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Approximating quantum channels by completely positive maps with small Kraus rank

Cécilia Lancien, Andreas Winter

Abstract

We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space $\mathrm{A}$ to states on some output Hilbert space $\mathrm{B}$ can be compressed into one with order $d\log d$ Kraus operators, where $d=\max(|\mathrm{A}|,|\mathrm{B}|)$, hence much less than $|\mathrm{A}||\mathrm{B}|$. In the case where the channel's outputs are all very mixed, this can be improved to order $d$. We discuss the optimality of this result as well as some consequences.

Approximating quantum channels by completely positive maps with small Kraus rank

Abstract

We study the problem of approximating a quantum channel by one with as few Kraus operators as possible (in the sense that, for any input state, the output states of the two channels should be close to one another). Our main result is that any quantum channel mapping states on some input Hilbert space to states on some output Hilbert space can be compressed into one with order Kraus operators, where , hence much less than . In the case where the channel's outputs are all very mixed, this can be improved to order . We discuss the optimality of this result as well as some consequences.

Paper Structure

This paper contains 9 sections, 14 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Quantum channel approximation: definitions and already known facts
  3. Statement of the main results
  4. Proofs of the main results
  5. Consequences and applications
  6. Approximation in terms of output entropies or fidelities
  7. Destruction of correlations with few resources and data hiding.
  8. Werner channels.
  9. Discussion

Key Result

Theorem \oldthetheorem

Fix $0<\varepsilon<1$ and let $\mathcal{N}:\mathcal{L}(\mathrm{A})\rightarrow\mathcal{L}(\mathrm{B})$ be a CPTP map with Kraus rank $|\mathrm{E}|\geqslant |\mathrm{A}|,|\mathrm{B}|$. Then, there exists a CP map $\widehat{\mathcal{N}}:\mathcal{L}(\mathrm{A})\rightarrow\mathcal{L}(\mathrm{B})$ with Kr

Theorems & Definitions (31)

  • Theorem \oldthetheorem
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem: Bernstein's inequality, see e.g. CGLP, Theorem 1.2.5
  • Proposition \oldthetheorem
  • Lemma \oldthetheorem
  • proof
  • proof : Proof of Proposition \ref{['prop:fixed']}
  • Proposition \oldthetheorem
  • proof
  • ...and 21 more