Qudits offer no advantages over dits for sending random messages

TL;DR

This paper proves that, for a simple message-sending scenario without entanglement, quantum qudits do not outperform classical dits, extending previous results to non-uniform distributions and mixed states.

Contribution

It establishes that qudits offer no advantage over dits for message transmission without entanglement, generalizing prior uniform distribution results to mixed states.

Findings

Qudits do not outperform dits in message guessing probability

The result extends to mixed states with a sharp upper bound

No advantage of quantum over classical in this scenario

Abstract

We consider the following simple scenario: Alice has one of many possible messages, drawn from a known distribution, and wants to maximize the probability that Bob guesses her message correctly. We prove that if Alice can send only a qudit to Bob, without preshared entanglement, there is never any advantage over sending him a classical dit. This result was previously known only for a uniform distribution. We also prove a mixed-state generalization of this result in the form of an upper bound on the success probability of discriminating between mixed quantum states with a single measurement. This bound is based solely on the dimension, probability distribution, and eigenvalues of the states and is sharp among such bounds.