The Margenau-Hill quasiprobability distribution (MH) 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 (quantities that cannot be precisely measured simultaneously). It is commonly used as a phase-space description of quantum states, similar to the Wigner quasiprobability distribution and KirkwoodâÂÂDirac quasiprobability distribution. It was introduced by Henry Margenau and Robert Nyden Hill in 1961.
A probability distribution is a non-negative function such that<blockquote></blockquote>A quasiprobability distribution is a real- or complex-valued function such that<blockquote></blockquote>where the integral is a definite integral over some relevant domain. Quasiprobability distributions are also known as signed probability measures (normalized signed 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 KirkwoodâÂÂDirac 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<blockquote>
</blockquote>for measuring and over state . This can be derived from the fact that and that the KirkwoodâÂÂDirac quasiprobability distribution gives correct marginals. This means that the MargenauâÂÂHill quasiprobability distribution can be regarded as a phase-space representation of the quantum state, similar to the Wigner function.
For a mixed state (positive-semidefinite and trace-one operator) 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.