Identification of Quantum Change Points for Hamiltonians

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.

Figure 1

Problem of discriminating change points in Hamiltonians. The Hamiltonian of a quantum system suddenly changes from $H_0(t)$ to $H_1(t)$ at a specific time $t=t^{\star }$. The exact values of $H_0(t)$ and $H_1(t)$, which may be time dependent, are known but the point of the transition occurrence (i.e., $t^{\star }$) is unknown. Assuming that candidate change points are given, we wish to identify $t^{\star }$ as accurately as possible by optimizing the state input to the system, the measurement for the output, and so on. Let us assume that the number of candidate change points is finite and that each candidate has an equal chance of becoming a change point.

Figure 2

(a) The most general protocol of change point discrimination for unitary channels. Each process ${\mathcal{E}}_n$ consists of a sequence of $N$ channels $({\mathcal{U}}_{n<1},\dots ,{\mathcal{U}}_{n<N})$. Any discrimination strategy is expressed as a collection of a state $\rho$, channels ${\sigma }_2,\dots ,{\sigma }_N$, and a measurement $\{{\mathrm{\Pi }}_m{\}}_{m=0}^N$. (b) The most general nonadaptive protocol, which consists of a state $\rho$ and a measurement $\{{\mathrm{\Pi }}_m{\}}_{m=0}^N$.

Figure 3

Probability of successful identification of change points for ${\lambda }_0=1$ and ${\lambda }_1=\exp (\mathbf{i}\pi /10)$. $P_{\mathrm{sep}}$ is the maximum success probability when the separable state $|+{⟩}^{\otimes N}$ is used as an input, which is obtained by numerically solving a semidefinite programming problem. $P^{(\infty )}=\gamma$ and $P_{\mathrm{sep}}^{(\infty )}$ are limits as $N\to \infty$. The analytical solution of $P_{\mathrm{sep}}^{(\infty )}$ is given in Ref. [15].