for instance.

In steering theory, systems are described by a collection of ensembles of quantum states on Bob’s side and a conditional probability distribution of measurement outcomes (outputs) given measurement settings (inputs) on Alice’s. Each input of Alice’s is associated with an ensemble of Bob’s and each output with a member state of such ensemble. That is, each pair of inputs and outputs of Alice’s is correlated with a state in an ensemble of Bob’s. Such systems are called assemblages Pusey13 (); Skrzypczyk13 (); Bowles14 (); Quintino14 (); Piani15 (); Belen15 (). The free operations for steering must thus arise from natural constrains native of a physical scenario where steerable assemblages are useful for some task. Up to now, no attempt for an operational framework of steering as a resource had been reported.

In this work we develop the resource theory of steering. First, we observe that one-way LOCCs (1W-LOCCs) from Bob to Alice are allowed operations –in the sense of not compromising the security– in 1S-DI-QKD protocols, the physical scenario where steering is a known resource Branciard12 (); He13 (). Then, for arbitrarily many inputs and outputs for Alice’s black box and arbitrary Hilbert-space dimension for Bob’s quantum system, we show that 1W-LOCCs from Bob to Alice do not create steering. These two facts give us clear physical motivations to take 1W-LOCCs as natural free operations for steering. We present the explicit parametrisation of a generic 1W-LOCC acting on assemblages. With this, we provide a formal definition of steering monotones. As an example thereof, we present the relative entropy of steering, for which we also introduce, on the way, the notion of relative entropy between assemblages. In addition, we prove 1W-LOCC monotonicity for two other recently proposed steering measures, the steerable weight Skrzypczyk13 () and the robustness of steering Piani15 (), and convexity for all three measures. To end up with, we prove two theorems on steering conversion under stochastic 1W-LOCCs for the lowest-dimensional case, i.e., qubits on Bob’s side and 2 inputs 2 outputs on Alice’s. In the first one, we show that it is impossible to transform via 1W-LOCCs, not even probabilistically, an assemblage composed of pairs of pure orthogonal states into another assemblage composed also of pairs of pure orthogonal states but with a different pair overlap, unless the latter is unsteerable. This yields infinitely many inequivalent classes of steering already for systems of lowest dimension. In the second one, we show that there exists no assemblage composed of pairs of pure states that can be transformed into any assemblage by stochastic 1W-LOCCs. This implies, in striking contrast to entanglement theory, that there exists no operationally well defined, measure-independent maximally steerable assemblage of minimal dimension.

The paper is organized as follows. In Sec. II we formally define assemblages and present their basic properties. In Sec. IV we discuss the role of 1W-LOCCs as the natural operations available to Alice and Bob in 1S-DI-QKD. In Sec. III we find an explicit parametrisation of all stochastic 1W-LOCCs from assemblages into assemblages and show that the resulting maps are steering non-increasing operations. In Sec. V we introduce the notion of convex steering monotones. In Sec. VI we present the relative entropy of steering. In Sec. VII we show convexity and 1W-LOCC-monotonicity of the steerable weight and the robustness of steering. In Sec. VIII we study, for minimal-dimensional systems, assemblage conversions under 1W-LOCCs and prove the in existence of pure-assemblage steering bits. Finally, in Sec. IX we present our conclusions and mention some future research directions that our results offer.

Ii Assemblages and steering

We consider two distant parties, Alice and Bob, who have each a half of a bipartite system. Alice holds a so-called black-box device, which, given a classical input , generates a classical output , where and are natural numbers and the notation , for , is introduced. Bob holds a quantum system of dimension (qudit), whose state he can perfectly characterize tomographically via trusted quantum measurements. The joint state of their system is thus fully specified by an assemblage

(1)

of normalized quantum states , with the set of linear operators on Bob’s subsystem’s Hilbert space , each one associated to a conditional probability of Alice getting an output given an input . We denote by the corresponding conditional probability distribution.

Equivalently, each pair can be univocally represented by the unnormalized quantum state

(2)

In turn, an alternative representation of the assemblage is given by the set of quantum states

(3)

where is an orthonormal basis of an auxiliary extension Hilbert space of dimension . The states do not describe the system inside Alice’s box, they are just abstract flag states to represent its outcomes with a convenient bra-ket notation. Expression (3) gives the counterpart for assemblages of the so-called extended Hilbert space representation used for ensembles of quantum states Oreshkov09 (). We refer to for short as the quantum representation of and use either notation upon convenience.

We restrict throughout to no-signaling assemblages, i.e., those for which Bob’s reduced state does not depend on Alice’s input choice :

(4)

The assemblages fulfilling the no-signaling condition (4) are the ones that possess a quantum realization. That is, they can be obtained from local quantum measurements by Alice on a joint quantum state shared with Bob, where is the Hilbert space of the system inside Alice’s box. For any no-signaling assemblage , we refer as the trace of the assemblage to the -independent quantity

(5)

and say that the assemblage is normalized if and unnormalized if .

An assemblage , being unnormalised states, is called unsteerable if there exist a probability distribution , a conditional probability distribution , and normalized states such that

(6)

Such assemblages can be obtained by sending a shared classical random variable to Alice, correlated with the state sent to Bob, and letting Alice classically post-process her random variable according to , with so that condition (4) holds. The variable is called a local-hidden variable and the decomposition (6) is accordingly referred to as a local-hidden state (LHS) model. We refer to the set of all unsteerable assemblages as . Any assemblage that does not admit a LHS model as in Eq. (6) is called steerable. An assemblage is compatible with classical correlations if, and only if, it is unsteerable.

To end up with, a comment on steering as a property of assemblages, as opposed to quantum states, is in order. In the pioneering works Wiseman07 (); Reid09 (), steering is defined as a property of quantum states. Namely, there, a state is said to be steerable if it can give rise, under local measurements, to correlations without a LHS model, i.e., to a steerable assemblage. The definition considered here, directly in terms of assemblages, and to which our resource theory applies, follows the treatment of Refs. Pusey13 (); Skrzypczyk13 (); Bowles14 (); Quintino14 (); Piani15 (); Belen15 (), for instance, where one embeds a share of the bipartite quantum state in the untrusted measurement device (Alice’s, in our case) and treats the entire embedding as a black box with unknown behaviour. This is the scenario of 1S-DI-QKD, the very task for which steering is a known useful resource. There, quantum states alone are not useful, since, without a trusted measurement device, correlations without LHS model can in general not be obtained from them. Working with assemblages, in contrast, is advantageous precisely because it removes the need of measurement specification in the untrusted part footnote2 (). Finally, it is important to note that post-quantum steering, i.e., steerable assemblages that, in spite of satisfying the no-signalling principle, do not admit a quantum realisation, has been recently discovered Belen15 ().

Iii The operational framework

In this section, we show that stochastic 1W-LOCCs from Bob to Alice do not create steering and can, therefore, be taken as free operations for steering. We consider the general scenario of stochastic 1W-LOCCs, i.e., 1W-LOCCs that do not necessarily occur with certainty, which map the initial assemblage into a final assemblage . See Fig. 1. Bob’s generic quantum operation can be represented by an incomplete generalised measurement. This is described by a completely-positive non trace-preserving map defined by

(7a)
(7b)

where is the final Hilbert space, of dimension , and is the measurement operator corresponding to the -th measurement outcome. For any normalised , the trace of the map’s output represents the probability that the physical transformation takes place. In turn, the map describes the post-selection of the -th outcome, which occurs with a probability

(8)
Figure 1: Schematic representation of a map : The initial assemblage consists of a black-box, with inputs and outputs , governed by the probability distribution , in Alice’s hand, and a quantum subsystem in one of the states , in Bob’s hands. The final assemblage is given by a final black-box, represented by the dashed-lined rectangle, of inputs and outputs , and a final subsystem, represented outside the dashed-lined rectangle, in the state . To implement , first, Bob applies, with a probability , a stochastic quantum operation that leaves his subsystem in the state . He communicates to Alice. Then, Alice generates by processing the classical bits and with a local wiring described by a conditional distribution . She inputs to her initial device, upon which the bit is output. Finally, Alice generates the output of the final device by processing , , , and , with a local wiring described by a distribution .

Since Alice can only process classical information, the allowed one-way communication from Bob to her must be classical too. Thus, it can only consists of the outcome of his quantum operation. Classical bit processings are usually referred to as wirings Brunner13 (). Alice’s wirings map and into input and out bits and , respectively, of the final assemblage, where and are natural numbers. The most general wirings respecting the above constraints are described by conditional probability distributions and of generating from and and from , , , and , respectively, as sketched in Fig. 1. Finally, since, as mentioned, her wirings must be deterministic, and must be normalised probability-preserving distributions.

All in all, the general form of the resulting maps is parametrised in the following definition (see App. A for details).

Definition 1 (Stochastic assemblage 1W-LOCCs).

We define the class of (stochastic) 1W-LOCCs as the set of (stochastic) maps that take an arbitrary assemblage into a final assemblage , where

(9)

being a deterministic wiring map given by

(10)

with and short-hand notations for the conditional probabilities and , respectively.

Note that the final assemblage (9) is in general not normalised: Introducing

(11)

such that , we obtain, using Eqs. (3), (4), (5), (8), (9), and (10), that

(12)

As with quantum operations, the trace (12) of represents the probability that the physical transformation takes place. Analogously, the map describes the assemblage transformation that takes place when Bob post-selects the -th outcome, which occurs with probability . In the particular case where is trace-preserving, we refer to it as a deterministic 1W-LOCC.

Finally, we prove in App. B the following theorem.

Theorem 1 ( invariance of ).

Any map of the class takes every unsteerable assemblage into an unsteerable assemblage.

Iv Physical motivation for free operations: 1W-LOCCs as safe operations in 1S-DI-QKD

As shown in the previous section, stochastic 1W-LOCCs from Bob to Alice satisfy the basic requirement of mapping every unsteerable assemblage into an unsteerable assemblage. However, there may in general exist other sets of operations with this feature. In entanglement theory, for instance, apart from the LOCCs, the separable operations Vedral97 (), the entanglement-assisted (catalytic) LOCCs Jonathan99 (), or simply the local operations, as well as any of these supplemented with particle swapping Horodecki09 (), are known not to create entanglement either. Each of the these classes of operations leads, strictly speaking, to a valid resource theory of entanglement. The choice of a given class over others is based upon actual constraints from the physical scenario in question. In what follows, we discuss the role of 1W-LOCCs as the allowed operations in 1S-DI-QKD, in the sense of being those that do not compromise the security. This gives us a physical motivation to choose 1W-LOCC as free operations for steering over other classes of operations that may also map into itself. To this end, and for pedagogic reasons, we first discuss the allowed safe operations for QKD and fully DI QKD.

QKD consists of the extraction of a secret key from the correlations of local-measurement outcomes on a bipartite quantum state. The most fundamental constraint to which any generic QKD protocol is subject is, of course, the lack of a private safe classical-communication channel between distant labs. If such a channel were available, the whole enterprise of QKD would be pointless. This imposes restrictions on the operations allowed, so as not to break the security of the protocol. For instance, clearly, the local-measurement outcomes cannot be communicated, as they can be intercepted by potential eavesdroppers who could, with them, crack the key. Of particular relevance for this work are the assumptions on the measurement devices. In non-DI QKD protocols, entanglement is the resource and security is proven under the assumption that the users have a specific quantum state and perfectly characterised measurement devices Ecker91 (). Knowledge of the state by an eavesdropper does not compromise the security. Therefore, prior to the measurements producing the key, the users are allowed to preprocess the state in any way and exchange information about it, for instance with LOCCs, or even to discard the state aborting the protocol run. Pre-processing LOCCs or abortions can at most provide an eavesdropper with knowledge about the state, not about the key, and therefore do not affect the security.

The situation is different in DI-QKD Deviceindependent (). There, the resource is given by Bell non-local correlations and no assumption is made either on the quantum state or the measurement devices. The users effectively hold black-box devices, whose inputs and outputs are all to which they have access. Since such inputs and outputs are precisely the bits with which the key is established, both classical communication and abortions are forbidden. Communication of outputs can directly reveal the key, as mentioned, whereas abortions and communication of inputs can, due to the locality and detection loopholes, respectively, be maliciously exploited by an eavesdropper to obtain information about the key too. Hence, the security constrains of DI-QKD naturally yield local classical information processing assisted by shared randomness or prior-to-input classical communication as the class of allowed operations Gallego12 (); deVicente14 ().

In 1S-DIQKD, in contrast, while no assumption is made on the bipartite quantum state or Alice’s apparatus, Bob’s measurement device is perfectly characterised. This is effectively described by assemblages of the form given in Eq. (1). The asymmetry in Alice and Bob’s devices leads to an asymmetry in the operations allowed to each of them. Alice is subject to the same restrictions as in both-sided DI QKD, while Bob, to those of non-DI QKD. Hence, Alice cannot abort or transmit any information, but, before measuring, Bob is allowed to implement arbitrary pre-processing quantum operations to his subsystem, including stochastic ones with possible abortions, and send any classical feedback about them to Alice. Altogether, this singles out a natural set of operations that do not compromise the security: all the assemblage transformations involving only deterministic classical maps on Alice’s side and arbitrary –possibly stochastic– quantum operations on Bob’s, assisted by one-way classical communication from Bob to Alice footnote1 (). These are, namely, the stochastic 1W-LOCCs from Bob to Alice (see Fig. 1). Note that shared randomness or prior-to-input classical communication from Alice to Bob Gallego12 (); deVicente14 (), which also do not introduce any security compromise, can always be recast as 1W classical communication from Bob to Alice and need, therefore, not be explicitly considered.

Finally, we emphasise that it is one of the measurement apparatuses what is assumed untrusted, not the users. Both users are trusted and reliably operate on their systems, carrying out the allowed assemblage transformations.

V Steering monotonicity

Once the free operations for steering are established, the natural next step is to introduce an axiomatic approach to define steering measures, i.e., a set of postulates that a bona fide quantifier of steering should fulfil.

Definition 2 (1W-LOCC-monotonicity and convexity).

A function , from the space of assemblages into , is a steering monotone if it fulfils the following two axioms:

  1. for all .

  2. does not increase, on average, under deterministic 1W-LOCCs, i.e.,

    (13)

    for all , with and .

Besides, is a convex steering monotone if it additionally satisfies the property:

  1. Given any real number , and assemblages and , then

    (14)

Condition reflects the basic fact that unsteerable assemblages should have zero steering. Condition formalizes the intuition that, analogously to entanglement, steering should not increase –on average– under 1W-LOCCs, even if the flag information produced in the transformation is available. Finally, condition states the desired property that steering should not increase by probabilistically mixing assemblages. The first two conditions are taken as mandatory necessary conditions, the third one only as a convenient property. Importantly, there exists a less demanding definition of monotonicity. There, the left-hand side of Eq. (13) is replaced by . That is, it is demanded only that steering itself, instead of its average over , is non-increasing under the free operations. The latter is actually the most fundamental necessary condition for a measure. However, monotonicity is in many cases (including the present work) easier to prove and, together with condition , implies monotonicity . Hence, we focus throughout on monotonicity as defined by Eq. (13) and refer to it simply as 1W-LOCC monotonicity. All three known quantifiers of steering, the two ones introduced in Refs. Skrzypczyk13 (); Piani15 () as well as the one we introduce in the next section, turn out to be convex steering monotones in the sense of Definition 2.

Vi The relative entropy of steering

In this section, we introduce a convex steering monotone called the relative entropy of steering. To this end, we first define the notion of relative entropy between assemblages. For any two density operators and , we first recall the quantum von-Neumann relative entropy

(15)

of with respect to and, for any two probability distributions and , the classical relative entropy, or Kullback-Leibler divergence,

(16)

of with respect to . The quantum and classical relative entropies (15) and (16) measure the distinguishability of states and distributions, respectively. To find an equivalent measure for assemblages, we note, for given by Eq. (3) and , that

(17)

where and are respectively the distributions over obtained from the conditional distributions and for a fixed . That is, the distinguishability between the states and equals the sum of the distinguishabilities between and and between and , weighted by and averaged over .

The entropy (VI), which depends on , does not measure the distinguishability between the assemblages and . Since the latter are conditional objects, i.e., with inputs, a general strategy to distinguish them must allow for Alice choosing the input for which the assemblages’ outputs are optimally distinguishable. Furthermore, Bob can first apply a generalised measurement on his subsystem and communicate the outcome to her, which she can then use for her input choice. This is the most general procedure within the allowed 1W-LOCCs. Hence, a generic distinguishing strategy under 1W-LOCCs involves probabilistically chosen inputs that depend on . Note, in addition, that the statistics of generated, described by distributions or , encode differences between and too and must therefore also be accounted for by a distinguishability measure. The following definition incorporates all these considerations.

Definition 3 (Relative entropy between assemblages).

Given any two assemblages and , we define the assemblage relative entropy of with respect to as

(18)

where are generalised-measurement operators such that , is a conditional probability distribution of given , the short-hand notation has been used, and

(19a)
(19b)

where is Bob’s reduced state for the assemblage .

In App. C, we show that does not increase –on average– under deterministic 1W-LOCCs and, as its quantum counterpart , is jointly convex. Hence, is a proper measure of distinguishability between assemblages under 1W-LOCCs footnote0 (). The first term inside the maximisation in Eq. (18) accounts for the distinguishability between the distributions of measurement outcomes and the second one for that between the distributions of Alice’s outputs and Bob’s states resulting from each , averaged over all inputs and measurement outcomes. In turn, the maximisation over and ensures that these output distributions and states are distinguished using the optimal 1W-LOCC-compatible strategy.

We are now in a good position to introduce a convex steering monotone. We do it with a theorem.

Theorem 2 (1W-LOCC-monotonicity and convexity of ).

The relative entropy of steering , defined for an assemblage as

(20)

is a convex steering monotone.

The theorem is proven in App. C.

Vii Other convex steering monotones

Apart from two other quantifiers of steering have been recently proposed: the steerable weight Skrzypczyk13 () and the robustness of steering Piani15 (). In this section, we show that these are also convex steering monotones.

Definition 4 (Steerable weight Skrzypczyk13 ()).

The steerable weight of a normalised assemblage is the minimum such that

(21)

with an arbitrary normalised assemblage and normalised .

Definition 5 (Robustness of steering Piani15 ()).

The robustness of steering of a normalised assemblage is the minimum such that the normalised assemblage

(22)

belongs to , with an arbitrary normalised assemblage.

In App. D, we prove the following theorem.

Theorem 3 (1W-LOCC-monotonicity and convexity of and ).

Both and are convex steering monotones.

To end up with, we note that a steering measure for assemblages containing continuous-variable (CV) bosonic systems in Gaussian states has very recently appeared Kogias14 (). Even though our formalism can be straightforwardly extended to CV systems, such extension is outside the scope of the present paper.

Viii Assemblage conversions and no steering bits

We say that and are pure assemblages if they are of the form

(23a)
(23b)

where and , and pure orthogonal assemblages if, in addition, for all . Note that pure orthogonal assemblages are the ones obtained when Alice and Bob share a pure maximally entangled state and Alice performs a von-Neumann measurement on her share. We present two theorems about assemblage conversions under 1W-LOCCs.

The first one, proven in App. E, establishes necessary and sufficient conditions for stochastic-1W-LOCC conversions between pure orthogonal assemblages, therefore playing a similar role here to the one played in entanglement theory by Vidal’s theorem Vidal99 () for stochastic-LOCC pure-state conversions.

Theorem 4 (Criterion for stochastic-1W-LOCC conversion).

Let and be any two pure orthogonal assemblages with . Then, can be transformed into by a stochastic 1W-LOCC iff: either or and

(24)

for some .

In other words, no pure orthogonal assemblage of minimal dimension can be obtained via a 1W-LOCC, not even probabilistically, from a pure orthogonal assemblage of minimal dimension with a different state-basis overlap (except for trivial relabellings of , given by ) unless the former is unsteerable. Hence, each state-basis overlap defines an inequivalent class of steering, there being infinitely many of them. This is in a way reminiscent to the inequivalent classes of entanglement in multipartite Duer99 () or infinite-dimensional bipartite Owari04 () systems, but here the phenomenon is found already for bipartite systems of minimal dimension.

The second theorem, proven in App. F, rules out the possibility of there being a (non-orthogonal) minimal-dimension pure assemblage from which all assemblages can be obtained.

Theorem 5 (Non-existence of steering bits).

There exists no pure assemblage with that can be transformed into any assemblage by stochastic 1W-LOCCs.

Hence, among the minimal-dimension assemblages there is no operationally well defined unit of steering, or steering bit, i.e., an assemblage from which all assemblages can be obtained for free and can therefore be taken as a measure-independent maximally steerable assemblage. This is again in striking contrast to entanglement theory, where pure maximally entangled states can be defined without the need of entanglement quantifiers and each one can be transformed into any state by deterministic LOCCs Vidal99 (); Nielsen99 ().

Ix Discussion and outlook

We have introduced the resource theory of Einstein-Podolsky-Rosen steering. The free operations of the theory are the 1W-LOCCs from the quantum part to the black box, i.e., all the assemblage transformations involving deterministic bit wirings on Alice’s side and stochastic quantum operations on Bob’s assisted by 1-way classical communication from Bob to Alice. These operations satisfy the basic requirement of mapping all unsteerable assemblages into unsteerable assemblages and are, besides, also the allowed operations that naturally arise from the basic security constraints of one-sided device-independent QKD, where steering is a physical resource. With these operations, we introduced the notion of convex steering monotones, presented the relative entropy of steering as a convenient example thereof, and proved monotonicity and convexity of two other previously proposed steering measures. In addition, for minimal-dimensional systems, we established necessary and sufficient conditions for stochastic-1W-LOCC conversions between pure-state assemblages and proved the non-existence of steering bits.

It is instructive to emphasise that the derived 1W-LOCCs are hybrids between the operations that map separable states into separable states, stochastic LOCCs, and those that map Bell local correlations into Bell local correlations, local wirings assisted by prior-to-input classical communication Gallego12 (); Navascues14 (); deVicente14 (). In fact, our findings are also potentially useful for the quantification of Bell non-locality. In addition, our work offers a number of challenges for future research. Namely, for example, the non-existence of steering bits of minimal dimension can be seen as an impossibility of steering dilution of minimal-dimension assemblages in the single-copy regime. We leave as open questions what the rules for steering dilution and distillation are for higher-dimensional systems, mixed-state assemblages, or in asymptotic multi-copy regimes, and what the steering classes are for mixed-state assemblages. Moreover, other fascinating questions are whether one can formulate a notion of bound steering or an analogue to the positive-partial-transpose criterion for assemblages.

Acknowledgements

We would like to thank Antonio Acín, Daniel Cavalcanti, Paul Skrzypczyk and Marco Túlio Quintino for discussions and the EU (RAQUEL, SIQS) for support. RG acknowledges support from the Alexander von Humboldt Foundation and LA from the EU (REQS - Marie Curie IEF No 299141).

References

  • (1) E. Schrödinger, Discussion of probability relations between separated systems, Proc. Camb. Phil. Soc. 31, 555 (1935).
  • (2) H. M. Wiseman, S. J. Jones and A. C. Doherty, Steering, Entanglement, Nonlocality, and the Einstein-Podolsky-Rosen Paradox, Phys. Rev. Lett. 98, 140402 (2007); S. J. Jones et al., Phys. Rev. A 76, 052116 (2007).
  • (3) M. D. Reid, P. D. Drummond, W. P. Bowen, E. G. Cavalcanti, P. K. Lam, H. A. Bachor, U. L. Andersen, and G. Leuchs, Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications, Rev. Mod. Phys. 81, 1727 (2009).
  • (4) R. Horodecki, P. Horodecki, M. Horodecki, and K. Horodecki, Quantum entanglement, Rev. Mod. Phys. 81, 865 (2009).
  • (5) N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani and S. Wehner, Bell nonlocality, Rev. Mod. Phys. 86, 419 (2014).
  • (6) E. G. Cavalcanti, S. J. Jones, H. M. Wiseman and M. D. Reid, Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox, Phys. Rev. A 80, 032112 (2009).
  • (7) Z. Y. Ou, S. F. Pereira, H. J. Kimble, and K. C. Peng, Realization of the Einstein-Podolsky-Rosen paradox for continuous variables, Phys. Rev. Lett. 68, 3663 (1992); W. P. Bowen, R. Schnabel, and P. K. Lam, Experimental Investigation of Criteria for Continuous Variable Entanglement, Phys. Rev. Lett. 90, 043601 (2003); D.-H. Smith et al., Conclusive quantum steering with superconducting transition-edge sensors, Nat. Commun. 3, 625 (2012); A. J. Bennet et al.,Arbitrarily Loss-Tolerant Einstein-Podolsky-Rosen Steering Allowing a Demonstration over 1 km of Optical Fiber with No Detection Loophole, Phys. Rev. X 2, 031003 (2012); V. Händchen et al., Observation of one-way Einstein-Podolsky-Rosen steering, Nat. Phot. 6, 598 (2012); S. Steinlechner, J. Bauchrowitz, T. Eberle, and R. Schnabel, Strong Einstein-Podolsky-Rosen steering with unconditional entangled states, Phys. Rev. A 87, 022104 (2013).
  • (8) D. J. Saunders, S. J. Jones, H. M. Wiseman, and G. J. Pryde, Experimental EPR-steering using Bell-local states, Nat. Phys. 6, 845 (2010).
  • (9) B. Wittmann, S. Ramelow, F. Steinlechner, N. K. Langford, N. Brunner, H. Wiseman, R. Ursin, A. Zeilinger, Loophole-free Einstein-Podolsky-Rosen experiment via quantum steering, New J. Phys. 14, 053030 (2012).
  • (10) C. Branciard, E. G. Cavalcanti, S. P. Walborn, V. Scarani and H. M. Wiseman, One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering, Phys. Rev. A 85, 010301(R) (2012).
  • (11) Q. Y. He and M. D. Reid, Genuine Multipartite Einstein-Podolsky-Rosen Steering, Phys. Rev. Lett. 111, 250403 (2013).
  • (12) J. Barrett, L. Hardy, and A. Kent, No Signaling and Quantum Key Distribution, Phys. Rev. Lett. 95, 010503 (2005); A. Acín, N. Gisin, and L. Masanes, From Bell’s Theorem to Secure Quantum Key Distribution, Phys. Rev. Lett. 97, 120405 (2006); A. Acín et al., Device-Independent Security of Quantum Cryptography against Collective Attacks, Phys. Rev. Lett. 98, 230501 (2007).
  • (13) V. Vedral and M. B. Plenio, Entanglement measures and purification procedures, Phys. Rev. A 57, 1619 (1998); M. B. Plenio and S. Virmani, An introduction to entanglement measures, Quant. Inf. Comput. 7, 1 (2007).
  • (14) F. G. S. L. Brandão and M. B. Plenio, Entanglement theory and the second law of thermodynamics, Nature Phys. 4, 873 (2008); F. G. S. L. Brandão and M. B. Plenio, A Reversible Theory of Entanglement and its Relation to the Second Law, Comm. Math. Phys. 295, 829 (2010).
  • (15) C. H. Bennett, D. P. DiVincenzo, J. Smolin, and W. K. Wootters, Mixed-state entanglement and quantum error correction, Phys. Rev. A 54, 3824 (1996).
  • (16) V. Vedral, M. B. Plenio, M. A. Rippin and P. L. Knight, Quantifying Entanglement, Phys. Rev. Lett 78, 2275 (1997)
  • (17) D. Jonathan and M. B. Plenio, Entanglement-Assisted Local Manipulation of Pure Quantum States, Phys. Rev. Lett. 83, 3566 (1999)
  • (18) F. G. S. L. Brandão, M. Horodecki, J. Oppenheim, J. M. Renes, and R. W. Spekkens, Resource Theory of Quantum States Out of Thermal Equilibrium, Phys. Rev. Lett. 111, 250404 (2013).
  • (19) M. Ahmadi, D. Jennings and T. Rudolph, The Wigner-Araki-Yanase theorem and the quantum resource theory of asymmetry, New J. Phys. 15, 013057 (2013).
  • (20) G. Gour and R. W. Spekkens, The resource theory of quantum reference frames: manipulations and monotones, New J. Phys. 10, 033023 (2008).
  • (21) F. Levi and F. Mintert, A quantitative theory of coherent delocalization, New J. Phys. 16, 033007 (2014).
  • (22) T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying Coherence, Phys. Rev. Lett. 113, 140401 (2014).
  • (23) M. F. Pusey, Negativity and steering: A stronger Peres conjecture, Phys. Rev. A 88, 032313 (2013).
  • (24) P. Skrzypczyk, M. Navascués, and D. Cavalcanti, Quantifying Einstein-Podolsky-Rosen Steering, Phys. Rev. Lett. 112, 180404 (2014).
  • (25) M. Piani and J. Watrous, Necessary and Sufficient Quantum Information Characterization of Einstein-Podolsky-Rosen Steering, Phys. Rev. Lett. 114, 060404 (2015).
  • (26) J. Bowles, T. Vértesi, M. T. Quintino and N. Brunner. One-way Einstein-Podolsky-Rosen Steering, Phys. Rev. Lett. 112, 200402 (2014).
  • (27) M. T. Quintino, T. Vértesi, N. Brunner. Joint Measurability, Einstein-Podolsky-Rosen Steering, and Bell Nonlocality, Phys. Rev. Lett. 113, 160402 (2014).
  • (28) A. B. Sainz, N. Brunner, D. Cavalcanti, P. Skrzypczyk, and T. Vértesi, Post-quantum steering, arXiv:1505.01430 (2015).
  • (29) One could in principle still attempt a resource theory of steerable quantum states as defined in Wiseman07 (); Reid09 (). However, since the mathematical condition that defines a quantum state as steerable involves an optimisation over its potential measurements, it is unclear what the precise resource to account for is. An analogous difficulty is found with Bell non-locality as a resource Gallego12 (); deVicente14 (). There, one could conceive a resource theory of “non-local quantum states”, defined as those able to yield, under local measurements, a non-local probability distribution. However, for the same reasons, non-locality defined directly in terms of probability distributions plays a more relevant role.
  • (30) O. Oreshkov and J. Calsamiglia, Distinguishability measures between ensembles of quantum states, Phys. Rev. A 79, 032336 (2009).
  • (31) A. K. Ekert, Quantum cryptography based on Bell’s theorem, Phys. Rev. Lett. 67, 661(1991).
  • (32) R. Gallego, L. E. Würflinger, A. Acín, and M. Navascués, Operational Framework for Nonlocality, Phys. Rev. Lett. 109, 070401 (2012).
  • (33) J. I. de Vicente, On nonlocality as a resource theory and nonlocality measures, J. Phys. A: Math. Theor. 47, 424017 (2014).
  • (34) B. Lang, T. Vértesi and M. Navascués, Closed sets of correlations: answers from the zoo, arXiv:1402.2850 (2014).
  • (35) The terminology “deterministic maps” refers throughout to probability (trace) preserving classical (quantum) maps, i.e. those that never cause an abortion. For classical maps, for instance, this should not be confused with maps where the output bit is a Kronecker delta function of the input bit. In turn, the term “stochastic” is used throughout to refer to non probability-preserving classical maps or non trace-preserving quantum transformations, which do not occur with certainty.
  • (36) A natural question (which we leave open) is whether there exists a definition of relative entropy between assemblages that is non-increasing under generic assemblage transformations instead of just 1W-LOCCs, so that it can be understood as measure of distinguishability under fully general strategies. For quantum states, that is the case of , for instance, which is non-increasing under not only LOCCs but also under any completely positive map. However, note that 1W-LOCC-monotonicity of suffices to introduce a steering monotone.
  • (37) I. Kogias, A. R. Lee, S. Ragy, and G. Adesso, Quantification of Gaussian Quantum Steering, Phys. Rev. Lett. 114, 060403 (2015).
  • (38) G. Vidal, Entanglement of Pure States for a Single Copy, Phys. Rev. Lett. 83, 1046 (1999).
  • (39) W. Dür, G. Vidal, and J. I. Cirac, Three qubits can be entangled in two inequivalent ways, Phys. Rev. A 62, 062314 (2000).
  • (40) M. Owari, K. Matsumoto, and M. Murao, Entanglement convertibility for infinite-dimensional pure bipartite states, Phys. Rev. A 70, 050301(R) (2004).
  • (41) M. A. Nielsen, Conditions for a Class of Entanglement Transformations, Phys. Rev. Lett. 83, 436 (1999).

Appendix A Parametrisation of the class

In this appendix, we show that any generic assemblage map involving stochastic local quantum operations on Bob’s side, one-way classical communication from Bob to Alice, and deterministic (probability-preserving) local wirings on Alice’s side is of the form given by Eqs. (9) and (10) and, therefore, belongs to the class the class of Definition 1.

Without loss of generality, such can be decomposed into the following sequence (see Fig. 1) of operations:

  1. Bob applies an arbitrary stochastic generalised measurement, described by a completely-positive non trace-preserving map , defined by Eqs. (7), to his quantum subsystem before Alice introduces an input to her device. Note that, since the non-signalling condition (4) is fulfilled, Bob has a well defined reduced quantum state , given by Eq. (4), independently of Alice still not having chosen her measurement input . Therefore, Bob’s measurement gives the outcome with the -independent probability given by Eq. (8).

  2. Bob sends the outcome to Alice. Alice applies a local wiring, described by the normalised conditional probability distribution , to the input of the final device and to , and uses the output of this wiring as the input of her initial device. For a given , her initial device outputs with a probability determined by the conditional distribution of the initial assemblage. In that case, Bob’s normalized state is given by the -independent density operator

    (25)
  3. Alice applies a local wiring, described by the normalised conditional probability distribution , to all the previously generated classical bits, , , , and , and uses the output of this wiring as the output of her final device. This final processing of the bit does not affect Bob’s state. Thus, Bob’s system ends up in the state .

We denote by the conditional distribution of given and , for chosen independently of (in contrast to Step 2 above), with elements . With this, the components of the final assemblage are explicitly given by:

(26)
(27)
(28)

Eq. (26) follows from basic properties of probability distributions and ensembles of states. Eq. (27) follows from the definition of . Eq. (28) follows from Bayes’ theorem together with the facts that (the output of Alice’s initial device only depends on the input and the measurement outcome ) and (the measurement outcome is independent of the input of Alice’s final device), and from the definition of . Next, note that, since the statistics of is fully determined by and regardless of whether and are independent or not, it holds that

(29)

where we have used Bayes’ theorem, that , and that, by definition, . Inserting Eq. (29) into Eq. (28), we obtain

(30)

where (30) follows from the fact that and the definition of . The right-hand side of Eq. (30) gives the most general expression of the components of explicitly as a function of the components of. The reader can straightforwardly verify that the quantum representation of the obtained final assemblage is given by the right-hand side of Eq. (9).

Appendix B Invariance of under maps

We now show that if then, for all , .

Proof of Theorem 1.

Replacing in Eq. (30) by the right-hand side of Eq. (6), we write

(31)

where the conditional probability and the normalized state have been introduced. Using that does not explicitly depend on , we see that

(32)

In turn, using the facts that is independent of and depends only on and , and Bayes’ theorem, we see that

(33)

Substituting into Eq. (31) yields

(34)
(35)

where Eq. (34) follows from Bayes’ theorem and summing over and , and Eq. (35) follows from defining the hidden variable governed by the normalized probability distribution . Eq. (35) manifestly shows that . ∎

Appendix C The relative entropy of steering

In this appendix we prove Theorem 2. The proof strategy is similar to that of the proof that the relative entropy of entanglement for quantum states is a convex entanglement monotone Vedral98 (). It relies on two Lemmas, which we state next but whose proofs we leave for App. G.

Lemma 1.

The assemblage relative entropy , defined by Eq. (18), does not increase, on average, under deterministic 1W-LOCCs. That is, for any map of the form given by Eqs. (9) and (10) but with and any two assemblages and , satisfies the inequality

(36)

where is the stochastic map defined in Eq. (11), , and , with .

Lemma 2.

The assemblage relative entropy , defined by Eq. (18), is jointly convex. That is, given two sets and of arbitrary assemblages each and positive real numbers such that , with , satisfies the inequality