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arXiv:quant-ph/0406086 · Physical Review Letters

Inequalities and Separations Among Assisted Capacities of Quantum Channels

Charles H. Bennett, Igor Devetak, Peter W. Shor, John A. Smolin

Published Apr 18, 2006• 40 citations
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Abstract

We exhibit quantum channels whose classical and quantum capacities, when assisted by classical feedback, exceed their unassisted classical Holevo capacity. These channels are designed to be noisy in a way that can be corrected with the help of the output and a reference system entangled with part of the input. A similar construction yields quantum channels whose classical capacity, when assisted by two-way classical communication independent of the source, exceeds their classical capacity assisted by feedback alone. We give a hierarchy of capacity inequalities and open questions.

DOI
10.1103/physrevlett.96.150502

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We exhibit quantum channels whose classical and quantum capacities, when assisted by classical feedback, exceed their unassisted classical Holevo capacity. These channels are designed to be noisy in a way that can be corrected with the help of the output and a reference system entangled with part of the input. A similar construction yields quantum channels whose classical capacity, when assisted by two-way classical communication independent of the source, exceeds their classical capacity assisted by feedback alone. We give a hierarchy of capacity inequalities and open questions.
In Shannon information theory, the capacity of a memoryless communication channel cannot be increased by the use of feedback from receiver to sender. In this correspondence, the use of classical feedback is shown to provide no increase in the unassisted classical capacity of a memoryless quantum channel when feedback is used across nonentangled input states, or when the channel is an entanglement-breaking channel. This gives a generalization of the Shannon theory for certain classes of feedback protocols when transmitting through noisy quantum communication channels.
In Shannon information theory the capacity of a memoryless communication channel cannot be increased by the use of feedback from receiver to sender. In this paper the use of classical feedback is shown to provide no increase in the unassisted classical capacity of a memoryless quantum channel when feedback is used across non-entangled input states, or when the channel is an entanglement--breaking channel. This gives a generalization of the Shannon theory for certain classes of feedback protocols when transmitting through noisy quantum communication channels.
We prove that the classical capacity of an arbitrary quantum channel assisted by a free classical feedback channel is bounded from above by the maximum average output entropy of the quantum channel. As a consequence of this bound, we conclude that a classical feedback channel does not improve the classical capacity of a quantum erasure channel, and by taking into account energy constraints, we conclude the same for a pure-loss bosonic channel. The method for establishing the aforementioned entropy bound involves identifying an information measure having two key properties: 1) it does not increase under a one-way local operations and classical communication channel from the receiver to the sender and 2) a quantum channel from sender to receiver cannot increase the information measure by more than the maximum output entropy of the channel. This information measure can be understood as the sum of two terms, with one corresponding to classical correlation and the other to entanglement.
We investigate the use of noisy entanglement as a resource in classical communication via a quantum channel. In particular, we are interested in the question whether for any entangled state, including bound entangled states, there exists a quantum channel, the classical capacity of which can be increased by providing the state as an additional resource. We partially answer this question by showing, for any entangled state, the existence of a quantum memory channel, the feedback-assisted classical capacity with product encodings of which can be increased by using the state as a resource. Using a different (memoryless) channel construction, we also provide a sufficient entropic condition for an advantage in classical communication (without feedback and for general encodings) and thus provide an example of a state that is not distillable by means of one-way local operations and classical communication but can provide an advantage in the classical capacity of a number of quantum channels. As separable states cannot provide an advantage in classical communication, our condition also provides an entropic entanglement witness.
Several relations between the Holevo capacity and the entanglement-assisted classical capacity of a quantum channel are proved, necessary and sufficient conditions for their coincidence are obtained. In particular, it is shown that these capacities coincide if (correspondingly, only if) the channel (correspondingly, the χχχ--essential part of the channel) belongs to the class of classical-quantum channels (the χχχ-essential part is a restriction of a channel obtained by discarding all states useless for transmission of classical information). The obtained conditions and their corollaries are generalized to channels with linear constraints. By using these conditions it is shown that the question of coincidence of the Holevo capacity and the entanglement-assisted classical capacity depends on the constraint (even for classical-quantum channels). Properties of the difference between the quantum mutual information and the χχχ-function (constrained Holevo capacity) of a quantum channel are explored.
We give a capacity formula for the classical communication over a noisy quantum channel, when local operations and global permutations allowed in the encoding and bipartite states preshared between the sender and the receiver. The two endpoints of this formula are the Holevo capacity (without entanglement assistance) and the entanglement assisted capacity (with unlimited entanglement assistance). What's more, we show that the capacity satisfies the strong converse property and thus the formula serves as a sharp dividing line between achievable and unachievable rates of communication. We prove that the difference between the assisted capacity and the Holevo capacity is upper bounded by the discord of formation of the preshared state. As examples, we derive analytically the classical capacity of various quantum channels of interests. Our result witnesses the power of random permutation in classical communication, whenever entanglement assistance is available.