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arXiv:1506.06479 · Journal of Physics A Mathematical and Theoretical

The classical-quantum channel with random state parameters known to the sender

Holger Boche, Ning Cai, Janis Nötzel

Published Apr 5, 2016• 19 citations
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Abstract

Abstract We study an analog of the well-known Gel’fand Pinsker channel which uses quantum states for the transmission of the data. We consider the case where both the sender’s inputs to the channel and the channel states are to be taken from a finite set (the cq-channel with state information at the sender). We distinguish between causal and non-causal channel state information input by the sender. The receiver remains ignorant, throughout. We give a single-letter description of the capacity in the first case. In the second case we present two different regularized expressions for the capacity. It is an astonishing and unexpected result of our work that a simple change from causal to non-causal channel state information by the encoder causes the complexity of a numerical computation of the capacity formula to change from trivial to seemingly difficult. Still, even the non-single letter formula allows one to draw nontrivial conclusions, for example regarding the continuity of the capacity with respect to changes in the system parameters. The direct parts of both coding theorems are based on a special class of positive operator valued measurements (POVMs) which are derived from orthogonal projections onto certain representations of the symmetric groups. This approach supports a reasoning that is inspired by the classical method of types. In combination with the non-commutative union bound these POVMs yield an elegant method of proof for the direct part of the coding theorem in the first case.

DOI
10.1088/1751-8113/49/19/195302

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Abstract We study an analog of the well-known Gel’fand Pinsker channel which uses quantum states for the transmission of the data. We consider the case where both the sender’s inputs to the channel and the channel states are to be taken from a finite set (the cq-channel with state information at the sender). We distinguish between causal and non-causal channel state information input by the sender. The receiver remains ignorant, throughout. We give a single-letter description of the capacity in the first case. In the second case we present two different regularized expressions for the capacity. It is an astonishing and unexpected result of our work that a simple change from causal to non-causal channel state information by the encoder causes the complexity of a numerical computation of the capacity formula to change from trivial to seemingly difficult. Still, even the non-single letter formula allows one to draw nontrivial conclusions, for example regarding the continuity of the capacity with respect to changes in the system parameters. The direct parts of both coding theorems are based on a special class of positive operator valued measurements (POVMs) which are derived from orthogonal projections onto certain representations of the symmetric groups. This approach supports a reasoning that is inspired by the classical method of types. In combination with the non-commutative union bound these POVMs yield an elegant method of proof for the direct part of the coding theorem in the first case.
We study an analog of the well-known Gel'fand Pinsker Channel which uses quantum states for transmission of data. We consider the case where both the sender's inputs to the channel and the channel states are elements of a finite set (cq-channel with state information at the sender). While the receiver has no information about the channel states, we distinguish between two cases at the sender: he either gets causal or non-causal channel state information. We give a single-letter description of the capacity in the first case and present two different regularized expressions of the capacity for the second. It turns out that the change from causal to non-causal channel state information at the encoder causes the complexity of numerical computation of the capacity formula to change from simple to seemingly difficult. Still, even in the difficult non-causal case we draw nontrivial conclusions, for example regarding continuity of the capacity with respect to changes in the system parameters.
Communication over a random-parameter quantum channel when the decoder is required to reconstruct the parameter sequence is considered. We study scenarios that include either strictly-causal, causal, or non-causal channel side information (CSI) available at the encoder, and also when CSI is not available. This model can be viewed as a form of quantum metrology, and as the quantum counterpart of the classical rate-and-state channel with state estimation at the decoder. Regularized formulas for the capacity-distortion regions are derived. In the special case of measurement channels, single-letter characterizations are derived for the strictly-causal and causal settings. Furthermore, in the more general case of entanglement-breaking channels, a single-letter characterization is derived when CSI is not available. As a consequence, we obtain regularized formulas for the capacity of random-parameter quantum channels with CSI, generalizing previous results by Bocheet al., 2016, on classical-quantum channels. Bosonic dirty paper coding is introduced as a consequence, where we demonstrate that the optimal coefficient is not necessarily that of minimum mean-square error estimation as in the classical setting.
We consider the problem of communicating over a classical-quantum (CQ) multiple access channel with random classical states non-causally available at the transmitter, referred to as a QMSTx. QMSTx is a classical-quantum multiple access analogue of the channel considered by Gelfand and Pinsker in 1980. We undertake a Shannon-theoretic study and focus on the problem of characterizing inner bounds to the capacity region of a QMSTx. We propose a new coding scheme based on \textit{union coset codes} - codes possessing algebraic properties and derive a new inner bound that subsumes the inner based on IID random coding. We identify examples for which the derived inner bound is strictly larger.
We study the hybrid classical-quantum version of the channel coding problem for the famous Gel'fand-Pinsker channel. In the classical setting for this channel the conditional distribution of the channel output given the channel input is a function of a random parameter called the channel state. We study this problem when a rate limited version of the channel state is available at the encoder for the classical-quantum Gel'fand-Pinsker channel. We establish the capacity region for this problem in the information-spectrum setting. The capacity region is quantified in terms of spectral-sup classical mutual information rate and spectral-inf quantum mutual information rate.
In this thesis we analyse the type of states and ensembles which achieve the capacity for certain quantum channels carrying classical information. We first concentrate on the product-state capacity of a particular quantum channel, that is, the capacity which is achieved by encoding the output states from a source into codewords comprised of states taken from ensembles of non-entangled states and sending them over copies of the quantum channel. Using the "single-letter" formula proved independently by Holevo and by Schumacher and Westmoreland we obtain the product-state capacity of the qubit quantum amplitude-damping channel, which is determined by a transcendental equation in a single real variable and can be solved numerically. We demonstrate that the product-state capacity of this channel can be achieved using a minimal ensemble of non-orthogonal pure states. Next we consider the classical capacity of two quantum channels with memory, namely a periodic channel with quantum depolarising channel branches and a convex combination of quantum channels. We prove that the classical capacity for each of the classical memory channels mentioned above is, in fact, equal to the respective product-state capacities. For those channels this means that the classical capacity is achieved without the use of entangled input-states. Next we introduce the channel coding theorem for memoryless quantum channels, providing a known proof by Winter for the strong converse of the theorem. We then consider the strong converse to the channel coding theorem for a periodic quantum channel.
We consider ergodic causal classical-quantum channels (cq-channels) which additionally have a decaying input memory. In the first part we develop some structural properties of ergodic cq-channels and provide equivalent conditions for ergodicity. In the second part we prove the coding theorem with weak converse for causal ergodic cq-channels with decaying input memory. Our proof is based on the possibility to introduce joint input-output state for the cq-channels and an application of the Shannon-McMillan theorem for ergodic quantum states. In the last part of the paper it is shown how this result implies coding theorem for the classical capacity of a class of causal ergodic quantum channels.