PaperlayerView exact revision on arXiv

Source-linked AI summary

All sets of incompatible measurements give an advantage in quantum state discrimination

Paul Skrzypczyk, Ivan Šupić, Daniel Cavalcanti

arXiv:1901.00816v1quant-ph

TL;DR

The paper asks how measurement incompatibility relates to usefulness in quantum state discrimination. It shows that the Robustness of Incompatibility exactly quantifies the optimal advantage in a suitably chosen task, and that QSD comparisons characterize the resource-theory ordering of measurement sets.

Abstract

from arXiv · show

Some quantum measurements can not be performed simultaneously, i.e. they are incompatible. Here we show that every set of incompatible measurements provides an advantage over compatible ones in a suitably chosen quantum state discrimination task. This is proven by showing that the Robustness of Incompatibility, a quantifier of how much noise a set of measurements tolerates before becoming compatible, has an operational interpretation as the advantage in an optimally chosen discrimination task. We also show that if we take a resource-theory perspective of measurement incompatibility, then the guessing probability in discrimination tasks of this type forms a complete set of monotones that completely characterize the partial order in the resource theory. Finally, we make use of previously known relations between measurement incompatibility and Einstein-Podolsky-Rosen steering to also relate the later with quantum state discrimination.

INTRODUCTION

Measurement incompatibility arises when measurements cannot be performed simultaneously, a feature linked to applications in quantum information. For general POVMs, joint measurability provides the relevant notion, and the paper connects incompatibility to advantages in state discrimination.

INTRODUCTION

  • Incompatible observables cannot yield precise information simultaneously, yet this feature underlies applications including quantum-key-distribution security and device-independent applications [1–3].Source: INTRODUCTION · p. 1
  • For general POVMs, incompatibility is captured by joint measurability rather than commutation alone.Source: INTRODUCTION · p. 1
  • A set is jointly measurable when a single parent measurement and classical post-processing can implement all its measurements.Source: INTRODUCTION · p. 1
  • The paper shows that incompatibility is equivalent to an advantage over compatible measurements in a suitably chosen discrimination task with multiple state ensembles.Source: INTRODUCTION · p. 1

INCOMPATIBILITY AND ADVANTAGE IN QUANTUM STATE DISCRIMINATION

The section establishes that the Robustness of Incompatibility exactly quantifies the greatest relative advantage a measurement set can achieve over a single-measurement strategy in an appropriately chosen quantum state discrimination task. It also sketches the matching upper bound and construction of a task attaining it.

INCOMPATIBILITY AND ADVANTAGE IN QUANTUM STATE DISCRIMINATION

RESOURCE THEORY OF INCOMPATIBILITY

The section treats compatible measurements as free objects and measurement simulations as allowed transformations, then relates those transformations to performance in QSD games. The games provide operational criteria that completely characterize the resource-theory partial order.

RESOURCE THEORY OF INCOMPATIBILITY

EPR STEERING AND ENTANGLEMENT-BASED QSD

The paper connects measurement incompatibility and EPR steering to an entanglement-based state-discrimination game. In the 1SDI setting, steering robustness lower-bounds the guessing advantage, and the bound is tight for pure entangled states.

EPR STEERING AND ENTANGLEMENT-BASED QSD

  • Steering robustness measures the minimum added noise needed for an assemblage to admit a local-hidden-state decomposition.The paper describes a modified steering robustness whose noise must have the same reduced state as the input assemblage.Source: EPR STEERING AND ENTANGLEMENT-BASED QSD · p. 4
  • The entanglement-based game asks Alice to guess Bob’s measurement outcome using her measurement on the shared state.Bob’s measurement can be viewed as remotely preparing the state Alice receives.Source: EPR STEERING AND ENTANGLEMENT-BASED QSD · p. 4
  • For pure entangled states, the 1SDI lower bound on the guessing advantage is tight.The steering robustness equals the incompatibility robustness in this setting.Source: EPR STEERING AND ENTANGLEMENT-BASED QSD · p. 4
  • In the 1SDI paradigm, steering robustness provides a lower bound on the advantage from using incompatible rather than compatible measurements.The measurements are treated as a black box, without assuming knowledge of their form or the system dimension.Source: EPR STEERING AND ENTANGLEMENT-BASED QSD · p. 4

CONCLUSIONS

The conclusions connect measurement incompatibility with quantum-state discrimination and identify discrimination-game guessing probabilities as complete criteria for measurement simulation. The paper also leaves extensions to partial incompatibility and multiple parent measurements for future work.

CONCLUSIONS

  • Discrimination-game guessing probabilities form a complete set of criteria for whether one measurement set can simulate another.The paper frames these criteria as monotones in a resource theory of measurement incompatibility.Source: CONCLUSIONS · p. 4
  • The paper gives measurement incompatibility an operational interpretation through its connection to quantum-state discrimination.It relates robustness of incompatibility to usefulness in a natural quantum-information game.Source: CONCLUSIONS · p. 4
  • Future extensions include characterizing pairwise-compatible but jointly incompatible sets and allowing multiple parent measurements.The paper leaves both questions open for future work.Source: CONCLUSIONS · p. 4

Note added

The note added identifies two related papers that became known to the authors while preparing the manuscript.

Note added

  • Two related papers are identified as [26] and [27].The authors list C. Carmeli, T. Heinosaari, and A. Toigo [26], and R. Uola et al. [27].Source: Note added · p. 4

Incompatibility Robustness – primal SDP formulation

The appendix shows how the incompatibility-robustness constraints yield a primal semidefinite program. Deterministic decompositions make the parent measurement explicit and leave linear constraints in the optimization.

Incompatibility Robustness – primal SDP formulation

Incompatibility Robustness – dual formulation

The dual SDP is derived from the primal problem through its Lagrangian. Under the stated strict-feasibility condition, the primal and dual optimal values coincide.

Incompatibility Robustness – dual formulation

  • The Lagrangian introduces positive-semidefinite dual variables ω_ax and y_a, alongside an unrestricted variable X.These variables are constrained so the Lagrangian lower-bounds the primal objective whenever the primal constraints hold.Source: Incompatibility Robustness – dual… · p. 6
  • The Lagrangian becomes independent of primal variables when tr X = 1 and X = Σ_a,x ω_ax D_a(a|x) + y_a for every a.Under these constraints, the Lagrangian equals the dual objective.Source: Incompatibility Robustness – dual… · p. 6
  • Strict dual feasibility and finite primal and dual problems ensure strong duality and equivalence of their optimal values.One stated strictly feasible choice is X = 1/d and ω_ax = α1, with 1/(nd) > α > 0.Source: Incompatibility Robustness – dual… · p. 6

Upper bound on the advantage in QSD from the primal SDP

The Robustness of Incompatibility upper-bounds the advantage of incompatible measurements over a single measurement in every quantum state discrimination game considered. The proof compares the game’s success probability with that attainable using a single measurement.

Upper bound on the advantage in QSD from the primal SDP

  • For each game, the proof bounds a success expression by the optimal single-measurement success probability.The intermediate expression is no larger because it restricts the strategy available to the single measurement.Source: Upper bound on the advantage in QSD… · p. 7
  • Because the bound holds for all ensemble collections and allowed strategies, maximizing over them yields the bound for every game considered.The right-hand side becomes the optimal success with incompatible measurements.Source: Upper bound on the advantage in QSD… · p. 7
  • 1 + IR({M_x}) upper-bounds the advantage in any quantum state discrimination game considered.The comparison is with the optimal success probability achievable using a single measurement.Source: Upper bound on the advantage in QSD… · pp. 6–7

Lower bound

A carefully chosen discrimination game attains the Robustness of Incompatibility bound. The paper also shows that guessing probabilities across these games completely characterize measurement simulation, with the robustness itself monotone under simulation.

Lower bound

  • A carefully chosen game attains the bound, giving an advantage of 1 + IR({M_x}) over compatible measurements.The construction uses a strategy that can be reduced without loss to a valid POVM.Source: Lower bound · pp. 7–8

Monotones for measurement simulation

  • Guessing probabilities across all discrimination games form a complete set of monotones characterizing the partial order of measurement simulation.The paper also establishes that the Robustness of Incompatibility is monotone under measurement simulation.Source: Monotones for measurement simulation · p. 10
Loading 1901.00816v1…