Universal first-passage time statistics for quantum diffusion

TL;DR

This work solves the quantum first-passage problem for diffusion driven by continuous measurement by mapping quantum trajectories to a drift-free, multiplicative-noise diffusion in Hilbert space. The diffusion has a state-dependent coefficient $D(x)=2\gamma^{2}x^{2}(1-x)^{2}$ with $x=|\mathcal{Q}_1(t)|^{2}$ and $\gamma=c_1-c_2$, enabling an exact solution of the associated Fokker–Planck equation and closed-form first-passage-time distributions that are universal (independent of $H$ and $L$). The authors provide explicit expressions for the two-sided first-passage distributions and moments, validated in quantum nondemolition and quantum synchronization scenarios, and show that reaching a decoherence-free subspace occurs with probability $1-\varepsilon$ in finite time while the mean hitting time diverges logarithmically as $\varepsilon\to0$, signaling a fundamental time–fidelity trade-off.

Abstract

First-passage phenomena play a fundamental role in classical stochastic processes. We here exactly solve a quantum first-passage time problem for quantum diffusion driven by measurement noise, a generalization of classical Brownian motion. Such continuous monitoring may trap the measured quantum system in a decoherence-free subspace, a fraction of the available state space that is isolated from the surroundings, and thus plays an important role in quantum information science. We analytically determine the first-passage time distribution, whose form neither depends on the system Hamiltonian nor on the measurement operator, and is therefore universal. These results provide a general framework to investigate the first-passage statistics of diffusive quantum trajectories.

Figures (3)

• Figure 1: Quantum nondemolition measurement of a two-qubit system. a) The magnetizations of the two spins along a single noisy trajectory are identical, even though only the first qubit is measured. They stay constant after the system has reached the decoherence-free subspace $\mathcal{Q}_1 = \text{span}(\ket{00})$. The inset shows the probability to be in $\mathcal{Q}_1$. b) First-passage time distributions $f_i(\tau|x_0)$, Eqs. (7), to be in subspace $\mathcal{Q}_i$$(i=1,2)$ with probability $1-\varepsilon$, for $\varepsilon = 0.003$ and initial overlap $x_0=0.1$ ($h_0=h_1=1$). Good agreement is obtained with the simulations of the stochastic master equation (1) with $3\times 10^4$ trajectories. c) Mean and variance (inset) as a function of the initial probability $x_0$, Eqs. (8)-(9), and simulated for $5\times 10^3$ trajectories.
• Figure 2: Measurement-induced quantum synchronization for a five-qubit ring. a) The magnetizations of qubit 2 and qubit 3 transition from a noisy unsynchronized state to a noise-free antisynchronized state after the system has reached the decoherence-free subspace $\mathcal{Q}_1$, when the first qubit is measured. The inset shows the corresponding probability to be in $\mathcal{Q}_1$. b) First-passage time distributions $f_i(\tau|x_0)$, Eqs. \ref{['7']}, to be in subspace $\mathcal{Q}_i$$(i=1,2)$ with probability $1-\varepsilon$, for $\varepsilon = 0.003$ and initial overlap $x_0=0.1$ ($h_0=h_1=1$). Good agreement is obtained with the simulations of the stochastic master equation \ref{['1']} with $3\times 10^4$ trajectories. c) Mean and variance (inset) as a function of the overlap $x_0$, \ref{['8', '9']}, and simulated for $5\times 10^3$ trajectories.
• Figure S1: First-passage time distribution $f(\tau \vert x_0)$ of leaving the interval ${\left(\varepsilon, 1-\varepsilon\right)}$ at either end. a) Repeated quantum nondemolition measurements of a two-qubit system (Illustration 1 of the main text). b) Measurement-induced quantum synchronization of a five-qubit ring (Illustration 2 of the main text). Excellent agreement between the analytical prediction, \ref{['eq:fpt-distribution-full']}, and the numerical simulation of the stochastic master equation \ref{['eq:diffusive-trajectory']} is obtained. Same parameters as in the main text.