This thesis will be about explaining a proof of a theorem about entropy presented in a scientific article by Arstein, Barthe Ball and Naor [1] in detail. The original proof is complex, especially for bachelor-level students. The goal of this thesis is to break down that proof, add mathematics and make the ideas understandable for students at this level. The theorem is about mathematical entropy, a concept in probability theory, which is a measure for uncertainty or chaos in an event or outcome. For instance, consider a coin toss: the outcome is uncertain, and this uncertainty is measured by entropy. The greater the uncertainty, the higher the entropy. There is also a concept of entropy in the world of physics, where it is a measure for describing the uncertainty or randomness in which systems evolve. One of the goals of this theorem is its demonstration that these two types of entropy, though defined in different contexts, exhibit analogous behavior. In particular, it shows that the mathematical entropy behaves like the second law of thermodynamics, which states that entropy in an isolated system increases over time. As an illustrative example, when a glass of water is spilled on a table, the
water gradually spreads out, increasing the disorder of the system. The theorem explains that if the entropy of a normalised sum of independent events is taken, then this entropy will increase with the amount of events that are summed. A normalized sum refers to the average obtained by summing independent events and dividing by their count. This seems logical at first, since the uncertainty of for example two separate events seems bigger than that of one event. However, the proof is complex and requires advanced analysis to prove.

The theorem is thus about the monotonicity of entropy of normalised sums. This thesis will connect the entropy to Fisher information, which enjoys nicer analytical properties to use. The concept is clear, entropy is a measure for uncertainty, while Fisher information quantifies the amount of information a random variable carries. This thesis will thus first connect the Fisher information and entropy and show that proving an increase of entropy can be done by proving a decrease in Fisher information. The proof of the decrease in Fisher information will need another theorem. This theorem is about connecting the Fisher information to the world of analysis. This requires some advanced analysis, like Green-Gauss on infinite surfaces and integrating the divergence of functions over hyperplanes. With this connection to the world of analysis eventually the decrease in Fisher information can be reached with some help from a lemma about commuting orthogonal projections in Hilbert spaces. All supporting theorems and lemmas will also be proved in detail in this thesis. In conclusion, this thesis will show a lot about the behaviour of entropy under normalised summation of independent events and show that these monotonically increase, which implies that the mathematical form of entropy behaves like the second law of thermodynamics.

Understanding the behavior of norms on Schur multiplier operators is of significant interest in functional analysis and applications in physics, particularly in quantummechanics. In this study, we focus on the Schur multiplier induced by the divided difference matrix of the absolute value function f (x) = |x|, all set in the finite space Mn(C). The primary objective is to approximate the operator norm ∥TBf ∥ and to analyze its behavior as a function of the Schatten normparameter p and the matrix size n. To achieve this, various numerical methods, including brute-force sampling and gradient ascent algorithms, are explored. Among these, the Adam optimization algorithm, combining momentum and RMSProp techniques, is found to be effective in achieving accurate approximations. The results are analyzed within a theoretical framework, revealing insights into the growth of the norm as well as the effectiveness of the chosen optimization method. Future research directions are suggested, particularly in the study of multilinear Schur multipliers, which present an intriguing challenge yet to be tackled.

The topic of this dissertation lies in the field of operator algebras and non-commutative functional analysis. The dissertation studies structural properties of C*-algebras and von Neumann algebras, with a focus on the latter. New rigidity results are obtain for von Neumann algebras coming from Coxeter groups and/or graph products. Furthermore, new results are obtained on approximation properties of von Neumann algebras and C*-algebras coming from graph products. Last, this dissertation obtains sharp estimates on the norms of commutators in factors.

This dissertation contributes to three areas of research in quantum information theory. 

1.    Cipriani and Sauvageot have shown that the generator of the L² implementation of a tracially symmetric quantum Markov semigroup can be expressed as the square of a derivation with values in a complete bimodule. Here, we study the existence of (twisted) derivations corresponding to quantum Markov semigroups which satisfy symmetry properties with respect to a non-tracial state. We show that this construction of a derivation can in general not be generalised to quantum Markov semigroups that are symmetric with respect to a non-tracial state. Next, we prove that the generator of the L² implementation of a KMS-symmetric quantum Markov semigroup can be expressed as the square of a twisted derivation with values in a complete bimodule, extending a subsequently released result by Wirth for GNS-symmetric semigroups. This result hinges on the introduction of a new completely positive map on the algebra of bounded operators on the GNS Hilbert space. This transformation maps symmetric Markovian operators to symmetric Markovian operators and is essential to obtain the required inner product on the complete bimodule. This transformation was discovered by adapting the approach in the first part to the twisted derivations introduced by Wirth.

     2.    Robust self-testing in non-local games allows a classical referee to certify that two untrustworthy players are able to perform a specific quantum strategy up to high precision. Proving robust self-testing results becomes significantly easier when one restricts the allowed strategies to symmetric projective maximally entangled (PME) strategies, which allow natural descriptions in terms of tracial von Neumann algebras. This has been exploited in the celebrated MIP*=RE paper and related articles to prove robust self-testing results for synchronous games when restricting to PME strategies. However, the PME assumptions are not physical, so these results need to be upgraded to make them physically relevant. In this chapter, we do just that: we prove that any perfect synchronous game which is a robust self-test when restricted to PME strategies, is in fact a robust self-test for all strategies. We then apply our result to the Quantum Low Degree Test to find an efficient n-qubit test.

3.    Families of expander graphs were first constructed by Margulis from discrete groups with property (T). Within the framework of quantum information theory, several authors have generalised the notion of an expander graph to the setting of quantum channels. In this chapter, we use discrete quantum groups with property (T) to construct quantum expanders in two ways. The first approach obtains a quantum expander family by constructing the requisite quantum channels directly from finite-dimensional irreducible unitary representations, extending earlier work of Harrow using groups. The second approach directly generalises Margulis' original construction and is based on a quantum analogue of a Schreier graph using the theory of coideals. To obtain examples of quantum expanders, we apply our machinery to discrete quantum groups with property (T) coming from compact bicrossed products.

In 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.

In 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 $n$-linear Fourier multiplier symbols.

It was first shown by D. Potapov and F. Sukochev in 2009 that Lipschitz functions are also operator-Lipschitz on Schatten class operators S^p, 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.

This thesis deals with multipliers and transference on noncommutative Lp-spaces. It falls within the realm of noncommutative harmonic analysis, i.e. harmonic analysis on noncommutative spaces. Such spaces appear in several areas of mathematics, such as non-abelian groups, quantum groups, noncommutative geometry and the general theory of operator algebras. Already the pioneering work of von Neumann, Gelfand and Dixmier showed that there are close connections between operator theory and representation theory of locally groups. These results formed the foundations for the theory of noncommutative harmonic analysis.....

In the field of quantum information theory, it is well-known that the purely quantum phenomenon called quantum entanglement can boost the capacity of a quantum channel, which is called the superadditivity of the capacity. Shor showed in his breakthrough paper on the equivalence of additivity conjectures that this superadditivity is equivalent to the subadditivity of the minimum output entropy of quantum channels, and Hastings gave a high-dimensional and probabilistic counterexample to the minimum output entropy additivity conjecture. Since then, researchers have endeavored to classify which quantum channels have subadditive minimum output entropy and to find deterministically constructed quantum channels with subadditive minimum output entropy. It is well-known that determining the minimum output entropy of an arbitrary quantum channel is a hard problem (in fact, it has been shown to be NP-complete), which in turn makes it difficult to determine whether a quantum channel has subadditive minimum output entropy.

In this thesis, we utilize the representation theory of compact (partition) quantum groups to construct so-called covariant Clebsch-Gordan quantum channels and analyze their minimum output entropy. We review the work of Brannan and Collins in the case of the free orthogonal quantum groups $O_N^+$, and introduce a similar analysis for the quantum permutation group $S_N^+$. Afterwards, we specialize to the subfamily of lowest-weight Clebsch-Gordan channels, and we show that, in the case where one embeds the fundamental representation of $O_N^+$ in the tensor product of two irreducible representations of $O_N^+$, the associated lowest-weight quantum channels have sufficient additional structure to analytically compute their minimum output entropy in terms of a recurrence relation.

Lastly, we present an analysis of certain numerical methods that can be utilized to bound the minimum output entropy from below and from above in low dimensions: we use epsilon-covers for lower bounds, and we use a modified version of the derivative-free optimization method called Particle Swarm Optimization over the unit sphere, hybridized with gradient descent, to find upper bounds. We show a proof-of-concept by applying the epsilon-cover to three $S_N^+$ channels with small input dimensions, but also note that the exponential scaling of epsilon-covers with the input dimension makes them intractable with higher input dimensions. We benchmark the Particle Swarm Optimization enriched with gradient-descent on the highest-weight Clebsch-Gordan channels for which it can be shown that the minimum output entropy is zero, and we see that the optimization technique performs adequately. We also apply the optimization scheme to the tensor product of two $S_N^+$-channels, but do not find any violation of the minimum output entropy additivity conjecture.

In this thesis, we use a variation of a commutator technique to prove that l^p-stability is independent of p, for p greater than or equal to one, and for convolution-dominated matrices indexed by relatively separated sets in groups of polynomial growth. Moreover, from the inverse-closedness of the Schur matrices we deduce a Wiener type Lemma for the matrices in the intersection of the weighted convolution-dominated matrices, over all polynomial weights.
Finally, applications of the convolution-dominated matrices are presented. We prove the inverse-closedness of a non-commutative space generated by a discrete series representation restricted to a lattice in a nilpotent Lie group. In addition, we apply the aforementioned result on l^p-stability to show that if \pi(\Lambda)g is a p-frame for the coorbit space Co(L^p) for some p in [1,\infty], then \pi(\Lambda)g is a q-frame for the coorbit space Co(L^q) for each q in [1,\infty], where (\pi, H) is a discrete series representation of a group G of polynomial growth, \Lambda is a relatively separated set in G, and g is a vector in H such that the matrix coefficient V_g g is in the Amalgam space W_{w_a}(G). Moreover, we prove that the frame operator of the frame \pi(\Lambda)g is invertible on the coorbit spaces Co(L^p) for each p in [1,\infty], under the assumption that g is a vector in H, such that V_g g belongs in W_{w_a}(G) for each polynomial weight w_a.

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 <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.

In this thesis we present a study of quantum Markov semigroups. In particular, we mainly consider quantum Markov semigroups with detailed balance that are defined on finite-dimensional C*-algebras. They have an invariant density matrix ρ. Carlen and Maas showed that the evolution on the set of invertible density matrices that is given by such a semigroup is gradient flow for the relative entropy with respect to ρ for some Riemannian metric. This result is a non-commutative analog of certain diffusion equations that are gradient flow in the second order Wasserstein space. We provide a self-contained and accessible account to these issues. Moreover, we give a complete introduction to Tomita-Takesaki theory which has a close relation with quantum Markov semigroups satisfying detailed balance. Finally, we present some examples of these semigroups that arise from quantum theory.

In 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.

Quantum computers promise an exponential speed-up over their classical counterparts for certain tasks relevant to various fields including science, technology, and finance. To unlock this potential, the technology must be scaled up and the errors at play must be reduced. As developments in scalable quantum computation devices advance, the demand for scalable benchmarking techniques that are able to reliably assess the fidelity – the complement of the error rate – of a device has increased significantly. Randomized benchmarking offers a single, concise number that reflects the average fidelity of multi-qubit operations performed on a quantum device. While this method is robust against state preparation and measurement errors, it still suffers from scalability issues. In this thesis, we present a protocol that efficiently predicts the multi-qubit fidelity obtained from randomized benchmarking by only benchmarking single- and two-qubit subspaces, greatly increasing the scalability. The protocol uses simultaneous randomized benchmarking with the aim of catching cross-talk effects while at the same time reducing the number of required benchmarking sequences. We have run numerical simulations of the protocol under two noise models, one depolarizing and one dephasing, to verify its performance. The results of these noisy simulations are promising and suggest that our protocol could offer a valuable tool on the road to developing large-scale quantum computers.

In 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.

Bell inequalities are certain probabilistic inequalities that should hold in the context of quantum measurement under assumption of a local hidden variable model. These inequalities can be violated according to the theory of quantum mechanics and have also been violated experimentally. Bell inequalities were therefore historically used to disprove local hidden variable models as an interpretation of quantum mechanics. An interesting question is to what degree Bell inequalities can be violated according to the theory of quantum mechanics. The degree to which a particular Bell inequality can be violated is quantified by the largest violation of the Bell inequality. A central question we address in this thesis is how large the largest violation can become when considering all possible Bell inequalities. In particular the CHSH-inequality, a well-known Bell inequality, has a largest violation equal to the square root of 2 and we want to find a Bell inequality with largest violation exceeding the square root of 2. This thesis consists of two main parts. In the first part we formally define Bell inequalities and the largest violation and prove several theorems about Bell inequalities. In the second part we search for Bell inequalities with largest violation exceeding the square root of 2. We do this by first approximating the largest violation of a given Bell inequality using numerical optimization. Next, using this approximation, we use numerical optimization to maximize the largest violation. Due to run-time restrictions we were only able to consider Bell inequalities with a relatively small number of terms and were unable to find any among them with largest violation exceeding the square root of 2.

In 2011 Avsec showed strong solidity of the q-Gaussian algebras, building upon previous results of Houdayer and Shlyakhtenko, and Ozawa and Popa. In this work we study this result as well as the necessary literature and q-mathematics needed to replicate the proof. The literature is combined within this thesis, whilst additionally filling in gaps in Avsec's proofs. Overall, the thesis aims to present an improved and more accessible proof of the strong solidity of the q-Gaussian algebras.

Quantum error correction is needed for future quantum computers. Classical error correcting codes are not suitable for this due to the nature of quantum mechanics. Therefore, new codes need to be developed. A promising candidate is the toric code, a surface code, because of its locality and its high error correcting capability and thresholds (the error probability below which increasing the size of the code decreases the failure rate). This thesis provides an introduction to quantum error correction and the stabilizer formalism, which is used to introduce the toric code. Several decoders are looked at, including the Minimum Weight Perfect Matching (MWPM) decoder and a recently developed decoder, the Union-Find (UF) decoder. The UF decoder is useful because of its almost-linear time complexity, while only reducing the threshold by a marginal amount compared to the MWPM decoder. In this thesis, the time complexity of the weighted growth version of the UF decoder is analysed. The toric code and the decoders have been implemented and simulated, and the thresholds have been determined. For the MWPM decoder, the threshold was determined to be 11.5±0.2%, which is not in agreement with the threshold in literature of 10.3%, and should not be possible according to the optimal threshold of about 11%. This probably is a result of the low grid sizes (up to a gridsize of 11) of the code used due to the time needed to simulate the MWPM decoder. For the UF decoder, the threshold found was 9.7 ± 0.9%, which is in agreement with the threshold of 9.9% found by N. Delfosse and N. Nickerson. The time complexity of the UF decoder has also been determined, and is indeed almost-linear as expected by the analysis.