Related papers: Information-Theoretic Lower Bounds for Approximati…
Minimization problems with respect to a one-parameter family of generalized relative entropies are studied. These relative entropies, which we term relative $\alpha$-entropies (denoted $\mathscr{I}_{\alpha}$), arise as redundancies under…
Estimating entropies from limited data series is known to be a non-trivial task. Naive estimations are plagued with both systematic (bias) and statistical errors. Here, we present a new 'balanced estimator' for entropy functionals Shannon,…
This thesis focuses on the intersection of mathematical and computational optimization and quantum information. Main contributions are open-source software code: A hybrid approach mixing "traditional" nonconvex and convex methods can make…
This dissertation investigates relative entropies, also called generalized divergences, and how they can be used to characterize information-theoretic tasks in quantum information theory. The main goal is to further refine characterizations…
Maximum entropy inference and learning of graphical models are pivotal tasks in learning theory and optimization. This work extends algorithms for these problems, including generalized iterative scaling (GIS) and gradient descent (GD), to…
One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to $\frac43$. For 40 years, the best known upper bound…
In the framework of quantum information geometry, we derive, from quantum relative Tsallis entropy, a family of quantum metrics on the space of full rank, N level quantum states, by means of a suitably defined coordinate free differential…
Deriving formulations for computing and estimating tight worst-case size increases for conjunctive queries with various constraints has been at the core of theoretical database research. If the problem has no constraints or only one…
The problem of quantum state inference and the concept of quantum entanglement are studied using a non-additive measure in the form of Tsallis entropy indexed by the positive parameter q. The maximum entropy principle associated with this…
We investigate the computational complexity of estimating the trace of quantum state powers $\text{tr}(\rho^q)$ for an $n$-qubit mixed quantum state $\rho$, given its state-preparation circuit of size $\text{poly}(n)$. This quantity is…
The paper suggests the concepts of an upper entropy and a lower entropy. We propose a new axiomatic definition, namely, upper entropy axioms, inspired by axioms of metric spaces, and also formulate lower entropy axioms. We also develop weak…
After reviewing unnormalized and normalized information distances based on incomputable notions of Kolmogorov complexity, we discuss how Kolmogorov complexity can be approximated by data compression algorithms. We argue that optimal…
We describe a quantum algorithm to estimate the $\alpha$-Renyi entropy of an unknown density matrix $\rho\in\mathcal{C}^{d\times d}$ for $\alpha\neq 1$ by combining the recent technique of quantum singular value transformations with the…
We define and investigate a notion of entropy for quantum error correcting codes. The entropy of a code for a given quantum channel has a number of equivalent realisations, such as through the coefficients associated with the Knill-Laflamme…
Quantum computation has been growing rapidly in both theory and experiments. In particular, quantum computing devices with a large number of qubits have been developed by IBM, Google, IonQ, and others. The current quantum computing devices…
This thesis is a multidisciplinary contribution to the information theory of single-particle Coulomb systems in their relativistic and not relativistic description, to the theory of special functions of mathematical physics with the…
Quantum mechanics and information theory are among the most important scientific discoveries of the last century. Although these two areas initially developed separately it has emerged that they are in fact intimately related. In this…
We present an information-theoretic approach to lower bound the oracle complexity of nonsmooth black box convex optimization, unifying previous lower bounding techniques by identifying a combinatorial problem, namely string guessing, as a…
Two new information-theoretic methods are introduced for establishing Poisson approximation inequalities. First, using only elementary information-theoretic techniques it is shown that, when $S_n=\sum_{i=1}^nX_i$ is the sum of the (possibly…
One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the…