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Detectors for local discrimination of sets of generalized Bell states

Cai-Hong Wang, Jiang-Tao Yuan, Ying-Hui Yang, Mao-Sheng Li, Shao-Ming Fei, Zhi-Hao Ma

TL;DR

This work develops a detector framework based on maximum commutative sets (MCSs) to decide LOCC distinguishability of generalized Bell state (GBS) sets in bipartite systems. It provides a concrete method (via Theorem 3.1) to determine all detectors for a given GBS set and demonstrates that detectors are not always required by constructing detector-free 4-GBS sets in even dimensions, while showing that detectors become almost necessary for one-way discrimination in 6 × 6 systems. The authors completely resolve the one-way LOCC discrimination problem for 4-GBS sets in C^6 ⊗ C^6, proving that a detector exists unless the difference set ΔS equals {(0,3),(3,0),(3,3)} up to LU-equivalence. The results illuminate the structural role of detectors in local discrimination, offer practical tools for analyzing LOCC distinguishability, and suggest further exploration for higher-dimensional GBS sets and LU-equivalence classes.

Abstract

A fundamental problem in quantum information processing is the discrimination among a set of orthogonal quantum states of a composite system under local operations and classical communication (LOCC). Corresponding to the LOCC indistinguishable sets of four ququad-ququad orthogonal maximally entangled states (MESs) constructed by Yu et al. [Phys. Rev. Lett. 109, 020506 (2012)], the maximum commutative sets (MCSs) were introduced as detectors for the local distinguishability of the set of generalized Bell states (GBSs), for which the detectors are sufficient to determine the LOCC distinguishability. In this work, we show how to determine all the detectors for a given GBS set. We construct also several 4-GBS sets without detectors, most of which are one-way LOCC indistinguishable and only one is one-way LOCC distinguishable, indicating that the detectors are not necessary for LOCC distinguishability. Furthermore, we show that for 4-GBS sets in quantum system $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$, the detectors are almost necessary for one-way LOCC distinguishability, except for one set in the sense of local unitary equivalence. The problem of one-way LOCC discrimination of 4-GBS sets in $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$ is completely resolved.

Detectors for local discrimination of sets of generalized Bell states

TL;DR

This work develops a detector framework based on maximum commutative sets (MCSs) to decide LOCC distinguishability of generalized Bell state (GBS) sets in bipartite systems. It provides a concrete method (via Theorem 3.1) to determine all detectors for a given GBS set and demonstrates that detectors are not always required by constructing detector-free 4-GBS sets in even dimensions, while showing that detectors become almost necessary for one-way discrimination in 6 × 6 systems. The authors completely resolve the one-way LOCC discrimination problem for 4-GBS sets in C^6 ⊗ C^6, proving that a detector exists unless the difference set ΔS equals {(0,3),(3,0),(3,3)} up to LU-equivalence. The results illuminate the structural role of detectors in local discrimination, offer practical tools for analyzing LOCC distinguishability, and suggest further exploration for higher-dimensional GBS sets and LU-equivalence classes.

Abstract

A fundamental problem in quantum information processing is the discrimination among a set of orthogonal quantum states of a composite system under local operations and classical communication (LOCC). Corresponding to the LOCC indistinguishable sets of four ququad-ququad orthogonal maximally entangled states (MESs) constructed by Yu et al. [Phys. Rev. Lett. 109, 020506 (2012)], the maximum commutative sets (MCSs) were introduced as detectors for the local distinguishability of the set of generalized Bell states (GBSs), for which the detectors are sufficient to determine the LOCC distinguishability. In this work, we show how to determine all the detectors for a given GBS set. We construct also several 4-GBS sets without detectors, most of which are one-way LOCC indistinguishable and only one is one-way LOCC distinguishable, indicating that the detectors are not necessary for LOCC distinguishability. Furthermore, we show that for 4-GBS sets in quantum system , the detectors are almost necessary for one-way LOCC distinguishability, except for one set in the sense of local unitary equivalence. The problem of one-way LOCC discrimination of 4-GBS sets in is completely resolved.

Paper Structure

This paper contains 10 sections, 27 equations, 2 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. F-equivalence and discriminant set
  4. Detectors
  5. Detectors of a GBS set in arbitrary dimensional systems
  6. Some 4-GBS sets without detectors
  7. A 4-GBS set in $\mathbb{C}^{d}\otimes\mathbb{C}^{d}$ with even $d$
  8. Three 4-GBS sets in $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$
  9. Detector is almost necessary for local distinguishability of 4-GBS sets in $\mathbb{C}^{6}\otimes\mathbb{C}^{6}$
  10. Conclusions

Figures (2)

  • Figure 1: Schematic diagram of the relationship between F-equivalence, discriminant sets and detectors: The larger elliptical region represents the set of all GBS sets with detectors.
  • Figure 2: Schematic diagram of the set $\Omega$ of all one-way LOCC distinguishable 4-GBS sets $\mathcal{S}$ in Theorem \ref{['th5.1']}.

Theorems & Definitions (6)

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