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Physical Review Letters

Identification of Quantum Change Points for Hamiltonians

Kenji Nakahira

Published Nov 22, 2023• 4 citations
Publisher

Abstract

Detecting sudden changes in the environment is crucial in many statistical applications. We mainly focus on identifying sudden changes in weak signals transmitted by electromagnetic or gravitational waves. Assuming that the Hamiltonians representing the signals before and after the change are known, we aim to find a discrimination strategy that can detect the change point with the best possible accuracy. This problem has potential applications in accurately detecting the precise timing of events such as stellar explosions, foreign object intrusions, specific chemical bonds, and phase transitions. We formulate this problem as a quantum process discrimination problem by discretizing the time evolution of a quantum system as a sequence of unitary channels. However, due to the complexity of the dynamics, solving such a multiple process discrimination problem is typically challenging. We demonstrate that the maximum success probability for the Hamiltonian change point problem with any finite number of candidate change points can be determined and has a simple analytical form.

DOI
10.1103/physrevlett.131.210804

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Detecting sudden changes in the environment is crucial in many statistical applications. We mainly focus on identifying sudden changes in weak signals transmitted by electromagnetic or gravitational waves. Assuming that the Hamiltonians representing the signals before and after the change are known, we aim to find a discrimination strategy that can detect the change point with the best possible accuracy. This problem has potential applications in accurately detecting the precise timing of events such as stellar explosions, foreign object intrusions, specific chemical bonds, and phase transitions. We formulate this problem as a quantum process discrimination problem by discretizing the time evolution of a quantum system as a sequence of unitary channels. However, due to the complexity of the dynamics, solving such a multiple process discrimination problem is typically challenging. We demonstrate that the maximum success probability for the Hamiltonian change point problem with any finite number of candidate change points can be determined and has a simple analytical form.
The identification of environmental changes is crucial in many fields. The present research is aimed at investigating the optimal performance for detecting change points in a quantum system when its Hamiltonian suddenly changes at a specific time. Assume that the Hamiltonians before and after the change are known and that the prior probability of each prospective change point is identical. These Hamiltonians can be time-dependent. The problem considered in this study is an extension of the problem of discriminating multiple quantum processes that consist of sequences of quantum channels. Although it is often extremely difficult to find an analytical solution to such a problem, we demonstrate that the maximum success probability for the Hamiltonian change point problem can be determined analytically and has a simple form.
Sudden changes are ubiquitous in nature. Identifying them is crucial for a number of applications in biology, medicine, and social sciences. Here we take the problem of detecting sudden changes to the quantum domain. We consider a source that emits quantum particles in a default state, until a point where a mutation occurs that causes the source to switch to another state. The problem is then to find out where the change occurred. We determine the maximum probability of correctly identifying the change point, allowing for collective measurements on the whole sequence of particles emitted by the source. Then, we devise online strategies where the particles are measured individually and an answer is provided as soon as a new particle is received. We show that these online strategies substantially underperform the optimal quantum measurement, indicating that quantum sudden changes, although happening locally, are better detected globally.
The detection of change points is a pivotal task in statistical analysis. In the quantum realm, it is a new primitive where one aims at identifying the point where a source that supposedly prepares a sequence of particles in identical quantum states starts preparing a mutated one. We obtain the optimal procedure to identify the change point with certainty-naturally at the price of having a certain probability of getting an inconclusive answer. We obtain the analytical form of the optimal probability of successful identification for any length of the particle sequence. We show that the conditional success probabilities of identifying each possible change point show an unexpected oscillatory behavior. We also discuss local (online) protocols and compare them with the optimal procedure.
Identifying the precise moment when a quantum channel undergoes a change is a fundamental problem in quantum information theory. We study how accurately one can determine the time at which a channel transitions to another. We investigate the quantum limit of the average success probability in unambiguous discrimination, in which errors are completely avoided by allowing inconclusive results with a certain probability. This problem can be viewed as a quantum process discrimination task, where the process consists of a sequence of quantum channels; however, obtaining analytical solutions for quantum process discrimination is generally extremely challenging. In this paper, we propose a method to derive lower and upper bounds on the maximum average success probability in unambiguous discrimination. In particular, when the channels before and after the change are unitary, we show that the maximum average success probability can be analytically expressed in terms of the length of the channel sequence and the discrimination limits for the two channels.
I study the problem of optimally discriminating between two open quantum dynamical processes in a single-shot scenario, with the aim of minimizing the error probability of identification. This task requires jointly optimising the input state—potentially entangled with an ancillary system isolated from the dynamics—and the time at which the resulting time-dependent quantum channels, generated by the two dynamical maps, are most distinguishable. As an illustrative case, I focus on Pauli dynamical maps and their associated families of time-dependent Pauli channels. I identify a regime in which separable strategies require waiting indefinitely until the dynamics reaches the stationary state, whereas entangled input states enable optimal discrimination at a finite time, with a strict reduction in error probability. These results highlight the crucial interplay between entanglement and timing in enhancing the distinguishability of quantum dynamical processes.
Detecting abrupt changes in data streams is crucial because they are often triggered by events that have important consequences if left unattended. Quickest change-point detection has become a vital sequential analysis primitive that aims at designing procedures that minimize the expected detection delay of a change subject to a bounded expected false alarm time. We put forward the quantum counterpart of this fundamental primitive on streams of quantum data. We give a lower bound on the mean minimum delay when the expected time of a false alarm is asymptotically large, under the most general quantum detection strategy, which is given by a sequence of adaptive collective (potentially weak) measurements on the growing string of quantum data. In addition, we give particular strategies based on repeated measurements on independent blocks of samples that asymptotically attain the lower bound and thereby establish the ultimate quantum limit for quickest change-point detection. Finally, we discuss online change-point detection in quantum channels.