Margenau-Hill quasiprobability distribution

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 Kirkwood–Dirac quasiprobability distribution. It was introduced by Henry Margenau and Robert Nyden Hill in 1961.

Definition

A probability distribution $p(a):X\mapsto [0,1]$ is a non-negative function such that

${\int }_{X}p(a)\mathrm{d}a=1.$

A quasiprobability distribution is a real- or complex-valued function $Q(a)$ such that

$\int Q(a)\mathrm{d}a=1,$

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 Kirkwood–Dirac quasiprobability distribution:$$Q_{\mathrm{M}\mathrm{H}}(a,b)=\mathrm{Re}(⟨a|\psi ⟩⟨\psi |b⟩⟨b|a⟩)=\mathrm{Re}{Q}_{\mathrm{K}\mathrm{D}}(a,b),$$where $|\psi ⟩$ is a quantum state that describe the status of a quantum system, and $|a⟩$ and $|b⟩$ are two normalized vectors corresponding to the projective measurement ${\mathrm{\Pi }}_a=|a⟩⟨a|,{\mathrm{\Pi }}_b=|b⟩⟨b|$. It is real-valued and can take negative values, and it is called a quasiprobability distribution because it is normalized; that is,$$\sum _{a,b}{Q}_{\mathrm{M}\mathrm{H}}(a,b)=1.$$The marginals gives correct quantum-mechanical probabilities

$p(a)=⟨\psi |{\mathrm{\Pi }}_a|\psi ⟩=\sum _b{Q}_{\mathrm{M}\mathrm{H}}(a,b),$ $p(b)=⟨\psi |{\mathrm{\Pi }}_b|\psi ⟩=\sum _a{Q}_{\mathrm{M}\mathrm{H}}(a,b)$

for measuring ${\mathrm{\Pi }}_a$ and ${\mathrm{\Pi }}_b$ over state $|\psi ⟩$. This can be derived from the fact that $\sum _{a,b}{Q}_{\mathrm{K}\mathrm{D}}(a,b)=1$ 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 $\rho$ (positive-semidefinite and trace-one operator) that describes the status of an open quantum system, the definition can be extended as$$Q_{\mathrm{M}\mathrm{H}}(a,b,\cdots ,c)=\mathrm{Re}{Q}_{\mathrm{K}\mathrm{D}}(a,b,\cdots ,c)=\mathrm{Re}\mathrm{Tr}(\rho {\mathrm{\Pi }}_a{\mathrm{\Pi }}_b\cdots {\mathrm{\Pi }}_c),$$where ${\mathrm{\Pi }}_a=|a⟩⟨a|,$ ${\mathrm{\Pi }}_b=|b⟩⟨b|,$ $\cdots$, ${\mathrm{\Pi }}_c=|c⟩⟨c|$ 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.

See also

• Wigner quasiprobability distribution
• Kirkwood–Dirac quasiprobability distribution
• Glauber–Sudarshan P representation
• Husimi Q representation
• Phase space formulation
• Negative probability
• Signed measure
• Generalized probabilistic theory