Margenau-Hill quasiprobability distribution

The Margenau-Hill quasiprobability distribution^[1] (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^[2] and Kirkwood–Dirac quasiprobability distribution.^[3][4] It was introduced by Henry Margenau and Robert Nyden Hill in 1961.^[1]

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

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

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

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

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:$${\displaystyle Q_{\rm {MH}}(a,b)=\operatorname {Re} (\langle a|\psi \rangle \langle \psi |b\rangle \langle b|a\rangle )=\operatorname {Re} Q_{\rm {KD}}(a,b),}$$where ${\displaystyle |\psi \rangle }$ is a quantum state that describe the status of a quantum system, and ${\displaystyle |a\rangle }$ and ${\displaystyle |b\rangle }$ are two normalized vectors corresponding to the projective measurement ${\displaystyle \Pi _{a}=|a\rangle \langle a|,\Pi _{b}=|b\rangle \langle b|}$. It is real-valued and can take negative values, and it is called a quasiprobability distribution because it is normalized; that is,$${\displaystyle \sum _{a,b}Q_{\rm {MH}}(a,b)=1.}$$The marginals gives correct quantum-mechanical probabilities

${\displaystyle p(a)=\langle \psi |\Pi _{a}|\psi \rangle =\sum _{b}Q_{\rm {MH}}(a,b),}$ ${\displaystyle p(b)=\langle \psi |\Pi _{b}|\psi \rangle =\sum _{a}Q_{\rm {MH}}(a,b)}$

for measuring ${\displaystyle \Pi _{a}}$ and ${\displaystyle \Pi _{b}}$ over state ${\displaystyle |\psi \rangle }$. This can be derived from the fact that ${\displaystyle \sum _{a,b}Q_{\rm {KD}}(a,b)=1}$ 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 ${\displaystyle \rho }$ (positive-semidefinite and trace-one operator) that describes the status of an open quantum system, the definition can be extended as$${\displaystyle Q_{\rm {MH}}(a,b,\cdots ,c)=\operatorname {Re} Q_{\rm {KD}}(a,b,\cdots ,c)=\operatorname {Re} \operatorname {Tr} (\rho \Pi _{a}\Pi _{b}\cdots \Pi _{c}),}$$where ${\displaystyle \Pi _{a}=|a\rangle \langle a|,}$ ${\displaystyle \Pi _{b}=|b\rangle \langle b|,}$ ${\displaystyle \cdots }$, ${\displaystyle \Pi _{c}=|c\rangle \langle c|}$ 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.

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