Memory effects in attenuation and amplification quantum processes

With increasing communication rates via quantum channels, memory effects become unavoidable whenever the use rate of the channel is comparable to the typical relaxation time of the channel environment. We introduce a model of a bosonic memory channel, describing correlated noise effects in quantum-optical processes via attenuating or amplifying media. To study such a channel model, we make use of a proper set of collective field variables, which allows us to unravel the memory effects, mapping the $n$-fold concatenation of the memory channel to a unitarily equivalent, direct product of $n$ single-mode bosonic channels. We hence estimate the channel capacities by relying on known results for the memoryless setting. Our findings show that the model is characterized by two different regimes, in which the cross correlations induced by the noise among different channel uses are either exponentially enhanced or exponentially reduced.

Figure 1

Basic building blocks of the Gaussian memory channel, mixing the signal (represented by the horizontal lines) and the idler mode (vertical lines): (left) the circuital representation of the beam-splitter transformations in Eqs. (1) and (right) the circuital representation of the linear amplifier in Eqs. (2).

Figure 2

(left) A single use of the attenuating memory channel is described as an elementary transformation which is the concatenation of two beam splitters, respectively characterized by the transmissivities $\mu$ and $\kappa$. The first beam splitter couples the memory mode with the local environment, and the second one mixes the memory mode with the input signal. (right) $n$ uses of the memory channel are described as the $n$-fold concatenation of the elementary transformation [2, 13]. The concatenation is obtained by identifying, for any $j$, the outgoing memory mode at the $j$th channel use with ingoing memory mode at the $(j+1)$th channel use. The memoryless limit is obtained when there is no information flow via the memory mode (i.e., for $\mu =0$).

Figure 3

(left) A single use of the amplifying memory channel is described as an elementary transformation which is the concatenation of a linear amplifier with gain $\kappa >1$ and a beam splitter with transmissivity $\mu$. (right) The $n$-fold concatenation of the amplifying memory channel is obtained by identifying, for any $j$, the outgoing memory mode at the $j$th channel use with ingoing memory mode at the $(j+1)$th channel use [2, 13].

Figure 4

A circuital representation of Eq. (18). Provided that the environmental modes and the initial memory mode are in the vacuum state, $n$ uses of the memory channel are unitarily equivalent to the direct product of $n$ channels independently acting on the collective modes.

Figure 5

A circuital representation of the law of composition of attenuating or amplifying bosonic channels. To fix the ideas, let us consider a sequence of $n=3$ uses of the memory channel, associated with three parameters ${\eta }_j$. For the attenuating channel, let us assume ${\eta }_1⩽{\eta }_2⩽{\eta }_3⩽1$. It follows that (a) the rate of information transmission cannot be greater than the one of the channel with all the transmissivities equal to ${\eta }_3$ and (b) it cannot be smaller than the one with all the transmissivities equal to ${\eta }_1$. The same bounds hold true for the amplifying channel, assuming ${\eta }_1⩾{\eta }_2⩾{\eta }_3⩾1$.

Figure 6

The contour plot shows the quantum capacity (measured in qubits per channel use) of the memory channel as function of the amplifying or attenuating factor $\kappa$ and of the memory parameter $\mu$. The quantum capacity is computed using Eq. (37), which diverges logarithmically for $\mu \to 1$ and for $\kappa \to 1$. The dashed lines denote the boundaries among different regions, from left to right: attenuating channel, amplifying channel below the threshold, and amplifying channel above the threshold.

Figure 7

The contour plot refers to the classical capacity (measured in bits per channel use) of the memory channel as a function of the amplifying or attenuating factor $\kappa$ and of the memory parameter $\mu$, under the constraint (32) with $N=8$. For the attenuation channel ($\kappa ⩽1$), the classical capacity of the memory channel is plotted using Eq. (43). For the amplifying channel ($\kappa >1$), the plotted quantity is a lower bound on the classical capacity. The dashed lines denote the boundaries among different regions, from left to right: attenuating channel, amplifying channel below the threshold, and amplifying channel above the threshold.