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A quasiprobability distribution is a mathematical object similar to a probability distribution but which relaxes some of Kolmogorov's axioms of probability theory. Although quasiprobabilities share several of general features with ordinary probabilities, such as, crucially, ''the ability to yield expectation values with respect to the weights of the distribution'', they all violate the third probability axiom, because regions integrated under them do not represent probabilities of mutually exclusive states. To compensate, some quasiprobability distributions also counterintuitively have regions of negative probability density, contradicting the first axiom. Quasiprobability distributions arise naturally in the study of quantum mechanics when treated in phase space formulation, commonly used in quantum optics, time-frequency analysis,〔L. Cohen (1995), ''Time-frequency analysis: theory and applications'', Prentice-Hall, Upper Saddle River, NJ, ISBN 0-13-594532-1 〕 and elsewhere. == Introduction == (詳細はOptical phase spaceを参照) In the most general form, the dynamics of a quantum-mechanical system are determined by a master equation in Hilbert space: an equation of motion for the density operator (usually written ) of the system. The density operator is defined with respect to a ''complete'' orthonormal basis. Although it is possible to directly integrate this equation for very small systems (i.e., systems with few particles or degrees of freedom), this quickly becomes intractable for larger systems. However, it is possible to prove〔E. C. G. Sudarshan "Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams", ''Phys. Rev. Lett.'',10 (1963) pp. 277–279. 〕 that the density can always be written in a ''diagonal'' form, provided that it is with respect to an ''overcomplete'' basis. When the density operator is represented in such an overcomplete basis, then it can be written in a way more like an ordinary function, at the expense that the function has the features of a quasiprobability distribution. The evolution of the system is then completely determined by the evolution of the quasiprobability distribution function. The coherent states, i.e. right eigenstates of the annihilation operator serve as the overcomplete basis in the construction described above. By definition, the coherent states have the following property: : They also have some additional interesting properties. For example, no two coherent states are orthogonal. In fact, if |''α'' 〉 and |''β'' 〉 are a pair of coherent states, then : Note that these states are, however, correctly normalized with 〈''α''|''α''〉 = 1. Owing to the completeness of the basis of Fock states, the choice of the basis of coherent states must be overcomplete.〔J. R. Klauder, The action option and a Feynman quantization of spinor fields in terms of ordinary c-numbers, ''Ann. Physics'' 11 (1960) 123–168. 〕 Click to show an informal proof. ^\sum_^ e^ \int_0^ \sum_^\sum_^ e^e^}^ \int_0^ \sum_^ \int_0^ e^e^}^ \int_0^ \sum_^ e^\delta(n-k)}^ \int e^} |n\rangle \langle n| \, dr \\ &= \pi \sum_^ \int e^ \cdot \frac |n\rangle \langle n| \, du \\ &= \pi \sum_^ |n\rangle \langle n| \\ &= \pi \hat.\end Clearly we can span the Hilbert space by writing a state as : On the other hand, despite correct normalization of the states, the factor of π>1 proves that this basis is overcomplete. |} In the coherent states basis, however, it is always possible〔 to express the density operator in the diagonal form : where ''f'' is a representation of the phase space distribution. This function ''f'' is considered a quasiprobability density because it has the following properties: : * (normalization) : *If is an operator that can be expressed as a power series of the creation and annihilation operators in an ordering Ω, then its expectation value is ::: (optical equivalence theorem). The function ''f'' is not unique. There exists a family of different representations, each connected to a different ordering . The most popular in the general physics literature and historically first of these is the Wigner quasiprobability distribution,〔E.P. Wigner, "On the quantum correction for thermodynamic equilibrium", ''Phys. Rev.'' 40 (June 1932) 749–759. 〕 which is related to symmetric operator ordering. In quantum optics specifically, often the operators of interest, especially the particle number operator, is naturally expressed in normal order. In that case, the corresponding representation of the phase space distribution is the Glauber–Sudarshan P representation.〔R. J. Glauber "Coherent and Incoherent States of the Radiation Field", ''Phys. Rev.'',131 (1963) pp. 2766–2788. 〕 The quasiprobabilistic nature of these phase space distributions is best understood in the representation because of the following key statement: This sweeping statement is unavailable in other representations. For example, the Wigner function of the EPR state is positive definite but has no classical analog.〔O. Cohen "Nonlocality of the original Einstein-Podolsky-Rosen state", ''Phys. Rev. A'',56 (1997) pp. 3484–3492. 〕〔K. Banaszek and K. Wódkiewicz "Nonlocality of the Einstein-Podolsky-Rosen state in the Wigner representation", ''Phys. Rev. A'',58 (1998) pp. 4345–4347. 〕 In addition to the representations defined above, there are many other quasiprobability distributions that arise in alternative representations of the phase space distribution. Another popular representation is the Husimi Q representation,〔Kôdi Husimi (1940). "Some Formal Properties of the Density Matrix", ''Proc. Phys. Math. Soc. Jpn.'' 22: 264-314 .〕 which is useful when operators are in ''anti''-normal order. More recently, the positive representation and a wider class of generalized representations have been used to solve complex problems in quantum optics. These are all equivalent and interconvertible to each other, viz. Cohen's class distribution function. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「quasiprobability distribution」の詳細全文を読む スポンサード リンク
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