  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.

Quantum Physics