Adaptive identification of coherent states

We present methods for efficient characterization of an optical coherent state $|\alpha \rangle$. We choose measurement settings adaptively and stochastically, based on data while it is collected. Our algorithm divides the estimation into two distinct steps: (i) before the first detection of a vacuum state, the probability of choosing a measurement setting is proportional to detecting vacuum with the setting, which makes using too similar measurement settings twice unlikely; and (ii) after the first detection of vacuum, we focus measurements in the region where vacuum is most likely to be detected. In step (i) [(ii)] the detection of vacuum (a photon) has a significantly larger effect on the shape of the posterior probability distribution of $\alpha$. Compared to nonadaptive schemes, our method makes the number of measurement shots required to achieve a certain level of accuracy smaller approximately by a factor proportional to the area describing the initial uncertainty of $\alpha$ in phase space. While this algorithm is not directly robust against readout errors, we make it such by introducing repeated measurements in step (i).

Figure 1

Measurement setup. The unknown coherent state within the cavity is denoted by $|\alpha \rangle$. The microwave source emits a pulse that displaces the coherent state by $\beta$ in phase space. The cavity is capacitively coupled to a Josephson photomultiplier (JPM) that registers one of two possible measurement outcomes: the displaced state is either vacuum or a state containing photons.

Figure 2

Illustration of an adaptive Bayesian inference scheme.

Figure 3

Exemplary evolution of the particles (see text) describing the probability distribution of the unknown coherent state $|\alpha \rangle$ in phase space. (a) Uniform initial prior within the disk $\left|\alpha \right|<R_0$. Here, $R_0=10$. (b) Posterior after ten measurement shots. The “holes” at positions $\left\{{\alpha }^{\prime }\right\}$ have been created by photon detections with pulse parameters $\beta =-{\alpha }^{\prime }$. (c) Posterior after the first detection of a vacuum state (here the 11th measurement shot). The weight of the posterior is concentrated near $\alpha =-{\beta }_1$, with ${\beta }_1$ the pulse parameter corresponding to the first vacuum detection.

Figure 4

Illustration of the policy of Eq. (12). The vertical and horizontal axes describe the number of vacuum detections and the number of measurement shots performed after the first vacuum detection, respectively. Different regions (i)–(iv) correspond the actions on the first to the fourth row of Eq. (12), respectively.

Figure 5

Median of the normalized squared error $2|{\alpha }_{\mathrm{true}}-\tilde{\alpha }{|}^2/R_0^2$ calculated from an ensemble of $15000$ simulated samples (see text) through the policies of Eq. (10). Different curves are for different radii $r$ denoted in the inset. Black curve is for $r\left(C\right)$ calculated through Eq. (11).

Figure 6

Median of the normalized squared error $2|{\alpha }_{\mathrm{true}}-\tilde{\alpha }{|}^2/R_0^2$ calculated from an ensemble of $15000$ simulated samples (see text) through the policies of Eq. (12) when the probability of a readout error is $P_{\mathrm{e}}=0.1$. Different curves are for different radii $r$ denoted in the inset.