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Localization of joint quantum measurements on $\mathbb{C}^d \otimes \mathbb{C}^d$ by entangled resources with Schmidt number at most $d$

Seiseki Akibue, Jisho Miyazaki

TL;DR

This work provides a protocol-independent algebraic characterization of localizable joint measurements on bipartite quantum systems. It shows that a rank-1 PVM containing at least one element with maximal Schmidt rank is localizable with Schmidt number at most $d$ if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis; this reveals strict limits imposed by non-adaptive local operations. The authors also fully resolve the two-qubit case, showing that any rank-1 PVM localizable with a two-qubit entangled resource must be LU-equivalent to a product, Bell, or BB84-type basis, addressing a conjecture by Gisin and Del Santo. Extending to ideal measurements, localization occurs in nice Bell bases, tying locality constraints to the structure of unitary-error-basis families and Bell-type bases. Overall, the paper advances the understanding of what joint measurements can be performed with limited entanglement under non-adaptive LOCC and contributes algebraic tools for LOSR-oriented quantum information theory.

Abstract

Localizable measurements are joint quantum measurements that can be implemented using only non-adaptive local operations and shared entanglement. We provide a protocol-independent characterization of localizable projection-valued measures (PVMs) by exploiting algebraic structures that any such measurement must satisfy. We first show that a rank-1 PVM on $\mathbb{C}^d\otimes\mathbb{C}^d$ containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most $d$ if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. This reveals strong limitations imposed by non-adaptive local operations, in contrast to the adaptive setting where any joint measurement is implementable. We then completely characterize two-qubit rank-1 PVMs that can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo, and finally extend our characterization to ideal two-qudit measurements, strengthening earlier results.

Localization of joint quantum measurements on $\mathbb{C}^d \otimes \mathbb{C}^d$ by entangled resources with Schmidt number at most $d$

TL;DR

This work provides a protocol-independent algebraic characterization of localizable joint measurements on bipartite quantum systems. It shows that a rank-1 PVM containing at least one element with maximal Schmidt rank is localizable with Schmidt number at most if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis; this reveals strict limits imposed by non-adaptive local operations. The authors also fully resolve the two-qubit case, showing that any rank-1 PVM localizable with a two-qubit entangled resource must be LU-equivalent to a product, Bell, or BB84-type basis, addressing a conjecture by Gisin and Del Santo. Extending to ideal measurements, localization occurs in nice Bell bases, tying locality constraints to the structure of unitary-error-basis families and Bell-type bases. Overall, the paper advances the understanding of what joint measurements can be performed with limited entanglement under non-adaptive LOCC and contributes algebraic tools for LOSR-oriented quantum information theory.

Abstract

Localizable measurements are joint quantum measurements that can be implemented using only non-adaptive local operations and shared entanglement. We provide a protocol-independent characterization of localizable projection-valued measures (PVMs) by exploiting algebraic structures that any such measurement must satisfy. We first show that a rank-1 PVM on containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. This reveals strong limitations imposed by non-adaptive local operations, in contrast to the adaptive setting where any joint measurement is implementable. We then completely characterize two-qubit rank-1 PVMs that can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo, and finally extend our characterization to ideal two-qudit measurements, strengthening earlier results.
Paper Structure (17 sections, 12 theorems, 61 equations, 2 figures, 1 table)

This paper contains 17 sections, 12 theorems, 61 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Localizable POVMs
  4. Linear operators and bipartite vectors
  5. Simplifying localizations
  6. Generally applicable simplification
  7. Localizability of rank-1 PVMs
  8. Localizability of rank-1 PVM by entangled state with bounded Schmidt rank
  9. Nice unitary error basis
  10. rank-1 PVMs with maximal Schmidt rank
  11. Two-qubit POVMs
  12. Ideal measurements in nice Bell bases
  13. Discussion
  14. Proof of Lemma \ref{['lem:rank-1_non-redundant']}
  15. Proof of Lemma \ref{['lem:closeness_localizable_POVM']}
  16. ...and 2 more sections

Key Result

Lemma 1

If POVM $\{ M_c \}_{c \in Z}$ on ${\mathcal{S}_A} \otimes {\mathcal{S}_B}$ can be localized by ${\psi_\mathcal{R}}$ on ${\mathcal{R}_A} \otimes {\mathcal{R}_B}$, there is a non-redundant rank-1 localization with ${\psi_\mathcal{R}}$.

Figures (2)

  • Figure 1: The localization scheme for a POVM measurement on ${\mathcal{S}_A} \otimes {\mathcal{S}_B}$, depicted by a circuit with transformations applied from bottom to top. The solid and dashed lines represent quantum and classical registers, respectively.
  • Figure 2: The localization scheme for the ideal measurement in the basis (\ref{['eq:twisted_nice_Bell']}). The shaded area corresponds to the sub-scheme for localizing the ideal measurement in the nice Bell basis $\{ {\left[ {M_i} \right]} \}_{i=1,\ldots,d^2}$.

Theorems & Definitions (16)

  • Definition 1: localization
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Definition 2: rank-1 localization
  • Lemma 5
  • Theorem 1
  • Theorem 2
  • Definition 3
  • ...and 6 more