Characterization of Gram matrices of multimode coherent states

TL;DR

This paper characterizes the Gram matrices of multi-mode coherent states to identify which quantum communication protocols are feasible with these states, revealing limitations and approximation capabilities relevant for experimental quantum communication.

Contribution

It provides a detailed characterization of Gram matrices of multi-mode coherent states and their approximation limits, advancing understanding of protocol implementation with these states.

Findings

Hadamard exponential of Euclidean distance matrices is positive semidefinite

Characterized the closure of Gram matrices realizable by multi-mode coherent states

Mutually unbiased bases Gram matrices cannot be approximated arbitrarily well

Abstract

Quantum communication protocols are typically formulated in terms of abstract qudit states and operations, leaving the question of an experimental realization open. Direct translation of these protocols, say into single photons with some d-dimensional degree of freedom, are typically challenging to realize. Multi-mode coherent states, on the other hand, can be easily generated experimentally. Reformulation of protocols in terms of these states has been a successful strategy for implementation of quantum protocols. Quantum key distribution and the quantum fingerprinting protocol have both followed this route. In this paper, we characterize the Gram matrices of multi-mode coherent states in an attempt to understand the class of communication protocols, which can be implemented using these states. As a side product, we are able to use this characterization to show that the Hadamard exponential of a Euclidean distance matrix is positive semidefinite. We also derive the closure of the Gram matrices, which can be implemented in this way, so that we also characterize those matrices, which can be approximated arbitrarily well using multi-mode coherent states. Using this we show that Gram matrices of mutually unbiased bases cannot be approximated arbitrarily well using multi-mode coherent states.