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Probabilistic Entanglement Distillation and Cost under Approximately Nonentangling and Dually Nonentangling Instruments

Xian Shi

TL;DR

This paper studies probabilistic entanglement distillation and entanglement cost under $\delta$-approximately nonentangling ($\mathcal{NE}$) and $\delta$-approximately dually nonentangling ($\mathcal{DNE}$) quantum instruments. It develops an analytic characterization of the asymptotic distillation error exponent by linking the task to postselected quantum hypothesis testing against the set of separable states, and extends the connection to measurements restricted to be separable in the DNE setting. The authors establish an exact (and tight) relation between probabilistic entanglement costs under NE and DNE instruments, including one-shot bounds and asymptotic equalities, and provide a Werner-state example to illustrate the exponent expressions. Overall, the work yields a unified information-theoretic framework that leverages postselected hypothesis testing to analyze probabilistic entanglement manipulation under these relaxed resource theories, with implications for computability and further extensions to other resource theories.

Abstract

Entanglement distillation and entanglement cost are fundamental tasks in quantum entanglement theory. This work studies both in the probabilistic setting and focuses on the asymptotic error exponent of probabilistic entanglement distillation when the operational model is $δ$-approximately nonentangling(ANE) and $δ$-approximately dually nonentangling(ADNE) quantum instruments. While recent progress has clarified limitations of probabilistic transformations in general resource theories, an analytic formula for the error exponent of probabilistic entanglement distillation under approximately (dually) nonentangling operations has remained unavailable. Building on the framework of postselected quantum hypothesis testing, we establish a direct connection between probabilistic distillation and postselected hypothesis testing against the set of separable states. In particular, we derive an analytical characterization of the distillation error exponent under ANE. Besides, we relate the exponent to postselected hypothesis testing with measurements restricted to be separable. We further investigate probabilistic entanglement dilution and establish a relation between probabilistic entanglement costs under approximately nonentangling and approximately dually nonentangling instruments, together with a bound on the probabilistic entanglement cost under nonentangling instruments

Probabilistic Entanglement Distillation and Cost under Approximately Nonentangling and Dually Nonentangling Instruments

TL;DR

This paper studies probabilistic entanglement distillation and entanglement cost under -approximately nonentangling () and -approximately dually nonentangling () quantum instruments. It develops an analytic characterization of the asymptotic distillation error exponent by linking the task to postselected quantum hypothesis testing against the set of separable states, and extends the connection to measurements restricted to be separable in the DNE setting. The authors establish an exact (and tight) relation between probabilistic entanglement costs under NE and DNE instruments, including one-shot bounds and asymptotic equalities, and provide a Werner-state example to illustrate the exponent expressions. Overall, the work yields a unified information-theoretic framework that leverages postselected hypothesis testing to analyze probabilistic entanglement manipulation under these relaxed resource theories, with implications for computability and further extensions to other resource theories.

Abstract

Entanglement distillation and entanglement cost are fundamental tasks in quantum entanglement theory. This work studies both in the probabilistic setting and focuses on the asymptotic error exponent of probabilistic entanglement distillation when the operational model is -approximately nonentangling(ANE) and -approximately dually nonentangling(ADNE) quantum instruments. While recent progress has clarified limitations of probabilistic transformations in general resource theories, an analytic formula for the error exponent of probabilistic entanglement distillation under approximately (dually) nonentangling operations has remained unavailable. Building on the framework of postselected quantum hypothesis testing, we establish a direct connection between probabilistic distillation and postselected hypothesis testing against the set of separable states. In particular, we derive an analytical characterization of the distillation error exponent under ANE. Besides, we relate the exponent to postselected hypothesis testing with measurements restricted to be separable. We further investigate probabilistic entanglement dilution and establish a relation between probabilistic entanglement costs under approximately nonentangling and approximately dually nonentangling instruments, together with a bound on the probabilistic entanglement cost under nonentangling instruments
Paper Structure (13 sections, 18 theorems, 152 equations, 1 figure)

This paper contains 13 sections, 18 theorems, 152 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary Knowledge
  3. Main results
  4. Conclusion
  5. Acknowledgement
  6. Appendix
  7. Entanglement
  8. Quantum Relative Entropies
  9. Quantum Postselected Hypothesis Testing
  10. Two classes of subchannels
  11. Probabilistic entanglement distillation exponents under (approximately) $\mathcal{DNE}$ instruments
  12. Probabilistic entanglement cost under (Approximatly) $\mathcal{NE}$ and $\mathcal{DNE}$ instruments
  13. Some properties of the Hilbert projective metric

Key Result

Theorem 1

Assume $\rho_{AB}$ is a bipartite state, the asymptotic error exponent of probabilistic entanglement distillation under $\mathcal{NE}_{\delta}$ is equal to the postselected hypothesis testing of the set of separable states and $\rho_{AB}$,

Figures (1)

  • Figure 1: The asymptotic error exponent of probabilistic entanglement distillation of $\rho_p$ under NE.

Theorems & Definitions (19)

  • Theorem 1
  • Example 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • Corollary 9
  • Lemma 10
  • ...and 9 more