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arXiv:quant-ph/0403211

Some Effects of Classical Feedback on the Classical Capacity of a Memoryless Quantum Channel

G.V. Klimovitch

Published Mar 29, 2004• 0 citations
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Abstract

Classical feedback is defined here as the knowledge by the transmitter of the quantum state of the qubit received by the receiver. Such classical feedback doubles capacities of certain memoryless quantum channels without preexisting entanglement between transmitter and receiver. The increase in capacity, which is absent on classical memoryless channels, occurs because we can transform an entangled qubit pair into any other entangled state by applying a unitary operator to only one of the qubits.

DOI
10.48550/arxiv.quant-ph/0403211

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Classical feedback is defined here as the knowledge by the transmitter of the quantum state of the qubit received by the receiver. Such classical feedback doubles capacities of certain memoryless quantum channels without preexisting entanglement between transmitter and receiver. The increase in capacity, which is absent on classical memoryless channels, occurs because we can transform an entangled qubit pair into any other entangled state by applying a unitary operator to only one of the qubits.
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.
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. In quantum information theory, the no-cloning theorem means that noiseless copying and feedback of quantum information cannot be achieved. In this correspondence, quantum feedback is defined as the unlimited use of a noiseless quantum channel from receiver to sender. Given such quantum feedback, it is shown to provide no increase in the entanglement-assisted capacities of a memoryless quantum channel, in direct analogy to the classical case. It is also shown that in various cases of nonassisted capacities, feedback may increase the capacity of memoryless quantum channels.
For any quantum discrete memoryless channel, we define a quantity called quantum entanglement capacity with classical feedback (EB)({E}_{B})(EB​), and we show that this quantity lies between two other well-studied quantities. These two quantities---namely the quantum capacity assisted by two-way classical communication (Q2)({Q}_{2})(Q2​) and the quantum capacity with classical feedback (QB)({Q}_{B})(QB​)---are widely conjectured to be different: There exists a quantum discrete memoryless channel for which Q2>QB{Q}_{2}>{Q}_{B}Q2​>QB​. We then present a general scheme to convert any quantum error-correcting codes into adaptive protocols for this newly defined quantity of the quantum depolarizing channel, and illustrate with the repetition code and Shor code. We contrast the present notion with entanglement purification protocols by showing that, whilst the Leung-Shor protocol can be applied directly, recurrence methods need to be supplemented with other techniques but at the same time offer a way to improve the aforementioned repetition code. For the quantum depolarizing channel, we prove a formula that gives lower bounds on the quantum capacity with classical feedback from any EB{E}_{B}EB​ protocols. We then apply this formula to the EB{E}_{B}EB​ protocols that we discuss to obtain lower bounds on the quantum capacity with classical feedback of the quantum depolarizing channel.
We investigate whether the use of a noiseless, classical feedback channel will increase the capacity of a quantum discrete memoryless channel to transmit classical information. This problem has been previously analyzed by Bowen and Nagarajan for the case of protocols restricted to product input states. They showed that feedback did not increase the information capacity. In this paper we introduce a quantum analogue of classical causality and prove a capacity theorem (in regularized form) for the transmission of classical information.