Error-Minimizing Measurements in Postselected One-Shot Symmetric Quantum State Discrimination and Acceptance as a Performance Metric

TL;DR

This paper addresses postselected quantum hypothesis testing for a two-state problem by defining and optimizing acceptance within the class of measurements that minimize the postselected symmetric error $e_s$. It provides a complete parametric characterization of all error-minimizing measurements in the equal-support case, leveraging a subspace transformation and a construction based on $ ext{ψ}_{ ext{max}}$, $ ext{ψ}_{ ext{min}}$, and scaling parameters, and derives closed-form expressions for the maximum achievable acceptance. USD is recovered as a special case when the supports differ, with explicit USD-like constraints and the corresponding maximum acceptance analyzed. The work reveals that among optimal measurements, the acceptance can vary, motivating explicit optimization of acceptance and highlighting tradeoffs between decision-making frequency and reliability. These results open paths to broader postselected frameworks, including asymptotic limits and postselection-enabled capacities for quantum channels and unitary operations.

Abstract

In hypothesis testing with quantum states, given a black box containing one of the two possible states, measurement is performed to detect in favor of one of the hypotheses. In postselected hypothesis testing, a third outcome is added, corresponding to not selecting any of the hypotheses. In postselected scenario, minimum error one-shot symmetric hypothesis testing is characterized in literature conditioned on the fact that one of the selected outcomes occur. We proceed further in this direction to give the set of all possible measurements that lead to the minimum error. We have given an arbitrary error-minimizing measurement in a parametric form. Note that not selecting any of the hypotheses decimates the quality of testing. We further give an example to show that these measurements vary in quality. There is a need to discuss the quality of postselected hypothesis testing. We then characterize the quality of postselected hypothesis testing by defining a new metric acceptance and give expression of acceptance for an arbitrary error-minimizing measurement in terms of some parameters of the measurement. On the set of measurements that achieve minimum error, we have maximized the acceptance, and given an example which achieves that, thus giving an example of the best possible measurement in terms of acceptance.

Figures (5)

• Figure 1: In all three subfigures, the black boxes on the left represent an unknown object in the state $\rho$ or $\sigma$. The colored boxes on the right represent the possible outcomes, where the green box corresponds to the accepted outcome, and the red box corresponds to the rejected outcome. The green line represents correct detection, the blue line shows incorrect detection, and the red line shows rejecting the test. Note that, in conventional hypothesis testing, only accepted outcomes were considered. Furthermore, from (b), it is clear that outcomes corresponding to blue lines are not allowed in USD.
• Figure 2: Figure showing connection among theorems in Section \ref{['sec: rhoEqsigma']} and Section \ref{['sec: rhoNeqsigma']}
• Figure 3: An illustration showing various sets of measurements. $\mathcal{M}$ is set of all measurements. $\mathcal{E}_s(\rho,\sigma,p)$ is set of all error-minimizing measurements. Maximum acceptance measurements comprise the set that achieves the maximum acceptance over $\mathcal{E}_s(\rho,\sigma,p)$.
• Figure 4: An illustration showing possible outcomes for different prior probability values of $p_\rho$ and $p_\sigma$. The figure shows the possible outcomes when $p_\rho\ne p_\rho^*$ (equivalently $p_\sigma\ne p_\sigma^*$) for any error minimizing measurement where (a) $\sigma$ never declared, (b) $\rho$ never declared. Further, (c) shows that if $p_\rho = p_\rho^*$ (equivalently $p_\sigma = p_\sigma^*$), all the outcomes are possible.
• Figure 5: The figure shows possible outcomes for an arbitrary measurement from the set $\mathcal{E}^1_{s}(\rho,\sigma)$, $\mathcal{E}^2_{s}(\rho,\sigma)$ and $\mathcal{E}^3_{s}(\rho,\sigma)$.

Theorems & Definitions (76)

• Definition 1: The minimum postselected symmetric error
• Definition 2: Set of error-minimizing measurements
• Definition 3: The maximum acceptance
• Theorem 3.1
• proof
• Remark 1: Notes on $\mathrm{T}^{\max}$ and $\mathrm{T}^{\min}$
• Remark 2: Notes on $\mathcal{P}(\mathrm{T}^{\max})$ and $\mathcal{P}(\mathrm{T}^{\min})$
• Corollary 1
• proof
• Remark 3