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

Abstract

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 3 or 4.

A mild update. In particular, I. D. Ivanovic's earlier derivation of the expansion is properly acknowledged. 16 pages, one PS figure, 1 table, written in RevTeX

References in corpus (5)

• Coherent control of macroscopic quantum states in a single-Cooper-pair box
• Quantum cryptography with 3-state systems
• Quantum Cryptography using larger alphabets
• Nuclear Magnetic Resonance Quantum Computing Using Liquid Crystal Solvents
• Reconstructing the density matrix of a spin s through Stern-Gerlach measurements (II)

Cited by in corpus (5)

• On the structure of the sets of mutually unbiased bases for N qubits
• Six-qubit permutation-based decoherence-free orthogonal basis
• Mutually unbiased bases and discrete Wigner functions
• Mutually unbiased bases and generalized Bell states
• Discrete phase-space mappings, tomographic condition and permutation invariance