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arXiv:2508.19085

Pretty Good Bounds on the worst-case Pretty Good Measurement

Sergio Escobar, Austin Pechan

Published Aug 26, 2025• 0 citations
arXivHTMLPDF

Abstract

We derive a new lower bound on the success probability of the Pretty Good Measurement (PGM) for worst-case quantum state discrimination among $m$ pure states. Our bound is strictly tighter than the previously known Gram-matrix-based bound for $m\geq 4$. The proof adapts techniques from Barnum and Knill's analysis of the average-case PGM, applied here to the worst-case scenario. By comparing the PGM to the sequential measurement algorithm, we obtain a guarantee showing that, in the low-fidelity regime, the PGM's success probability decreases quadratically with respect to the maximum pairwise overlap, rather than linearly as in earlier bounds.

DOI
10.48550/arxiv.2508.19085

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We derive a new lower bound on the success probability of the Pretty Good Measurement (PGM) for worst-case quantum state discrimination among mmm pure states. Our bound is strictly tighter than the previously known Gram-matrix-based bound for m≥4m\geq 4m≥4. The proof adapts techniques from Barnum and Knill's analysis of the average-case PGM, applied here to the worst-case scenario. By comparing the PGM to the sequential measurement algorithm, we obtain a guarantee showing that, in the low-fidelity regime, the PGM's success probability decreases quadratically with respect to the maximum pairwise overlap, rather than linearly as in earlier bounds.
We determine the optimal universal coefficient in a reduction of multiple to binary quantum hypothesis testing. Specifically, for every finite ensemble in any Hilbert space dimension (finite or infinite), we prove that the error probability of the global pretty good measurement (PGM) is at most four times the sum of the optimal binary error probabilities. By constructing a family of regular-simplex ensembles, we further show that the coefficient four is optimal, even for arbitrary global measurements. This result improves on the pairwise bounds established by Cheng and Liu [arXiv:2606.06246 (2026)] and entails an explicit, sharp guarantee for the standard PGM itself. Our proof is also simpler: rather than constructing sequential measurements and applying a union bound, our proof relies on purely matrix-analytic techniques, combining a direct block Gram matrix analysis of the PGM error and a refined inequality between quantum Hellinger distance and Bures χ2χ^2χ2-divergence. Our analysis also yields a refined, one-shot pairwise Chernoff bound and explicit sufficient number of copies for a desired discrimination error.
We show that the quantum measurement known as the pretty good measurement can be used to identify an unknown quantum state picked from any set of nnn mixed states that have pairwise fidelities upper-bounded by a constant below 1, given O(log⁡n)O(\log n)O(logn) copies of the unknown state, with high success probability in the worst case. If the unknown state is promised to be pure, there is an explicit measurement strategy which solves this worst-case quantum state discrimination problem with O~(∥G∥)\widetilde{O}(\|G\|)O(∥G∥) copies, where GGG is the Gram matrix of the states.
Pretty Good Measurement (PGM) is a near-optimal strategy for quantum state discrimination, but its practical realization becomes unstable when the ensemble operator is singular or ill-conditioned. We introduce a numerically robust PGM formulation based on the Moore-Penrose pseudoinverse, replacing the standard inverse square root with a threshold-regularized variant that remains well-defined across different spectral regimes. We develop a hybrid classical-quantum framework that combines pseudoinverse-based spectral preprocessing with quantum circuit realizations using block-encoding and spectral-transformation techniques. The framework incorporates support awareness, yielding physically meaningful measurement operators even in rank-deficient cases, and employs oblivious amplitude amplification to improve circuit-level success probabilities. Extensive numerical and circuit-level simulations show close agreement between theoretical predictions and quantum circuit outputs. Experiments on synthetic and real datasets, including ill-conditioned and degenerate scenarios, demonstrate stable discrimination performance where standard PGM becomes numerically unstable. The results establish a practical hybrid classical-quantum framework for robust quantum state discrimination and extend previous circuit-based implementations of the PGM testing stage toward pseudoinverse-aware measurement design.
The Barnum-Knill theorem states that the optimal success probability in the multiple state discrimination task is not more than the square root of the success probability when the pretty good or square-root measurement is used for this task. An assumption of the theorem is that the underlying ensemble consists of finitely many quantum states over a finite-dimensional quantum system. Motivated in part by the fact that the success probability is not a relevant metric for continuous ensembles, in this paper we provide a generalization of the notion of pretty good measurement and the Barnum-Knill theorem for general quantum ensembles, including those described by a continuous parameter space and an infinite-dimensional Hilbert space. To achieve this, we also design a general metric of performance for quantum measurements that generalizes the success probability, namely, the expected gain of the measurement with respect to a positive score function. A notable consequence of the main result is that, in a Bayesian estimation task, the mean square error of the generalized pretty good measurement does not exceed twice the optimal mean square error.
The quantum state discrimination problem has Alice sending a quantum state to Bob who wins if he correctly identifies the state. The pretty good measurement, also known as the square root measurement, performs pretty well at this task. We study the version of this problem where Bob tries to lose with the greatest probability possible (which is harder than it sounds). We define the pretty bad measurement, which performs pretty well at this task, or in other words, pretty poorly for the original task. We show that both the pretty good measurement and the pretty bad measurement are always no worse than blind guessing at their respective tasks. As an application, we apply the pretty bad measurement to the quantum state anomaly detection problem and show how to avoid pretty bad qubits.
Abstract A geometrically uniform (GU) ensemble is a uniformly weighted quantum state ensemble generated from a fixed state by a unitary representation of a finite group G. In this work we analyze the problem of discriminating GU ensembles from various angles. Assuming that the representation of G is irreducible, we first show that a particular optimal measurement can be understood as the limit of weighted ‘pretty good measurements’ (PGMs). This naturally provides examples of state discrimination for which the unweighted PGM is provably sub-optimal. We extend this analysis to certain reducible representations, and use Schur–Weyl duality to discuss two particular examples of GU ensembles in terms of Werner-type and permutation-invariant generator states. For the case of pure-state GU ensembles we give a streamlined proof of optimality of the PGM first proved in Eldar et al (2004 IEEE Trans. Inf. Theory 50 1198–207). We use this result to give simplified proofs of the optimality of the PGM, along with expressions for the corresponding success probabilities, for two tasks: the hidden subgroup problem (HSP) over semidirect product groups (first proved in Bacon et al (2005 46th Annual IEEE Symp. Foundations of Computer Science (FOCS’05) pp 469–78), and port-based teleportation (first proved in Mozrzymas et al (2019 New J. Phys. 20 053006) and Leditzky (2022 Lett. Math. Phys. 112 98). Finally, we consider the n-copy setting and adapt a result of Montanaro (2007 Commun. Math. Phys. 273 619–36) to derive a compact and easily evaluated lower bound on the success probability of the PGM for this task. This result can be applied to the HSP to obtain a new proof for an upper bound on the sample complexity by Hayashi et al (2006 Quantum Inf. Comput. 8 345–58).