Distinguishing two single-mode Gaussian states by homodyne detection: An information-theoretic approach

It is known that quantum fidelity, as a measure of the closeness of two quantum states, is operationally equivalent to the minimal overlap of the probability distributions of the two states over all possible positive-operator-valued measures (POVM’s); the POVM realizing the minimum is optimal. We consider the ability of homodyne detection to distinguish two single-mode Gaussian states and investigate to what extent it is optimal in this information-theoretic sense. We completely identify the conditions under which homodyne detection makes an optimal distinction between two single-mode Gaussian states of the same mean and show that, if the Gaussian states are pure, they are always optimally distinguished.

Figure 1

Parametrization for two Gaussian states with the same mean. Contours of the Wigner function are plotted: (a) the special case where state ${\rho }_1$ is radially symmetric in phase space $(s_1=1)$ and (b) the general case.

Figure 2

Normalized difference of the probability overlap $I_{\varphi }$ measured in homodyne detection and the fidelity $F$ in the case of pure states $({\gamma }_1={\gamma }_2=1)$, for $s_1=2$ and $\tilde{\theta }=\pi ∕3$: (a) as a function of $s_2$ and $\varphi$, (b) as a function of $\varphi$ for $s_2=1.5$ (solid line), $s_2=2$ (dotted line), and $s_2=4$ (dashed line). It is seen that there always exists an angle $\varphi$ for which $I_{\varphi }=F$.

Figure 3

Normalized difference of the probability overlap $I_{\varphi }$ measured in homodyne detection and the fidelity $F$ when one state is pure and the other is mixed $({\gamma }_1=1,{\gamma }_2=4)$, for $s_1=2$ and $\tilde{\theta }=\pi ∕3$. The plot shows that the equality $I_{\varphi }=F$ does not hold for any $s_2$.

Figure 4

Normalized difference of the probability overlap $I_{\varphi }$ measured in homodyne detection and the fidelity $F$ in the case of two mixed states $({\gamma }_1=2,{\gamma }_2=4)$, for $s_1=2$ and $\tilde{\theta }=\pi ∕3$: (a) as a function of $s_2$ and $\varphi$, (b) as a function of $\varphi$ for $s_2=1.4$ (solid line), $s_2=1.1$ (dotted line), and $s_2=3$ (dashed line). It is seen that there exists an angle $\varphi$ for which $I_{\varphi }=F$ only when Eq. (33) is satisfied—i.e., for $s_2=1.4$. The inset expands the region around the minimum.