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Characterising memory in quantum channel discrimination via constrained separability problems

Ties-A. Ohst, Shijun Zhang, Hai Chau Nguyen, Martin Plávala, Marco Túlio Quintino

TL;DR

The work addresses how limited quantum memory impacts the quality of quantum channel discrimination. By casting memory constraints as constrained separability problems and employing a converging SDP hierarchy plus seesaw techniques, it provides computable upper and lower bounds for both single-copy and multi-copy discrimination tasks. It reveals nuanced roles for memory types: quantum memory can be essential in some tasks, but classical memory can compensate in adaptive strategies, and there is no strict hierarchy between parallel and adaptive approaches. The results generalise to memory-constrained adaptive testers and classically adaptive schemes, offering practical tools for determining necessary memory in quantum protocols and certifying memory dimensions in experiments.

Abstract

Quantum memories are a crucial precondition in many protocols for processing quantum information. A fundamental problem that illustrates this statement is given by the task of channel discrimination, in which an unknown channel drawn from a known random ensemble should be determined by applying it for a single time. In this paper, we characterise the quality of channel discrimination protocols when the quantum memory, quantified by the auxiliary dimension, is limited. This is achieved by formulating the problem in terms of separable quantum states with additional affine constraints that all of their factors in each separable decomposition obey. We discuss the computation of upper and lower bounds to the solutions of such problems which allow for new insights into the role of memory in channel discrimination. In addition to the single-copy scenario, this methodological insight allows to systematically characterise quantum and classical memories in adaptive channel discrimination protocols. Especially, our methods enabled us to identify channel discrimination scenarios where classical or quantum memory is required, and to identify the hierarchical and non-hierarchical relationships within adaptive channel discrimination protocols.

Characterising memory in quantum channel discrimination via constrained separability problems

TL;DR

The work addresses how limited quantum memory impacts the quality of quantum channel discrimination. By casting memory constraints as constrained separability problems and employing a converging SDP hierarchy plus seesaw techniques, it provides computable upper and lower bounds for both single-copy and multi-copy discrimination tasks. It reveals nuanced roles for memory types: quantum memory can be essential in some tasks, but classical memory can compensate in adaptive strategies, and there is no strict hierarchy between parallel and adaptive approaches. The results generalise to memory-constrained adaptive testers and classically adaptive schemes, offering practical tools for determining necessary memory in quantum protocols and certifying memory dimensions in experiments.

Abstract

Quantum memories are a crucial precondition in many protocols for processing quantum information. A fundamental problem that illustrates this statement is given by the task of channel discrimination, in which an unknown channel drawn from a known random ensemble should be determined by applying it for a single time. In this paper, we characterise the quality of channel discrimination protocols when the quantum memory, quantified by the auxiliary dimension, is limited. This is achieved by formulating the problem in terms of separable quantum states with additional affine constraints that all of their factors in each separable decomposition obey. We discuss the computation of upper and lower bounds to the solutions of such problems which allow for new insights into the role of memory in channel discrimination. In addition to the single-copy scenario, this methodological insight allows to systematically characterise quantum and classical memories in adaptive channel discrimination protocols. Especially, our methods enabled us to identify channel discrimination scenarios where classical or quantum memory is required, and to identify the hierarchical and non-hierarchical relationships within adaptive channel discrimination protocols.

Paper Structure

This paper contains 32 sections, 22 theorems, 99 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Single-copy channel discrimination
  4. Unlimited memory case
  5. Limited memory case
  6. Constrained separability problems
  7. Multi-copy channel discrimination without memory constraints
  8. Parallel schemes without memory constraints
  9. Adaptive schemes without memory constraints
  10. Multi-copy channel discrimination with memory constraints and the role of classical memory
  11. Parallel schemes with memory constraints
  12. Adaptive schemes with memory constraints
  13. The role of classical memory
  14. Adaptive schemes with classical communication
  15. Optimisation over memory-constrained adaptive testers as constrained separability problem
  16. ...and 17 more sections

Key Result

Proposition 5

A collection of positive operators $\{T_{\rm IO}\}_{i=1}^{N}$ is a single-copy tester if and only if there exists a state $\sigma_{\rm I}$ such that Furthermore, the dimension $d_{\rm E}$ of the memory system $\rm{E}$ in the Definition def:sc_tester of a single-copy tester can always be chosen equal to $d_{\rm I}$, i.e., equal to the dimension of the input system.

Figures (2)

  • Figure 1: Simple inner polytopes $\mathcal{V}_{\rm A}$ of the unconstrained qubit state space (Bloch sphere) with reference state $\tau_{\rm A} = \frac{\mathds{1}_{2}}{2}$. (a) Polytope spanned by the $6$ eigenvectors of the three Pauli matrices. (b) Polytope spanned by the $8$ Bloch vectors $(\pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}}, \pm \frac{1}{\sqrt{3}})$. Although having a different number of vertices, both polytopes have the same approximation radius $l_{\tau_{\rm A}}(\mathcal{V}_{\rm A}) = \frac{1}{\sqrt{3}} \approx 0.577$ with respect to $\tau_{\rm A}$.
  • Figure 2: Venn diagram of the sets of different adaptive testers. Classically adaptive testers and parallel testers form intersecting subsets of the set of adaptive testers.

Theorems & Definitions (55)

  • Definition 1
  • Definition 2: Link product PhysRevA.80.022339PhysRevA.77.062112
  • Definition 3: Single-copy tester PhysRevA.80.022339PhysRevA.77.062112
  • Example 4: Superdense coding
  • Proposition 5: Ref. PhysRevA.77.062112PhysRevA.80.022339
  • Definition 6: Memory-$d_{\rm E}$ single-copy tester
  • Theorem 7
  • Remark 8
  • Theorem 9
  • Remark 10
  • ...and 45 more