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Asymptotic implementation of multipartite quantum channels and other quantum instruments using local operations and classical communication

Scott M. Cohen

TL;DR

The paper develops a geometric framework to test whether multipartite quantum channels and instruments can be approximated by LOCC, via zonoids generated from Kraus representations and monotone, locally piecewise product-operator paths. It proves a necessary condition for LOCC-approximability of channels and extends this to all quantum instruments, supported by a detailed two-qubit example showing LOCC is not closed for measurements. The results illuminate how asymptotic LOCC links to geometric objects and how limiting paths signal impossibility of exact LOCC implementation, while remaining amenable to arbitrarily close approximation. The work also generalizes the construction to P-qubit systems, clarifying how the LOCC-closure status can differ between measurements and channels and guiding future multipartite LOCC analyses.

Abstract

We prove a necessary condition that a quantum channel on a multipartite system may be approximated arbitrarily closely using local operations and classical communication (LOCC). We then extend those arguments to obtain a condition that applies to all quantum instruments, which range from the most refined case, a generalized measurement, to the most coarse-grained, which is a quantum channel. We illustrate these results by a detailed analysis of a quantum instrument that is known not to be implementable by LOCC, but which can be arbitrarily closely approximated within that framework. As one outgrowth of this analysis, we find a quantum measurement that falls into the same category: it cannot be implemented exactly by LOCC, but can be approximated by LOCC arbitrarily closely. This measurement has an infinite number of outcomes, leaving open the question as to whether or not there exists a measurement within this same category but having only a finite number of outcomes.

Asymptotic implementation of multipartite quantum channels and other quantum instruments using local operations and classical communication

TL;DR

The paper develops a geometric framework to test whether multipartite quantum channels and instruments can be approximated by LOCC, via zonoids generated from Kraus representations and monotone, locally piecewise product-operator paths. It proves a necessary condition for LOCC-approximability of channels and extends this to all quantum instruments, supported by a detailed two-qubit example showing LOCC is not closed for measurements. The results illuminate how asymptotic LOCC links to geometric objects and how limiting paths signal impossibility of exact LOCC implementation, while remaining amenable to arbitrarily close approximation. The work also generalizes the construction to P-qubit systems, clarifying how the LOCC-closure status can differ between measurements and channels and guiding future multipartite LOCC analyses.

Abstract

We prove a necessary condition that a quantum channel on a multipartite system may be approximated arbitrarily closely using local operations and classical communication (LOCC). We then extend those arguments to obtain a condition that applies to all quantum instruments, which range from the most refined case, a generalized measurement, to the most coarse-grained, which is a quantum channel. We illustrate these results by a detailed analysis of a quantum instrument that is known not to be implementable by LOCC, but which can be arbitrarily closely approximated within that framework. As one outgrowth of this analysis, we find a quantum measurement that falls into the same category: it cannot be implemented exactly by LOCC, but can be approximated by LOCC arbitrarily closely. This measurement has an infinite number of outcomes, leaving open the question as to whether or not there exists a measurement within this same category but having only a finite number of outcomes.
Paper Structure (17 sections, 11 theorems, 63 equations, 1 figure)

This paper contains 17 sections, 11 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quantum Channels, Measurements, and Instruments
  4. Local Operations and Classical Communication
  5. Asymptotic LOCC and convergence of quantum channels
  6. Zonotopes and Zonoids
  7. Notation
  8. Approximating quantum channels by LOCC
  9. Proof of Theorem \ref{['thm1']}
  10. Detailed example of a quantum channel
  11. Quantum instruments
  12. LOCC$\ne\overline{\textrm{LOCC}}$
  13. Generalization of ${\cal E}$ to any number of qubits
  14. Conclusions
  15. Convergence of channels implies convergence of the associated zonoids
  16. ...and 2 more sections

Key Result

Lemma 1

myLOCCbyFirstMeasEach node $n$ in a finite-round LOCC tree is equal to the sum of the collection of all leaf nodes that are descendant from that node $n$.

Figures (1)

  • Figure 1: Tree graph for the first several local measurements performed by Alice and Bob in any one of the sequence of LOCC protocols described in the text. Note that we have defined $\eta=1-\epsilon$.

Theorems & Definitions (11)

  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Theorem 2
  • Lemma 3
  • Lemma 4
  • Lemma 5
  • Theorem 3
  • Theorem 4
  • Lemma 6
  • ...and 1 more