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Extendible quantum measurements and limitations on classical communication

Vishal Singh, Theshani Nuradha, Mark M. Wilde

TL;DR

The paper introduces k-extendible POVMs as a semidefinite hierarchy approximating measurements achievable by one-way LOCC, linking measurement unextendibility to incompatibility and enabling tractable SDP formulations. This framework provides tighter upper bounds on the one-shot classical capacity of quantum channels than prior efficiently computable bounds and extends to n-shot capacities via permutation-symmetric reductions. By incorporating PPT constraints, the authors further tighten the relaxations beyond PPT alone. The approach promises improved analyses of communication protocols with forward classical assistance and paves the way for applications to restricted hypothesis testing and privacy in quantum information processing.

Abstract

Unextendibility of quantum states and channels is inextricably linked to the no-cloning theorem of quantum mechanics, it has played an important role in understanding and quantifying entanglement, and more recently it has found applications in providing limitations on quantum error correction and entanglement distillation. Here we generalize the framework of unextendibility to quantum measurements and define $k$-extendible measurements for every integer $k\ge 2$. Our definition provides a hierarchy of semidefinite constraints that specify a set of measurements containing every measurement that can be realized by local operations and one-way classical communication. Furthermore, the set of $k$-extendible measurements converges to the set of measurements that can be realized by local operations and one-way classical communication as $k\to \infty$. To illustrate the utility of $k$-extendible measurements, we establish a semidefinite programming upper bound on the one-shot classical capacity of a channel, which outperforms the best known efficiently computable bound from [Matthews and Wehner, IEEE Trans. Inf. Theory 60, pp. 7317-7329 (2014)] and also leads to efficiently computable upper bounds on the $n$-shot classical capacity of a channel.

Extendible quantum measurements and limitations on classical communication

TL;DR

The paper introduces k-extendible POVMs as a semidefinite hierarchy approximating measurements achievable by one-way LOCC, linking measurement unextendibility to incompatibility and enabling tractable SDP formulations. This framework provides tighter upper bounds on the one-shot classical capacity of quantum channels than prior efficiently computable bounds and extends to n-shot capacities via permutation-symmetric reductions. By incorporating PPT constraints, the authors further tighten the relaxations beyond PPT alone. The approach promises improved analyses of communication protocols with forward classical assistance and paves the way for applications to restricted hypothesis testing and privacy in quantum information processing.

Abstract

Unextendibility of quantum states and channels is inextricably linked to the no-cloning theorem of quantum mechanics, it has played an important role in understanding and quantifying entanglement, and more recently it has found applications in providing limitations on quantum error correction and entanglement distillation. Here we generalize the framework of unextendibility to quantum measurements and define -extendible measurements for every integer . Our definition provides a hierarchy of semidefinite constraints that specify a set of measurements containing every measurement that can be realized by local operations and one-way classical communication. Furthermore, the set of -extendible measurements converges to the set of measurements that can be realized by local operations and one-way classical communication as . To illustrate the utility of -extendible measurements, we establish a semidefinite programming upper bound on the one-shot classical capacity of a channel, which outperforms the best known efficiently computable bound from [Matthews and Wehner, IEEE Trans. Inf. Theory 60, pp. 7317-7329 (2014)] and also leads to efficiently computable upper bounds on the -shot classical capacity of a channel.

Paper Structure

This paper contains 20 sections, 6 theorems, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. k-Extendible measurements
  4. k-Extendible POVMs
  5. Adding PPT constraints
  6. Application to one-shot classical capacity
  7. Computing the classical capacity in the n-shot setting
  8. Conclusion and future works
  9. Convergence of k-extendible POVMs to 1WL
  10. Upper bound on the one-shot classical capacity of a channel
  11. Proof of Theorem \ref{['prop:class_cap_ub_SDP']}
  12. Efficient computability of the upper bound on n-shot classical capacity of a channel
  13. Constraint in (22)
  14. Constraint in 23
  15. Constraint in 24
  16. ...and 5 more sections

Key Result

Proposition 1

The set of $k$-extendible POVMs converges to $\operatorname{1WL}$ as $k\to \infty$.

Figures (1)

  • Figure 1: Upper bounds on the one-shot classical capacity of the channel described in \ref{['eq:spec_ch_defn']}, computed using the semidefinite program given in Theorem \ref{['prop:class_cap_ub_SDP']}. The solid curve corresponds to setting $k=1$, which is equivalent to PPT constraints, and the dotted curve to $k=2$ in Theorem \ref{['prop:class_cap_ub_SDP']}.

Theorems & Definitions (14)

  • Definition 1: $k$-extendible states Werner1989DPS02DPS04
  • Definition 2: $k$-extendible channels KDWW19KDWW21
  • Definition 3: Two-extendible POVM
  • Definition 4: $k$-extendible POVM
  • Remark 1
  • Proposition 1
  • Theorem 1
  • proof
  • Remark 2
  • Proposition 2: BBFS21
  • ...and 4 more