The rate of purification of quantum trajectories

TL;DR

The paper analyzes quantum trajectories under indirect, repeated measurements and proves that, in the absence of dark subspaces, trajectories purify exponentially fast in expectation using Lyapunov methods. It then shows that the fidelity between true and estimated trajectories converges exponentially fast when both are driven by the same measurement record, establishing exponential stability of quantum filters. The results yield explicit rates determined by contractions of two-dimensional subspaces under Kraus-operator words and provide a finite-step criterion for verifying the purification condition. The findings have implications for real-time state estimation and quantum-control tasks, and they point to future work on infinite-dimensional settings and connections to dynamical purification phenomena.

Abstract

We investigate the behavior of quantum trajectories conditioned on measurement outcomes. Under a condition related to the absence of so-called dark subspaces, Kümmerer and Maassen had shown that such trajectories almost surely purify in the long run. In this article, we first present a simple alternative proof of this result using Lyapunov methods. We then strengthen the conclusion by proving that purification actually occurs at an exponential rate in expectation, again using a Lyapunov approach. Furthermore, we address the quantum state estimation problem by propagating two trajectories under the same measurement record--one from the true initial state and the other from an arbitrary initial guess--and show that the estimated trajectory converges exponentially fast to the true one, thus quantifying the rate at which information is progressively revealed through the measurement process.

Figures (5)

• Figure 1: (a) Geometric action of a Kraus operator $V_I$ on the two--dimensional subspace $\mathop{\mathrm{span}}\nolimits(\psi_k,\psi_l)$ when this subspace is dark. For any word $I$, the images $V_I\psi_k$ and $V_I\psi_l$ remain orthogonal and have the same norm. (b) Geometric action of a Kraus word $V_I$ on the two--dimensional subspace $\mathop{\mathrm{span}}\nolimits(\psi_k,\psi_l)$ in the non--dark case. There exists at least one word $I$ such that the images $V_I\psi_k$ and $V_I\psi_l$ are not orthogonal and/or do not have the same norm, leading to a strict distortion of the associated two--dimensional area.
• Figure 2: Spin chain subject to a uniform magnetic field $\vec{B}$, with repeated projective measurements performed on the last spin along the $z$ axis.
• Figure 3: (a) 300 trajectories of $\widetilde{\mathcal{V}}(\rho_n)$ (blue) and the corresponding exponential decrease of the empirical average (thick red). (b) Logarithmic-scale plot of 300 trajectories of $\widetilde{\mathcal{V}}(\rho_n)$(blue) together with the empirical average (thick red) showing the expected linear behavior. Theoretical upper bounds with rates $\gamma_p$ defined in Equation \ref{['eq:exp_bound_with_arbitrary_p']} are plotted in blue dashed line for different values of $p=1,\dots , 5$.
• Figure 4: (a) The rate $\gamma_{\textrm{2}}/(2\tau)$ as a function of $J$ and $\tau$, for $B_x=1$ and $B_z=1$. (b) The empirical rate $\gamma_{\textrm{emp}}/\tau$ as a function of $J$ and $\tau$, for $B_x=1$ and $B_z=1$.
• Figure 5: (a) The rate $\gamma_{\textrm{2}}/2$ as a function of $(B_x,B_z)$, for $J=1$ and $\tau=1$. (b) The rate $\gamma_{\textrm{emp}}$ as a function of $(B_x,B_z)$, for $J=1$ and $\tau=1$.

Theorems & Definitions (21)

• Theorem 1: Kümmerer and Maassen, 2006
• Proposition 1: Constancy of the purity on a dark subspace
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Proposition 4: Almost sure purification
• proof
• Remark 1