Random projectors with continuous resolutions of the identity in a finite-dimensional Hilbert space

TL;DR

This paper introduces a novel approach to constructing continuous resolutions of the identity in finite-dimensional Hilbert spaces using random projectors, enabling new ways to analyze quantum states with continuous variables.

Contribution

It generalizes the concept of resolutions of the identity via random projectors, incorporating continuous variables into finite-dimensional quantum systems.

Findings

Constructed an infinite family of continuous resolutions of the identity.

Defined a phase-space-like function analogous to the Wigner function.

Demonstrated expansion of vectors using a continuum of components.

Abstract

Random sets are used to get a continuous partition of the cardinality of the union of many overlapping sets. The formalism uses M\"obius transforms and adapts Shapley's methodology in cooperative game theory, into the context of set theory. These ideas are subsequently generalized into the context of finite-dimensional Hilbert spaces. Using random projectors into the subspaces spanned by states from a total set, we construct an infinite number of continuous resolutions of the identity, that involve Hermitian positive semi-definite operators. The simplest one is the diagonal continuous resolution of the identity, and it is used to expand an arbitrary vector in terms of a continuum of components. It is also used to define the $F(x_1,x_2)$ function on the `probabilistic quadrant' $[0,\infty) \times [0,\infty)$, which is analogous to the Wigner function for the harmonic oscillator, on the phase-space plane. Systems with finite-dimensional Hilbert space (which are naturally described with discrete variables) are described here with continuous probabilistic variables.