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All $d\otimes d$ dimensional entangled states are useful for the antidiscrimination of quantum measurements when $d$ is even

Satyaki Manna

TL;DR

The paper advances antidistinguishability in quantum information by proving that every $d\otimes d$ entangled state with even $d$ is useful for antidistinguishing three measurements, while a product probe cannot achieve this. It formalizes AMS and AME to relate measurement antidistinguishability to state antidistinguishability in reduced subsystems and provides explicit constructive schemes for qubit-qubit, $m=4p$, and $n=4p+2$ dimensional cases. The result extends the spirit of Piani and Watrous to antidistinguishability and highlights a clear limitation to odd dimensions, offering a pathway for future exploration of antidistinguishability in broader channel classes. Overall, it demonstrates a broad entanglement advantage for ruling out measurement alternatives in even-dimensional quantum systems.

Abstract

Piani and Watrous [Phys. Rev. Lett.102, 250501 (2009)] proved that all entangled states are useful for discrimination of quantum channels. We pose the same question in the context of antidiscrimination of quantum channels. We partially answer this by showing that for every $d\otimes d$ entangled state (with even $d$), there exist three projective measurements which are antidiscriminable (but not discriminable) with that input state but those three measurements are not antidiscriminable with the product probe.

All $d\otimes d$ dimensional entangled states are useful for the antidiscrimination of quantum measurements when $d$ is even

TL;DR

The paper advances antidistinguishability in quantum information by proving that every entangled state with even is useful for antidistinguishing three measurements, while a product probe cannot achieve this. It formalizes AMS and AME to relate measurement antidistinguishability to state antidistinguishability in reduced subsystems and provides explicit constructive schemes for qubit-qubit, , and dimensional cases. The result extends the spirit of Piani and Watrous to antidistinguishability and highlights a clear limitation to odd dimensions, offering a pathway for future exploration of antidistinguishability in broader channel classes. Overall, it demonstrates a broad entanglement advantage for ruling out measurement alternatives in even-dimensional quantum systems.

Abstract

Piani and Watrous [Phys. Rev. Lett.102, 250501 (2009)] proved that all entangled states are useful for discrimination of quantum channels. We pose the same question in the context of antidiscrimination of quantum channels. We partially answer this by showing that for every entangled state (with even ), there exist three projective measurements which are antidiscriminable (but not discriminable) with that input state but those three measurements are not antidiscriminable with the product probe.

Paper Structure

This paper contains 9 sections, 6 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Antidiscrimination of quantum states
  3. Antidiscrimination of quantum measurements
  4. Antidiscrimination with single system probe
  5. Antidiscrimination with entangled probe
  6. Results
  7. Proof of Theorem 1
  8. Proof of Theorem 2
  9. Proof of Theorem 3

Key Result

Lemma 1

The sufficient condition for a set of qubit states $\{\rho_i\}_{i=1}^n$ to be anti-distinguishable is the following: $\sum_i\mu_i\rho_i=\mathbbm{1}$, for some positive real numbers $\mu_i$.

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1
  • Theorem 2
  • Theorem 3