Markov Approximations of the Evolution of Quantum Systems

The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system is studied. The statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution are described in terms of the law of large numbers for operator-valued random processes. For compositions of independent random semigroups of completely positive operators, the convergence of mean values to a semigroup described by the Gorini–Kossakowski–Sudarshan–Lindblad equation is established. Moreover, a sequence of random operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge in probability to an operator-valued function with values in the set of infinitely divisible operators.

Below, we discuss the convergence in probability of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum system. In doing this, we apply one-parameter families of completely positive mappings of the algebra of bounded linear operators into itself that satisfy the Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) equation [1][2][3]. Additionally, we examine the asymptotic properties of a sequence of compositions of independent random completely positive transformations of the algebra of linear operators acting in a finite-dimensional complex Hilbert space H. The results presented in this paper are obtained using the approach to the general theory of random semigroups developed in [4,5]. † Deceased.
( ) H + Statistical properties of the dynamics of open quantum systems with random generators of Markovian evolution were considered in [6,7]. In the approach proposed in this paper, such properties are described in terms of the law of large numbers for operator-valued random processes. We discuss random variables with values in the space of strongly continuous mappings of the real half-line to the cone of completely positive operators of the Banach space . In what follows, we use the terminology and notation from [8]. Given a locally convex space E, the set of linear continuous mapping from E to E is denoted by the symbol (rather than by ). Specifically, the linear space of linear continuous mappings from to is denoted by . For random completely positive semigroups, we establish the convergence of mean values to a semigroup of mappings described by the GKSL equation. Moreover, a sequence of operator-valued functions with values in the set of operators without the infinite divisibility property is shown to converge to an operator-valued function with values in the set of infinitely divisible operators.
Note that a similar problem, namely, the convergence in distribution of a sequence of products of independent random matrices was studied in [9]. However, the problem formulation and the results presented below differ considerably from the results of [9]. Specifically, Theorem 7 in [9] presents a law of large numbers stating the convergence in distribution of a sequence of products of random matrices to a deterministic limit matrix, while the law of large numbers in this paper establishes the convergence in probability of a sequence of random compositions to a limit semigroup and allows us to estimate the probability of deviation in the form of the Chebyshev inequality [4].
Note that the conditions we impose on random semigroups differ significantly from the conditions used in [4,10]. Moreover, we systematically apply the combinatorial approach proposed in [11].

QUANTUM DYNAMICAL SEMIGROUPS
Throughout this paper, , where . Since is a finite-dimensional space, it follows that any of two norms in each of the tensor products discussed below are equivalent and that these tensor products are complete with respect to the topologies defined by each of these norms.
In what follows, let be the Banach space of linear operators in H equipped with the standard operator norm, and let be the Banach space of linear mappings of into itself equipped with the standard operator norm. Recall that an element is called positive if for any such that . An element is called a completely positive mapping of the space into itself (see [12]) if the linear operator acting in the algebra according to the rule is positive in the operator algebra , where is the algebra of d × d matrices over the field of complex numbers.
Theorem [13]. If Recall that a quantum channel in a space of observables is defined as a linear completely positive mapping of the space into itself that preserves the identity operator . A one-parameter continuous semigroup of quantum channels in the algebra of Theorem (Gorini-Kossakowski-Sudarshan-Lindblad theorem) [14,15]. The generator of any oneparameter uniformly continuous semigroup , , of completely positive mappings of the space into itself is defined by the equality , is a set of at most d 2 -1 operators from the algebra and .
2. RANDOM SEMIGROUPS The generator of a quantum dynamical semigroup is random if the operators L a and H in (1) are random variables with values in the matrix algebra . A random variable with values in the space of linear mappings of into itself is defined as a measurable mapping of the probability space to equipped with the weak operator topology (which, in the case of a finite-dimensional space H, coincides with the operator norm topology). Compositions of random orthogonal transformations of finite-dimensional Euclidean spaces were studied in [17].
The task we address is to investigate the properties of compositions of independent random processes with values in the cone of completely positive mappings of . The set of uniformly continuous quantum dynamical semigroups acting in the Banach algebra is denoted by . By the Hille-Yosida and Lindblad theorems, there is a bijection between the set equipped with the topology of uniform convergence on each interval in the space of linear operators and the set of generators of continuous quantum dynamical semigroups equipped with the operator norm topology. Note that, in our case, uniform continuity follows from continuity. Let , , be a random quantum dynamical semigroup in the algebra defined on the probability space , i.e., : is an -measurable mapping with values in . The mean value of , , is its expectation defined by the equality , is a random quantum dynamical semigroup in , then its expectation is a one-parameter family of completely positive mappings, which may

LIMIT THEOREMS
Let A be a random variable with values in the Banach space of linear operators acting in the Banach algebra , and let A be defined as a weakly measurable mapping of the probability space to . In the case of a finite-dimensional space H, the weak measurability of A: (i.e., the measurability of functions for any , ) is equivalent to the measurability of this mapping to and implies the measurability of the real-valued random variable . The analogue of the law of large numbers for the composition of random semigroups (2) is established using an operator counterpart of the Chebyshev inequality [4,11], to derive which the operator-valued variance of a random operator is introduced. Since the spaces H and are finite-dimensional, we equip with a norm equivalent to , namely, with the norm of the space of Hilbert-Schmidt operators, which turns into a Hilbert space. The symbol denotes the linear space equipped with the Hilbert-Schmidt norm. Then and for each and each . Since is a Hilbert space, for compositions of random operators with values in the space equipped with the strong operator topology, we apply the approach proposed in [4,11]. The variance of the random operator-valued function (2) is defined as To estimate the variance of the random composition W n , we introduce the following random deviation from the expectation. For each , let V k (t) = , . Then , , and, in view of (3), for each with probability 1 we have the estimate (4) By virtue of (2), for each , it is true that Therefore, The product of 2n binomials in (5)   Consequently, the variance of the operator-valued random variable can be expressed as (6) Since , it follows from estimate (4) that Estimating the remaining terms in formula (6) in the same manner, by virtue of (3) and (4)      Since the random generators are independent, we have . By Theorem 1, for any and T > 0. Since the norms of and are equivalent, inequality (7) proves Theorem 2.