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Factorizability of multi-party quantum sequence discrimination under local operations and classical communication

Donghoon Ha, Jeong San Kim

TL;DR

This work investigates multi-party quantum sequence discrimination under LOCC constraints and derives conditions under which the optimal LOCC discrimination of a tensor-product sequence factorizes into the product of per-step optimizations. It introduces a tight characterization based on separable-operator duality (via $\mathrm{SEP}^*$) and Hermitian witnesses $H$ that certify when LOCC discrimination matches separable discrimination and, in certain cases, when it equals the global optimum. A key result provides a necessary-and-sufficient condition for when the optimal LOCC discrimination reduces to guessing the most probable state, with clear implications for quantum data hiding. The paper also demonstrates that factorization is not universal by presenting explicit non-factorizable examples, highlighting the nuanced role of entanglement and LOCC structure in multi-party discrimination tasks.

Abstract

We consider multi-party quantum sequence discrimination under local operations and classical communication(LOCC), and provide conditions under which the optimal LOCC discrimination of a multi-party quantum sequence ensemble can be factorized into that of each individual ensemble. In other words, the optimal LOCC discrimination of a multi-party quantum sequence ensemble can be achieved just by performing optimal LOCC discrimination independently at each step of the quantum sequence. We also illustrate through examples of multi-party quantum states that such factorizability of optimal LOCC discrimination is possible. We further establish a necessary and sufficient condition under which the optimal LOCC discrimination of a multi-party quantum state ensemble can be realized just by guessing the state with the largest probability. Our results can provide a useful application to investigate the fundamental limits of quantum data hiding.

Factorizability of multi-party quantum sequence discrimination under local operations and classical communication

TL;DR

This work investigates multi-party quantum sequence discrimination under LOCC constraints and derives conditions under which the optimal LOCC discrimination of a tensor-product sequence factorizes into the product of per-step optimizations. It introduces a tight characterization based on separable-operator duality (via ) and Hermitian witnesses that certify when LOCC discrimination matches separable discrimination and, in certain cases, when it equals the global optimum. A key result provides a necessary-and-sufficient condition for when the optimal LOCC discrimination reduces to guessing the most probable state, with clear implications for quantum data hiding. The paper also demonstrates that factorization is not universal by presenting explicit non-factorizable examples, highlighting the nuanced role of entanglement and LOCC structure in multi-party discrimination tasks.

Abstract

We consider multi-party quantum sequence discrimination under local operations and classical communication(LOCC), and provide conditions under which the optimal LOCC discrimination of a multi-party quantum sequence ensemble can be factorized into that of each individual ensemble. In other words, the optimal LOCC discrimination of a multi-party quantum sequence ensemble can be achieved just by performing optimal LOCC discrimination independently at each step of the quantum sequence. We also illustrate through examples of multi-party quantum states that such factorizability of optimal LOCC discrimination is possible. We further establish a necessary and sufficient condition under which the optimal LOCC discrimination of a multi-party quantum state ensemble can be realized just by guessing the state with the largest probability. Our results can provide a useful application to investigate the fundamental limits of quantum data hiding.

Paper Structure

This paper contains 7 sections, 7 theorems, 90 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Quantum state discrimination
  3. Quantum sequence discrimination
  4. LOCC discrimination of multi-party quantum sequences
  5. Multi-party quantum sequence discrimination
  6. Factorizable LOCC discrimination of quantum sequences
  7. Discussion

Key Result

Proposition 1

For a multi-party quantum state ensemble $\mathcal{E}=\{\eta_{i},\rho_{i}\}_{i=1}^{n}$ and a separable measurement $\mathcal{M}=\{M_{i}\}_{i=1}^{n}$, $\mathcal{M}$ provides $p_{\sf SEP}(\mathcal{E})$ if and only if there exists $H\in\mathbb{H}$ satisfying for all $i=1,\ldots,n$.

Figures (1)

  • Figure 1: Multi-party quantum sequence ensemble $\bigotimes_{l=1}^{L}\mathcal{E}^{l}=\{\eta_{\vec{c}},\rho_{\vec{c}}\}_{\vec{c}\in\mathbb{N}_{\vec{n}}}$. The $m$ parties, $\mathsf{A}_{1},\ldots,\mathsf{A}_{m}$, share a sequence of $L$ multi-party quantum states, $(\rho_{c_{1}}^{1},\ldots,\rho_{c_{L}}^{L})$. For each $l=1,\ldots,L$, the $l$th state $\rho_{c_{l}}^{l}$ is prepared from the ensemble $\mathcal{E}^{l}$ with the probability $\eta_{c_{l}}^{l}$, for $c_{l}=1,\ldots,n_{l}$. The quantum sequence $(\rho_{c_{1}}^{1},\ldots,\rho_{c_{L}}^{L})$ can be regarded as the tensor producted state $\rho_{\vec{c}}$ with the probability $\eta_{\vec{c}}$, for $\vec{c}=(c_{1},\ldots,c_{L})\in\mathbb{N}_{\vec{n}}$.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Proposition 1: ha20231
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Definition 3
  • Theorem 2
  • proof
  • ...and 8 more