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The resource theory of steering

Rodrigo Gallego, Leandro Aolita

arXiv:1409.5804v3quant-ph

TL;DR

The paper addresses the lack of an operational resource theory for steering by defining free operations and convex monotones directly for assemblages. It characterizes Bob-to-Alice 1W-LOCCs, introduces relative entropy of steering, proves monotonicity and convexity results for several measures, and establishes minimal-dimensional conversion constraints. The results include infinitely many inequivalent steering classes and no measure-independent maximally steerable assemblage.

  • Problem

    Steering lacked an operational resource framework specifying physically motivated free operations and corresponding quantifiers.

  • Method

    The paper models steering with assemblages and studies stochastic one-way LOCCs from Bob’s trusted quantum subsystem to Alice’s black box as free operations.

  • Results

    The framework proves that 1W-LOCCs preserve unsteerability, defines convex steering monotones, presents relative entropy of steering, and proves convex monotonicity for steerable weight and robustness.

  • Takeaways & Limitations

    Minimal-dimensional steering has infinitely many inequivalent pure-state classes, and no measure-independent maximally steerable assemblage exists from which all assemblages can be obtained.

Abstract

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We present an operational framework for Einstein-Podolsky-Rosen steering as a physical resource. To begin with, we characterize the set of steering non-increasing operations (SNIOs) --i.e., those that do not create steering-- on arbitrary-dimensional bipartite systems composed of a quantum subsystem and a black-box device. Next, 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 SNIOs 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 resource-theory approaches, with potential implications in Bell non-locality.

I. INTRODUCTION

The paper frames steering as a resource and develops an operational theory based on one-way LOCCs from Bob to Alice. It defines steering monotones and establishes conversion results showing that minimal-dimensional steering has infinitely many inequivalent classes and no universal maximally steerable state.

  • Motivation: Steering remotely prepares ensembles of quantum states and lies between entanglement and Bell non-locality in device-assumption requirements.It can certify entanglement with Alice’s measurements untrusted and has applications in one-sided device-independent QKD.
  • Research gap: Assemblages encode Alice’s black-box input-output statistics together with the corresponding quantum states on Bob’s side.Before this work, no operational framework had been reported for steering as a resource.
  • Contributions: The paper motivates 1W-LOCCs from Bob to Alice as natural free operations because they are allowed in one-sided device-independent QKD and do not create steering.The framework also parametrizes these operations and defines steering monotones.
  • Contributions: The paper introduces convex steering monotones, relative entropy of steering, and proves convex monotonicity for steerable weight and robustness of steering.These results provide several resource quantifiers within the proposed theory.
  • Conversion results: In minimal dimension, pure-state conversion results yield infinitely many inequivalent steering classes and rule out measure-independent maximally steerable assemblages.No pure assemblage can be transformed into every assemblage by stochastic 1W-LOCCs.

II. ASSEMBLAGES AND STEERING

The paper represents steering systems as assemblages: conditional ensembles of Bob’s quantum states indexed by Alice’s black-box inputs and outputs. It applies the resource theory directly to assemblages, including no-signaling and unsteerability conditions, rather than to quantum states alone.

  • Assemblages: An assemblage associates each Alice input with an ensemble on Bob’s system and each output with a member state of that ensemble.Alice’s device is treated as a black box, while Bob’s quantum subsystem has trusted measurements.
  • Assemblages: Each pair of Alice’s input and output is represented by a normalized quantum state together with its conditional output probability.The pair can equivalently be encoded as an unnormalized quantum state.
  • Representation: The quantum representation uses orthonormal auxiliary flag states to record Alice’s outputs without describing a physical system inside her black box.This provides a convenient extended-Hilbert-space notation for assemblages.
  • Constraints: The analysis restricts to no-signaling assemblages, whose reduced state on Bob’s side is independent of Alice’s input and therefore admits a quantum realization.Assemblages are normalized when their trace is one and unnormalized when their trace is at most one.
  • Unsteerability: Unsteerable assemblages are those admitting a local-hidden-state decomposition using a shared classical variable, Alice’s conditional processing, and normalized states on Bob’s side.The set of all such assemblages is denoted LHS.
  • Operational setting: The resource theory is defined directly on assemblages because this removes the need to specify measurements in the untrusted part, matching the one-sided device-independent QKD scenario.The framework also encompasses the possibility of post-quantum steering assemblages.

III. THE OPERATIONAL FRAMEWORK

The operational framework defines stochastic one-way LOCCs from Bob’s quantum subsystem to Alice’s black box as maps on assemblages. These maps combine Bob’s post-selected quantum operation with Alice’s classical wirings and preserve unsteerability.

  • Bob’s operation: Bob’s stochastic operation is an incomplete generalized measurement represented by a completely positive, non-trace-preserving map.The outcome ω occurs with a probability given by the trace of the post-selected output.
  • Classical wirings: Bob communicates only his classical outcome ω to Alice, who uses it in local wirings that transform the initial input-output pair into the final pair.The wirings are described by conditional distributions for generating x and af.
  • Map definition: Alice’s wirings are normalized probability-preserving distributions, reflecting deterministic classical processing of inputs and outputs.The final assemblage can nevertheless be unnormalized because the overall transformation is stochastic.
  • Map definition: The resulting stochastic maps are explicitly parametrized as assemblage 1W-LOCCs using Bob’s operation and Alice’s conditional probability distributions.The shorthand probabilities are P(x|xf,ω) and P(af|a,x,ω,xf).
  • Stochasticity: Post-selection on outcome ω produces a branch with probability Tr[Mω(ρ̂A|X)], while trace-preserving maps are called deterministic 1W-LOCCs.The normalized branch is obtained by dividing the transformed quantum representation by its trace.
  • Free operations: Every 1W-LOCC maps an unsteerable assemblage into an unsteerable assemblage, making the class steering non-increasing.This establishes the basic free-operation property required by the resource theory.

IV. PHYSICAL MOTIVATION FOR FREE OPERATIONS: 1W-LOCCS AS SAFE OPERATIONS IN 1S-DI-QKD

The paper motivates Bob-to-Alice 1W-LOCCs through the asymmetric security constraints of one-sided device-independent QKD. Alice remains subject to black-box restrictions, while Bob may perform trusted quantum preprocessing and communicate its classical outcome.

  • Free operations: These security-motivated restrictions support treating Bob-to-Alice 1W-LOCCs as safe operations for steering.The paper contrasts this physically motivated class with other possible unsteerability-preserving operation sets.
  • Comparison: Unlike non-device-independent QKD, one-sided device-independent QKD makes no assumptions about the shared state or Alice’s apparatus.Unlike fully device-independent QKD, Bob’s measurement device is characterized.
  • 1S-DI-QKD: In one-sided device-independent QKD, Alice’s apparatus is untrusted but Bob’s measurement device is trusted, producing asymmetric operational constraints.Assemblages describe this setting because they retain Alice’s black-box behavior while characterizing Bob’s quantum states.
  • 1S-DI-QKD: Alice cannot abort or transmit information, whereas Bob may perform arbitrary quantum preprocessing before measurement because his device is trusted.Bob’s allowed operations can therefore include one-way classical communication of his processing outcome to Alice.

V. STEERING MONOTONICITY

The paper formulates steering measures through axioms requiring vanishing on unsteerable assemblages, monotonicity under 1W-LOCCs, and convexity for convex steering monotones.

  • A steering monotone vanishes on all assemblages in LHS.
  • A steering monotone does not increase on average under deterministic 1W-LOCCs, even when transformation flags are available.
  • A convex steering monotone additionally cannot increase when assemblages are probabilistically mixed.
  • Average monotonicity together with convexity implies the weaker requirement that steering itself is non-increasing under free operations.

VI. THE RELATIVE ENTROPY OF STEERING

The paper constructs the relative entropy of steering by adapting quantum relative entropy to assemblages and optimizing over 1W-LOCC-compatible discrimination strategies. It proves that this quantity is a convex steering monotone.

  • The relative entropy of steering is introduced as a convex steering monotone for assemblages.
  • Assemblage relative entropy combines classical and quantum distinguishability for conditional outputs and Bob’s states.
  • 1W-LOCC discrimination optimizes over Bob’s generalized measurements and Alice’s input choices conditioned on Bob’s communicated outcome.
  • The distinguishability measure is jointly convex and does not increase on average under deterministic 1W-LOCCs.
  • The relative entropy of steering is defined by minimizing assemblage relative entropy over unsteerable assemblages.

VII. OTHER CONVEX STEERING MONOTONES

The paper shows that the steerable weight and robustness of steering satisfy the same convex steering-monotone axioms. Its formalism can extend to continuous-variable systems, but that extension is outside the paper’s scope.

  • The steerable weight is defined through the minimum steerable component in a decomposition with an unsteerable assemblage.
  • The robustness of steering is the minimum mixing weight that makes a mixture with another assemblage unsteerable.
  • The steerable weight and robustness of steering are both convex steering monotones.
  • The formalism can be straightforwardly extended to continuous-variable bosonic systems in Gaussian states, but that extension is outside the present paper’s scope.

VIII. ASSEMBLAGE CONVERSIONS AND NO STEERING BITS

For minimal-dimensional assemblages, the paper characterizes stochastic 1W-LOCC conversions between pure orthogonal assemblages and rules out a universal steering bit. Different state-basis overlaps define infinitely many inequivalent steering classes.

  • Assemblage conversions: Pure assemblages have component states |ψ(a, x)⟩, while pure orthogonal assemblages additionally require orthogonality between outputs for each input.Pure orthogonal assemblages arise when Alice and Bob share a pure maximally entangled state and Alice performs a von-Neumann measurement.
  • Assemblage conversions: The first theorem gives necessary and sufficient conditions for stochastic 1W-LOCC conversions between minimal-dimensional pure orthogonal assemblages.This result is presented as analogous to Vidal’s theorem for stochastic pure-state conversions under LOCC.
  • Assemblage conversions: Different state-basis overlaps cannot be connected by stochastic 1W-LOCCs, except for trivial output relabellings, unless the target assemblage is unsteerable.Consequently, each overlap defines an inequivalent steering class, yielding infinitely many classes already in minimal dimension.
  • No steering bits: No minimal-dimensional pure assemblage can be transformed into every assemblage by stochastic 1W-LOCCs.Thus, there is no operationally well-defined steering bit or measure-independent maximally steerable assemblage in this setting.

IX. DISCUSSION AND OUTLOOK

The discussion presents 1W-LOCCs from Bob’s quantum subsystem to Alice’s black box as the free operations and summarizes proofs that they preserve unsteerability and support convex steering monotones.

  • Free operations: The resource theory takes 1W-LOCCs from the quantum part to the black box as its free operations.The paper frames these operations as the central operational structure of the steering resource theory.
  • Free operations: A generic 1W-LOCC consists of Bob’s stochastic quantum operation, communication of its outcome to Alice, and Alice’s input and output wirings.The construction allows arbitrary stochastic measurements on Bob’s side and normalized conditional wirings on Alice’s side.
  • Free operations: The resulting assemblage maps preserve the LHS set, so 1W-LOCCs do not create steering.The proof identifies a combined hidden variable that gives the transformed assemblage an LHS decomposition.
  • Steering monotones: The assemblage relative entropy does not increase on average under deterministic 1W-LOCCs and is jointly convex.These properties provide the key ingredients for proving it is a convex steering monotone.

Appendix D: Proof of Theorem 3

Appendix D proves that the steerable weight and robustness of steering satisfy the convex steering-monotone conditions through normalized decompositions and analogous arguments.

  • Steerable weight: The steerable weight proof applies the stochastic map to the assemblage and its unsteerable component, then renormalizes the resulting terms.The construction yields normalized steerable and unsteerable assemblages for the transformed mixture.
  • Robustness of steering: For the robustness parameter, any decomposition feasible for the steerable-weight form is also feasible for the robustness form.This feasibility relation supplies one direction of the comparison used in the proof.
  • Steerable weight: Any decomposition witnessing a mixture’s steerable weight need not be optimal, so it provides an upper bound sufficient for proving convexity.The appendix explicitly notes that the constructed decomposition is not necessarily optimal before deriving the convexity inequality.
  • Robustness of steering: The robustness proof uses the same strategy as the steerable-weight proof for monotonicity and convexity.Its decomposition introduces a nonnegative parameter and corresponding normalized steerable and unsteerable assemblages.

Appendix E: Proof of Theorem 4

Appendix E analyzes stochastic 1W-LOCC conversions between minimal-dimensional pure orthogonal assemblages by restricting possible wirings and Kraus-operator assignments, deriving the overlap criterion.

  • Structural restrictions: No-signaling restricts minimal-dimensional pure orthogonal assemblages to deterministic or uniform output distributions for every input.The deterministic case is unsteerable, so the nontrivial analysis concerns uniform output distributions.
  • Branch analysis: For each nonzero stochastic branch, purity forces every contributing term to be zero or proportional to the same target-state projector.This constrains the possible input-output assignments generated by the wirings and Kraus operators.
  • Branch analysis: Unless the target is unsteerable, the input and final settings must be fully correlated or anticorrelated, with deterministic output processing.This follows from the restrictions imposed by the branch equations on the conditional distributions.
  • Case analysis: Up to relabellings, only three assignment types can contribute to a target assemblage.The proof checks these three cases and shows that each requires either an unsteerable target or the overlap relation.
  • Conversion criterion: The conversion condition preserves the pairwise state overlap up to an output relabelling α ∈ {0, 1}.The proof uses Bloch-sphere planar parametrization and Kraus-operator constraints to derive the relation or force the target to be unsteerable.

Appendix F: Non-existence of minimal-dimension steering bits

The appendix proves that no pure normalized minimal-dimensional assemblage can generate every assemblage through stochastic 1W-LOCCs. The proof proceeds by analyzing the constrained transformations and deriving contradictions for the relevant cases, while mixed-state generalization remains open.

  • Proof strategy: Assuming a pure normalized assemblage generates all minimal-dimensional assemblages via stochastic 1W-LOCCs leads to a contradiction.The assumption concerns d = s = r = 2 and transformations into every assemblage in the relevant family.
  • Proof strategy: For each admissible branch, normalization and nonzero Kraus operators force at least two output pairs to be proportional to the corresponding pure-state projectors.This structural constraint is used to restrict the possible assignments of inputs and outputs.
  • Case analysis: The first branch forces deterministic output distributions and full correlation or anticorrelation between input variables.The correlation pattern is encoded by a binary function fθ(ω).
  • Case analysis: The resulting angle condition cannot hold for two distinct members of the target family, yielding the required contradiction.The fixed parameters would require α01 = 0 = α11 and ϕ10 = ϕ01 = ϕ11, incompatible with the stated assumptions.
  • Case analysis: The remaining branch likewise produces a contradiction by restricting the support of the relevant Kraus operator and comparing two different output pairs.This completes the proof for pure assemblages.
  • Scope boundary: The argument does not straightforwardly extend to mixed assemblages because pure components in a convex decomposition need not individually satisfy no-signaling.The authors leave the existence of minimal-dimension steering bits among mixed assemblages as an open question.
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