An expectation value expansion of Hermitian operators in a discrete Hilbert space

• We discuss a real-valued expansion of any Hermitian operator defined in a Hilbert space of finite dimension N, where N is a prime number, or an integer power of a prime. The expansion has a direct interpretation in terms of the operator expectation values for a set of complementary bases. The expansion can be said to be the complement of the discrete Wigner function. We expect the expansion to be of use in quantum information applications since qubits typically are represented by a discrete, and finite-dimensional, physical system of dimension N = 2(p), where p is the number of qubits involved. As a particular example we use the expansion to prove that an intermediate measurement basis (a Breidbart basis) cannot be found if the Hilbert space dimension is three or four.

Nyckelord

quantum cryptography

Hermitian operators

state reconstruction

Breidbart basis

stern-gerlach measurements

quantum-state tomography

wigner-function

spin-s

cryptography

ensembles

mechanics

systems

factorization

entanglement

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