Quantum Markov Semigroups and the Lindblad Master Equation

Abstract

Quantum Markov Semigroups (QMS) describe the evolution of a
quantum system by evolving a projection or density operator in time. QMS are
generated by a generator obeying the well-known Lindblad equation. However,
this is a difficult equation. Therefore, the result that the Lindblad form
greatly simplifies in the case of the generator commuting with the modular
automorphisms group, is useful. Unfortunately, the proof only works for finite
dimensional Hilbert spaces, which is why the aim of this thesis is to
generalise this result to countably infinite dimensional Hilbert spaces. To this
end, the Lindblad equation is derived from both a mathematical and physical
perspective. Where the former relies on rigorous proof and the latter relies on
approximations.   In the rigorous case
the theory of unital completely positive maps is used. Furthermore, multiple
topologies are considered which put less stringent conditions on the operators
of interest than the norm topology. Additionally, the Haar measure is used on
the unitaries of the bounded linear operators to construct the explicit Lindblad
form. To derive the result by employing physical assumptions the interaction
picture is used. The physical derivation starts from the Von Neumann equation
and uses multiple assumptions to obtain the final Lindblad form. The most
important physical assumptions are: the Born approximation, the Markov
approximation and the rotating wave approximation.   Furthermore, the main result is the
generalisation of the simplified Lindblad form. This simplified form holds for
generators commuting with the modular automorphisms group in case the Hilbert
spaces are countably infinite dimensional. However, this requires the domain of
the generator to be restricted to trace class operators with the identity
operator artificially added. Additionally, the generator needs to map strongly
convergent sequences to weakly convergent sequences. It also needs to be
self-adjoint with respect to the Hilbert-Schmidt inner product. Lastly, the
generator is assumed to be self-adjoint with respect to the
Gelfand-Naimark-Segal (GNS) inner product =Tr(σ X*Y) for σ a
density operator. This last assumption implies that the generator commutes with
the modular automorphisms group, which is the symmetry we are considering.
Hence, the two previous assumptions are the additional requirements needed to
generalise the result, besides the restriction of the domain. Therefore, it is
recommended for further research to generalise the result for the domain
extended to the bounded operators B(H). It should be noted that the proof
heavily relies on the Hilbert space structure induced by the Hilbert-Schmidt
inner product. Consequently, the generalisation for the bounded operators would
probably require a different approach. Another recommendation is to try and
lift the sequence and self-adjoint requirements on the generator. In addition,
it is interesting to investigate which physical systems actually have the
symmetry of generators commuting with the modular automorphisms group.