Quantum Schrodinger bridges: large deviations and time-symmetric ensembles

TL;DR

The paper formulates quantum Schrödinger bridges as a large-deviations problem for pre- and post-selected ensembles, establishing a time-symmetric framework in which endpoint density matrices are connected by a modified Kraus-map dynamics. The core result is that the most likely joint endpoint distribution induces a quantum bridge with forward and backward evolutions that preserve a product structure akin to the classical $\varphi(t,x)\hat{\varphi}(t,x)$ form, generalized to density operators via $\upphi$ and $\hat{\upphi}$. The authors extend the framework to intervening projective and generalized measurements, derive expressions for the most likely intermediate outcomes and weak values, and illustrate the construction with a two-level amplitude-damping example. Overall, the work unifies prior quantum-bridge approaches, links large deviations and time-symmetric quantum measurement, and provides a practical algorithmic route to infer likely intermediate measurement results under quantum dynamics. The results offer insights into how entropy-regularized transport concepts extend to noncommutative quantum dynamics and suggest avenues for quantum control and inference in time-symmetric measurement scenarios.

Abstract

Quantum counterparts of Schrodinger's classical bridge problem have been around for the better part of half a century. During that time, several quantum approaches to this multifaceted classical problem have been introduced. In the present work, we unify, extend, and interpret several such approaches through a classical large deviations perspective. To this end, we consider time-symmetric ensembles that are pre- and post-selected before and after a Markovian experiment is performed. The Schrodinger bridge problem is that of finding the most likely joint distribution of initial and final outcomes that is consistent with obtained endpoint results. The derived distribution provides quantum Markovian dynamics that bridge the observed endpoint states in the form of density matrices. The solution retains its classical structure in that density matrices can be expressed as the product of forward-evolving and backward-evolving matrices. In addition, the quantum Schrodinger bridge allows inference of the most likely distribution of outcomes of an intervening measurement with unknown results. This distribution may be written as a product of forward- and backward-evolving expressions, in close analogy to the classical setting, and in a time-symmetric way. The derived results are illustrated through a two-level amplitude damping example.

Figures (2)

• Figure 1: Quantum analog of Schrödinger's $1931$ bridge problem.
• Figure 2: Probability $P_\tau(z')$ of measuring $z'\in\{0,1\}$ at different times $\tau$. The inferred most likely probability is plotted in solid, while the dashed line captures the prior probability. The parameter $\gamma$ has been set to $1.5$.