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arXiv:quant-ph/9809010 · IEEE Transactions on Information Theory

On quantum fidelities and channel capacities

H. Barnum, E. Knill, Michael A. Nielsen

Published Jul 1, 2000• 201 citations
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Abstract

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.

DOI
10.1109/18.850671

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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 kkk senders and mmm 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.
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We discuss the capacity of quantum channels for information transmission and storage. Quantum channels have dual uses: they can be used to transmit known quantum states which code for classical information, and they can be used in a purely quantum manner, for transmitting or storing quantum entanglement. We propose here a definition of the von Neumann capacity of quantum channels, which is a quantum-mechanical extension of the Shannon capacity and reverts to it in the classical limit. As such, the von Neumann capacity assumes the role of a classical or quantum capacity depending on the usage of the channel. In analogy to the classical construction, this capacity is defined as the maximum von Neumann mutual entropy processed by the channel, a measure which reduces to the capacity for classical information transmission through quantum channels (the ``Kholevo capacity'') when known quantum states are sent. The quantum mutual entropy fulfills all basic requirements for a measure of information, and observes quantum data-processing inequalities. We also derive a quantum Fano inequality relating the quantum loss of the channel to the fidelity of the quantum code. The quantities introduced are calculated explicitly for the quantum depolarizing channel. The von Neumann capacity is interpreted within the context of superdense coding, and an extended Hamming bound is derived that is consistent with that capacity.
The capacity of a quantum channel for transmission of classical information depends in principle on whether product states or entangled states are used at the input, and whether product or entangled measurements are used at the output. We show that when product measurements are used, the capacity of the channel is achieved with product input states, so that entangled inputs do not increase capacity. We show that this result continues to hold if sequential measurements are allowed, whereby the choice of successive measurements may depend on the results of previous measurements. We also present a new simplified expression which gives an upper bound for the Shannon capacity of a channel, and which bears a striking resemblance to the well-known Holevo bound.
We show that the capacity of a quantum channel demarcates a phase transition: while reliable transmission below capacity is always possible, any attempt to transmit information above it fails catastrophically. Specifically, we prove exponential strong converse theorems for unassisted quantum and classical communication over arbitrary finite-dimensional memoryless quantum channels. At rates beyond the respective capacity, the entanglement-generation fidelity and the success probability for classical communication decay exponentially with the number of channel uses. This rules out transmission above capacity even when one tolerates arbitrarily large errors. Our proof follows the classical Arimoto strategy, augmented by a crucial new ingredient: integral representations of Rényi information measures that lead to asymptotic continuity bounds for Rényi capacities.