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Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

Eric Chitambar, Debbie Leung, Laura Mancinska, Maris Ozols, Andreas Winter

TL;DR

The paper provides a rigorous, instrument-based framework to define and analyze LOCC, including finite- and infinite-round protocols, and studies the topological shape of LOCC within the space of quantum operations. It proves LOCC has a nonempty interior and an explicit open ball around the depolarizing map that remains LOCC, while also establishing that LOCC is not closed by constructing a bipartite instrument in the closure but not in LOCC. A key part of the work is a Fortescue-Lo–inspired construction showing convergence to a limiting LOCC-inaccessible instrument, together with a random-concurrence monotone on W-class states used to prove LOCC impossibility for a particular transformation. Overall, the results clarify the hierarchical relationships between LOCC, SEP, PPT, and their closures, and highlight fundamental limits of LOCC as a resource for quantum information tasks.

Abstract

In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.

Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

TL;DR

The paper provides a rigorous, instrument-based framework to define and analyze LOCC, including finite- and infinite-round protocols, and studies the topological shape of LOCC within the space of quantum operations. It proves LOCC has a nonempty interior and an explicit open ball around the depolarizing map that remains LOCC, while also establishing that LOCC is not closed by constructing a bipartite instrument in the closure but not in LOCC. A key part of the work is a Fortescue-Lo–inspired construction showing convergence to a limiting LOCC-inaccessible instrument, together with a random-concurrence monotone on W-class states used to prove LOCC impossibility for a particular transformation. Overall, the results clarify the hierarchical relationships between LOCC, SEP, PPT, and their closures, and highlight fundamental limits of LOCC as a resource for quantum information tasks.

Abstract

In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.

Paper Structure

This paper contains 13 sections, 13 theorems, 44 equations, 2 figures.

Table of Contents

  1. Introduction
  2. How to define LOCC?
  3. Quantum Instruments
  4. LOCC Instruments
  5. Relationships Between the Classes
  6. What is the shape of LOCC?
  7. Is LOCC closed?
  8. Proof of
  9. Digression: Random Concurrence Distillation
  10. LOCC Impossibility
  11. What did we learn?
  12. Proof of Theorem \ref{['Thm:Finite']}
  13. Proof of Lemma \ref{['Lem:Concurrence Monotone']}

Key Result

Lemma 1

If $\mathfrak{J}=(\mathcal{E}_1,\dotso,\mathcal{E}_m)$ is some $N$-partite separable instrument with $d$ being the total dimension of $\mathcal{H}$. Then $\mathfrak{J}$ can be implemented by SLOCC with a success probability at least $\frac{1}{md^2}$.

Figures (2)

  • Figure 1: The instrument $\mathfrak{J}'$ is LOCC linked to the instrument $\mathfrak{J}$. Conditional instruments $(\mathcal{B}_{1|1},\mathcal{B}_{2|1})$ and $(\mathcal{B}_{1|2},\mathcal{B}_{2|2})$ can be composed with the two elements of $\mathfrak{J}$ so that after coarse-graining, the resulting instrument is $\mathfrak{J}'$.
  • Figure 2: Construction of $\Omega$ with four rounds of LOCC according to Eq. \ref{['Eq:Choistate']}. From the state $\Omega$ the action of $\mathcal{E}[\mathfrak{J}]$ on $\rho$ can be obtained by inputting $\rho$ into wires $A_0 B_0$ and applying the projector $|\Phi\rangle$ onto each of the the pairs $A_0^{}A_1'$, $B_0^{}B_2'$, $A_1^{}A_3'$, and $B_2^{}B_4'$ as in Eq. \ref{['Eq:Choiequality']}.

Theorems & Definitions (23)

  • Example
  • Example
  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Theorem 1
  • Proposition 2: Barnum03
  • Corollary 1
  • proof
  • ...and 13 more