Quantum State Smoothing for Linear Gaussian Systems

Quantum state smoothing is a technique for assigning a valid quantum state to a partially observed dynamical system, using measurement records both prior and posterior to an estimation time. We show that the technique is greatly simplified for linear Gaussian quantum systems, which have wide physical applicability. We derive a closed-form solution for the quantum smoothed state, which is more pure than the standard filtered state, while still being described by a physical quantum state, unlike other proposed quantum smoothing techniques. We apply the theory to an on-threshold optical parametric oscillator, exploring optimal conditions for purity recovery by smoothing. The role of quantum efficiency is elucidated, in both low and high efficiency limits.

Figure 1

Various long-time states of the on-threshold OPO system in Eq. (24), represented by their 1-SD Wigner function contours in phase space, centered at the origin. The homodyne angles used by Alice and Bob (${\theta }_o$, ${\theta }_u$) are at the black dot in Fig. 2. The unconditional state (solid black) shows infinite and finite variances in $q$ and $p$, respectively, as a result of the damping and squeezing. Alice’s filtered and smoothed states, are blue (filled gray) and dashed-red ellipses, respectively. The dotted-black ellipse shows the (pure) true state, conditioned on both Alice’s and Bob’s results, while the dot-dashed green ellipse shows the SWV state.

Figure 2

(Top) Contour plots of the RPR, Eq. (23), for the OPO system for different values of observed and unobserved homodyne phases using ${\eta }_o=0.5$. The dashed line represents ${\theta }_o={\theta }_u$ and the solid line is the optimal ${\theta }_u$ (that giving the highest RPR for each value of ${\theta }_o$). The circle and the star relate to Figs. 1 and 3, respectively. (Bottom) Purity for the OPO’s filtered (solid blue) and smoothed (dashed red) states, choosing the optimal ${\theta }_u^{\mathrm{opt}}$ for each ${\theta }_o$.

Figure 3

Purities, and the RPR, Eq. (23), at the starred point in Fig. 2, for the full range of Alice’s measurement efficiency ${\eta }_o$, with the lower efficiencies plotted on a log scale and the higher efficiencies on a linear scale, where the dashed vertical line at ${\eta }_o=0.1$ indicates the split. On both sides, we plot the purities of the filtered (solid blue), smoothed (dashed red), and the SWV (dotted green) states, all on a log scale (left-hand-side axis); and the RPR ($\mathcal{R}$) (dot-dashed magenta), on a linear scale (the right-hand-side axis). For ${\eta }_o\to 0$, $P_F$ matches the simple analytic expression [50] $\sqrt{2|\cos {\theta }_o|}{\eta }_o^{1/4}$ (dashed black on left), and smoothing gives a factor of $\sqrt{2}$ improvement [50], as shown by the small $\updownarrow$ symbol. For ${\eta }_o\to 1$, the RPR is $\propto (1-{\eta }_o)$ (dashed black on right).