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arXiv:1512.03878 · IEEE International Symposium on Information Theory (ISIT)

Coding for classical-quantum channels with rate limited side information at the encoder: An information-spectrum approach

Naqueeb Ahmad Warsi, Justin P. Coon

Published Jul 1, 2016• 12 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/isit.2016.7541660

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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.
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 study and solve the problem of classical channel simulation with quantum side information at the receiver. This is a generalization of both the classical reverse Shannon theorem, and the classical-quantum Slepian-Wolf problem. The optimal noiseless communication rate is found to be reduced from the mutual information between the channel input and output by the Holevo information between the channel output and the quantum side information. Our main theorem has two important corollaries. The first is a quantum generalization of the Wyner-Ziv problem: rate-distortion theory with quantum side information. The second is an alternative proof of the tradeoff between classical communication and common randomness distilled from a quantum state. The fully quantum generalization of the problem considered isquantumstateredistribution. Here the sender and receiver share a mixed quantum state and the sender wants to transfer part of her state to the receiver using entanglement and quantum communication. We present outer and inner bounds on the achievable rate pairs.
The transmission of classical information over a classical channel gave rise to the classical capacity theorem with the optimal rate in terms of the classical mutual information. Despite classical information being a subset of quantum information, the rate of the quantum capacity problem is expressed in terms of the coherent information, which does not mathematically generalize the classical mutual information. Additionally, there are multiple capacity theorems with distinct formulas when dealing with transmitting information over a noisy quantum channel. This leads to the question of what constitutes a mathematically accurate quantum generalization of classical mutual information and whether there exists a quantum task that directly extends the classical capacity problem. In this paper, we address these inquiries by introducing a quantity called the generalized information, which serves as a mathematical extension encompassing both classical mutual information and coherent information. We define a transmission task, which includes as specific instances both classical information and quantum information capacity problems, and show that the transmission capacity of this task is characterized by the generalized information.