Multiple-copy distillation and purification of phase-diffused squeezed states

We provide a detailed theoretical analysis of multiple-copy purification and distillation protocols for phase-diffused squeezed states of light. The standard iterative distillation protocol is generalized to a collective purification of an arbitrary number of $N$ copies. We also derive a semianalytical expression for the asymptotic limit of the iterative distillation and purification protocol and discuss its properties.

Figure 1

(Color online) Setup for purification of single-mode squeezed states. Two single-mode squeezed states are sent through independent dephasing channels, which result in their decoherence. They are then mixed on a balanced beam splitter (BS) and homodyned in both output ports. The virtual beam splitters (VBS) are meant to simulate inefficient homodyning.

Figure 2

(Color online) Performance of the four-step iterative purification protocol. The variance (a) and purity (b) of the purified state as well as the total success rate of the purification (c) are plotted as functions of the phase noise $\sigma$. The curve for the initial phase-diffused state (blue dashed line) is plotted along curves of several iterations (green solid lines), where the arrows indicate the direction of an increasing number of steps. The parameters were $\eta =0.85$, $V_x=0.2$, $V_p=2$, and $X=0.45$.

Figure 3

(Color online) Trade-off between the success rate of the four-step purification protocol and the resulting squeezing (a) and purity (b) of the state. The parameters were $\eta =0.85$, $V_x=0.2$, $V_p=2$, and $\sigma =0.5$.

Figure 4

(Color online) Fidelity of the purified state with respect to a Gaussian state with the same covariance matrix (green solid lines); the arrow indicates an increasing number of iteration steps. The lowermost curve (blue dashed line) corresponds to the initial dephased state. The parameters were $\eta =0.85$, $V_x=0.2$, $V_p=2$, and $X=0.45$.

Figure 5

(Color online) Iterative purification using conditioning on $\mid p\mid <Q$ (a) or phase-randomized conditioning (b), with the parameter settings $\eta =0.85$, $V_x=0.2$, $V_p=2$, $Q=0.45$. The arrows indicate an increasing number of iterations (green solid lines). The uppermost curve (blue dashed line) corresponds to the initial phase-diffused state.

Figure 6

(Color online) Collective purification of $N$ copies of the state. The beams are combined on an array of $N-1$ beam splitters ${\mathrm{BS}}_j$ and all output modes except for the last one are monitored with balanced homodyne detectors ${\mathrm{BHD}}_k$.

Figure 7

(Color online) Variance of the $x$ quadrature of the state obtained by purification from 2, 3, and 4 copies is plotted in dependence on $\sigma$. The curve labeled “in” shows the variance for the initial state. The initial parameters were $V_x=0.2$, $V_p=2$, $\eta =1$, and $X=0$.

Figure 8

(Color online) Collective purification from three copies of the state. The figure shows the variance of the $x$ quadrature before (blue dashed line) and after purification. The quadratures measured by balanced homodyne detectors ${\mathrm{BHD}}_1$ and ${\mathrm{BHD}}_2$ were the following: $x_1$, $x_2$ (green solid line), $p_1$, $p_2$ (red dot-dashed line), and $x_1$, $p_2$ (black dotted line).

Figure 9

(Color online) Equivalence of the single step of the purification for homodyning (left) and projection onto vacuum (right) as a conditioning measurement.

Figure 10

(Color online) Equivalence of the iterative purification setup for homodyning (left) and projection onto vacuum (right) as a conditioning measurement.

Figure 11

(Color online) Performance of the iterative purification protocol (green line), compared to the initial phase-diffused state (blue dashed line), asymptotic limit (red circles), and state without phase diffusion (black plus signs). The arrows indicate an increasing number of iterations. The values shown are as if verified by an imperfect detector. The parameters were $V_x=0.2$, $V_p=2$, $\eta =0.85$, and $X=0$.

Figure 12

(Color online) Variance of the asymptotic limit state relative to amount of phase fluctuations, $\sigma$ and the angle of conditioning detection, $\theta$. The values shown are as if verified by an imperfect detector. The parameters were $V_x=0.2$, $V_p=2$, and $\eta =0.85$.

Figure 13

(Color online) Purity of the asymptotic limit state relative to the amount of phase fluctuations, $\sigma$ and the angle of conditioning detection, $\theta$. The values shown are as if verified by an imperfect detector. The parameters were $V_x=0.2$, $V_p=2$, and $\eta =0.85$.