From time-reversal symmetry to quantum Bayes' rules

TL;DR

The paper builds a unified framework that recasts quantum Bayes' rules as manifestations of a state-over-time construction tied to a quantum time-reversal map $\tau$, thereby encoding retrodictive structure for quantum dynamics. By axiomatizing state-over-time functions with marginal-preservation and a backwards-time Bayes relation $\mathcal{E}\star\rho = \tau(\mathcal{E}^{\star}_{\rho}\star\mathcal{E}(\rho))$, the authors show that many existing quantum Bayes rules (Petz, Fuchs, Leifer–Spekkens, rotated Petz, and variants) arise as special cases under different state-over-time choices. They extend the framework to non-positive Bayes maps via the Jordan (symmetric) bloom, derive explicit $(r,s)$-parametrized Bayes maps, and connect generalized conditional expectations to Bayes inverses, offering a robust notion of quantum retrodiction that preserves time-reversal symmetry. The work clarifies the role of time-reversal in quantum theory, resolves asymmetries between forward and backward views, and provides a versatile toolkit for inference, measurement, and dynamical updating in quantum systems.

Abstract

Bayes' rule $\mathbb{P}(B|A)\mathbb{P}(A)=\mathbb{P}(A|B)\mathbb{P}(B)$ is one of the simplest yet most profound, ubiquitous, and far-reaching results of classical probability theory, with applications in any field utilizing statistical inference. Many attempts have been made to extend this rule to quantum systems, the significance of which we are only beginning to understand. In this work, we develop a systematic framework for defining Bayes' rule in the quantum setting, and we show that a vast majority of the proposed quantum Bayes' rules appearing in the literature are all instances of our definition. Moreover, our Bayes' rule is based upon a simple relationship between the notions of state over time and a time-reversal symmetry map, both of which are introduced here.

Figures (3)

• Figure 1: If $\mathcal{E}:{{\mathcal{A}}}\to{{\mathcal{B}}}$ is viewed as a CPTP map describing some dynamics from initial time $t_0$ to final time $t_1$, then this figure depicts the two states over time associated with $(\mathcal{E},\rho)$ and a Bayesian inverse. If $\mathcal{E}\star\rho$ is interpreted as having a time orientation $t_{0}\rightarrow t_{1}$, then $\mathcal{E}^{\star}_{\rho}\star\mathcal{E}(\rho)$ has time orientation $t_{1}\rightarrow t_{0}$. The quantum time-reversal map $\tau$ simultaneously reverses the orientation of time and switches the two factors so that the resulting elements can be compared on an equal footing. Bayes' rule says that these two elements are the same. In this way, $\tau$ embodies a fundamental time-reversal symmetry for states over time associated with any input process and state $(\mathcal{E},\rho)$ that admits a Bayesian inverse (see Ref. PaBu22 for a closely related inferential form of time-reversal symmetry).
• Figure 2: Given a PEM scenario $(p,\mathcal{P},\mathcal{E},\mathcal{M})$, we can use the latter three maps to define a classical channel $f:{{\mathbb C}}^{X}\to{{\mathbb C}}^{Y}$ via $f=\mathcal{M}\circ\mathcal{E}\circ\mathcal{P}$. This equality is expressed by saying that the diagram on the left commutes. The diagram in the middle depicts the evolution of the state $p$ along preparation to $\rho$, evolution to $\sigma$, and measurement to $q$. One can then use the probability $p$ and the classical channel $f$ to define the (classical) Bayesian inverse $g:{{\mathbb C}}^{Y}\to{{\mathbb C}}^{X}$. However, one can compute another map ${{\mathbb C}}^{Y}\to{{\mathbb C}}^{X}$ via $\mathcal{P}^{\star}_{p}\circ\mathcal{E}^{\star}_{\rho}\circ\mathcal{M}^{\star}_{\sigma}$ by using the Leifer--Spekkens state over time to Bayesian invert each of the pairs $(\mathcal{M},\sigma)$, $(\mathcal{E},\rho)$, and $(\mathcal{P},p)$, where $\sigma=\mathcal{E}(\rho)$ and $\rho=\mathcal{P}(p)$, to arrive at the CPTP maps $\mathcal{M}^{\star}_{\sigma}$, $\mathcal{E}^{\star}_{\rho}$, and $\mathcal{P}^{\star}_{p}$ (note that the Bayesian inverse of a preparation is a measurement and vice-versa, so that $(q,\mathcal{M}^{\star}_{\sigma},\mathcal{E}_{\rho}^{\star},\mathcal{P}^{\star}_{p})$ defines another PEM scenario). The two stochastic channels $g$ and $\mathcal{P}^{\star}_{p}\circ\mathcal{E}^{\star}_{\rho}\circ\mathcal{M}^{\star}_{\sigma}$, are equal, i.e., the diagram on the right commutes, because the Petz recovery map is compositional. This compositionality can also be viewed a quantum generalization of Jeffrey's probability kinematics Je90PaBu22BSS22.
• Figure 3: In this figure, the algebras ${{\mathcal{A}}}_0,{{\mathcal{A}}}_1,$ and ${{\mathcal{A}}}_2$ are all equal to some fiducial ${{\mathcal{A}}}$, and the subscript is meant to label the time. A state $|\psi\rangle$ is initially prepared at time $t_{0}$. Then, it evolves from $t_{0}$ to $t_{2}$ via $U_{t_{2}\leftarrow t_{0}}$. Finally, a state $|\phi_{x}\rangle$ is measured at time $t_{2}$ via the POVM $\mathcal{E}$. The two-state of Refs. ReAh95AhVa08 at some intermediate time $t_{1}$ is obtained by forward-propagating $|\psi\rangle$ to $t_{1}$ via $U_{t_{1}\leftarrow t_{0}}$ and back-propagating $|\phi_{x}\rangle$ to $t_{1}$ via $U^{\dag}_{t_{2}\leftarrow t_{1}}$. Taking the outer product of these two defines the two-state $|\psi'\rangle\langle\phi'_{x}|$. This two-state is precisely the state over time associated with our right bloom $(\mathcal{E}\circ\mathrm{Ad}_{U_{t_{2}\leftarrow t_{0}}})\star\rho\in{{\mathcal{A}}}_{0}\otimes{{\mathcal{B}}}$ after forward-propagating the latter via $U_{t_{1}\leftarrow t_{0}}$ to get an element of ${{\mathcal{A}}}_{1}\otimes{{\mathcal{B}}}$.

Theorems & Definitions (28)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 4.1
• proof
• Theorem 4.2
• proof
• Definition 5.1
• Definition 5.2
• Remark 5.3