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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Belavkin equation ： ウィキペディア英語版 Belavkin equation In quantum probability, the Belavkin equation, also known as Belavkin-Schrödinger equation, quantum filtering equation, stochastic master equation, is a quantum stochastic differential equation describing the dynamics of a quantum system undergoing observation in continuous time. It was derived and henceforth studied by Viacheslav Belavkin in 1988. Unlike the Schrödinger equation, which describes deterministic evolution of wavefunction $\backslash psi(t)$ of a closed system (without interaction), the Belavkin equation describes stochastic evolution of a random wavefunction $\backslash psi(t,\backslash omega)$ of an open quantum system interacting with an observer: Here, $L$ is a self-adjoint operator (or a column vector of operators) of the system coupled to the external field, $H$ is the Hamiltonian, $i=\backslash sqrt$ is the imaginary unit, $\backslash hbar$ is the Planck constant, and $y(t)=\backslash int\_0^t\; dy(r)$ is a stochastic process representing the measurement noise that is a martingale with independent increments with respect to the input probability measure $P(0,d\backslash omega)=\backslash |\backslash psi(0,\backslash omega)\backslash |^2\backslash mu(d\backslash omega)$. Note that this noise has dependent increments with respect to the output probability measure $P(t,d\backslash omega)=\backslash |\backslash psi(t,\backslash omega)\backslash |^2\backslash mu(d\backslash omega)$ representing the output innovation process (the observation). For $L=0$, the equation becomes the standard Schrödinger equation. The stochastic process $y(t)$ can be a mixture of two basic types: the ''Poisson'' (or ''jump'') type $y(t)\backslash simeq\; n(t)-t$, where $n(t)$ is a Poisson process corresponding to counting observation, and the ''Brownian'' (or ''diffusion'') type $y(t)\backslash simeq\; w(t)$, where $w(t)$ is the standard Wiener process corresponding to continuous observation. The equations of the diffusion type can be derived as the central limit of the jump type equations with the expected rate of the jumps increasing to infinity. The random wavefunction $\backslash psi(t,\backslash omega)$ is normalized only in the mean-squared sense $\backslash int\backslash |\backslash psi(t,\backslash omega)\backslash |^2\backslash mu(d\backslash omega)=\backslash |\backslash psi\_0\backslash |=1$, but generally $\backslash psi(t,\backslash omega)$ fails to be normalized for each $\backslash omega$. The normalization of $\backslash psi(t,\backslash omega)$ for each $\backslash omega$ gives the random ''posterior state vector'' $\backslash varphi(t,\backslash omega)=\backslash psi(t,\backslash omega)/\backslash |\backslash psi(t,\backslash omega)\backslash |$, the evolution of which is described by the posterior Belavkin equation, which is nonlinear, because operators $L$ and $H$ depend on $\backslash varphi$ due to normalization. The stochastic process $y(t)$ in the posterior equation has independent increments with respect to the output probability measure $P(t,d\backslash omega)=\backslash |\backslash psi(t,\backslash omega)\backslash |^2\backslash mu(d\backslash omega)$, but not with respect to the input measure. Belavkin also derived linear equation for unnormalized density operator $\backslash varrho(t,\backslash omega)=\backslash psi(t,\backslash omega)\backslash psi^\backslash ast(t,\backslash omega)$ and the corresponding nonlinear equation for the normalized random posterior density operator $\backslash rho(t,\backslash omega)=\backslash varphi(t,\backslash omega)\backslash varphi^\backslash ast(t,\backslash omega)$. For two types of measurement noise, this gives eight basic quantum stochastic differential equations. The general forms of the equations include all types of noise and their representations in Fock space. The nonlinear equation describing observation of position of a free particle, which is a special case of the posterior Belavkin equation of the diffusion type, was also obtained by Diosi and appeared in the works of Gisin, Ghirardi, Pearle and Rimini, although with a rather different motivation or interpretation. Similar nonlinear equations for posterior denisty operators were postulated (although without derivation) in quantum optics and the quantum trajectories theory, where they are called ''stochastic master equations''. The averaging of the equations for the random density operators $\backslash rho(t,\backslash omega)$ over all random trajectories $\backslash omega$ leads to the Lindblad equation, which is deterministic. The nonlinear Belavkin equations for posteior states play the same role as the Stratonovich–Kushner equation in classical probability, while the linear equations correspond to the Zakai equation. The Belavkin equations describe continuous-time decoherence of initially pure state $\backslash rho(0)=\backslash psi\_0\backslash psi\_0^\backslash ast$ into a mixed posterior state $\backslash rho(t)=\backslash int\backslash varphi(t,\backslash omega)\backslash varphi^\backslash ast(t,\backslash omega)P(t,d\backslash omega)$ giving a rigorous description of the dynamics of the wavefunction collapse due to an observation or measurement. == Non-demolition Measurement and Quantum Filtering == Noncommutativity presents a major challenge for probabilistic interpretation of quantum stochastic differential equations due to non-existence of conditional expectations for general pairs of quantum observables. Belavkin resolved this issue by discovering the error-perturbation uncertainty relation and formulating the non-demolition principle of quantum measurement.〔 In particular, if the stochastic process $dy(t)$ corresponds to the error $e(t)dt=dw$ (white noise in the diffusive case) of a noisy observation $Y(t)=X(t)+e(t)$ of operator $X(t)=\backslash lambda()$ with the accuracy coefficient $\backslash lambda>0$, then the indirect observation perturbs the dynamics of the system by a stochastic force $f(t)$, called the ''Langevin force'', which is another white noise of intensity $(\backslash hbar\backslash lambda)^2$ that does not commute with the error $e(t)$. The result of such a perturbation is that the output process $Y(t)$ is commutative $()=0$, and hence $\backslash int\_0^tY(r)dr$ corresponds to a classical observation, while the system operators $X$ satisfy the non-demolition condition: all future observables must commute with the past observations (but not with the future observations): $()=0$ for all $s\backslash geq\; t$ (but not 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「Belavkin equation」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.