Abstract
A cornerstone of quantum mechanics is the characterization 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 with the dynamics of closed systems and the stochastic evolutions associated with 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.
Highlights
Symmetries play a central role in the modern approach to quantum mechanics [1,2]
We extend Wigner’s theorem from quantum states to quantum evolutions, including both the deterministic evolution associated with the dynamics of closed systems and the stochastic evolutions associated with 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
Summary
Symmetries play a central role in the modern approach to quantum mechanics [1,2] They provide powerful methods for solving problems in atomic physics, condensed matter, high-energy physics, and quantum information science [3–5]. Wigner established that every symmetry of the set of pure states can be represented by a transformation that is either unitary or antiunitary. In this paper we characterize the dynamical symmetries of quantum theory by extending Wigner’s theorem from quantum states to quantum evolutions. The third contribution is a time-symmetric variant of the quantum framework, obtained by restricting the set of allowed quantum evolutions to a suitable subset that preserves the maximally mixed state.
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