Practical purification scheme for decohered coherent-state superpositions via partial homodyne detection

We present a simple protocol to purify a coherent-state superposition that has undergone a linear lossy channel. The scheme constitutes only a single beam splitter and a homodyne detector, and thus is experimentally feasible. In practice, a superposition of coherent states is transformed into a classical mixture of coherent states by linear loss, which is usually the dominant decoherence mechanism in optical systems. We also address the possibility of producing a larger amplitude superposition state from decohered states, and show that in most cases the decoherence of the states are amplified along with the amplitude.

Figure 1

Schematic of the conditional purification scheme using partial measurement. BS: Beam splitter with transmittivity $T$.

Figure 2

Probability quadrature distributions of a CSS $P_{C_0}$ with $\phi =0$ (solid line) and a mixture of coherent states $P_0$ (dotted line). $T=0.5$ and $\alpha =1$. The purification protocol is successful when the measurement outcome falls within the interval indicated by the dashed vertical lines.

Figure 3

Probability quadrature distributions for a CSS $P_{C_{\phi }}$ with $\phi =\pi$ (solid line) and a mixture of coherent states $P_0$ (dotted line). $T=0.5$ and $\alpha =1$. The purification protocol is successful when the measurement outcome falls outside the interval indicated by the dashed vertical lines.

Figure 4

Purification efficiency for the CSS with $\phi =0$ conditioned on the homodyne detection outcomes. The phase of the local oscillator is chosen to measure the quadrature $k\equiv x_{\pi ∕2}$. $T=0.5$, $\alpha =1$, and $p_{\mathrm{in}}=0.5$.

Figure 5

Purification efficiency for the CSS with $\phi =\pi$ conditioned on the homodyne detection outcomes. The phase of the local oscillator is chosen to measure the quadrature $k\equiv x_{\pi ∕2}$. $T=0.5$, $\alpha =1$, and $p_{\mathrm{in}}=0.5$.

Figure 6

Dependence of the $p_{\mathrm{out}}$ on $p_{\mathrm{in}}$ for the CSS with $\phi =0$ (solid line) and $\phi =\pi$ (dashed line) where the outputs are conditioned on the measurement outcomes $k=0$ and $k=\pi ∕2$, respectively. $T=0.5$ and $\alpha =1$. The dotted line denotes $p_{\mathrm{out}}=p_{\mathrm{in}}$ as a reference.

Figure 7

Dependence of the purification efficiency on the initial amplitude for CSSs with $\phi =0$ (solid line) and $\phi =\pi$ (dashed line). $T=0.5$ and $p_{\mathrm{in}}=0.5$. The measurement outcome is selected to satisfy the optimal condition $\phi +\theta =0$.

Figure 8

Dependences of the purification efficiency $p_{\mathrm{out}}∕p_{\mathrm{in}}$ (left) and the probability density $P_{C_{\phi }}\left(k_{\mathrm{opt}}\right)$ (right) on the transmittance of the tapping beam splitter $T$ for $\phi =0$ (solid line) and $\phi =\pi$ (dashed line). $p_{\mathrm{in}}=0.5$ and $\alpha =1$. The measurement outcome $k_{\mathrm{opt}}$ is selected to satisfy the optimal condition $\phi +\theta =0$.

Figure 9

A schematic of the CSS-amplification process (see Ref. 11) with the decohered input and output.

Figure 10

Concatenation of the purification and amplification processes.