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arXiv:2403.17252 · Physical Review A

Quantum state exclusion through offset measurement

Caleb McIrvin, Ankith Mohan, Jamie Sikora

Published Oct 10, 2024• 2 citations
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Abstract

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.

DOI
10.1103/physreva.110.042211

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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.
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 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.
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.
Distinguishing assigned quantum states with assigned probabilities via quantum measurements is a crucial problem for the transmission of classical information through quantum channels. Measurement operators maximizing the probability of correct discrimination have been characterized by Helstrom, Holevo, and Yuen since the 1970s. On the other hand, closed-form solutions are available only for particular situations enjoying high degrees of symmetry. As a suboptimal solution to the problem, measurement operators, directly determined from states and probabilities and known as square-root measurements (SRMs), were introduced by Hausladen and Wootters. These operators were also recognized to be optimal for pure states equipped with geometrical uniform symmetry (GUS). In this paper we discuss the optimality of the SRM and find necessary and sufficient conditions in order that SRM maximize the correct decision probabilities for the set of states formed by several constellations of GUS states. The results are applied to some specific examples concerning double constellations of quantum phase-shift keying and pulse-position modulation states, with possible applications to practical systems of quantum communications.
The pretty good measurement is a fundamental analytical tool in quantum information theory, giving a method for inferring the classical label that identifies a quantum state chosen probabilistically from an ensemble. Identifying and constructing the pretty good measurement for the class of bosonic Gaussian states is of immediate practical relevance in quantum information processing tasks. Holevo recently showed that the pretty good measurement for a bosonic Gaussian ensemble is a bosonic Gaussian measurement that attains the accessible information of the ensemble [IEEE Trans. Inf. Theory 66(9) (2020) 5634]. In this paper, we provide an alternate proof of Gaussianity of the pretty good measurement for a Gaussian ensemble of multimode bosonic states, with a focus on establishing an explicit and efficiently computable Gaussian description of the measurement. We also compute an explicit form of the mean square error of the pretty good measurement, which is relevant when using it for parameter estimation. Generalizing the pretty good measurement is a quantum instrument, called the pretty good instrument. We prove that the post-measurement state of the pretty good instrument is a faithful Gaussian state if the input state is a faithful Gaussian state whose covariance matrix satisfies a certain condition. Combined with our previous finding for the pretty good measurement and provided that the same condition holds, it follows that the expected output state is a faithful Gaussian state as well. In this case, we compute an explicit Gaussian description of the post-measurement and expected output states. Our findings imply that the pretty good instrument for bosonic Gaussian ensembles is no longer merely an analytical tool, but that it can also be implemented experimentally in quantum optics laboratories.
Quantum state discrimination is a fundamental problem in quantum information science. We investigate the optimal distinguishability of orthogonal two-qubit (bipartite) quantum states. The scenario consists of three parties: Alice, Bob, and Charlie. Charlie prepares one of two orthogonal states and sends one qubit to Alice and the other to Bob. Their goal is to correctly identify which state Charlie sent. In most state discrimination scenarios, it is assumed that Alice and Bob can freely communicate with one another so as to collectively agree on the best guess. In this research, we consider a more restricted setting where only one-way classical communication is possible from Alice to Bob. Under this setting, we study two figures of merit (i) Alice's optimal probability, PPP, of identifying the state , and (ii) Alice's optimal probability, P⊥P^\perpP⊥, of identifying the state along with helping Bob identify the state perfectly. We show that in general P≠P⊥P\neq P^\perpP=P⊥ and we prove a theorem for when P=P⊥P=P^\perpP=P⊥. We also found that the maximum of P−P⊥P-P^\perpP−P⊥ can arbitrarily approach 1/21/21/2.