Weak values in continuous weak measurements of qubits

Abstract

For continuous weak measurement of qubits, we obtain exact expressions for weak values (WVs) from the post-selection restricted average of measurement outputs, by using both the quantumtrajectory-equation (QTE) and quantum Bayesian approach. The former is applicable to short-time weak measurement, while the latter can relax the measurement strength to finite. We find that even in the "very" weak limit the result can be essentially different from the one originally proposed by Aharonov, Albert and Vaidman (AAV), in a sense that our result incorporates non-perturbative correction which could be important when the AAV's WV is large. Within the Bayesian framework, we obtain also elegant expressions for finite measurement strength and find that the amplifier's noise in quantum measurement has no effect on the WVs. In particular, we obtain very useful result for homodyne measurement in circuit-QED system, which allows for measuring the real and imaginary parts of the AAV's WV by simply tuning the phase of the local oscillator. This advantage can be exploited as efficient state-tomography technique.

Probability distributions of continuous measurement results for conditioned quantum evolution

“…If we compare the distributions (24) and (34), we see that the results of the previous section are reproduced in the limit → 0, as well as in the limit of T ( 2 t a ) −1 if we take σ 2 = t a /4T . The distribution (34) thus generalizes (24) to the case where the Hamiltonian dynamics is relevant.…”

“…Indeed, one can relate the above result with weak value to conform to the definition [17] if one takes into account the evolution of the quantum state during the measurement [33]. However, we need to stress that the full distribution of the outputs cannot be obtained with the traditional weak value formalism and so far has not been obtained with its extensions [22][23][24] for continuous measurement. The method outlined here does not explicitly evoke the notion of weak values and provides a more elaborated description of a realistic measurement process.…”

“…Since that, the concept has been extended in various directions, e.g., to account for the intermediate measurement strength, the Hamiltonian evolution of the quantum states during the measurement (see [20,21] for review). In [18], the average measurement outputs have been investigated in the context of continuous weak measurement; this has been further elaborated in [22][23][24]. As to the detailed statistics of the measurement outcomes, in this context it has been considered only for simplified meter setups that correspond to measuring the light intensities in quantum optics [20,21].…”

Qubit state tomography in a superconducting circuit via weak measurements

“…In this work, we present an analysis of the weak-value-based scheme for qubit state tomography in the cQED system. In order to apply to generic parameter conditions, our study will focus on the finite strength of weak measurement [38,39]. This goes beyond the usual limit of vanishing strength, and thus results in a generalized pre-and post-selection (PPS) average, rather than the original AAV WV, as the shift of the pointer in the apparatus.…”

“…Moreover, very recent research has shown that stronger finite-strength measurement can give a better result in state tomography [40], while some efforts are devoted to developing new schemes of dispersive readout [41][42][43], which go beyond the standard approach used in the state-of-the-art experiments [28][29][30][31][32][35][36][37] and in our present work. To extract the AAV WV from the PPS average (raw signal), we propose to apply the analytic formula derived for the homodyne measurement in circuit QED [39]. By varying the phase of the local oscillator (LO), one can easily extract the complex weak value and determine the complex wavefunction by applying a simple iterative algorithm.…”