Rigidity of near-optimal superdense coding protocols

Abstract

Rigidity in quantum information theory refers to the stringent constraints underlying optimal or near-optimal performance in certain quantum tasks. This property plays a crucial role in verifying untrusted quantum devices and holds significance for secure quantum protocols. Previous work by Nayak and Yuen demonstrated that all optimal superdense coding protocols are locally equivalent to the canonical Bennett-Wiesner protocol. For higher-dimensional superdense coding protocols, Nayak and Yuen showed they may exist only in a relaxed form, and Farkas, Kaniewski and Nayak showed there are infinitely many dimensions $d\geq 4$ such that the rigidity does not exist even in the relaxed form.

Our work is dedicated to establishing the rigidity properties of near-optimal superdense coding protocols. Specifically, we explore scenarios where Alice can employ finite but arbitrary ancilla qubits for encoding, Bob can perform positive operator-valued measure (POVM) for decoding and can answer with error. In such contexts, we prove that any near-optimal superdense coding must be locally equivalent to a superdense coding protocol close to the canonical Bennett-Wiesner protocol.

In the search for extending the result to higher dimensional superdense coding protocols, we find a method to orthogonalize any two unitary matrices in the same space. However, the question of whether it is feasible to orthogonalize more than two $d\times d$ unitary matrices when $d>2$ remains an intriguing yet unresolved matter.