Suppressing correlated noise in signals transmitted over the Gaussian memory channels using a $2N$-port splitter and phase flips

A scheme for suppressing the correlated noise in signals transmitted over the bosonic Gaussian memory channels is proposed. This is a compromise solution rather than removing the noise completely. The scheme is based on linear optical elements, two $N$-port splitters and $N$ number of phase flips. The proposed scheme has the advantages that the correlated noise of the memory channels is greatly suppressed, and the input signal states can be protected excellently when transmitting over the noise channels. We examine the suppressing efficiency of the scheme for the correlated noise, both from quantum information of the states directly transmitted through the noise channel and also from the entanglement teleportation. The operation of phase flips in our scheme is important for the suppression of the correlated noise, which can diminish the effect of noise in quantum communication. Increasing the number of beam splitters also can improve the suppressing efficiency of the scheme in quantum communication.

Figure 1

The scheme for suppressing correlated noise when signals are transmitted over the noise channels. First, the input signals are encoded by the first $N$-port splitters and $N/2$ phase flips which are implemented immediately on the even or odd number channels. Then, the encoded signals are transmitted over the noise channels and contaminated by the noise unavoidably. Finally, the other $N/2$ phase flips and the seconde $N$-port splitters are implemented in the decoding procedure. Noise components are filtered out subtotally and the signals are purified. The short vertical lines between the noise channels denote the correlated noise.

Figure 2

The splitters in the encoding (a) and decoding (b) procedures. The pairs of input and output operators can be related by a unitary transformation with the the reflectivity $r_i$ and transmissivity $t_i$ for both splitters in the encoding and decoding procedures.

Figure 3

Bar charts of the magnitude for components which contribute in the output states. We have taken the transmissivity $\eta =0.6$ and memory factor $\epsilon =0.3$, respectively, for the lossy bosonic memory channel, Eq. (10). Phase flips have not been implemented in panels a(1) and b(1), while in a(2) and b(2) they have been implemented. The signs ${\mathrm{aux}}_i$ express the auxiliary components which correspond to the operators $b_{ini}$. In the suppressing scheme, when the phase flips have been implemented, the magnitude of the signal in the output state has been greatly improved, while all the other noise components have been suppressed.

Figure 4

The fidelity of the input coherent state transmitted over a lossy bosonic Gaussian memory channel in the suppressing scheme of the correlated noise. We have taken the values of the parameters $T=3$ and ${\left|\alpha \right|}^2=8$ in the numerical analysis. In order to compare the results, phase flips have been implemented in panels a(2) and b(2), while in panels a(1) and b(1) they have not been implemented.

Figure 5

The least symplectic eigenvalue ${\tilde{d}}_{-}$ of the partially transposed covariance matrix of the entangled states as a function of four parameters, ${\tilde{d}}_{-}={\tilde{d}}_{-}(\eta ,\epsilon ,T,\mu )$. Phase flips have been implemented in panels a(2) and b(2), while in panels a(1) and b(1) they have not been implemented. We assume $T=1$ and $\mu =0.6$ in the numerical analysis. In order to facilitate the comparison with the separability criterion, ${\tilde{d}}_{-}=0.5$ is also shown as dashed lines. Phase flips transform the memory factors, becoming positive ingredients for survival of entanglements when transmitting over the lossy bosonic Gaussian memory channels.

Figure 6

The density plot of ${\tilde{d}}_{-}$ as a function of the four parameters $\eta ,\epsilon ,T$, and $\mu$. In panels a(2) and b(2) phase flips have been implemented, while in a(1) and b(1) they have not been implemented. We assume $T=1$ and $\mu =0.6$ in the numerical analysis. In order to facilitate the comparison with the separability criterion, the density ${\tilde{d}}_{-}=0.5$ is also shown as dashed lines in the figures.