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arXiv:2305.18872

Identifying quantum change points for Hamiltonians

Kenji Nakahira

Published May 30, 2023• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2305.18872

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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.
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 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.
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.
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.
In the change point problem, we determine when the observed distribution has changed to another one. We expand this problem to a quantum case where copies of an unknown pure state are being distributed. That is, we estimate when the distributed quantum pure state is changed. As the most fundamental case, we treat the problem of deciding the true change point tc{t}_{c}tc​ between the two given candidates t1{t}_{1}t1​ and t2{t}_{2}t2​. Our problem is mathematically equal to identifying a given state with one of the two unknown states when multiple copies of the states are provided. The minimum of the averaged error probability is given and the optimal positive operator-valued measure (POVM) is given to obtain it when the initial and final quantum pure states are subject to the invariant prior. We also compute the error probability for deciding the change point under the above POVM when the initial and final quantum pure states are fixed. These analytical results allow us to calculate the value in the asymptotic case.