Past observable dynamics of a continuously monitored qubit

Abstract

Monitoring a quantum observable continuously in time produces a stochastic measurement record that noisily tracks the observable. For a classical process such noise may be reduced to recover an average signal by minimizing the mean squared error between the noisy record and a smooth dynamical estimate. We show that for a monitored qubit this usual procedure returns unusual results. While the record seems centered on the expectation value of the observable during causal generation, examining the collected past record reveals that it better approximates a moving-mean Gaussian stochastic process centered at a distinct (smoothed) observable estimate. We show that this shifted mean converges to the real part of a generalized weak value in the time-continuous limit without additional postselection. We verify that this smoothed estimate minimizes the mean squared error even for individual measurement realizations. We go on to show that if a second observable is weakly monitored concurrently, then that second record is consistent with the smoothed estimate of the second observable based solely on the information contained in the first observable record. Moreover, we show that such a smoothed estimate made from incomplete information can still outperform estimates made using full knowledge of the causal quantum state.Over the past decade, time-continuous quantum measurements [1-9] of superconducting qubits (such as transmons [10]) have become an important and increasingly well-controlled component of emerging quantum computing technology . Indeed, the primary method for extracting information from a superconducting transmon is to dispersively couple it to a pumped microwave resonator, then amplify and mix the leaked microwave field with a local oscillator to perform a homodyne measurement of the traveling field, which produces a stochastic time-dependent voltage that encodes information about the transmon energy basis [33][34][35]. Understanding what information is contained in the resulting stochastic readout is thus an essential theoretical issue.In simple terms, a continuous measurement can be understood as a sequence of weak measurements [36,37] on the qubit. In the superconducting case, each temporal segment of the steady-state traveling coherent microwave field acts as an independent and approximately Gaussian meter that becomes entangled with the qubit and later measured [35]. During the measurement of the field, the finite bandwidth of the circuitry typically discretizes the field into time bins of size dt. Provided that dt is longer than the correlation timescale of the traveling field, the statistics of the averaged homodyne voltage collected in each independent time bin are approximately Gaussian, producing a discrete temporal sequence of Gaussian-distributed measurement results {r j } with a wide variance that inversely depends upon the time step size dt. All information about the qubit must be extracted by processing this stochastic time series.For convenience, this time series is traditionally interpolated ...

Continuous-monitoring measured signals bounded by past and future conditions in enlarged quantum systems

“…These conventional weak values can be obtained by both methods as shown in Eqs. (8,10), where the enlarged quantum state at time t is given by Eq. ( 15) and propagates forward in time.…”

“…In this case, a time-dependent weak value can be defined by incorporating both a forward-evolving state and a backward-evolving state at the same given time [7,8]. However, it neither be obtained directly nor be monitored continuously because the quantum trajectories of the forward-and backwardevolving states are obtained separately [1][2][3][4][9][10][11]. It implies that the measured system does not evolve causally from the past to the future.…”

“…(e) and (f) represent the conventional weak values of the atom population σz w, the photon number n w , and the fluorescence signal σ− w correspond to the system prepared in the state ρ0 at time t = 0 and postselected onto the state ET at time T . These conventional weak values can be obtained by both methods as shown in Eqs (8,10),. where the enlarged quantum state at time t is given by Eq.…”

Unifying theory of quantum state estimation using past and future information

“…However, for estimating quantities that are inherently "quantum", such as quantum states or observables, naively applying classical techniques can lead to strange results if operators representing the quantities of interest and the observed operators do not commute. This occurs when we include both past and future information in the estimation [7,[13][14][15][16], as a quantum system's observables at time t do not commute with operators representing measurement results of the system at later time [17]. It is precisely this class of estimation problems, for quantum states and observables, involving the use of past-future information that we are concerned with in this paper.…”

Quantum state smoothing: why the types of observed and unobserved measurements matter

“…This allows one to assign a quantum state in the usual sense to a system of interest, conditioned on measurement records both prior and posterior to the estimation time. Inspired by the semiclassical 'posterior decoding' of [30], Guevara and Wiseman introduced quantum state smoothing in [29] by considering a situation in which a quantum system of interest is coupled to several baths (see also the subsequent related work of [31,32]). An observer, denoted by Alice, can measure only some of the baths, yielding an observed record O.…”