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Journal of Mathematical Physics

Decoherence for positive semigroups on M2(${\mathbb C}$C)

Raffaella Carbone, Emanuela Sasso, Veronica Umanità

Published Mar 1, 2011• 11 citations
Publisher

Abstract

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 $\mathcal {B}(\mathsf {h})$B(h). In particular, in the case $\mathsf {h}=\mathbb {C}^2$h=C2, we completely characterize the decomposition $\mathcal {B}(\mathsf {h}) =\mathcal {M}_1\oplus \mathcal {M}_2$B(h)=M1⊕M2 of $\mathcal {B}(\mathsf {h})$B(h) in the sum of a decoherence-free part $\mathcal {M}_1$M1 and of a space $\mathcal {M}_2$M2 on which the semigroup vanishes with time.

DOI
10.1063/1.3560478

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Decoherence for positive semigroups on M2(C{\mathbb C}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)\mathcal {B}(\mathsf {h})B(h)B(h). In particular, in the case h=C2\mathsf {h}=\mathbb {C}^2h=C2h=C2, we completely characterize the decomposition B(h)=M1⊕M2\mathcal {B}(\mathsf {h}) =\mathcal {M}_1\oplus \mathcal {M}_2B(h)=M1​⊕M2​B(h)=M1⊕M2 of B(h)\mathcal {B}(\mathsf {h})B(h)B(h) in the sum of a decoherence-free part M1\mathcal {M}_1M1​M1 and of a space M2\mathcal {M}_2M2​M2 on which the semigroup vanishes with time.
The mean evolution of an open quantum system in continuous time is described by a time continuous semigroup of quantum channels (completely positive and trace-preserving linear maps). Baumgartner and Narnhofer presented a general decomposition of the underlying Hilbert space into a sum of invariant subspaces, also called enclosures. We propose a new reading of this result, inspired by the work of Carbone and Pautrat. In addition, we apply this decomposition to a class of open quantum random walks and to quantum trajectories, where we study its uniqueness.
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.
The Extension of Unital Completely Positive Semigroups on Operator Systems to Semigroups on C∗\boldsymbol{C^{*}}C∗-Algebras
The study of open quantum systems relies on the notion of unital completely positive semigroups on C∗C^{*}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∗C^{*}C∗ -algebra, which is an injective envelope of the matricial system. In case the semigroup is invertible, this extension is unique.
Each semigroup describing time evolution of an open quantum system on a finite dimensional Hilbert space is related to a special structure of this space. It is shown how the space can be decomposed into orthogonal subspaces: One part is related to decay, some subspaces of the other subspace are ranges of the stationary states. Specialities are highlighted where the complete positivity of evolutions is actually needed for analysis, mainly for evolution of coherence. Decompositions are done the same way for discrete as for continuous time evolutions, but they may show differences: Only for discrete semigroups there may appear cases of sudden decay and of perpetual oscillation. Concluding the analysis, we identify the relation of the state space structure to the processes of decay, decoherence, dissipation and dephasing.
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).