44 research outputs found
Patient participation in rheumatology research: A four level responsive evaluation
No full textAbma, T.A. [Promotor]Kirwan, J.R. [Promotor
Quantum Markov Semigroups and the Lindblad Master Equation: A generalisation to countably infinite dimensional Hilbert spaces of the Lindblad form for generators commuting with the modular automorphism group
No full textQuantum 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 <X, Y>=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. Applied Mathematic
Multilinear Fourier multipliers of a locally compact group
No full textIn his paper "Group C*-algebras without the completely bounded approximation property", Haagerup proves several important results about the weak amenability of locally compact groups. Among these, is the result that a lattice in a second-countable, unimodular, locally compact group is weakly amenable if and only if the surrounding group itself is weakly amenable. A key ingredient in his proof is a method of using (linear) completely bounded Fourier multipliers on the lattice to construct (linear) completely bounded Fourier multipliers on the surrounding group. We use a similar approach to construct multilinear completely bounded Fourier multipliers on the group from multilinear completely bounded Fourier multipliers on the lattice. Our construction is both bounded in the norm of completely bounded Fourier multipliers and preserves uniform convergence on compact sets for bounded nets. We also prove an equivalent characterization of weak amenability where the Fourier algebra is replaced by the space of continuous and compactly supported -linear Fourier multiplier symbols.Applied Mathematic
Constructing gradient-S<sub>p</sub> quantum Markov semi-groups to obtain strong solidity results for von Neumann algebras
No full textIn this thesis we will for a quantum Markov semi-group (Φt)t≥0 on a finite von Neumann algebra N with a trace τ , investigate the property of the semi-group being gradient-Sp for some p ∈ [1, ∞]. This property was introduced in [12] (see also [9]) and has been studied in [9, 10, 12] for quantum Markov semi-groups on compact quantum groups and on q-Gaussian algebras. Beyond these classes the property gradient-Sp has not been studied; in particular for groups and their operator algebras no (non-trivial) examples were known before this thesis. The main aim of this thesis is therefore to construct interesting examples of quantum Markov semi-groups that possess the gradient-Sp property. The reason why we are interested in constructing such semi-groups, is because they can be used to obtain properties like the Akemann-Ostrand property (AO+) and strong solidity for the underlying von Neumann algebra. Over the last decade, these properties have become a topic of interest and have been studied for several von Neumann algebras, see [3, 8, 9, 10, 12, 23, 32, 33, 37, 41]. In this thesis we shall focus on group von Neumann algebras (L(Γ), τ ) for certain discrete groups Γ that possess the Haagerup property. Namely, for such groups there exists a proper, conditionally negative definite function ψ on Γ. We can then define an unbounded operator ∆ψ on the GNS-Hilbert space L2(L(Γ), τ ) as ∆ψ(λv) = ψ(v)λv and consider the corresponding quantum Markov semi-group (e−t∆ )t≥0. For this semi-group we can investigate for what p it has the gradient-Sp property. In particular we will be considering group von Neumann algebras of Coxeter groups. Namely, a Coxeter group W possesses the Haagerup property by [4], and a proper conditionally negative function ψ on W is given by the minimal word length ψ(w) = |w| w.r.t some set of generators. We will ‘almost completely’ characterize for what types of Coxeter systems the semi-group corresponding to the word length is gradient-Sp. Moreover, in the cases that we get the gradient-S2 property, we obtain the Akemann-Ostand property (AO+) and strong solidity for L(W). Hereafter, we will also consider other quantum Markov semi-groups on L(W). We consider word lengths that arise by putting different weights on the generators, and consider the semi-groups associated to these proper, conditionally negative functions. From this we obtain (AO+) and strong solidity for L(W) for some other cases. Thereafter, we will generalize some of our results obtained for L(W) to the Hecke algebras Nq(W), which are q-deformations of L(W). For the case of group von Neumann algebras L(Γ) for general groups, we shall examine for semi-groups induced by a proper, conditionally negative function ψ, how the gradient-Sp property of the semi-group (Φt)t≥0 := (e−t∆ )t≥0 relates to the gradient-Sq property of the semi-group that is generated by the αth-root ∆α of the generator. Last, we will also show a method that allows us, for right-angled word hyperbolic Coxeter groups, to obtain (AO+) and strong solidity for L(W) without building a gradient-Sp quantum Markov semi-group, but by using a slightly different method. Applied Mathematic
Schur Multipliers of Divided Differences and Multilinear Harmonic Analysis
No full textIt was first shown by D. Potapov and F. Sukochev in 2009 that Lipschitz functions are also operator-Lipschitz on Schatten class operators Sp, 1<p<∞, which is related to a conjecture by M. Krein. Their proof combined Schur multiplication, a generalisation of component-wise matrix multiplication, with the so-called first order divided difference of a function, an approximation of its derivative. Showing that the Schur multipier associated with a divided difference function is bounded relies on a so-called transference technique, the boundedness of certain Schur multipliers can be inferred from the boundedness of associated Fourier multipliers. Soon after, this boundedness result was extended by D. Potapov, A. Skripka, and F. Sukochev to multilinear Schur multipliers of divided differences of arbitrary order, i.e. approximations of higher derivatives. In this thesis, we offer an alternative boundedness proof for bilinear Schur multipliers of second order divided differences, in which we use recent results of multilinear harmonic analysis towards a multilinear transference proof, as well as recently found sufficient conditions for the boundedness of linear Schur multipliers which cannot be studied by transference. These methods were not known at the time Potapov, Skripka, and Sukochev proved their result. Moreover, we show that this new proof improves the growth of the bound on the norm of the considered Schur multiplier for p→∞ significantly. Finally, we give an outlook on further steps towards an alternative boundedness proof of multilinear Schur multipliers of divided differences of arbitrary order.Applied Mathematic
Triangle inequalities of quantum Wasserstein distances on noncommutative tori
No full textIn 2022, Golse and Paul defined a pseudometric for quantum optimal transport that extends the classical Wasserstein distance. They proved that the pseudometric satisfies the triangle inequality in certain cases. This thesis reviews their proof in the case where the middle point is a classical density. Motivated by that proof, we formulate the optimal transport problem and propose the quantum Wasserstein distance on the noncommutative 2-torus. This thesis also proves that the proposed quantum Wasserstein distance satisfies the triangle inequality in the case where the middle point is a classical density on the 2-torus.Applied Mathematic
Non-commutative differentiation and estimates on operator integrals
No full textIn 2017 Martijn Caspers, Fedor Sukochev and Dmitriy Zanin published a paper which generalises the proof of Davies' 1988 paper, and thus resolves the Nazarov-Peller conjecture. The proofs of these papers have been presented in this thesis. They have been expanded with a proof that generalises the conjecture to arbitrary Schatten classes. The optimality of the estimates in the conjecture is also studied, following the example of the 2016 paper by Coine et al. In addition, the quantum mechanical context is provided to interpret the presented results.Applied Mathematic
Single-qubit dynamics: Determining the density matrix of a qubit in closed and open quantum systems when considering free evolution and weak measurements
No full textQuantum technology is evolving faster than ever. Currently, all eyes are on the quantum computer, the promising computer that can solve problems which are unsolvable for regular computers. In order to understand this new technology, it is necessary to understand the qubit, the basic unit of quantum information. This can be done by means of the density matrix: a mathematical representation of the state of a system. The aim of this thesis is to find the density matrix of a single qubit in closed (isolated) and open quantum systems. In the case of a closed system, an alternating sequence of two processes with different Hamiltonians is considered, which both last a fixed amount of time after one another. These systems have been solved using a direct formula for a 2 × 2 matrix with distinct eigenvalues raised to the power n for any n ∈ N. In the case of an open system, dissipation is taken into account compared to a single process in a closed system. These open systems have been solved numerically using the Lindblad master equation and the spin-boson model to model the environment as a bath of bosons. However, the use of multiple approximations and assumptions questions the validity of the results for ’strong’ interactions. Suggestions for further research include investigating quantitatively when the Lindblad equation is valid to use and solve open quantum systems using different models for the environment.Applied Mathematics | Applied Physic
