Removing correlations in signals transmitted over a quantum memory channel

We consider a model of a bosonic memory channel, which induces correlations among the transmitted signals. The application of suitable unitary transformations at the encoding and decoding stages allows the complete removal of correlations, thereby mapping the memory channel into a memoryless one. However, such transformations, being global over an arbitrarily large number of bosonic modes, are not realistically implementable. We then introduce a family of efficiently realizable transformations, which can be used to partially remove correlations among errors, and we quantify the reduction of the gap with memoryless channels.

Figure 1

Left: A single use of the memory channel is described as an elementary transformation, which is the concatenation of two beam splitters, respectively characterized by the transmissivities $\epsilon$ and $\eta$. The first beam splitter couples the memory mode with the local environment; the second one mixes the memory mode with the input mode. Right: $n$ uses of the memory channel are described as the concatenation of the elementary transformation [3, 11]. The concatenation is obtained by identifying, for any $k$, the outgoing memory mode at the $k$th channel use, with ingoing memory mode at the $(k+1)$th. The memoryless limit is obtained by cutting the information flow through the memory mode, i.e., setting $\epsilon =0$.

Figure 2

Preprocessing and postprocessing of depth $\ell =2$, involving unitary transformations which couple two consecutive bosonic modes at the input of the memory channel (preprocessing) and two consecutive output modes (postprocessing).

Figure 3

Preprocessing and postprocessing of depth $\ell =3$, involving unitary transformations which couple three consecutive bosonic modes at the input of the memory channel (preprocessing) and three consecutive output modes (postprocessing). The generalization to unitaries of higher depth is straightforward.

Figure 4

Left: A depth-2 canonical unitary realized as beam-splitter transformation. Right: A depth-3 canonical unitary realized as a network of three beam splitters, according to Euler decomposition.

Figure 5

The mutual information $I$ vs $\epsilon$, for $N=1$. From bottom to top: without preprocessing and postprocessing, with preprocessing and postprocessing of depth $\ell =2$, and with preprocessing and postprocessing of depth $\ell =3$. From left to right, the three plots refer to $\eta =0.1$, $0.5$, $0.9$.

Figure 6

The mutual information $I^{\prime }$ vs $\epsilon$, for $N=1$. From top to bottom: without preprocessing and postprocessing, with preprocessing and postprocessing of depth $\ell =2$, and with preprocessing and postprocessing of depth $\ell =3$. From left to right, the three plots refer to $\eta =0.1$, $0.5$, $0.9$.