Classical and Quantum Mechanical State Reconstruction

TL;DR

This paper reviews classical and quantum state reconstruction methods, including tomography and Mutually Unbiased Bases, highlighting their applications in continuous and finite-dimensional systems.

Contribution

It compares classical and quantum state reconstruction techniques and extends the Mutually Unbiased Bases method to finite-dimensional quantum systems.

Findings

Classical state reconstruction uses tomography similar to medical imaging.

Quantum state reconstruction involves Wigner functions that can be negative.

Mutually Unbiased Bases provide an alternative approach, applicable to finite-dimensional systems.

Abstract

We review the problem of state reconstruction in classical and in quantum physics, which is rarely considered at the textbook level. We review a method for retrieving a classical state in phase space, similar to that used in medical imaging known as Computer Aided Tomography. We explain how this method can be taken over to quantum mechanics, where it leads to a description of the quantum state in terms of the Wigner function which, although may take on negative values, plays the role of the probability density in phase space in classical physics. We explain another approach to quantum state reconstruction based on the notion of Mutually Unbiased Bases, and indicate the relation between these two approaches. Both are for a continuous, infinite-dimensional Hilbert space. We then study the finite-dimensional case and show how the second method, based on Mutually Unbiased Bases, can be used for state reconstruction.