The study of open quantum systems relies on the notion of unital completely positive semigroups on $$C^{*}$$ -algebras representing physical systems. The natural generalisation would be to consider the unital completely positive semigroups on operator systems. We show that any continuous unital completely positive semigroup on matricial system can be extended to a semigroup on a finite-dimensional $$C^{*}$$ -algebra, which is an injective envelope of the matricial system. In case the semigroup is invertible, this extension is unique.
The Extension of Unital Completely Positive Semigroups on Operator Systems to Semigroups on C∗-Algebras
The study of open quantum systems relies on the notion of unital completely positive semigroups on C∗ -algebras representing physical systems. The natural generalisation would be to consider the unital completely positive semigroups on operator systems. We show that any continuous unital completely positive semigroup on matricial system can be extended to a semigroup on a finite-dimensional C∗ -algebra, which is an injective envelope of the matricial system. In case the semigroup is invertible, this extension is unique.
We investigate some particular completely positive maps which admit a stable commutative Von Neumann subalgebra. The restriction of such maps to the stable algebra is then a Markov operator. In the first part of this article, we propose a recipe in order to find a quantum extension of a given Markov operator in the above sense. We show that the existence of such an extension is linked with the existence of a special form of dilation for the Markov operator studied by Attal in \cite{Att1}, reducing the problem to the extension of dynamical system. We then apply our method to the same problem in continuous time, proving the existence of a quantum extension for Lévy processes. In the second part of this article, we focus on the case where the commutative algebra is isomorphic to $\Acal=l^\infty(1,...,N)$ with N either finite or infinite. We propose a classification of the CP maps leaving $\Acal$ stable, producing physical examples of each classes.
Motivated by existence problems for dissipative systems arising naturally in lattice models from quantum statistical mechanics, we consider the following C*-algebraic setting: A given Hermitian dissipative mapping δ is densely defined in a unital C*-algebra A. The identity element in A is also in the domain of δ. Completely dissipative maps δ are defined by the requirement that the induced maps, (aij)→(δ(aij)), are dissipative on the n-by-n complex matrices over A for all n. We establish the existence of different types of maximal extensions of completely dissipative maps. If the enveloping von Neumann algebra of A is injective, we show the existence of an extension of δ which is the infinitesimal generator of a quantum dynamical semigroup of completely positive maps in the von Neumann algebra. If δ is a given well-behaved *-derivation, then we show that each of the maps ±δ is completely dissipative.
Decoherence for positive semigroups on M2(CC)
We extend Blanchard and Olkiewicz's definition of decoherence to open quantum systems whose dynamics are described by semigroups of positive (and not necessarily completely positive) operators on B(h)B(h). In particular, in the case h=C2h=C2, we completely characterize the decomposition B(h)=M1⊕M2B(h)=M1⊕M2 of B(h)B(h) in the sum of a decoherence-free part M1M1 and of a space M2M2 on which the semigroup vanishes with time.
Background: Operator algebras and quantum dynamical semigroups are the mathematical tools of choice for understanding open quantum systems, especially in infinite-dimensional spaces. Such tools are extremely important for the time development of quantum systems out of equilibium under the influence of environment, where conventional finite dimensional methods do not include all possibilities for interaction. Objective: The objects of the present research are the r-properties and the decompositions of generators of quantum dynamical semigroups on infinite-dimensional spaces including their Hamiltonian and dissipative part, semigroup continuity, stationary states, and positivity improvement. The aim is to set a fundament in order to think about long-time dynamics in open quantum systems. Methods: The theorem Hille–Yosida is used to get conditions for the semigroup strongly continuous generated by unbounded operators. The time-evolution operator of the system is approximated with both Gaussian quadrature and Monte Carlo integration during infinitesimal time slices, by using Inf-LCHS in conjunction. Generators were studied under decomposition of their Hamiltonian and dissipative parts, while commutant algebras were investigated for irreducibility and positivity enhancement. Results: Most systems studied for these properties presented a singular decomposition, strong and weak semigroup continuity, existence of stationary states and positivity improvement. The counterexamples were the systems with in a lack of verification for dissipative parts, and they lost smoothing property, uniqueness of decomposition, partial semigroup continuity or large errors in approximation when we perform numerically. Conclusions: The combination of the methods of operator algebra with state-of-the-art simulation techniques yields a powerful approach to study infinite-dimensional quantum dynamics. Although dissipative components are not yet completely characterized, the results improve our understanding of generator structures and long-term system behavior which will help future work in quantum statistical mechanics and quantum information theory.
In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).
Almost-idempotent quantum channels and approximate C∗-algebras
Let Φ be a unital completely positive (UCP) map on the space of operators on some Hilbert space. We assume that Φ is η-idempotent, namely, ∥Φ2−Φ∥cb≤η, and construct an associated ε-C∗ algebra (of almost-invariant observables) for ε=O(η). This type of structure has the axioms of a unital C∗ algebra but the associativity and other axioms involving the multiplication and the unit hold up to ε. We prove that any finite-dimensional ε-C∗ algebra A is O(ε)-isomorphic to some genuine C∗ algebra B. These bounds are universal, i.e. do not depend on the dimensionality or other parameters. When A comes from a finite-dimensional η-idempotent UCP map Φ, the O(η)-isomorphism and its inverse can be realized by UCP maps. This gives an approximate factorization of the quantum channel Φ∗ into a decoding channel, producing a state on B, and an encoding channel.