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 @math do not depend on inputs at times @math 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.
@cite_34: The capacity of a classical-quantum channel (or, in other words, the classical capacity of a quantum channel) is considered in the most general setting, where no structural assumptions such as the stationary memoryless property are made on a channel. A capacity formula as well as a characterization of the strong converse property is given just in parallel with the corresponding classical results of Verdu-Han (1994) which are based on the so-called information-spectrum method. The general results are applied to the stationary memoryless case with or without cost constraint on inputs, whereby a deep relation between the channel coding theory and the hypothesis testing for two quantum states is elucidated.
@cite_45: A formula for the capacity of arbitrary single-user channels without feedback (not necessarily information stable, stationary, etc.) is proved. Capacity is shown to equal the supremum, over all input processes, of the input-output inf-information rate defined as the liminf in probability of the normalized information density. The key to this result is a new converse approach based on a simple new lower bound on the error probability of m-ary hypothesis tests among equiprobable hypotheses. A necessary and sufficient condition for the validity of the strong converse is given, as well as general expressions for spl epsiv -capacity. >
All the memory channels discussed in this Section are causal quantum channels, and thus the Structure Theorem applies. A completely different approach has been taken by Hayashi and Nagaoka @cite_34 , who refrain from imposing any structural assumption on the quantum channels they consider, and apply the information-spectrum method to obtain a coding theorem for the classical product state capacity, following work by Verd ' u and Han @cite_45 on classical channels with memory.