Time-reversal symmetry of quantum channels: the no-go conjecture without temporal-order assumptions

Chiribella, Aurell and Życzkowski (Phys. Rev. Research 3, 033028 (2021)) extended Wigner's theorem from quantum states to quantum evolutions and proved that the time reversal of unitary dynamics, $U\mapsto U^\dagger$ or $U\mapsto U^T$, cannot be extended to a symmetry of the full set of quantum operations. They left open one issue: a channel-level no-go theorem was known only under an additional structural assumption (that time reversal reverses the temporal order of processes in every sequence, from Chiribella and Liu's work on the quantum time flip); the authors conjecture that the no-go persists for the restricted set of quantum channels without this assumption but state that the extension is beyond their scope. The paper also observes, in its discussion of relaxations of the notion of symmetry, that if linearity is given up, time reversals of all operations exist (Crooks/Petz recovery maps parametrised by a reference state, plus a complementary-channel choice for general operations) but are non-canonical, resting on arbitrary choices. The source's channel-level no-go conjecture remains open.

Background

Wigner's theorem characterises the symmetries of the quantum state space: every transformation of the density matrices that preserves measurement statistics is implemented by a unitary operator, or by an antiunitary operator (a norm-preserving antilinear map, the type that implements time reversal in this framework). Chiribella, Aurell and Życzkowski (2021) extended this question from states to evolutions. The most general evolutions of standard quantum theory are quantum operations: linear, completely positive maps $Q$ on density matrices satisfying the trace-non-increasing condition $\mathrm{Tr}[Q(\rho)]\le\mathrm{Tr}[\rho]$, where the trace is interpreted as the probability that the operation takes place in an experiment. Deterministic evolutions, called quantum channels, are the trace-preserving operations. In the Choi representation, a finite-dimensional channel is encoded by a positive semidefinite matrix $J$ with $\mathrm{Tr}_{\mathrm{out}}J=I$, which makes positivity and trace preservation concretely checkable. A bistochastic (or unital) channel is one that also maps the identity to the identity, so that it preserves the maximally mixed state.

The paper defines a symmetry of a set of quantum operations as a one-to-one transformation that (i) maps random mixtures of operations into random mixtures with the same probabilities and (ii) maps impossible (zero-probability) operations into impossible operations. Its Wigner theorem for evolutions states that every symmetry of the full operation space decomposes into two state-space symmetries acting before and after the evolution, which are either both unitary or both antiunitary. Time reversal of closed-system dynamics is the map $U\mapsto U^\dagger$, or the transpose $U\mapsto U^T$ (defined relative to a fixed but otherwise arbitrary basis); both extend canonically to the bistochastic channels. Building on the decomposition theorem, the paper proves its main no-go result (Theorem 4): no symmetry of the full set of quantum operations acts as $U\mapsto U^\dagger$ on every unitary channel, and likewise none acts as $U\mapsto U^T$. Consequently, any time-symmetric operational formulation of quantum theory must either restrict the allowed evolutions (the paper constructs one such restriction, containing unitary dynamics and projective measurements but excluding deterministic preparation of pure states) or modify the operational interpretation of states and processes.

One issue is left open, and a related observation points to a second direction. The no-go is proven for the full space of operations, including probabilistic (trace-decreasing) ones. Chiribella and Liu (2022), in their work on the quantum time flip (a higher-order construction that coherently superposes an evolution with its time-reversed counterpart), had proven a no-go for the restricted set of quantum channels, but under an additional assumption: their input-output inversion is required to reverse the order in which processes take place in any time sequence (their Requirement 1). Whether the no-go survives on the set of channels without any order-reversing assumption is conjectured, but not proven, in the source paper. The paper further observes, in its discussion of relaxations of the notion of symmetry, that if one gives up linearity, a time reversal of all operations exists but is non-canonical. Crooks' time reversal (2008), which coincides with the Petz recovery map of quantum information theory, sends a channel $\mathcal{C}$ with fixed point $\rho_0$ to the channel $\mathcal{C}_{\rho_0}(\rho)=\rho_0^{1/2}\mathcal{C}^\dagger(\rho_0^{-1/2}\rho\rho_0^{-1/2})\rho_0^{1/2}$, where $\rho_0^{-1/2}$ denotes the inverse on the support; the source paper extends it to arbitrary input/output dimensions via an arbitrary reference state $\rho_0$ (its Eq. (12)) and to arbitrary quantum operations via an arbitrary complementary operation $\mathcal{C}_0$ — an operation whose sum with $Q$ restores trace preservation, operationally the environment's share of the dynamics — together with a state $\rho_0$ (its Eq. (13)). Both constructions depend on arbitrary choices. Aurell, Zakrzewski and Życzkowski (2015) had earlier stressed that time reversal of irreversible quantum maps is intrinsically non-unique.

A substantial later literature — the quantum retrodiction program, retrodiction being the inference of past states from present data through a reversed channel — develops exactly this prior-dependent reversal. Parzygnat and Buscemi (2023) axiomatised retrodiction with priors and proved that the original Petz recovery map is the unique retrodiction functor — the unique assignment of a reversed channel to every channel that also respects sequential composition — among the candidate families (rotated and averaged Petz maps, and the universal recovery map of Junge–Renner–Sutter–Wilde–Winter, all of which fail this composition requirement). Aw, Zaw, Balanzó-Juandó and Scarani (2024) proved that reversing a process through any of its dilations (a dilation is a unitary extension to a bath, which is then reversed and the bath discarded) coincides with Bayesian retrodiction, collapsing the dilation freedom to the choice of prior. Bai, Buscemi and Scarani (2025) derived the Petz transpose map from a minimum-change principle. Parzygnat and Fullwood (2023, 2025) built the quantum Bayes' rule framework and showed time-symmetric two-time measurement correlations for open systems evolving by general channels. Ouyang, Fullwood and Wu (2026) characterised exactly which reference states admit Bayesian inverses for unital qubit channels. In all of these, the reference state (the prior) remains a free input; no canonical, prior-free selection is known.

Problem Statement

The objective is to delimit where the time reversal of quantum evolutions is structurally impossible: whether the impossibility proven for the full operation space persists for the deterministic core of the theory (the channels).

Prove or disprove the conjecture of Chiribella, Aurell and Życzkowski: for every finite-dimensional quantum system of nontrivial dimension (the two-dimensional case may require separate treatment, as Chiribella and Liu found degenerate behaviour there under the order-reversal assumption), every symmetry of the set of quantum channels — that is, every bijection of the set of trace-preserving completely positive maps onto itself compatible with random mixtures (equivalently, every affine automorphism of this convex set), with no assumption about reversing the temporal order of processes in sequences — fails to act as the adjoint $U\mapsto U^\dagger$ on all unitary channels, and likewise fails to act as the transpose $U\mapsto U^T$. An affirmative answer must cover both the adjoint and the transpose extensions; a negative answer consists of an explicit counterexample symmetry together with a proof of its required properties.

Related background, not an additional completion requirement: the source paper observes that, if linearity is given up, non-linear state-dependent time reversals exist (the Petz/Crooks maps of its Eqs. (12)–(13), parametrised by a reference state $\rho_0$ and, for general quantum operations, a complementary channel $\mathcal{C}_0$) but are non-canonical; whether any canonical, prior-free selection principle exists for them is a connected question developed by the retrodiction literature summarised in the Background section.

References

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