Limits on the ability of quantum states to convey classical messages

Ashwin Nayak, Julia Salzman
2006 Journal of the ACM  
We revisit the problem of conveying classical messages by transmitting quantum states, and derive new, optimal bounds on the number of quantum bits required for this task. Much of the previous work on this problem, and on other communication tasks in the setting of bounded error entanglement-assisted communication, is based on sophisticated information theoretic arguments. Our results are derived from first principles, using a simple linear algebraic technique. A direct consequence is a tight
more » ... wer bound for the Inner Product function that has found applications to privacy amplification in quantum key distribution protocols. * Theorem 1.1 Suppose one party, Alice, wishes to convey n bits to the other, Bob, by communicating over a quantum channel. In any protocol, possibly two-way, in which for any x ∈ {0, 1} n the probability that Bob correctly recovers x is at least p ∈ (0, 1], the total number of qubits m exchanged by the two parties over all the rounds of communication is at least n − log 1 p . An application of the extended Holevo bound, along with Fano's inequality [14, Section 2.11], would result in the weaker bound m ≥ pn − H(p). Remark: The error in the decoding of the classical message referred to above arises from the probabilistic nature of the measurement process, rather than any noise introduced into the quantum states during the communication. Indeed, in this paper, we restrict ourselves to the study of a noiseless quantum channel. Central to the proof of Theorem 1.1 is the idea of bounding the probability of correct decoding, also known as the fidelity, when a random variable X is transmitted over a quantum channel using m quantum bits. We give a tight bound on this decoding probability by a direct argument which allows us to infer lower bounds for m without appealing to Holevo's theorem. An additional resource that may be available to parties communicating over a quantum channel is "shared entanglement": the two parties may be given some number of quantum bits jointly prepared in a fixed superposition, prior to communicating with each other. For example, they may jointly hold some number of EPR pairs, which consist of pairs of qubits prepared in the maximally entangled state 1 √ 2 (|00 + |11 ). The quantum communication channel is then said to be "entanglement-assisted."
doi:10.1145/1120582.1120587 fatcat:uxn2p35mczbcrififlw7eagbdi