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Capacities of Quantum Channels and How to Find Them

Peter W. Shor

Published Jan 1, 2003• 38 citations
OpenAlex

Abstract

Abstract: We survey what is known about the information transmitting capacities of quantum channels, and give a proposal for how to calculate some of these capacities using linear programming. 1

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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.
Quantum information processing exploits the quantum nature of information. It offers fundamentally new solutions in the field of computer science and extends the possibilities to a level that cannot be imagined in classical communication systems. For quantum communication channels, many new capacity definitions were developed in comparison to classical counterparts. A quantum channel can be used to realize classical information transmission or to deliver quantum information, such as quantum entanglement. Here we review the properties of the quantum communication channel, the various capacity measures and the fundamental differences between the classical and quantum channels.
We study the properties of the quantum information transmission channel that emerges from the quantum dynamics of particles interacting with a black hole horizon. We calculate the quantum channel capacity in two limiting cases where a single-letter capacity is known to exist: the limit of perfectly reflecting and perfectly absorbing black holes. We find that the perfectly reflecting black hole channel is closely related to the Unruh channel and that its capacity is non-vanishing, allowing for the perfect reconstruction of quantum information outside of the black hole horizon. We also find that the complementary channel (transmitting entanglement behind the horizon) is entanglement-breaking in this case, with vanishing capacity. We then calculate the quantum capacity of the black hole channel in the limit of a perfectly absorbing black hole and find that this capacity vanishes, while the capacity of the complementary channel is non-vanishing instead. Rather than inviting a new crisis for quantum physics, this finding instead is in accordance with the quantum no-cloning theorem, because it guarantees that there are no space-like surfaces that contain both the sender’s quantum state and the receiver’s reconstructed quantum state.