Tradeoff between energy and error in the discrimination of quantum-optical devices

We address the problem of energy-error tradeoff in the discrimination between two linear passive quantum optical devices with a single use. We provide an analytical derivation of the optimal strategy for beamsplitters and an iterative algorithm converging to the optimum in the general case. We then compare the optimal strategy with a simpler strategy using coherent input states and homodyne detection. It turns out that the former requires much less energy in order to achieve the same performances.

Figure 1

Geometrical representation of the optimization problem. For fixed values $n_1$ and $n_2$, each state of the form (18) is represented by a point of the line segment $l_{n_1,n_2}$ which extrema are $(\text{cos}(\delta n_1),n_1)$ and $(\text{cos}(\delta n_2),n_2)$. The x and y coordinates represent the expectation values $\langle \psi \left|U\right|\psi \rangle$ and $\langle \psi \left|N\right|\psi \rangle$, respectively. From the picture it is clear that the line segment $l_{0,n^{*}}$ lies below any other line segment $l_{n_1,n_2}$, and so it identifies the family of optimal states, as we prove in Prop. 1.

Figure 2

Optimal tradeoff between the energy $E$ and the probability of error $P_e$ in the discrimination of $I$ and $U=\exp [i(\delta a_1^{†}{a}_1-\delta a_2^{†}{a}_2)]$ for various values of $\delta$.

Figure 3

Optimal tradeoff between the energy $E$ and the probability of error $P_e$ in the discrimination of two linear passive optical devices $I$ and $U$. We consider the case in which $U$ is a two mode device, namely $U=\exp [i({\delta }_1{a}_1^{†}{a}_1+{\delta }_2{a}_2^{†}{a}_2)]$.

Figure 4

Optimal tradeoff between the energy $E$ and the probability of error $P_e$ in the discrimination of $I$ and $U=\exp [i(\delta a_1^{†}{a}_1-\delta a_2^{†}{a}_2)]$ [$\delta =\pi /4$ in (a) and $\delta =\pi /12$ in (b)]. The upper line represents the discrimination with coherent states and homodyne detections, while the lower line represents the optimal discrimination. Comparing (a) and (b) we notice that the improvement provided by the optimal strategy increases as $\delta$ decreases.