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A PRIORI ESTIMATES AND EXISTENCE THEOREMS FOR THE LINDBLAD EQUATION WITH UNBOUNDED TIME-DEPENDENT COEFFICIENTS(Recent Trends in Infinite Dimensional Non-Commutative Analysis)

Alexander Mikhailovich Chebotarev, Julio C. García, Roberto B. Quezada

Published Apr 1, 1998• 5 citations
OpenAlex

Abstract

We prove new a priori estimates for the resolvent of a class of minimal Quantum Dynamical Semigroup (QDS).These estimates simplify the proof of the unital property of QDS and suggest a continuity condition for time-dependent in- finitesimal generators to ensure existence and conservativity of the Markov master evolution equation. \S 1. INTRODUCTION.The theory of the Markov $\mathrm{m}\mathrm{a}s$ ter equation has been intensively studied dur- ing the recent years [1][2][3][4][5][6][7][8][9][10].Important applications of this theory were contributed to qu.antum chemistry [3] and quantum optics [4-6].Numerical Monte-Carlo and Runge-Kutta algorithms for these equations are discussed in [7-8].One of efficient analytical tools in the theory of master equation is the interaction representation technique [6-7], where the Markov $\mathrm{m}.$.aster equations with time-dependent coeffi- cients arise in a natural way.To study Markov $\mathrm{m}\mathrm{a}s$ ter equations with unbounded time-dependent coefficients describing evolution of observables from the algebra $B(\mathcal{H})$ of bounded operators or states from the algebra $\mathcal{T}(\mathcal{H})$ of operators with finite trace, we introduce the continuity conditions for the generator.We recall that one parameter contraction semigroup $P_{t}(\cdot)$ acting in $B(\mathcal{H})$ is called a Quantum Dynamical Semigroup (QDS) if it is completely positive, normal, conservative, and ultraweak continuous $[2, 11]$ .Normal and ultraweak continuity properties mean respectively that l.u.b. $P_{t}(xn)=P_{t}(1.\mathrm{u}.\mathrm{b}.x_{n})$ and $\mathrm{T}\mathrm{r}\{\rho(P_{t}(B)-$ $B)\}arrow 0$ as $tarrow \mathrm{O}$ for any $\rho\in \mathcal{T}(\mathcal{H})$ and $B\in B(\mathcal{H})$ .In the Heisenberg rep- resentation, the conservative (or unital) property means the conservation of the unit operator $I$ in the algebra $B(\mathcal{H})$ of observables: $P_{t}(I)\equiv I\forall t\geq 0$ ; in the Schr\"odinger representation, it means the conservation of the trace of an initial state $\rho\in \mathcal{T}(\mathcal{H})$ during the evolution: $\mathrm{T}\mathrm{r}T_{t}(\beta)\equiv \mathrm{b}\rho t\geq 0$ , where $T_{t}$ : $\mathcal{T}(\mathcal{H})arrow \mathcal{T}(\mathcal{H})$ and $P_{t}$ : $B(\mathcal{H})arrow B(\mathcal{H})$ are dual semigroups: $\mathrm{T}\mathrm{r}\{\tau_{t}(\rho)B\}=\mathrm{T}\mathrm{r}\{\rho P_{t}(B)\}$ .The semigroup $T_{t}=P_{t}^{1}$ is called the predual of the semigroup $P_{t}$ .The dagger "\dagger '' is used to denote predual operators and maps.The map. $P(\cdot)$ with the predual $P^{\uparrow}(\cdot)$ is referred to as a completely positive $(CP)$ if $\sum_{i,j}\mathrm{T}\mathrm{r}\{P(B_{i}^{*}B_{j})\sigma j\sigma_{i}\}*\geq 0$ or equivalently $\sum_{i,j}\mathrm{T}\mathrm{r}\{B^{*}iBjP^{\mathrm{t}}(\sigma_{j}\sigma_{i})*\}\geq 0$ (1.1)

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Sufficient conditions for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS.The unital property of a minimal quantum dynamical semigroup implies the uniqueness of the solution of the corresponding Markov master equation in the class of quantum dynamical semigroups and, in the corresponding representation, it ensures preservation of the trace or unit operator. We recall that only in the unital case the formal generator of MME determines uniquely the corresponding QDS.

arXiv:quant-ph/0106094•Birkhäuser Basel eBooks•Jan 1, 2003•6 citations

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In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.

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The main aim of this work is to provide a criterion for conservativity of minimal solutions of time-dependent master equations with unbounded coefficients (Quantum Dynamical Evolution System, QDES). In Section 2 we prove a criterion for conservativity of a QDES without using resolvents and in Section 3 we give sufficient conditions for conservativity of the class of QDES whose completely positive part is a time-dependent bounded sesquilinear form on a Hilbert space K t densely and continuously embedded in H and satisfies an extension property.

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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.

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Nonstandard derivation of the Gorini-Kossakowski-Sudarshan-Lindblad master equation of a quantum dynamical semigroup from the Kraus representation

Yui Kuramochi

We give a new nonstandard proof of the well-known theorem that the generator $L$ of a quantum dynamical semigroup $\exp(tL)$ on a finite-dimensional quantum system has a specific form called a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator (also known as a Lindbladian) and vice versa. The proof starts from the Kraus representation of the quantum channel $\exp (δt L)$ for an infinitesimal hyperreal number $δt>0$ and then estimates the orders of the traceless components of the Kraus operators. The jump operators naturally arise as the standard parts of the traceless components of the Kraus operators divided by $\sqrt{δt}$. We also give a nonstandard proof of a related fact that close completely positive maps have close Kraus operators.

arXiv:2406.03775•Jun 6, 2024•0 citations
We prove new a priori estimates for the resolvent of a class of minimal Quantum Dynamical Semigroup (QDS).These estimates simplify the proof of the unital property of QDS and suggest a continuity condition for time-dependent in- finitesimal generators to ensure existence and conservativity of the Markov master evolution equation. \S 1. INTRODUCTION.The theory of the Markov mas\mathrm{m}\mathrm{a}smas ter equation has been intensively studied dur- ing the recent years [1][2][3][4][5][6][7][8][9][10].Important applications of this theory were contributed to qu.antum chemistry [3] and quantum optics [4-6].Numerical Monte-Carlo and Runge-Kutta algorithms for these equations are discussed in [7-8].One of efficient analytical tools in the theory of master equation is the interaction representation technique [6-7], where the Markov m.\mathrm{m}.m..aster equations with time-dependent coeffi- cients arise in a natural way.To study Markov mas\mathrm{m}\mathrm{a}smas ter equations with unbounded time-dependent coefficients describing evolution of observables from the algebra B(H)B(\mathcal{H})B(H) of bounded operators or states from the algebra T(H)\mathcal{T}(\mathcal{H})T(H) of operators with finite trace, we introduce the continuity conditions for the generator.We recall that one parameter contraction semigroup Pt(⋅)P_{t}(\cdot)Pt​(⋅) acting in B(H)B(\mathcal{H})B(H) is called a Quantum Dynamical Semigroup (QDS) if it is completely positive, normal, conservative, and ultraweak continuous [2,11][2, 11][2,11] .Normal and ultraweak continuity properties mean respectively that l.u.b. Pt(xn)=Pt(1.u.b.xn)P_{t}(xn)=P_{t}(1.\mathrm{u}.\mathrm{b}.x_{n})Pt​(xn)=Pt​(1.u.b.xn​) and Tr{ρ(Pt(B)−\mathrm{T}\mathrm{r}\{\rho(P_{t}(B)-Tr{ρ(Pt​(B)− B)}arrow0B)\}arrow 0B)}arrow0 as tarrowOtarrow \mathrm{O}tarrowO for any ρ∈T(H)\rho\in \mathcal{T}(\mathcal{H})ρ∈T(H) and B∈B(H)B\in B(\mathcal{H})B∈B(H) .In the Heisenberg rep- resentation, the conservative (or unital) property means the conservation of the unit operator III in the algebra B(H)B(\mathcal{H})B(H) of observables: Pt(I)≡I∀t≥0P_{t}(I)\equiv I\forall t\geq 0Pt​(I)≡I∀t≥0 ; in the Schr"odinger representation, it means the conservation of the trace of an initial state ρ∈T(H)\rho\in \mathcal{T}(\mathcal{H})ρ∈T(H) during the evolution: TrTt(β)≡bρt≥0\mathrm{T}\mathrm{r}T_{t}(\beta)\equiv \mathrm{b}\rho t\geq 0TrTt​(β)≡bρt≥0 , where TtT_{t}Tt​ : T(H)arrowT(H)\mathcal{T}(\mathcal{H})arrow \mathcal{T}(\mathcal{H})T(H)arrowT(H) and PtP_{t}Pt​ : B(H)arrowB(H)B(\mathcal{H})arrow B(\mathcal{H})B(H)arrowB(H) are dual semigroups: Tr{τt(ρ)B}=Tr{ρPt(B)}\mathrm{T}\mathrm{r}\{\tau_{t}(\rho)B\}=\mathrm{T}\mathrm{r}\{\rho P_{t}(B)\}Tr{τt​(ρ)B}=Tr{ρPt​(B)} .The semigroup Tt=Pt1T_{t}=P_{t}^{1}Tt​=Pt1​ is called the predual of the semigroup PtP_{t}Pt​ .The dagger "\dagger '' is used to denote predual operators and maps.The map. P(⋅)P(\cdot)P(⋅) with the predual P↑(⋅)P^{\uparrow}(\cdot)P↑(⋅) is referred to as a completely positive (CP)(CP)(CP) if ∑i,jTr{P(Bi∗Bj)σjσi}∗≥0\sum_{i,j}\mathrm{T}\mathrm{r}\{P(B_{i}^{*}B_{j})\sigma j\sigma_{i}\}*\geq 0∑i,j​Tr{P(Bi∗​Bj​)σjσi​}∗≥0 or equivalently ∑i,jTr{B∗iBjPt(σjσi)∗}≥0\sum_{i,j}\mathrm{T}\mathrm{r}\{B^{*}iBjP^{\mathrm{t}}(\sigma_{j}\sigma_{i})*\}\geq 0∑i,j​Tr{B∗iBjPt(σj​σi​)∗}≥0 (1.1)
Sufficient conditions for a quantum dynamical semigroup (QDS) to be unital are proved for a class of problems in quantum optics with Hamiltonians which are self-adjoint polynomials of any finite order in creation and annihilation operators. The order of the Hamiltonian may be higher than the order of completely positive part of the formal generator of a QDS.The unital property of a minimal quantum dynamical semigroup implies the uniqueness of the solution of the corresponding Markov master equation in the class of quantum dynamical semigroups and, in the corresponding representation, it ensures preservation of the trace or unit operator. We recall that only in the unital case the formal generator of MME determines uniquely the corresponding QDS.
In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.
The main aim of this work is to provide a criterion for conservativity of minimal solutions of time-dependent master equations with unbounded coefficients (Quantum Dynamical Evolution System, QDES). In Section 2 we prove a criterion for conservativity of a QDES without using resolvents and in Section 3 we give sufficient conditions for conservativity of the class of QDES whose completely positive part is a time-dependent bounded sesquilinear form on a Hilbert space K t densely and continuously embedded in H and satisfies an extension property.
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.
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.
We give a new nonstandard proof of the well-known theorem that the generator LLL of a quantum dynamical semigroup exp⁡(tL)\exp(tL)exp(tL) on a finite-dimensional quantum system has a specific form called a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) generator (also known as a Lindbladian) and vice versa. The proof starts from the Kraus representation of the quantum channel exp⁡(δtL)\exp (δt L)exp(δtL) for an infinitesimal hyperreal number δt>0δt>0δt>0 and then estimates the orders of the traceless components of the Kraus operators. The jump operators naturally arise as the standard parts of the traceless components of the Kraus operators divided by δt\sqrt{δt}δt​. We also give a nonstandard proof of a related fact that close completely positive maps have close Kraus operators.