Source-linked AI summary
Quantifying Einstein-Podolsky-Rosen steering
Paul Skrzypczyk, Miguel Navascues, Daniel Cavalcanti
TL;DR
The paper quantifies steerability through the steerable weight and its dual operational interpretation via optimal steering inequalities. It applies this framework to pure entangled states, the antisymmetric state, and erasure states, establishing maximal steerability and one-way steering results.
Problem
A quantitative, operational assessment of steerability is needed to relate steerable weight to violations of steering inequalities.
Method
The paper uses steerable weight and dual semidefinite-program formulations, constructing steering inequalities and local hidden state models for selected quantum states.
Results
Every pure entangled state and the normalized antisymmetric projector have steerable weight one, while erasure states exhibit one-way steering under specified measurement conditions.
Takeaways & Limitations
Steering inequalities operationally bound steerable weight, and erasure states provide directional steering asymmetry with local hidden state models in the reverse direction.
Abstract
from arXiv · showhide
Einstein-Podolsky-Rosen (EPR) steering is a form of bipartite quantum correlation that is intermediate between entanglement and Bell nonlocality. It allows for entanglement certification when the measurements performed by one of the parties are not characterised (or are untrusted) and has applications in quantum key distribution. Despite its foundational and applied importance, EPR steering lacks a quantitative assessment. Here we propose a way of quantifying this phenomenon and use it to study the steerability of several quantum states. In particular we show that every pure entangled state is maximally steerable, the projector onto the anti-symmetric subspace is maximally steerable for all dimensions, we provide a new example of one-way steering, and give strong support that states with positive-partial-transposition are not steerable.
A: Local Hidden State Assemblages
The paper reduces local hidden state models to finite mixtures indexed by Alice’s deterministic strategies, with positive sub-normalised states assigned to each strategy.
- Any probability distribution over Alice’s outputs can be decomposed into deterministic strategies for fixed measurement settings and outcomes.Each deterministic strategy specifies one definite outcome for every measurement choice.
- A local hidden state model can therefore be written using one sub-normalised Bob state for each deterministic strategy.The resulting finite model reproduces the assemblage without loss of generality.
- The operators assigned to hidden strategies are positive semidefinite, and their traces give the probabilities of the corresponding hidden variables.
- The hidden-state weights sum to one, completing the local hidden state characterization.
B: Deriving the SDP for steerable weight
The steerable-weight optimization is transformed into a semidefinite program by eliminating redundant variables and expressing assemblage constraints through positive semidefinite operators.
- The derivation begins from the optimization problem defining the steerable weight and rewrites it using the sets of unsteerable and steerable assemblages.
- The remaining optimization variables are the operators σλ associated with deterministic hidden strategies.
- Positivity constraints on the assemblage components allow the problem to be expressed through positive semidefinite operator inequalities.
- Consistency of the input assemblage ensures that the marginal-state constraint is automatically satisfied.
- After redefining variables and collecting the constraints, the optimization reaches the final SDP form.
C: Dual SDP for steerable weight: bounding SW by steering inequality violations.
The dual SDP interprets steerable weight through optimized steering-inequality violations: unsteerable assemblages attain the unviolated bound, while maximal violations identify maximal steerability.
- Dual variables Fa|x and Gλ are introduced for the two constraint sets, yielding a dual program for the steerable-weight optimization.
- The operators Fa|x define a linear steering inequality whose objective value is the value obtained by the input assemblage.
- The dual searches for the steering inequality giving the maximal violation for a specified assemblage.
- Strong duality gives steerable weight the operational interpretation 1 − µ∗, where µ∗ is determined by the optimal standardized steering inequality.
- A steering-inequality violation supplies an upper bound on µ∗ and therefore a lower bound on steerable weight.
D: All pure entangled states are maximally steerable
The paper proves that every entangled pure state can generate a maximally steerable assemblage using only two d-outcome projective measurements, via an explicitly constructed steering inequality.
- D: All pure entangled states are maximally steerable: Every pure entangled state of arbitrary dimension is shown to be steerable, with only 2d assemblage states needed from two d-outcome measurements.
- D: All pure entangled states are maximally steerable: The proof uses the dual steerable-weight characterization by constructing a steering inequality that is maximally violated by the generated assemblage.
- D: All pure entangled states are maximally steerable: Alice performs two maximal projective von Neumann measurements, producing an assemblage whose conditional states for Bob are pure.
- D: All pure entangled states are maximally steerable: The constructed operators Fa|x are positive semidefinite, have rank d −1, and give zero value on the corresponding assemblage elements.
- D: All pure entangled states are maximally steerable: For the two d-outcome measurements, the dual constraints must hold for all d2 deterministic pairs of outcomes.
- D: All pure entangled states are maximally steerable: Entanglement ensures the maximum overlap ξ is below one, so states prepared by the two measurements are nonparallel and the inequality constraints can be satisfied.
- D: All pure entangled states are maximally steerable: The resulting construction establishes maximal steerability for projective-measurement assemblages from pure entangled states, while broader mixed-state generalization is left open.
E: Local Hidden State model for 3 × 3 Werner state with MUBs
For the 3 × 3 Werner state, four mutually unbiased bases admit an explicit local hidden state model, reproducing Alice’s outcomes and Bob’s conditional states.
- Explicit model: Four mutually unbiased bases on the d = 3 Werner state are described by an explicit local hidden state model.The construction focuses on the normalized antisymmetric projector and extends to general Werner states by mixing with the identity assemblage.
- Hidden variables: The hidden variable takes one of 9 values with equal probability, while Alice uses deterministic outputs and Bob holds states specified by Table I.Each hidden-variable value determines Alice’s output for every measurement and a corresponding state for Bob.
- Reproducing the assemblage: For every input and outcome, exactly 3 hidden-variable values are compatible, and their associated Bob states are orthogonal to Alice’s projected state.Mixing those three states with equal probability produces the required state on Bob’s orthogonal subspace.
F: The antisymmetric state is maximally steerable
The normalized antisymmetric projector has steerable weight one: suitable projective measurements generate an assemblage that maximally violates a steering inequality.
- Main result: The normalized antisymmetric projector has steerable weight equal to one and is shown to be maximally steerable.The proof follows the strategy used for entangled pure states by constructing a maximally violated steering inequality.
- Assemblage construction: Projective von Neumann measurements with rank-one elements generate Bob’s assemblage from the antisymmetric state.The measurement elements are one-dimensional projectors onto vectors indexed by outcome and measurement choice.
- Steering inequality: The steering-inequality operators are chosen positive semidefinite and satisfy tr Fa|xσa|x = 0, so the assemblage attains value 0.The remaining task is selecting α and measurement vectors so the operators satisfy the inequality constraints.
- Constraint satisfaction: Choosing bases such that one vector from each basis always spans the whole space allows sufficiently large α to satisfy the inequality constraints.The paper argues that d random bases have this spanning property.
G: One way steering of the erasure state
The erasure state exhibits one-way steering: Alice-to-Bob assemblages admit local hidden state models under a measurement bound, whereas Bob-to-Alice steering is demonstrated with two projective measurements.
- State construction: The erasure state is a qutrit-qubit state obtained by sending Alice’s singlet subsystem through an erasure channel with parameter p and flag state |2⟩.This flag structure underlies the directional steering behavior.
- Alice-to-Bob direction: For p ≤ 1/k, every assemblage from k POVM measurements by Alice has a local hidden state model, regardless of the number of outcomes.This follows from a k-symmetric extension on Alice’s side.
- LHS construction: A k-symmetric extension yields the local hidden state model by assigning Alice’s k measurements to different extension copies and combining their outputs into one deterministic strategy.The construction uses a single POVM whose outcomes are strings containing the outputs of all k measurements.
- Bob-to-Alice direction: When Bob steers Alice, two Pauli measurements X and Z produce an assemblage that violates a steering inequality for every p > 0.The inequality operators satisfy the unsteerable bound, while the generated assemblage takes a lower value.
- One-way steering: At p = 1/k, the state is unsteerable from Alice for k or fewer POVMs but steerable from Bob with two projective measurements.The construction fails only in the limit k →∞, where infinite symmetric extension implies separability.