On Feedback and the Classical Capacity of a Noisy Quantum Channel

Abstract

Abstract-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.Index Terms-Quantum information, channel capacity, quantum channels, feedback.

Quantum Feedback Channels

“…The right hand side of (26) is equivalent to the expression for the entanglement-assisted capacity C E in (4), thus demonstrating the required inequality in (19).…”

“…Whether or not classical feedback can increase the unassisted classical capacity of any noisy quantum channel is still not known, although some partial results have been obtained [19]. The classical capacities of memoryless quantum channels are therefore related by the inequalities,…”

Preface to the Second Edition

“…Shor (2002a) established the additivity of the Holevo information for entanglementbreaking channels. Bowen and Nagarajan (2005) proved that the classical capacity of an entanglement-breaking channel is not enhanced by a classical feedback channel from receiver to sender. King (2002) first proved additivity of the Holevo information for unital qubit channels and later showed it for the depolarizing channel (King, 2003).…”

Feedback in Quantum Communication

“…Indeed, it is known that if the two parties do not generate entanglement through the channel then the use of classical feedback cannot increase the capacity, and C FB = C. 12 This criterion obviously includes all product (or non-entangled) coding schemes as well as all entanglement-breaking channels.…”