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Bounds in Sequential Unambiguous Discrimination of Multiple Pure Quantum States

Jordi Pérez-Guijarro, Alba Pagès-Zamora, Javier R. Fonollosa

Abstract

Sequential methods for quantum hypothesis testing offer significant advantages over fixed-length approaches, which rely on a predefined number of state copies. Despite their potential, these methods remain underexplored for unambiguous discrimination. In this work, we derive performance bounds for such methods when applied to the discrimination of a set of pure states. The performance is evaluated based on the expected number of copies required. We establish a lower bound applicable to any sequential method and an upper bound on the optimal sequential method. The upper bound is derived using a novel and simple non-adaptive method. Importantly, the gap between these bounds is minimal, scaling logarithmically with the number of distinct states.

Bounds in Sequential Unambiguous Discrimination of Multiple Pure Quantum States

Abstract

Sequential methods for quantum hypothesis testing offer significant advantages over fixed-length approaches, which rely on a predefined number of state copies. Despite their potential, these methods remain underexplored for unambiguous discrimination. In this work, we derive performance bounds for such methods when applied to the discrimination of a set of pure states. The performance is evaluated based on the expected number of copies required. We establish a lower bound applicable to any sequential method and an upper bound on the optimal sequential method. The upper bound is derived using a novel and simple non-adaptive method. Importantly, the gap between these bounds is minimal, scaling logarithmically with the number of distinct states.
Paper Structure (13 sections, 13 theorems, 67 equations, 1 algorithm)

This paper contains 13 sections, 13 theorems, 67 equations, 1 algorithm.

Table of Contents

  1. Quantum Hypothesis Testing and Unambiguous Discrimination
  2. Lower bounds on the expected number of copies
  3. Lower Bound in unambiguous discrimination
  4. Lower bound in quantum hypothesis testing
  5. Upper bound in unambiguous discrimination
  6. Performance of the proposed method
  7. Removing the requirement for linear independence
  8. Analysis for Mixed States
  9. Conclusions
  10. Acknowledgments
  11. Proof of Lemma \ref{['ref_sequential_to_fixed']}
  12. Proof of Lemma \ref{['lemma_3']}
  13. Proof of Lemma \ref{['ref_sequential_to_fixed_probability_of_error']}

Key Result

Theorem 1

For any sequential method that distinguishes unambiguously a set of pure states $\{\ket{\psi_i}\}_{i=1}^N$, the maximum expected value of state copies $\max_{s\in[N]}\mathbb{E}[L|s]$ satisfies

Theorems & Definitions (26)

  • Definition 1
  • Theorem 1
  • Lemma 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 16 more