Margenau-Hill quasiprobability distribution

The Margenau-Hill quasiprobability distribution is a mathematical tool used in quantum mechanics, particularly in quantum information science, quantum optics, and quantum thermodynamics, to describe the joint "quasiprobability" of outcomes for measurements of multiple, potentially non-commuting observables. It is commonly used as a phase-space description of quantum states, similar to the quasiprobability distribution and Kirkwood–Dirac quasiprobability distribution. It was introduced by Henry Margenau and Robert Nyden Hill in 1961.

Definition

A probability distribution is a non-negative function such that

A quasiprobability distribution is a real- or complex-valued function such that

where the integral is a definite integral over some relevant domain. Quasiprobability distributions are also known as signed probability measures in measure theory. They have applications in many fields, especially in phase-space descriptions of quantum mechanics.
The Margenau-Hill quasiprobability distribution is a real-valued generalization of the classical joint probability distribution, obtained by taking the real part of the complex-valued Kirkwood–Dirac quasiprobability distribution:where is a quantum state that describe the status of a quantum system, and and are two normalized vectors corresponding to the projective measurement. It is real-valued and can take negative values, and it is called a quasiprobability distribution because it is normalized; that is,The marginals gives correct quantum-mechanical probabilities

for measuring and over state. This can be derived from the fact that and that the Kirkwood–Dirac quasiprobability distribution gives correct marginals. This means that the Margenau–Hill quasiprobability distribution can be regarded as a phase-space representation of the quantum state, similar to the Wigner function.
For a mixed state that describes the status of an open quantum system, the definition can be extended aswhere, are projective measurements.
The ability to take negative values is often seen as a mathematical indicator of the "non-classical" nature of the system it describes, reflecting phenomena like the Heisenberg uncertainty principle.