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Factorizability of optimal quantum sequence discrimination under maximum-confidence measurements

Donghoon Ha, Jeong San Kim

TL;DR

The paper addresses discriminating quantum sequences under maximum-confidence measurements and proves that the optimal discrimination of a sequence ensemble factorizes into independent optimizations at each step, i.e., $p_G(\mathcal{E})=\prod_{l=1}^L p_G(\mathcal{E}^l)$ and $C_{\vec{x}}(\mathcal{E})=\prod_{l=1}^L C_{x_l}(\mathcal{E}^l)$. It provides a necessary and sufficient condition for optimal quantum state discrimination under maximum-confidence measurements and develops a projection-based duality framework for maximum-confidence measurements. The results imply that memory or collective measurements offer no advantage for such tasks and connect naturally to unambiguous discrimination, where factorization reduces to per-step unambiguous discrimination when possible. These findings have implications for quantum information processing tasks like quantum cryptography and quantum teleportation, clarifying fundamental limits on sequence discrimination with maximum confidence.

Abstract

We consider the discrimination of quantum sequences under maximum-confidence measurements and show that the optimal discrimination of a quantum sequence ensemble can always be factorized into that of each individual ensemble. In other words, the optimal quantum sequence discrimination under maximum-confidence measurements can be achieved just by performing a maximum-confidence discrimination independently at each step of the quantum sequence. We also show that the maximum confidence of identifying a quantum sequence is to achieve the maximum confidence of identifying each state comprising the quantum sequence. We further provide a necessary and sufficient condition for the optimal quantum state discrimination under maximum-confidence measurements.

Factorizability of optimal quantum sequence discrimination under maximum-confidence measurements

TL;DR

The paper addresses discriminating quantum sequences under maximum-confidence measurements and proves that the optimal discrimination of a sequence ensemble factorizes into independent optimizations at each step, i.e., and . It provides a necessary and sufficient condition for optimal quantum state discrimination under maximum-confidence measurements and develops a projection-based duality framework for maximum-confidence measurements. The results imply that memory or collective measurements offer no advantage for such tasks and connect naturally to unambiguous discrimination, where factorization reduces to per-step unambiguous discrimination when possible. These findings have implications for quantum information processing tasks like quantum cryptography and quantum teleportation, clarifying fundamental limits on sequence discrimination with maximum confidence.

Abstract

We consider the discrimination of quantum sequences under maximum-confidence measurements and show that the optimal discrimination of a quantum sequence ensemble can always be factorized into that of each individual ensemble. In other words, the optimal quantum sequence discrimination under maximum-confidence measurements can be achieved just by performing a maximum-confidence discrimination independently at each step of the quantum sequence. We also show that the maximum confidence of identifying a quantum sequence is to achieve the maximum confidence of identifying each state comprising the quantum sequence. We further provide a necessary and sufficient condition for the optimal quantum state discrimination under maximum-confidence measurements.
Paper Structure (11 sections, 11 theorems, 119 equations, 1 figure)

This paper contains 11 sections, 11 theorems, 119 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Maximum-confidence discrimination of quantum states
  3. Quantum state discrimination and confidence of measurement
  4. Optimal quantum state discrimination under maximum-confidence measurements
  5. Maximum-confidence discrimination of quantum sequences
  6. Quantum sequence discrimination and confidence of measurement
  7. Optimal quantum sequence discrimination under maximum-confidence measurements
  8. Discussion
  9. PROOF OF THEOREM \ref{['THM:PQGE']}
  10. PROOF OF LEMMA \ref{['LEM:WXYZ']}
  11. PROOF OF LEMMA \ref{['LEM:PPEE']}

Key Result

Lemma 1

For an ensemble $\mathcal{E}=\{\eta_{i},\rho_{i}\}_{i\in\mathbb{N}_{n}}$ and $x\in\mathbb{N}_{n}$, we have where $\rho_{0}$ is defined in Eq. eq:rhzd and the maximum is taken over all possible $E\in\mathbb{H}_{+}$ with $\mathop{\mathrm{Tr}}\nolimits(\rho_{0}E)=1$.

Figures (1)

  • Figure 1: Quantum sequence ensemble $\bigotimes_{l=1}^{L}\mathcal{E}^{l}=\{\eta_{\vec{c}},\rho_{\vec{c}}\}_{\vec{c}\in\mathbb{N}_{\vec{n}}}$. For each $l=1,\ldots,L$, the $l$th state $\rho_{c_{l}}^{l}$ is prepared from the ensemble $\mathcal{E}^{l}$ with the probability $\eta_{c_{l}}^{l}$, for $c_{l}=1,\ldots,n_{l}$. The quantum sequence $(\rho_{c_{1}}^{1},\ldots,\rho_{c_{L}}^{L})$ can be regarded as the tensor producted state $\rho_{\vec{c}}$ with the probability $\eta_{\vec{c}}$, for $\vec{c}=(c_{1},\ldots,c_{L})\in\mathbb{N}_{\vec{n}}$.

Theorems & Definitions (29)

  • Definition 1
  • Lemma 1
  • proof
  • Definition 2
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • Theorem 3
  • ...and 19 more