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 ↦ [ 0 , 1 ] {\displaystyle p(a):X\mapsto [0,1]} is a non-negative function such that ∫ X p ( a ) d a = 1. {\displaystyle \int _{X}p(a)\,\mathrm {d} a=1.} A quasiprobability distribution is a real- or complex-valued function Q ( a ) {\displaystyle Q(a)} such that ∫ Q ( a ) d a = 1 , {\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: Q M H ( a , b ) = Re ( ⟨ a | ψ ⟩ ⟨ ψ | b ⟩ ⟨ b | a ⟩ ) = Re Q K D ( a , b ) , {\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 | ψ ⟩ {\displaystyle |\psi \rangle } is a quantum state that describe the status of a quantum system, and | a ⟩ {\displaystyle |a\rangle } and | b ⟩ {\displaystyle |b\rangle } are two normalized vectors corresponding to the projective measurement Π a = | a ⟩ ⟨ a | , Π b = | b ⟩ ⟨ b | {\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, ∑ a , b Q M H ( a , b ) = 1. {\displaystyle \sum _{a,b}Q_{\rm {MH}}(a,b)=1.} The marginals gives correct quantum-mechanical probabilities p ( a ) = ⟨ ψ | Π a | ψ ⟩ = ∑ b Q M H ( a , b ) , {\displaystyle p(a)=\langle \psi |\Pi _{a}|\psi \rangle =\sum _{b}Q_{\rm {MH}}(a,b),}
p ( b ) = ⟨ ψ | Π b | ψ ⟩ = ∑ a Q M H ( a , b ) {\displaystyle p(b)=\langle \psi |\Pi _{b}|\psi \rangle =\sum _{a}Q_{\rm {MH}}(a,b)} for measuring Π a {\displaystyle \Pi _{a}} and Π b {\displaystyle \Pi _{b}} over state | ψ ⟩ {\displaystyle |\psi \rangle } . This can be derived from the fact that ∑ a , b Q K D ( a , b ) = 1 {\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 ρ {\displaystyle \rho } (positive-semidefinite and trace-one operator) that describes the status of an open quantum system, the definition can be extended as Q M H ( a , b , ⋯ , c ) = Re Q K D ( a , b , ⋯ , c ) = Re Tr ( ρ Π a Π b ⋯ Π c ) , {\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 Π a = | a ⟩ ⟨ a | , {\displaystyle \Pi _{a}=|a\rangle \langle a|,} Π b = | b ⟩ ⟨ b | , {\displaystyle \Pi _{b}=|b\rangle \langle b|,} ⋯ {\displaystyle \cdots } , Π c = | c ⟩ ⟨ c | {\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.
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
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