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FUDMA Journal of Sciences

STRONGLY CONTINUOUS SEMIGROUP ON DECOHERENCE-FREE SUBALGEBRA (DFS) OF QUANTUM MARKOV SEMIGROUP (QMS)

Ezekiel Abiodun Oluwafemi, Micheal O. Ogundiran, U.S. Idiong, Bankole Vincent Akinremi

Published Feb 15, 2026• 0 citations
Publisher

Abstract

This study examines the strongly continuous semigroup on the decoherence-free subalgebra (DFS) of the QMS. Initially, we derived the infinitesimal generator (IG) for the DFS. A theorem characterizing a strongly continuous semigroup on the decoherence-free subalgebra was established. Strongly continuous semigroups offer a natural framework for modeling the time evolution of quantum systems influenced by a quantum Markov semigroup. When paired with the concept of decoherence-free subalgebras, they enable the modeling of systems that are immune to decoherence due to the specific nature of their interaction with the environment. The dynamics on these subalgebras are governed by the restricted semigroup, which can be analyzed through operator theory and the tools of quantum Markov processes. This approach is particularly relevant for applications in quantum error correction, quantum computing, and quantum control, where isolating parts of a system from decoherence effects is crucial.

DOI
10.33003/fjs-2026-1004-4302

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This study examines the strongly continuous semigroup on the decoherence-free subalgebra (DFS) of the QMS. Initially, we derived the infinitesimal generator (IG) for the DFS. A theorem characterizing a strongly continuous semigroup on the decoherence-free subalgebra was established. Strongly continuous semigroups offer a natural framework for modeling the time evolution of quantum systems influenced by a quantum Markov semigroup. When paired with the concept of decoherence-free subalgebras, they enable the modeling of systems that are immune to decoherence due to the specific nature of their interaction with the environment. The dynamics on these subalgebras are governed by the restricted semigroup, which can be analyzed through operator theory and the tools of quantum Markov processes. This approach is particularly relevant for applications in quantum error correction, quantum computing, and quantum control, where isolating parts of a system from decoherence effects is crucial.
Let [Formula: see text] be a quantum Markov semigroup on [Formula: see text] with a faithful normal invariant state ρ. The decoherence-free subalgebra [Formula: see text] of [Formula: see text] is the biggest subalgebra of [Formula: see text] where the completely positive maps [Formula: see text] act as homomorphisms. When [Formula: see text] is the minimal semigroup whose generator is represented in a generalised GKSL form [Formula: see text], with possibly unbounded H, Lℓ, we show that [Formula: see text] coincides with the generalised commutator of [Formula: see text] under some natural regularity conditions. As a corollary we derive simple sufficient algebraic conditions for convergence towards a steady state based on multiple commutators of H and Lℓ. We give examples of quantum Markov semigroups [Formula: see text], with h infinite-dimensional, having a non-trivial decoherence-free subalgebra.
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).
We establish the structure of uniformly continuous quantum Markov semigroups with atomic decoherence-free subalgebra and apply the result to derive a natural decomposition of a Markovian open quantum system into its noiseless (decoherence-free) and irreducible (ergodic) components. We deduce the structure of invariant states and a method for finding decoherence-free subsystems and subspaces. The relationship between environment induced decoherence and the decoherence-free subalgebra is also discussed.
Quantum Markov Semigroups (QMSs) originally arose in the study of the evolutions of irreversible open quantum systems. Mathematically, they are a generalization of classical Markov semigroups where the underlying function space is replaced by a non-commutative operator algebra. In the case when the QMS is uniformly continuous, theorems due to Lindblad \cite{lindblad}, Stinespring \cite{stinespring}, and Kraus \cite{kraus} imply that the generator of the semigroup has the form L(A)=∑n=1∞Vn∗AVn+GA+AG∗L(A)=\sum_{n=1}^{\infty}V_n^*AV_n +GA+AG^*L(A)=∑n=1∞​Vn∗​AVn​+GA+AG∗ where VnV_nVn​ and G are elements of the underlying operator algebra. In the present paper we investigate the form of the generators of QMSs which are not necessarily uniformly continuous and act on the bounded operators of a Hilbert space. We prove that the generators of such semigroups have forms that reflect the results of Lindblad and Stinespring. We also make some progress towards forms reflecting Kraus' result. Lastly we look at several examples to clarify our findings and verify that some of the unbounded operators we are using have dense domains.
Abstract We demonstrate a method for finding the decoherence-free subalgebra N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) of a Gaussian quantum Markov semigroup on the von Neumann algebra B(Γ(Cd)){\mathcal {B}}(\Gamma (\mathbb {C}^d))B(Γ(Cd)) B ( Γ ( C d ) ) of all bounded operator on the Fock space Γ(Cd)\Gamma (\mathbb {C}^d)Γ(Cd) Γ ( C d ) on Cd\mathbb {C}^dCd C d . We show that N(T){\mathcal {N}}({\mathcal {T}})N(T) N ( T ) is a type I von Neumann algebra L∞(Rdc;C)⊗‾B(Γ(Cdf))L^\infty (\mathbb {R}^{d_c};\mathbb {C}){\overline{\otimes }}{\mathcal {B}}(\Gamma (\mathbb {C}^{d_f}))L∞(Rdc​;C)⊗​B(Γ(Cdf​)) L ∞ ( R d c ; C ) ⊗ ¯ B ( Γ ( C d f ) ) determined, up to unitary equivalence, by two natural numbers dc,df≤dd_c,d_f\le ddc​,df​≤d d c , d f ≤ d . This result is illustrated by some applications and examples.