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Quantum steering: a review with focus on semidefinite programming

D. Cavalcanti, P. Skrzypczyk

arXiv:1604.00501v2quant-ph

TL;DR

Quantum steering studies entanglement tests in which one party’s measurements are uncharacterised, creating an asymmetric setting between entanglement and Bell nonlocality. This review develops semidefinite-programming techniques for steering detection, quantification, and applications, while surveying related results and providing code. It presents SDP-based methods as efficient tools for these problems and notes open practical directions for asymmetric implementations and experimentally friendly inequalities.

  • Problem

    Determining whether states are unsteerable is hard because all assemblages generated by arbitrary measurements must be checked for local-hidden-state models.

  • Method

    The review characterises steering through semidefinite programming and applies related SDPs to detection, quantification, applications, and unsteerable-state searches.

  • Results

    The review provides SDP-based methods for steering detection, quantification, and applications, together with a collection of codes for studying these topics.

  • Takeaways & Limitations

    SDP techniques provide a framework for studying steering and its connections to asymmetric quantum-information tasks and other quantum properties.

Abstract

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Quantum steering refers to the non-classical correlations that can be observed between the outcomes of measurements applied on half of an entangled state and the resulting post-measured states that are left with the other party. From an operational point of view, a steering test can be seen as an entanglement test where one of the parties performs uncharacterised measurements. Thus, quantum steering is a form of quantum inseparability that lies in between the well-known notions of Bell nonlocality and entanglement. Moreover, quantum steering is also related to several asymmetric quantum information protocols where some of the parties are considered untrusted. Because of these facts, quantum steering has received a lot of attention both theoretically and experimentally. The main goal of this review is to give an overview of how to characterise quantum steering through semidefinite programming. This characterisation provides efficient numerical methods to address a number of problems, including steering detection, quantification, and applications. We also give a brief overview of some important results that are not directly related to semidefinite programming. Finally, we make available a collection of semidefinite programming codes that can be used to study the topics discussed in this article

D. Codes

The review frames steering through uncharacterised measurements on one side and develops semidefinite-programming methods for testing assemblages and related quantum states. It also connects these methods to steering witnesses, measurement incompatibility, and practical applications.

  • II. DEFINITION OF QUANTUM STEERING: Steering uses black-box measurements for Alice while Bob reconstructs the conditional-state assemblage through tomography.The assemblage consists of unnormalised states σa|x = p(a|x)ρa|x.
  • A. Quantum steering as the impossibility of local-hidden-state models: A steering assemblage is one that lacks a local-hidden-state decomposition; a state is unsteerable when every local measurement produces an LHS assemblage.Finite measurement and outcome sets make this membership problem computable through semidefinite programming.
  • II. DEFINITION OF QUANTUM STEERING: Steering is asymmetric and can certify both shared entanglement and incompatibility of the uncharacterised party’s measurements.A state may be steerable from Alice to Bob but unsteerable in the reverse direction.
  • A. The membership problem: Finite LHS models replace integrals over hidden variables with finite sums over deterministic strategies, simplifying SDP-based steering tests.The deterministic distributions fix one outcome for each measurement choice.
  • A. The membership problem: The SDP test returns a negative optimum for steering and a non-negative optimum when the assemblage admits an LHS model.The latter case corresponds to all auxiliary operators being positive semidefinite.

B. Optimal steering inequalities

The dual of the assemblage-membership SDP produces optimal steering inequalities. The review illustrates this construction for Pauli measurements on a two-qubit Werner state and relates the result to a known linear inequality.

  • A. The membership problem: The SDP dual returns Hermitian operators Fa|x that define a steering inequality obeyed by all LHS assemblages.The observed assemblage violates the inequality when the dual objective is negative.
  • 1. Example: the 2-qubit Werner state: For two Pauli measurements on a two-qubit Werner state, the dual constraints reduce to four positivity conditions over deterministic strategies.These strategies correspond to the four assignments of outcomes for the two measurements.
  • 1. Example: the 2-qubit Werner state: The symmetric ansatz sets αa|x = 1/16 and ma|x = m, with the constraints saturated at m = 1/8.The vectors are aligned or anti-aligned with the measurement directions.
  • 1. Example: the 2-qubit Werner state: The two-measurement Werner-state example yields a violation whenever w > 1/2.This matches the stated two-measurement steering threshold for the example.
  • 1. Example: the 2-qubit Werner state: With three Pauli measurements, the optimal construction coincides with the well-known linear steering inequality first derived in Ref..The corresponding observables are the Pauli directions X, Y, and Z.

C. Further results on steering detection

Steering detection extends beyond SDP tests through steering inequalities and finite-measurement constructions. SDP-based methods can certify unsteerability, including for entangled states, by replacing infinitely many measurements with finite checks and noise transformations.

  • Steering inequalities: Detection-efficiency threshold 1/mA is tight for Alice’s measurements: below it, no steering can be demonstrated.The result applies to the cited inequality and mA measurement setting.
  • Steering inequalities: Some steering inequalities exhibit violations that increase indefinitely with the dimension of the measured system.One construction uses mutually unbiased bases; other examples allow more general measurements and state-preparation or measurement errors.
  • SDP-based state tests: SDP tests certify unsteerability either for a specific state or for states found within a chosen region of state space.These tests provide sufficient, rather than generally complete, certification methods.
  • SDP-based state tests: The finite-measurement method replaces infinitely many measurements with a finite LHS-model search on a potentially unphysical operator.A polytope containing a noisy-measurement ball enables this reduction.
  • SDP-based state tests: The resulting operator need not be a physical state; after white noise is applied on one subsystem, it must become the target state ρAB.If it has an LHS model for the finite measurements, the target is unsteerable for all projective measurements.
  • SDP-based state tests: Random entanglement-witness SDPs found thousands of new unsteerable entangled states.Negative SDP solutions certify entanglement through witness violation while establishing unsteerability through the model constraints.

D. Further results on LHS models

LHS models connect unsteerability with local hidden-variable descriptions and motivate SDP-based steering quantifiers. The steering weight admits a dual interpretation as an optimized steering inequality, with strong duality guaranteed.

  • LHS models: Some entangled states are nevertheless unsteerable because local hidden-variable models can also serve as local-hidden-state models.Examples include Werner states and other state classes with established LHV or LHS models.
  • Steering weight: The steering weight quantifies the minimum non-LHS fraction required when an assemblage is decomposed into a generic assemblage and an LHS assemblage.It is motivated by analogous decompositions for entanglement and nonlocality.
  • Steering weight: The steering-weight optimization can be rewritten as an SDP by combining positivity, decomposition, and deterministic-strategy constraints.The derivation introduces variables associated with the LHS components and steering-function coefficients.
  • Steering weight: Strong duality holds for the steering-weight SDP because the dual is strictly feasible.Choosing Fa|x = α11 for sufficiently large α satisfies the dual constraints strictly.
  • Steering weight: The dual program searches for the optimal positive linear steering functional for the supplied assemblage, with LHS bound βLHS = 1.This gives the steering weight an operational interpretation through steering-inequality optimization.

B. Robustness-based steering quantifiers

Robustness-based quantifiers measure the noise needed to make an assemblage LHS and can be formulated as SDPs. The framework supports several noise sets and can also optimize steering over measurement strategies for a fixed state.

  • Robustness quantifiers: Steering robustness measures how much noise must be added to an assemblage for it to admit an LHS model.The noise set determines the specific robustness quantifier.
  • Robustness quantifiers: Robustness programs become SDPs after rescaling hidden states and replacing the noise set with its conic hull.Positive-semidefinite constraints and linear matrix inequalities are preserved under this reformulation.
  • Robustness quantifiers: The dual of Steering Robustness has an interpretation in terms of steering-inequality violation, and strong duality holds.Strict feasibility of the dual underlies the equality between primal and dual formulations.
  • Robustness quantifiers: The Steering Robustness uses all valid assemblages as the noise set, while LHS-Robustness and Random Steering Robustness use LHS assemblages and the maximally mixed assemblage, respectively.These choices correspond to distinct operational noise models.
  • State-level quantification: Optimizing a steering quantifier for a fixed state ranges over Alice’s measurement strategies, potentially including infinitely many measurements and arbitrary outcome numbers.With fixed measurement and outcome counts, a see-saw algorithm can heuristically search for strategies.
  • State-level quantification: For d-dimensional systems, POVMs with at most d^2 outcomes suffice when extremal POVMs can be used under convexity.This restricts the outcome count in the fixed-state optimization.

VI. QUANTUM STEERING IN MULTIPARTITE SCENARIOS

Multipartite steering generalizes one-sided device independence to asymmetric networks with multiple uncharacterised parties and multiple separability notions. In the tripartite case, SDP outer approximations test whether assemblages are compatible with non-steering structures.

  • Tripartite scenarios: Tripartite steering has one-sided and two-sided device-independent variants, depending on whether Alice alone or Alice and Bob use uncharacterised measurements.The remaining parties’ assemblages are then analysed.
  • Tripartite scenarios: Multipartite steering covers asymmetric network scenarios intermediate between entanglement and Bell nonlocality, with some devices characterised and others uncharacterised.Each scenario has a corresponding multipartite assemblage capturing observable data.
  • Separability structures: Multipartite steering depends on the chosen separability notion, including fully separable, bi-separable, and genuinely multipartite entangled states.States that are not bi-separable are genuinely multipartite entangled.
  • Separability structures: With one uncharacterised device, separable-state assemblages require no steering between Alice and Bob-Charlie and separable states between Bob and Charlie.A steerable Alice-Bob state can violate this structure even when Charlie is uncorrelated.
  • Separability structures: With two uncharacterised devices, bi-separable assemblages separate cases where only Bob, only Alice, or neither Alice and Bob can steer Charlie.Alice and Bob may still share local or nonlocal correlations depending on the term.
  • SDP detection: SDPs provide outer approximations to the non-steering assemblage set; lying outside such an approximation certifies steering.The method tests necessary conditions for assemblages compatible with non-steering models.

B. Quantumness testing

The steering-NPA hierarchy uses SDP feasibility tests to approximate whether multipartite assemblages have quantum realisations. Related SDPs test multipartite and genuine multipartite steering, with infeasibility certifying incompatibility with specified separability structures.

  • Quantum realisations: The steering-NPA hierarchy generalises NPA moment-matrix tests to assemblages and converges as the hierarchy level k increases.Each level tests necessary conditions for a quantum realisation; convergence occurs as k →∞.
  • Quantum realisations: The hierarchy constructs operator strings up to length k and maps them through a completely positive map whose output must be positive semidefinite.Projective measurement orthogonality removes vanishing strings, while POVMs can be handled through a projective extension.
  • Multipartite steering: Multipartite steering tests relax non-directly applicable constraints into approximate SDP feasibility problems for specified patterns of uncharacterised devices.The formulations use k-symmetric extensions for components of local-hidden-state models.
  • Multipartite steering: An infeasible test certifies multipartite steering because the assemblage is incompatible with the structure produced by fully separable states.Passing the test yields no conclusion at that hierarchy level.
  • Genuine multipartite steering: An infeasible genuine-steering SDP certifies incompatibility with biseparable states and therefore demonstrates genuine multipartite steering.The same framework treats scenarios with one or two uncharacterised devices.
  • Examples: The review illustrates these tests with white-noise robustness bounds for tripartite GHZ and W states under multipartite and genuine multipartite steering criteria.The table considers three measurements by one or two parties using uncharacterised devices.

D. Multipartite steering inequalities

Multipartite steering inequalities arise as duals of SDP tests and provide witnesses for steering. The framework extends to general multipartite patterns, but increasing the number of parties makes the resulting numerical tests difficult to solve efficiently.

  • Multipartite steering inequalities: SDP duality converts multipartite steering tests into steering inequalities whose operators satisfy PSD constraints or linear matrix inequalities.These constraints ensure non-steering assemblages obey the inequality conditions.
  • Multipartite steering inequalities: A steering functional value β = tr Σ F_a|xσ^BC_a|x below zero certifies multipartite steering.The inequality operators are chosen according to the tested entanglement class.
  • Multipartite steering inequalities: The same dual construction applies when two devices are uncharacterised, with coefficients indexed by both parties’ outcomes and settings.The corresponding operator constraints exclude assemblages lacking the tested multipartite steering property.
  • Generalisations: For general N-partite systems, one specifies an entanglement class and a characterised-device pattern, then relaxes non-directly applicable constraints into an approximate SDP.Possible classes include full separability, fixed bipartition separability, and convex combinations across partitions.
  • Generalisations: As the number of parties increases, quantum-realisation and separability constraints become difficult enough that numerical techniques may no longer solve the tests efficiently.The steering-NPA hierarchy and multipartite k-extendibility provide relaxations, but the multipartite setting soon becomes intractable.
  • Post-quantum steering: In tripartite settings, SDP methods can test post-quantum steering by combining quantum-realisation bounds with locality conditions for additional measurements.A steering-NPA bound can certify that an assemblage is post-quantum.
  • Post-quantum steering: Post-quantum steering can produce only quantum-realizable nonlocal behaviours, making it distinct from post-quantum nonlocality.The review contrasts this with assemblages built directly from non-quantum nonsignalling behaviours, such as Popescu-Rohrlich boxes.

G. Further results on multipartite steering

The review extends semidefinite-programming methods to continuous-variable steering and constructs moment-matrix tests from experimentally accessible moments. It also gives a feasibility criterion whose infeasibility certifies steering, with a homodyne example.

  • Continuous-variable systems: Continuous-variable steering requires moment-matrix methods because the finite-dimensional SDP methods are not suitable for infinite-dimensional systems.The approach covers continuous outcomes, infinite-dimensional assemblages, and multipartite continuous-variable systems.
  • Experimental asymmetry: Bob’s characterised measurements provide access to expectation values of operators on his system, while Alice’s measurements remain uncharacterised and even her Hilbert-space dimension is unspecified.This preserves the asymmetric, one-sided device-independent structure of the steering scenario.
  • Moment-matrix construction: The moment matrix is positive semidefinite and obeys constraints from accessible Alice–Bob moments, Bob-only moments, and algebraic relations among known observables.These conditions are combined with additional structure corresponding to separable states and commuting Alice observables.
  • Steering test: A steering test asks whether a valid moment matrix consistent with the available data exists under the no-steering constraints.If no such matrix exists, the data certify steering and imply non-commuting Alice observables on an entangled state.
  • Example: For the single-photon NOON-state example, homodyne measurements demonstrate steering when η > 2/3 using the second-level test Γ(2).Bob estimates his local state in this construction.

1. Example: Quadrature measurements

The quadrature example reduces the steering test to a partially specified moment matrix with one unknown entry. Positive-semidefinite feasibility distinguishes data compatible with no steering from data that demonstrate steering.

  • Moment-matrix setup: For k = 1, the moment-matrix operator set contains the identity, Alice’s two observables, and Bob’s position and momentum observables.Bob’s observables are B0 = q̂ and B1 = p̂.
  • Accessible moments: All entries except ⟨A0A1⟩ and ⟨A1A0⟩ are accessible or constrained, leaving one unknown element when Alice’s observables commute.The equality ⟨A0A1⟩ = ⟨A1A0⟩ follows from the no-steering assumption.
  • Feasibility test: If a value z makes the partially specified matrix positive semidefinite, the data are compatible with no steering; otherwise, the feasibility SDP certifies steering.The unknown is identified as z = ⟨A0A1⟩ = ⟨A1A0⟩.
  • Relation to prior criteria: For Gaussian continuous-variable states in standard form, the moment-matrix test coincides with earlier criteria for Gaussian steering by Gaussian measurements.This correspondence was established for the reviewed test.

B. Further results on continuous-variable quantum steering

The review surveys continuous-variable steering, entanglement quantification, and robustness-based constructions beyond the basic SDP framework. Moment matrices and assemblage mappings yield one-sided device-independent lower bounds on several entanglement measures.

  • Continuous-variable steering: Continuous-variable steering can be studied through uncertainty-relation violations, general steering inequalities, multipartite constructions, and experimental tests.Conditioning on Alice’s results can reveal an apparent violation of Bob’s uncertainty relation that signals shared entanglement.
  • Robustness quantifiers: Steering robustness provides a one-sided device-independent lower bound on entanglement robustness, and steering weight similarly lower-bounds the Best-Separable-Approximation quantifier.The bound follows by mapping states in the entanglement-noise set to assemblages in the corresponding steering set.
  • PPT and moment matrices: Moment matrices provide necessary conditions for assemblages arising from PPT states by adding partial-transpose structure to the positive-semidefinite completion problem.This framework was used to study whether PPT entangled states can demonstrate steering.
  • PPT steering: PPT entangled states can indeed demonstrate steering, despite the earlier technique not reaching a definite conclusion on that question.The later result used a different technique.
  • Convex-roof measures: The moment-matrix approach also lower-bounds convex-roof entanglement measures, including linear entropy of entanglement, through SDP relaxations.Trace-preserving local completely positive maps are required for this construction.

B. One-sided device-independent estimation of measurement incompatibility

Assemblages can quantify the incompatibility of Alice’s uncharacterised measurements through modified steering quantifiers. These quantities provide one-sided device-independent lower bounds on several robustness-based measurement-incompatibility measures.

  • Measurement incompatibility: Steering certifies that Alice’s measurements are incompatible, meaning they cannot be performed jointly.The assemblage also contains quantitative information about the amount of measurement incompatibility.
  • Incompatibility quantifiers: Incompatibility weight quantifies the maximal jointly measurable component in a convex decomposition of Alice’s measurements.The decomposition combines an arbitrary measurement set with jointly measurable measurements.
  • Incompatibility quantifiers: Incompatibility robustness is the minimum noise parameter t for which mixing Alice’s measurements with another set yields jointly measurable measurements.Random robustness fixes the added measurements to uniform effects, while jointly-measurable robustness restricts them to a jointly measurable set.
  • Device-independent estimation: Modified steering quantifiers provide lower bounds on incompatibility weight, incompatibility robustness, and related robustness measures.The listed quantities include Consistent Steering Weight, Consistent Steering Robustness, Reduced-State Steering Robustness, and Consistent LHS Steering Robustness.
  • Steering quantifiers: Consistent and reduced-state assemblage robustnesses minimize added noise while preserving the reduced state for Bob.The definitions require the relevant mixture to become an LHS assemblage.

C. Sub-channel discrimination

Steering is connected to sub-channel discrimination and randomness certification through assemblage-based semidefinite programs. These formulations bound Eve’s guessing probability and relate steerable states to advantages in discrimination tasks.

  • Sub-channel discrimination: Steerable states can provide an advantage in sub-channel discrimination when Alice and Bob use local measurements assisted by one-way communication.For every steerable state, a suitable one-sided channel exists; the cited passage states that the advantage is exact but truncates its value.
  • Local randomness: Local randomness certification quantifies how well Eve can guess Alice’s outcome for a chosen measurement.The guessing probability is modeled using a tripartite state, Alice’s measurements, and Eve’s measurement on her subsystem.
  • Local randomness: If Pg(x*) < 1, Eve cannot predict Alice’s outcome with certainty, certifying intrinsic randomness.Perfect guessing corresponds to Pg(x*) = 1; lower values indicate unpredictability of the selected outcome.
  • Local randomness: An SDP maximizes Eve’s guessing probability over quantum realizations consistent with Bob’s observed assemblage, no-signalling, and positivity.The first constraint expresses Bob’s assemblage as an average over Eve-conditioned assemblages, while normalization follows automatically.
  • Steering inequalities: Steering duality expresses guessing-probability bounds through steering inequalities whose values upper-bound the determinism of the selected outcomes.The constraint is interpreted by applying the inequality to an arbitrary assemblage and comparing it with Eve’s guessing probability.
  • Global randomness: For joint randomness, an SDP maximizes Eve’s probability of correctly guessing both Alice’s and Bob’s outcomes for specified measurements.The optimization includes Bob’s known POVM and imposes consistency with the observed assemblage, no-signalling, and positivity.

G. Further applications in the steering scenario

The review applies steering methods to key distribution, secret sharing, teleportation, experimental optimization, and detection efficiency. It also identifies computational, practical, foundational, and multipartite directions that remain open.

  • Applications: One-sided device-independent quantum key distribution bounds secret-key rates using uncertainty relations for Bob’s measurements without assumptions about Alice’s implementation.The review presents this as an early application of steering to asymmetric cryptographic settings.
  • Applications: Steering has qualitative and formal connections to quantum secret sharing and secure quantum teleportation.The review reports that Gaussian states useful for secure teleportation are necessarily steerable in both directions.
  • Experimental optimization: See-saw optimization searches alternately for steering inequalities and states or measurements suited to a fixed experimental setting.The review considers fixed measurements with optimized states, fixed states with optimized measurements, and detection-efficiency constraints.
  • Experimental optimization: Bound entangled states can violate steering inequalities, showing that positive-partial-transpose states may nevertheless be steerable.The review identifies this as the first demonstration of steering by bound entangled states.
  • Experimental optimization: Random projective measurements are reported as good candidates for demonstrating steering with highly noisy quantum states.This result concerns the setting where the state is fixed and the measurements are optimized or selected.
  • Detection efficiency: Detection-efficiency analyses treat the no-click event as an additional measurement outcome in the observed assemblage.For mA projective measurements on any pure entangled state, steering can be demonstrated when η > 1/mA; no steering can be demonstrated when η ≤ 1/mA.
  • Open questions: The review identifies missing necessary-and-sufficient steerability criteria and computational difficulty when measurement inputs or outputs become large.It also highlights the need for experimentally friendly inequalities and applications where genuine multipartite steering is the key resource.
  • Open questions: Foundational open questions include the extent of post-quantum steering and whether it offers advantages beyond quantum mechanics.The review also points to generalized scenarios with quantum inputs or promised dimensional constraints.

Appendix A: Semidefinite Programming Basics

Semidefinite programs optimize linear objectives over positive semidefinite operators subject to linear matrix constraints, with dual formulations that provide bounds and often equal optimal values. The review extends this framework through see-saw optimization and SDP hierarchies for steering-related problems.

  • SDP formulation: Semidefinite programs are convex optimization problems that can be solved efficiently in many cases of interest.The review introduces them as the mathematical foundation for its steering-characterization framework.
  • SDP formulation: The primal SDP maximizes tr[AX] over positive semidefinite X satisfying the linear matrix equality Φ(X) = B.The objective is a real linear function, and feasibility requires both positivity and the matrix equality constraint.
  • Duality: The dual introduces Lagrange multipliers and minimizes tr[YB] subject to Z = Φ†(Y) − A and Z ≥ 0.The dual variables enforce the relation between the primal objective, the constraint map, and positivity.
  • Duality: Weak duality guarantees that the dual optimum upper-bounds the primal optimum, while strict feasibility commonly yields strong duality and equality.Under strong duality, primal and dual optimal values coincide.
  • See-saw optimization: When an optimization variable enters the SDP constraints, the resulting problem can be handled heuristically by alternating SDP optimizations over the coupled variables.Holding one variable fixed makes the objective and constraints suitable for optimizing the other, producing an iterated see-saw sequence.
  • See-saw optimization: The see-saw procedure iterates until the generated variable sequence converges up to numerical precision.The review applies this approach to optimal measurements and examples of post-quantum steering.
  • Separability testing: The DPS hierarchy supplies SDP outer approximations to the separable set that converge to separability in the limit.An assemblage has a k-symmetric PPT extension when every member has such an extension, yielding a feasibility SDP for testing this relaxation.
  • Implementation: The accompanying MATLAB code implements the review’s SDPs using CVX and QETLAB, including assemblage generation and steering certification.The examples also provide routines for normalization, non-signalling checks, steering inequalities, and see-saw optimization.
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