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arXiv:quant-ph/0603106 · Physical Review A

Quantum channel capacities: Multiparty communication

Maciej Demianowicz, Paweł Horodecki

Published Oct 27, 2006• 11 citations

Abstract

We analyze different aspects of multiparty communication over quantum memoryless channels and generalize some of the key results known from bipartite channels to the multiparty scenario. In particular, we introduce multiparty versions of subspace and entanglement transmission fidelities. We also provide alternative, local, versions of fidelities and show their equivalence to the global ones in context of capacity regions defined. An equivalence of two different capacity notions with respect to two types of fidelities is proven. In analogy to the bipartite case it is shown, via sufficiency of isometric encoding theorem, that additional classical forward side channel does not increase capacity region of any quantum channel with $k$ senders and $m$ receivers which represents a compact unit of general quantum networks theory. The result proves that recently provided capacity region of a multiple access channel [M. Horodecki et al., Nature 436, 673 (2005); J. Yard et al., e-print quant-ph/0501045], is optimal also in a scenario of an additional support of forward classical communication.

DOI: 10.1103/physreva.74.042336

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We analyze different aspects of multiparty communication over quantum memoryless channels and generalize some of the key results known from bipartite channels to the multiparty scenario. In particular, we introduce multiparty versions of subspace and entanglement transmission fidelities. We also provide alternative, local, versions of fidelities and show their equivalence to the global ones in context of capacity regions defined. An equivalence of two different capacity notions with respect to two types of fidelities is proven. In analogy to the bipartite case it is shown, via sufficiency of isometric encoding theorem, that additional classical forward side channel does not increase capacity region of any quantum channel with

k

senders and

m

receivers which represents a compact unit of general quantum networks theory. The result proves that recently provided capacity region of a multiple access channel [M. Horodecki et al., Nature 436, 673 (2005); J. Yard et al., e-print quant-ph/0501045], is optimal also in a scenario of an additional support of forward classical communication.

We show equivalence of different capacity notions in case of multipartite communication and discuss some of its consequences. We also provide quantum capacity region for a quantum channel with

k

senders and

m

receivers in the scenario in which receivers are communicated by single senders. We also point out natural generalizations to the case of two-way capacity regions.

For the discrete memoryless quantum channel, we show the equivalence of two different notions of quantum channel capacity: that which uses the entanglement fidelity as its criterion for success in transmission, and that which uses the minimum fidelity of pure states in a subspace of the input Hilbert space as its criterion. As a corollary, any source with entropy rate less than the capacity may be transmitted with high entanglement fidelity. We also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel. This enables us to simplify a known upper bound for the channel capacity. It also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity.

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.

Abstract — We show the equivalence of two different notions of quantum channel capacity: that which uses the entanglement fidelity as its criterion for success in transmission, and that which uses the minimum fidelity of pure states in a subspace of the input Hilbert space as its criterion. As a corollary, any source with entropy less than the capacity may be transmitted with high entanglement fidelity. We also show that a restricted class of encodings is sufficient to transmit any quantum source which may be transmitted on a given channel. This enables us to simplify a known upper bound for the channel capacity. It also enables us to show that the availability of an auxiliary classical channel from encoder to decoder does not increase the quantum capacity.

This work contains results concerning transmission of entanglement and subspaces as well as generation of entanglement in the limit of arbitrary many uses of compound- and arbitrarily varying quantum channels (CQC, AVQC). In both cases, the channel is described by a set of memoryless channels. Only forward communication between one sender and one receiver is allowed. A code is said to be „good“ only, if it is „good“ for every channel out of the set. Both settings describe a scenario, in which sender and receiver have only limited channel knowledge. For different amounts of information about the channel available to sender or receiver, coding theorems are proven for the CQC. For the AVQC, both deterministic and randomised coding schemes are considered. Coding theorems are proven, as well as a quantum analogue of the Ahlswede-dichotomy. The connection to zero-error capacities of stationary memoryless quantum channels is investigated. The notion of symmetrisability is defined and used for both classes of channels.

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.