A Time-Symmetric Formulation of Quantum Measurement: Reinterpreting the Arrow of Time as Information Flow

TL;DR

This paper introduces a time-symmetric quantum measurement framework that models measurement as a bidirectional information update, reconciling reversibility with causality and thermodynamics, and unifying pre- and post-selected statistics.

Contribution

It presents a novel operator-based formalism for quantum measurement that is time-symmetric, preserves key physical principles, and connects quantum and classical estimation methods.

Findings

Preserves complete positivity, normalization, no-signalling.

Satisfies Spohn's inequality, ensuring non-negative entropy production.

Reduces to classical Kalman filter and RTS smoother in the classical limit.

Abstract

This study proposes a time-symmetric framework for quantum measurement that restores microscopic reversibility at the level of the dynamical description while remaining compatible with causality and thermodynamic consistency. Instead of invoking a stochastic wavefunction collapse, the measurement process is modeled as a bidirectional informational update between a forward-evolving state and a backward-propagating effect, governed by a completely positive generator and its adjoint. Within this operator-based formalism, pre- and post-selected statistics are treated on an equal footing, yielding a unified description of both. The proposed scheme rigorously preserves complete positivity, normalization, and the no-signalling principle, and it is shown to satisfy Spohn's inequality for the associated quantum Markov semigroup, thereby ensuring non-negative entropy production within this setting. The framework admits a direct experimental interpretation across a range of scenarios, including weak measurements, EPR-Bell tests, homodyne detection, and photon counting. Furthermore, in the classical limit, the bidirectional update is demonstrated to reduce to the well-established Kalman filter and Rauch-Tung-Striebel (RTS) smoother used in classical estimation theory. These results support the view that the apparent temporal asymmetry of quantum measurement arises not from fundamental dynamical irreversibility, but from informational conditioning, specifically, the one-sided way in which measurement outcomes are incorporated into our description. In this sense, the arrow of time in measurement theory may be understood as an arrow of information.