Weak values in continuous weak measurements of qubits

For continuous weak measurements of qubits, we obtain exact expressions for weak values (WVs) from the postselection restricted average of measurement outputs, by using both the quantum-trajectory equation (QTE) and the 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 the sense that our result incorporates nonperturbative correction which could be important when the AAV 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 results for homodyne measurement in a circuit-QED system, which allows for measuring the real and imaginary parts of the AAV WV by simply tuning the phase of the local oscillator. This advantage can be exploited as an efficient state-tomography technique.

Figure 1

Nonperturbative correction of Eq. (16) to the canonical AAV WV (blue line). In this plot, we specify the initial state as $|{\psi }_i\rangle =\frac{1}{\sqrt{2}}\left(\right|1\rangle +|2\rangle )$ and the postselection state as $|{\psi }_f\rangle =\left[(cos\frac{\theta }{2}+sin\frac{\theta }{2})\right|1\rangle +(cos\frac{\theta }{2}-sin\frac{\theta }{2})|2\rangle ]/\sqrt{2}$, which result in the AAV WV ${\sigma }_w^z=tan(\theta /2)$. We see that, in contrast to the AAV WV, a different trend appears in the large ${\sigma }_w^z$ regime.