Symmetries of quantum evolutions

TL;DR

This paper extends Wigner's theorem to quantum evolutions, showing symmetries decompose into unitary or antiunitary parts, and demonstrates the impossibility of extending time reversal symmetry to all quantum evolutions, proposing a restricted time symmetric formulation.

Contribution

It generalizes Wigner's theorem to quantum evolutions and establishes a no-go theorem for extending time reversal symmetry to all evolutions, proposing a new time symmetric framework.

Findings

Symmetries of quantum evolutions decompose into two state space symmetries.

Impossible to extend time reversal symmetry to the full set of quantum evolutions.

A new time symmetric formulation restricts allowed evolutions, including unitary and projective measurements.

Abstract

A cornerstone of quantum mechanics is the characterisation of symmetries provided by Wigner's theorem. Wigner's theorem establishes that every symmetry of the quantum state space must be either a unitary transformation, or an antiunitary transformation. Here we extend Wigner's theorem from quantum states to quantum evolutions, including both the deterministic evolution associated to the dynamics of closed systems, and the stochastic evolutions associated to the outcomes of quantum measurements. We prove that every symmetry of the space of quantum evolutions can be decomposed into two state space symmetries that are either both unitary or both antiunitary. Building on this result, we show that it is impossible to extend the time reversal symmetry of unitary quantum dynamics to a symmetry of the full set of quantum evolutions. Our no-go theorem implies that any time symmetric formulation of quantum theory must either restrict the set of the allowed evolutions, or modify the operational interpretation of quantum states and processes. Here we propose a time symmetric formulation of quantum theory where the allowed quantum evolutions are restricted to a suitable set, which includes both unitary evolution and projective measurements, but excludes the deterministic preparation of pure states. The standard operational formulation of quantum theory can be retrieved from this time symmetric version by introducing an operation of conditioning on the outcomes of past experiments.