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arXiv:funct-an/9706006

The Quantum Stochastic Differential Equation Is Unitarily Equivalent to a Symmetric Boundary Value Problem for the Schrödinger Equation

Alexander Mikhailovich Chebotarev

Published Jun 19, 1997• 0 citations
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Abstract

We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schrödinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the ``master equation'') is derived from the boundary value problem for the Schrödinger equation.

DOI
10.48550/arxiv.funct-an/9706006

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We prove that the solution of the Hudson-Parthasarathy quantum stochastic differential equation in the Fock space coincides with the solution of a symmetric boundary value problem for the Schrödinger equation in the interaction representation generated by the energy operator of the environment. The boundary conditions describe the jumps in the phase and the amplitude of the Fourier transforms of the Fock vector components as any of its arguments changes the sign. The corresponding Markov evolution equation (the Lindblad equation or the ``master equation'') is derived from the boundary value problem for the Schrödinger equation.
We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of the Schrödinger equations in Fock space: the strong resolvent limit is unitarily equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitarily equivalent to the symmetric (or Stratonovich) form of QSDE. We also prove that QSDE is unitarily equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps in phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.
We show a new remarkable connection between the symmetric form of a quantum stochastic differential equation (QSDE) and the strong resolvent limit of Schrödinger equations in Fock space: the strong resolvent limit is unitary equivalent to QSDE in the adapted (or Ito) form, and the weak limit is unitary equivalent to the symmetric (or Stratonovich) form of QSDE. We prove that QSDE is unitary equivalent to a symmetric boundary value problem for the Schrödinger equation in Fock space. The boundary condition describes standard jumps of the phase and amplitude of components of Fock vectors belonging to the range of the resolvent. The corresponding Markov evolution equation (the Lindblad or Markov master equation) is derived from the boundary value problem for the Schrödinger equation.}
We develop systematically a new unifying approach to the analysis of linear stochastic, quantum stochastic and even deterministic equations in Banach spaces. Solutions to a wide class of these equations (in particular those describing the processes of continuous quantum measurements) are proved to coincide with the interaction representations of the solutions to certain Dirac type equations with boundary conditions in pseudo-Fock spaces. The latter are presented as the semiclassical limit of an appropriately dressed unitary evolutions corresponding to a boundary-value problem for rather general Schrödinger equations with bounded below Hamiltonians.
Quite a few researchers have contributed to the development of quantum stochastic differential equations driven by quantum noises, initiated by Hudson and Parthasarathy [HP84a]. A nonexhaustive list of contributors includes Fagnola and Wills [FW00], [FW03], Obata [Oba97], and Linsay and Skalski [LS10]. Based on the results presented in Chapter 3, this chapter derives and considers a general form of linear (left) and (right) Hudson-Parthasarathy quantum stochastic differential equations driven by quantum noises in symmetric Fock space with operator-valued matrices as coefficients. Specifically, it studies the existence and uniqueness of the strong and mild solution processes for both the left and right quantum stochastic differential equations and conditions under which the solution processes are unitary, isometry, co-isometry, and regular. These results make extensive use of the properties of stochastic integral driven by quantum noise and its quantum stochastic calculus in the context of a symmetric Fock space, which was explored in Chapter 3. Various discrete approximation schemes, including Bouten et al. [BvH07], of the left Hudson-Parthasarathy QSDE are also explored for numerical computation. Specifically, it is shown that the solution of the Hudson- Parthasarathy QSDE can be approximated by a sequence of discrete interaction models with decreasing time step. In order to study this problem, discrete interaction models are embedded in a limiting space. This allows one to present strong convergence of the embedded discrete cocycles to the solution of the Hudson-Parthasarathy QSDE. It will be pointed out that the way in which the embedding is done does not affect the proof of the main result for discrete approximations. Examples of a QSDE that is analytically solvable and a QSDE that possesses a mild solution but not a strong solution are also given. QSDE of Hudson-Parthasarathy Type It has been mentioned earlier in Chapter 3 that an open quantum systems treated in this monograph are to be formulated as a quantum system in which the underlined (or the initial) quantum system is perturbed by quantum noises that represent the environment.
We consider the quantum stochastic differential equation introduced by Hudson and Parthasarathy to describe the stochastic evolution of an open quantum system together with its environment. We study the (unbounded) Hamiltonian operator generating the unitary group connected, as shown by Frigerio and Maassen, to the solution of the equation. We find a densely defined restriction of the Hamiltonian operator; in some special cases we prove that this restriction is essentially self-adjoint and in one particular case we get the whole Hamiltonian with its full domain.