Characterizing a Statistical Arrow of Time in Quantum Measurement Dynamics

In both thermodynamics and quantum mechanics, the arrow of time is characterized by the statistical likelihood of physical processes. We characterize this arrow of time for the continuous quantum measurement dynamics of a superconducting qubit. By experimentally tracking individual weak measurement trajectories, we compare the path probabilities of forward and backward-in-time evolution to develop an arrow of time statistic associated with measurement dynamics. We compare the statistics of individual trajectories to ensemble properties showing that the measurement dynamics obeys both detailed and integral fluctuation theorems, thus establishing the consistency between microscopic and macroscopic measurement dynamics.

Figure 1

Forward and reversed trajectories of a continuously monitored superconducting qubit. (a) The experiment setup consists of a superconducting qubit dispersively coupled to the fundamental mode of a waveguide cavity. The output signal reflected from the cavity acquires a qubit-state-dependent phase shift. (b) In a single update step, a measurement record $r_k$ of duration $\delta t$ from a continuous cavity probe induces backaction on the quantum state. Upon time reversal of this update step, the state responds to backaction of a measurement result of opposite sign $-r_k$ by returning to the initial state. (c) Schematic of the state and measurement labels for forward ($M_{r_k}$) and backward (${\tilde{M}}_{r_k}$) state update procedures.

Figure 2

Measurement inefficiency as multiple observers. (a) Finite quantum efficiency can be modeled as a beam splitter, where the cavity probe is split between two observers, Alice and Bob. (b) We model the beam splitter as a time segmented splitter, which directs the signal to Alice or Bob at each time step with probabilities $\eta$ and $1-\eta$, respectively. The measurement record of a third observer, Charlie, who has access to both Alice’s and Bob’s records, can be constructed by taking either Alice’s record or Bob’s record at each time step. (c) By sampling several possible measurement records for Bob, where Bob performs measurements on his signal in the same quadrature as Alice, we construct an ensemble of possible pure state trajectories for Charlie. The average of these unraveled trajectories (black solid line) corresponds to the finite quantum efficiency trajectory based only on Alice’s record (dashed line). (d) If Bob instead measures the cavity probe in an orthogonal quadrature to Alice’s measurement, the resulting backaction on the qubit causes state evolution outside the $X–Z$ plane.

Figure 3

Statistical arrow of time for quantum measurement evolution. (a) The distributions $\mathcal{P}(\mathcal{Q})$ for different propagation times which are obtained from $2.8\times {10}^5$ runs of experiments. Here the solid curves correspond to Alice’s arrow of time when Bob measures in the same quadrature as Alice, and the dashed curve indicates the case where Bob measures in the orthogonal quadrature, constraining a range of possible values for Alice’s arrow of time. (b) Calculation of the detailed fluctuation theorem. The distributions of $\mathcal{Q}$ at time $t=0.32\mu \mathrm{s}$ (blue curves) is used to calculate the quantity $\ln [\mathcal{P}(\mathcal{Q})/\mathcal{P}(-\mathcal{Q})]$. The detailed fluctuation theorem in Eq. (3) for both of Bob’s measurement bases agrees well with the theory prediction (black line). Error bars indicate the statistical uncertainty in quantity $\ln [\mathcal{P}(\mathcal{Q})/\mathcal{P}(-\mathcal{Q})]$.

Figure 4

Distribution of $\mathcal{Q}$ and absolute irreversibility. (a) The distribution of $\mathcal{Q}$ for different propagation times obtained from $2.8\times {10}^5$ experimental trajectories. As the time increases, the distribution is more biased to large positive values of $\mathcal{Q}$. (b) Calculation of the absolute irreversibility in the case Bob measures in the same quadrature as Alice. The integral fluctuations theorem gives a quantity less than unity as a consequence of the initial state of the trajectory. Here, the dynamics of state projection favor measurement records correlated with the qubit state, resulting in a surplus of state updates when $\mathcal{Q}>0$, causing an overall longer arrow of time.