Quantum channels with memory

We present a general model for quantum channels with memory and show that it is sufficiently general to encompass all causal automata: any quantum process in which outputs up to some time $t$ do not depend on inputs at times $t^{\prime }>t$ can be decomposed into a concatenated memory channel. We then examine and present different physical setups in which channels with memory may be operated for the transfer of (private) classical and quantum information. These include setups in which either the receiver or a malicious third party have control of the initializing memory. We introduce classical and quantum channel capacities for these settings and give several examples to show that they may or may not coincide. Entropic upper bounds on the various channel capacities are given. For forgetful quantum channels, in which the effect of the initializing memory dies out as time increases, coding theorems are presented to show that these bounds may be saturated. Forgetful quantum channels are shown to be open and dense in the set of quantum memory channels.

Figure 1

Left: a quantum memory channel with input register $\mathcal{A}$, output register $\mathcal{B}$, and memory system $\mathcal{M}$. Right: a threefold concatenation $S_3$ of memory channels, with time running from left to right and coded information running from bottom to top.

Figure 2

By the structure theorem, a causal automaton $T$ can be decomposed into a chain of concatenated memory channels $S$ plus some input initializer $R$. Evaluation with the identity operator $\mathbf{1}$ means that the corresponding output is ignored.

Figure 3

Two signal states are encoded into three input registers, sent through the concatenated memory channel and then decoded into two output states. If the overall channel is (in some sense to be specified in Sec. 3B) close to the ideal channel on two inputs, the transmission rate of the above scheme is $\frac{2}{3}$. Capacity is the largest such rate in the limit of long messages and optimal encoding and decoding. In the above setup, the initial memory input can be thought of as being controlled by either the sender or a malicious third party. Similarly, the receiver may or may not be able to read out the final memory state.

Figure 4

An unmodulated spin chain as a quantum channel with memory: Alice $\left(\mathcal{A}\right)$ places the input signal on the first spin of the chain and lets it propagate to Bob $\left(\mathcal{B}\right)$, who controls the spin at the opposite end of the chain.

Figure 5

In a micromaser, a stream of two-level atoms is injected into a high-quality superconducting cavity. The field modes introduce correlations between consecutive atoms.