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_29: We show that entanglement is a useful resource to enhance the mutual information of the depolarizing channel when the noise on consecutive uses of the channel has some partial correlations. We obtain a threshold in the degree of memory above which a higher amount of classical information is transmitted with entangled signals.
@cite_21: We study the information transmission through a quantum channel, defined over a continuous alphabet and losing its energy en route, in the presence of correlated noise among different channel uses. We then show that entangled inputs improve the rate of transmission of such a channel.
@cite_1: We prove that a general upper bound on the maximal mutual information of quantum channels is saturated in the case of Pauli channels with an arbitrary degree of memory. For a subset of such channels we explicitly identify the optimal signal states. We show analytically that for such a class of channels entangled states are indeed optimal above a given memory threshold.
@cite_9: We present an experiment demonstrating the entanglement enhanced capacity of a quantum channel with correlated noise, modeled by a fiber optic link exhibiting fluctuating birefringence. In this setting, introducing entanglement between two photons is required to maximize the amount of information that can be encoded into their joint polarization degree of freedom. We demonstrated this effect using a fiber-coupled source of entagled photon pairs based on spontaneous parametric down-conversion, and a linear-optics Bell state measurement. The obtained experimental classical capacity with entangled states is equal to @math per a photon pair, and it exceeds approximately 2.5 times the theoretical upper limit when no quantum correlations are allowed.
@cite_27: We reduce the number of open additivity problems in quantum information theory by showing that four of them are equivalent. Namely, we show that the conjectures of additivity of the minimum output entropy of a quantum channel, additivity of the Holevo expression for the classical capacity of a quantum channel, additivity of the entanglement of formation, and strong superadditivity of the entanglement of formation, are either all true or all false.
@cite_49: We develop a model for a noisy communication channel in which the noise affecting consecutive transmissions is correlated. This model is motivated by fluctuating birefringence of fiber optic links. We analyze the role of entanglement of the input states in optimizing the classical capacity of such a channel, with a constraint that in every use of the channel at most one photon is being transmitted. Assuming a general form of the ensemble for two consecutive transmissions, we derive tight bounds on the classical channel capacity depending on whether the input states used for communication are separable or entangled across different temporal slots. This result demonstrates that by an appropriate choice, the channel capacity may be notably enhanced by exploiting entanglement.
@cite_2: The additivity of both the entanglement of formation and the classical channel capacity is known to be a consequence of the strong superadditivity conjecture. We show that, conversely, the strong superadditivity conjecture follows from the additivity of the entanglement of formation; this means that the two conjectures are equivalent and that the additivity of the classical channel capacity is a consequence of them.
@cite_51: We discuss a Bosonic channel model with memory effects. It relies on a multimode squeezed (entangled) environment's state. The case of lossy Bosonic channels is analyzed in detail. We show that in the absence of input energy constraints the memory channels are equivalent to their memoryless counterparts. In the case of input energy constraint we provide lower and upper bounds for the memory channel capacity.
Apparently, the first model of a quantum channel with memory was introduced by in 2001: they gave an example of a qubit channel with Markovian correlated noise @cite_29 @cite_1 in which entangled input states may increase the transmission rate for classical information. These results have recently been extended to some bosonic Gaussian channels @cite_51 @cite_21 . Such an effect has been demonstrated experimentally for optical fiber channels with fluctuating birefringence, in which consecutive light pulses undergo strongly correlated polarization transformation @cite_49 @cite_9 . (Whether such examples exist in the memoryless setting is still an open question, and presently considered one of the most eminent open problems of quantum information theory, with wide implications for other problems in the field @cite_27 @cite_2 .)