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arXiv:cs/0508031 · IEEE International Symposium on Information Theory (ISIT)

Capacity theorems for quantum multiple access channels

Jon Yard, I. Devetak, Patrick M. Hayden

Published Jan 1, 2005• 17 citations
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Abstract

We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multi-letter characterizations of two different two-dimensional capacity regions. The first region characterizes the rates at which it is possible for one sender to send classical information while the other sends quantum information. The second region gives the rates at which each sender can send quantum information. We give an example of a channel for which each region has a single-letter description, concluding with a characterization of the rates at which each user can simultaneously send classical and quantum information

DOI
10.1109/isit.2005.1523464

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We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multi-letter characterizations of two different two-dimensional capacity regions. The first region characterizes the rates at which it is possible for one sender to send classical information while the other sends quantum information. The second region gives the rates at which each sender can send quantum information. We give an example of a channel for which each region has a single-letter description, concluding with a characterization of the rates at which each user can simultaneously send classical and quantum information
We consider quantum channels with two senders and one receiver. For an arbitrary such channel, we give multiletter characterizations of two different two-dimensional capacity regions. The first region comprises the rates at which it is possible for one sender to send classical information, while the other sends quantum information. The second region consists of the rates at which each sender can send quantum information. For each region, we give an example of a channel for which the corresponding region has a single-letter description. One of our examples relies on a new result proved here, perhaps of independent interest, stating that the coherent information over any degradable channel is concave in the input density operator. We conclude with connections to other work and a discussion on generalizations where each user simultaneously sends classical and quantum information.
The rates at which classical and quantum information can be simultaneously transmitted from two spatially separated senders to a single receiver over an arbitrary quantum channel are characterized. Two main results are proved in detail. The first describes the region of rates at which one sender can send classical information while the other sends quantum information. The second describes those rates at which both senders can send quantum information. For each of these situations, an example of a channel is given for which the associated region admits a single-letter description. This is the author's Ph.D. dissertation, submitted to the Department of Electrical Engineering at Stanford University in March, 2005. It represents an expanded version of the paper quant-ph/0501045, containing a number of tutorial chapters which may be of independent interest for those learning about quantum Shannon theory.
The depolarizing channel is one of the basic quantum channels.For simplicity and generality, quantum channels with two senders and one receiver are considered to discuss the capacity of the quantum depolarizing channel.Two different capacity regions,classical-quantum region and quantum-quantum re- gion,are deduced in theory for the quantum depolarizing channels with two senders and one receiver.The first region is comprised of the rates at which it is achievable for one sender to transmit classical information, while the other to send quantum information.The other region consists of the rates at which each sender sends quantum information.At the same time,the performance of the quantum multiple access depolarizing channel is analyzed.
We introduce and analyze an information theoretical task that we call the quantum multiple-access one-time pad. Here, a number of senders initially share a correlated quantum state with a receiver and an eavesdropper. Each sender performs a local operation to encode a classical message and sends their system to the receiver, who subsequently performs a measurement to decode the messages. The receiver will be able to decode the messages almost perfectly, while the eavesdropper must not be able to extract information about the messages even if they have access to the quantum systems transmitted. We consider a “conditional” scenario in which a portion of the receiver’s side information is also accessible to the eavesdropper. We investigate the maximum amount of classical information that can be encoded by each of the senders. We derive a single-letter characterization for the achievable rate region in an asymptotic limit of infinitely many copies and vanishingly small error.
We analyze the quantum binary adder channel, i.e. the quantum generalization of the classical, and well-studied, binary adder channel: in this model qubits rather than classical bits are transmitted. This of course is as special case of the general theory of quantum multiple access channels, and we may apply the established formulas for the capacity region to it. However, the binary adder channel is of particular interest classically, which motivates our generalizing it to the quantum domain. It turns out to be a very nice case study not only of multi-user quantum information theory, but also on the role entanglement plays there. It turns out that the analogous classical situation, the multi-user channel supported by shared randomness, is not distinct from the channel without shared randomness, as far as rates are concerned. However, we discuss the effect the new resource has on error probabilities, in an appendix. We focus specially on the effect entanglement between the senders as well as between senders and receiver has on the capacity region. Interestingly, in some of these cases one can devise rather simple codes meeting the capacity bounds, even in a zero-error model, which is in marked difference to code construction in the classical case.
Communication networks have multiple users, each sending and receiving messages. A multiple access channel (MAC) models multiple senders transmitting to a single receiver, such as the uplink from many mobile phones to a single base station. The optimal performance of a MAC is quantified by a capacity region of simultaneously achievable communication rates. We study the two-sender classical MAC, the simplest and best-understood network, and find a surprising richness in both a classical and quantum context. First, we find that quantum entanglement shared between senders can substantially boost the capacity of a classical MAC. Second, we find that optimal performance of a MAC with bounded-size inputs may require unbounded amounts of entanglement. Third, determining whether a perfect communication rate is achievable using finite-dimensional entanglement is undecidable. Finally, we show that evaluating the capacity region of a two-sender classical MAC is in fact NP-hard.