Time-symmetric correlations for open quantum systems

TL;DR

The paper shows that time-symmetric two-time correlations extend to open quantum systems evolving under general quantum channels when analyzed through the quantum Bayes' rule. It proves an equivalence between Bayesian inverses and $\mathscr{S}$-operational inverses for this setting, enabling explicit construction of time-reversed two-time statistics for light-touch observables. A detailed amplitude-damping example demonstrates the symmetry and contrasts the Bayesian inverse with the Petz recovery map, which generally fails to reproduce the symmetry. An experimentally feasible protocol is proposed to verify these predictions using Pauli measurements and amplitude-damping dynamics, highlighting potential tests in current quantum platforms.

Abstract

Two-time expectation values of sequential measurements of dichotomic observables are known to be time symmetric for closed quantum systems. Namely, if a system evolves unitarily between sequential measurements of dichotomic observables $\mathscr{O}_{A}$ followed by $\mathscr{O}_{B}$, then it necessarily follows that $\langle\mathscr{O}_{A}\,,\mathscr{O}_{B}\rangle=\langle\mathscr{O}_{B}\,,\mathscr{O}_{A}\rangle$, where $\langle\mathscr{O}_{A}\,,\mathscr{O}_{B}\rangle$ is the two-time expectation value corresponding to the product of the measurement outcomes of $\mathscr{O}_{A}$ followed by $\mathscr{O}_{B}$, and $\langle\mathscr{O}_{B}\,,\mathscr{O}_{A}\rangle$ is the two-time expectation value associated with the time reversal of the unitary dynamics, where a measurement of $\mathscr{O}_{B}$ precedes a measurement of $\mathscr{O}_{A}$. In this work, we show that a quantum Bayes' rule implies a time symmetry for two-time expectation values associated with open quantum systems, which evolve according to a general quantum channel between measurements. Such results are in contrast with the view that processes associated with open quantum systems -- which may lose information to their environment -- are not reversible in any operational sense. We give an example of such time-symmetric correlations for the amplitude-damping channel, and we propose an experimental protocol for the potential verification of the theoretical predictions associated with our results.

Figures (5)

• Figure 1: Visualizations of the image (in blue) of the Bloch ball (in translucent orange) after one application of the amplitude-damping channel $\mathcal{E}$ with $\gamma=0.2$ and $\gamma=0.6$, respectively. The north pole on the Bloch ball corresponds to the pure state $|0\rangle\langle0|$, which is fixed under $\mathcal{E}$.
• Figure 2: A visualization of the image (in blue) of the Bloch ball (in translucent orange) after one application of the Bayesian inverse $\mathcal{F}$ of the amplitude-damping channel with $r_3=0.2$ and $\gamma=0.6$ is shown on the left. Meanwhile, the right shows the analogous situation but with the Petz recovery map $\mathcal{R}$ for the same values of $r_{3}$ and $\gamma$. While both channels are bit-flipped amplitude-damping channels, the Bayesian inverse has an additional dephasing effect.
• Figure 3: Quantum circuits are read from left to right in time qiskit2024. The unitary operators $U_{\gamma}$ associated with the measurement of the Pauli operators $\sigma_{\gamma}$ are given by $U_{1}=H$, $U_{2}=HR_{z}(\pi/2)$, and $U_{3}=\mathds{1}_{2}$, where $H=\frac{1}{\sqrt{2}}\left(111-1\right)$ is the Hadamard gate, $R_{y}(\theta)=\left(\cos(\theta/2)-\sin(\theta/2)\sin(\theta/2)\cos(\theta/2)\right)$ is the $SU(2)$ rotation operator about the $y$-axis for $\theta\in[0,4\pi)$, and $R_{z}(\delta)=\left(100e^{i\delta}\right)$ is the phase-shift gate for $\delta\in [0,2\pi)$BCS04. Since $\sigma_{0}=\mathds{1}_{2}$ corresponds to no measurement, this case is not illustrated. (a) The circuit depicts the action of a sequential measurement of $\sigma_{\alpha}$ followed by $\sigma_{\beta}$ with an amplitude-damping channel $\mathcal{E}$ between measurements. The parameter $\theta$ is related to the amplitude-damping parameter $\gamma$ by $\cos\left(\frac{\theta}{2}\right)=\sqrt{1-\gamma}$. (b) The circuit depicts the action of a sequential measurement of $\sigma_{\beta}$ followed by $\sigma_{\alpha}$ with the Bayesian inverse $\mathcal{F}$ of $(\mathcal{E},\rho)$ between measurements. The parameters $\vartheta$ and $\varphi$ are related to $r_{3}$ and $\gamma$ by $\cos\left(\frac{\vartheta}{2}\right)=\sqrt{\frac{1+r_{3}}{1+s_{3}}}$ and $\cos\left(\frac{\varphi}{2}\right)=\sqrt{\frac{(1-\gamma)(1+s_3)}{1+r_3}}$, where $s_{3}$ is as in \ref{['eq:s3']}.
• Figure 4: Visualizations of the image of the Bloch ball after one application of the Bayesian inverse $\mathcal{F}$ of the amplitude-damping channel. The original Bloch ball is in translucent orange, while the image after one application is shown in blue. On the left, $r_3=-0.5$ and $\gamma=0.15$ so that $r_{3}<\frac{\gamma}{\gamma-2}$, which guarantees the map $\mathcal{F}$ is not completely positive. Indeed, the image of the Bloch ball on the left partly escapes the original Bloch ball, illustrating that $\mathcal{F}$ is not even positive. Meanwhile, the right shows the analogous situation for $r_{3}=-0.5$ and $\gamma=0.7$, which satisfies $r_{3}>\frac{\gamma}{\gamma-2}$ so that $\mathcal{F}$ is completely positive. It is visible that the image of the Bloch ball remains inside the original Bloch ball.
• Figure 5: The set of values for $\epsilon\in(0,1)$, $\gamma\in(0,1)$, and $z=r_{3}\in(0,1)$ for which the Bayesian inverse of the map $\mathcal{E}$ given by \ref{['eq:ADCmixCD']} and state $\rho=\frac{1}{2}(\mathds{1}_{2}+r_{3}\sigma_{3})$ is a valid quantum channel.

Theorems & Definitions (14)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Theorem 1
• Lemma 1: FuPa24a Theorem 5.3
• Lemma 2: FuPa24a Proposition 5.8
• proof : Proof of Theorem \ref{['MTX87']}