Time reversal symmetry of generalized quantum measurements with past and future boundary conditions

Abstract

We expand the time reversal symmetry arguments of quantum mechanics, originally proposed by Wigner in the context of unitary dynamics, to contain situations including generalized measurements for monitored quantum systems. We propose a scheme to derive the time reversed measurement operators by considering the Schrödinger picture dynamics of a qubit coupled to a measuring device, and show that the time reversed measurement operators form a Positive Operator Valued Measure (POVM) set. We present three particular examples to illustrate time reversal of measurement operators : (1) the Gaussian spin measurement, (2) a dichotomous POVM for spin, and (3) the measurement of qubit fluorescence. We then propose a general rule to unravel any rank two qubit measurement, and show that the backward dynamics obeys the retrodicted equations of the forward dynamics starting from the time reversed final state. We demonstrate the time reversal invariance of dynamical equations using the example of qubit fluorescence. We also generalize the discussion of a statistical arrow of time for continuous quantum measurements introduced by Dressel et al. [Phys. Rev. Lett. 119, 220507 (2017)]: we show that the backward probabilities can be computed from a process similar to retrodiction from the time reversed final state, and extend the definition of an arrow of time to ensembles prepared with pre-and post-selections, where we obtain a non-vanishing arrow of time in general. We discuss sufficient conditions for when time's arrow vanishes and show our method also captures the contributions to time's arrow due to natural physical processes like relaxation of an atom to its ground state. As a special case, we recover the time reversibility of the weak value as its complex conjugate using our method, and discuss how our conclusions differ from the time-symmetry argument of Aharonov-Bergmann-Lebowitz (ABL) rule. I. INTRODUCTIONAlthough most 1 of the microscopic laws of physics are invariant under a suitable time reversal symmetry operation, there seems to exist a preferred ordering in which events are more likely to happen than otherwise, in the macroscopic world. This is true for a box of an ideal gas -where one can practically keep track of the dynamics of every single molecule given their initial conditions, and knowing all the microscopic interactions, while the second law of thermodynamics dictates that the gas molecules within the box re-distribute themselves and evolve towards a final state where the entropy is a maximum [4] -and in cosmology, from the observation of an expanding universe [5]. These apparent asymmetrical notions of time are also manifest in our everyday experiences as conscious observers; According to Wheeler, our notion of a past corresponds to experiencing a definite, informative, thus recordable set of events [6,7], and from an information theory perspective, they correspond to processes where entropy always increases or remains constant [7]. Understanding how a definite arrow of time emerges from a time reversal invar...

Fluctuation theorems for continuous quantum measurements and absolute irreversibility

“…While previous studies [39,40] focused on the average of the defined arrow of time measure, we show here that it is constrained by a FT analogous to those previously derived for the entropy produced in contact with a heat bath. We demonstrate that continuous quantum measurement leads to absolutely irreversible dynamics: just as the free expansion of a gas, the wavefunction collapse generates backward trajectories without forward counterparts.…”

“…Reversibility, i.e. a zero average entropy production, is impossible for such processes, no matter the speed at which one implements the transformation under study.Recently, stochastic thermodynamics was extended to include quantum system undergoing quantum measurement, in the absence of any thermal reservoir [36][37][38][39][40][41]. Indeed this situation leads to quantum trajectories of the measured system that are analogous to the stochastic trajectories in phase space of classical stochastic thermodynamics.…”

“…The equivalent of the first law and the second law have been derived for generic form of measurements [37], leading to applications such as an engine fueled by the quantum measurement process [42][43][44]. In [39,40], a new arrow of time measure was introduced to describe the irreversibility of continuous quantum measurement on qubits. Such weak measurements do not completely project the qubit's wavefunction on an eigenstate of the measured observable and therefore gen- Bottom: dispersive spin measurement, having a single readout r(t).…”

“…They have been studied intensively [45][46][47][48][49][50][51] and provide a wide range of applications exploiting their low invasiveness with respect to strong (projective) measurements [52][53][54][55][56][57][58][59], which justifies to extend quantum stochastic thermodynamics to describe them. The approach followed here relies on the fact that, just as the dynamics of classical systems, continuous measurements on qubit are microscopically reversible and can be undone [39,40,52,59], but yet a statistical arrow of time can be identified for the set of quantum trajectories.…”

Photonic realization of erasure-based nonlocal measurements

“…Our scheme is actually a nondemolition measurement of the nonlocal observable where quantum coherence is preserved during the process. As a consequence, our result can be extended to more complex protocols where further measurements are needed, for example, investigating causal roles and extracting nonlocal information in quantum networks and other distributed systems [39][40][41]. In addition, we employed this scheme for performing a complete Bell measurement which is free of classical communication.…”