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Entanglement cost of bipartite quantum channel discrimination under positive partial transpose operations

Chengkai Zhu, Shuyu He, Gereon Koßmann, Xin Wang

Abstract

Quantum channel discrimination is a fundamental task in quantum information processing. In the one-shot regime, discrimination between two candidate channels is characterized by the diamond norm. Beyond this basic setting, however, many scenarios in distributed quantum information processing remain unresolved, motivating notions of distinguishability that capture the power of the available resources. In this work, we formulate a theory of testers for bipartite channel discrimination, leading to the concept of the entanglement cost of bipartite channel discrimination: the minimum Schmidt rank $k$ of a shared maximally entangled state required for local protocols to achieve the globally optimal success probability. We introduce $k$-injectable testers as a tester-based description of entanglement-assisted local discrimination and, in particular, study the class of $k$-injectable positive-partial-transpose (PPT) testers, which constitutes a numerically tractable relaxation of the practically relevant class of LOCC testers. For every $k$, we derive a semidefinite program (SDP) for the optimal success probability, which in turn yields an efficiently computable one-shot PPT entanglement cost. To render these optimization problems numerically feasible, we prove a symmetry-reduction principle for covariant channel pairs, thereby reducing the effective dimension of the associated SDPs. Finally, by dualizing the SDP, we derive bounds on the composite channel-discrimination problem and illustrate our framework with proof-of-principle examples based on the depolarizing channel, the depolarized SWAP channel, and the Werner--Holevo channels.

Entanglement cost of bipartite quantum channel discrimination under positive partial transpose operations

Abstract

Quantum channel discrimination is a fundamental task in quantum information processing. In the one-shot regime, discrimination between two candidate channels is characterized by the diamond norm. Beyond this basic setting, however, many scenarios in distributed quantum information processing remain unresolved, motivating notions of distinguishability that capture the power of the available resources. In this work, we formulate a theory of testers for bipartite channel discrimination, leading to the concept of the entanglement cost of bipartite channel discrimination: the minimum Schmidt rank of a shared maximally entangled state required for local protocols to achieve the globally optimal success probability. We introduce -injectable testers as a tester-based description of entanglement-assisted local discrimination and, in particular, study the class of -injectable positive-partial-transpose (PPT) testers, which constitutes a numerically tractable relaxation of the practically relevant class of LOCC testers. For every , we derive a semidefinite program (SDP) for the optimal success probability, which in turn yields an efficiently computable one-shot PPT entanglement cost. To render these optimization problems numerically feasible, we prove a symmetry-reduction principle for covariant channel pairs, thereby reducing the effective dimension of the associated SDPs. Finally, by dualizing the SDP, we derive bounds on the composite channel-discrimination problem and illustrate our framework with proof-of-principle examples based on the depolarizing channel, the depolarized SWAP channel, and the Werner--Holevo channels.
Paper Structure (23 sections, 17 theorems, 149 equations, 4 figures, 1 table)

This paper contains 23 sections, 17 theorems, 149 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Contributions
  3. Organization
  4. Preliminaries
  5. Notation
  6. One-shot quantum channel discrimination
  7. Bipartite channel discrimination under locality constraints
  8. Problem formulation
  9. Tester under locality constraints
  10. One-shot PPT entanglement cost for optimal discrimination
  11. Entanglement injectable testers
  12. Entanglement-assisted success probability and entanglement cost
  13. Composite quantum channel discrimination
  14. Symmetry and examples
  15. Covariant channel
  16. ...and 8 more sections

Key Result

Theorem 4.1

Let ${\cal N}_{A_0B_0\to A_1B_1}$ and ${\cal M}_{A_0B_0\to A_1B_1}$ be two bipartite quantum channels given with prior probabilities $\lambda$ and $1-\lambda$, respectively, where $\lambda\in(0,1)$. The maximal $k$-ebit-assisted average success probability of discriminating ${\cal N}_{A_0B_0\to A_1B

Figures (4)

  • Figure 1: Protocols for entanglement-assisted discrimination of bipartite quantum channels. (a) A framework in which both parties prepared states and performed joint measurements to achieve diamond norm distance. (b) A general framework in which shared entanglement ($\Phi_k$) assists both the state-preparation and measurement stages. (c) A reduction to the point-to-point setting where entanglement specifically enhances the LOCC measurement. (d) A reduction in which entanglement is used to enhance probe-state preparation.
  • Figure 2: The general framework for bipartite channel discrimination is presented. Compared with the well-understood setting of channel discrimination in Section \ref{['sec:qcd']}, several refinements are introduced. First, the channels in the ensemble $\Omega=\{(p_j,{\cal N}_{A_0B_0\to A_1B_1}^{(j)})\}_j$ have bipartite inputs $A_0B_0$ and bipartite outputs $A_1B_1$, shared between Alice and Bob. Second, Alice and Bob are provided with local entanglement assistance in their laboratories, denoted by $A'$ and $B'$, respectively. By means of LOCC operations with respect to the bipartition $A\!:\!B$, they encode input states $\rho_A$ and $\sigma_B$ into states on $A_0A'$ and $B_0B'$. The unknown channel is then applied. Finally, restricting again to LOCC operations on $A_1A'\!:\!B_1B'$, Alice and Bob output a guess $j$, indicating that the channel ${\cal N}_{A_0B_0\to A_1B_1}^{(j)}$ was applied.
  • Figure 3: The adapted channel discrimination setup is shown. Comparing with Figure \ref{['fig:locc_qcd']}, an additional resource of entanglement $\Phi_k\in{\cal D}(A_RB_R)$ between Alice and Bob is added. In order to formally add them, we introduce the notion of $k$-injectable testers. It has explicit resource ports $A_RB_R$ such that, upon injecting $\Phi_k$, they induce an effective tester on the channel ports $A_0B_0A_1B_1$.
  • Figure 4: Maximal average success probability for discriminating the equiprobable ensemble $\{(\frac{1}{2}, {\cal A}_{\gamma}), (\frac{1}{2}, {\cal A}_{1-\gamma})\}$ of amplitude damping channels over the parameter range $\gamma \in [0,0.2]$. Panels (a), (b), and (c) correspond to single, two, and three parallel uses of the unknown channel. We benchmark the globally optimal success probability against the performance of PPT testers in the unassisted regime ($k=1$, $0$ ebits) and the entanglement-assisted regime ($k=2$, $1$ ebit). While single and three parallel uses achieve the global optimum without any shared entanglement, the two-copy scenario strictly requires $1$ ebit to close the performance gap between unassisted local strategies and the global optimum.

Theorems & Definitions (42)

  • Definition 1: SEP tester
  • Definition 2: PPT tester
  • Definition 3: $k$-injectable PPT tester
  • Definition 4: Entanglement-assisted average success probability
  • Definition 5: One-shot entanglement cost of channel discrimination
  • Theorem 4.1
  • proof
  • Corollary 4.2
  • Remark 4.3
  • Theorem 4.4
  • ...and 32 more