Multi-Unitary Complex Hadamard Matrices

We analyze the set of real and complex Hadamard matrices with additional symmetry constrains. In particular, we link the problem of existence of maximally entangled multipartite states of $2k$ subsystems with $d$ levels each to the set of complex Hadamard matrices of order $N=d^k$. To this end, we investigate possible subsets of such matrices which are, dual, strongly dual ($H=H^{\rm R}$ or $H=H^{\rm\Gamma}$), two-unitary ($H^R$ and $H^{\Gamma}$ are unitary), or $k$-unitary. Here $X^{\rm R}$ denotes reshuffling of a matrix $X$ describing a bipartite system, and $X^{\rm \Gamma}$ its partial transpose. Such matrices find several applications in quantum many-body theory, tensor networks and classification of multipartite quantum entanglement and imply a broad class of analytically solvable quantum models in $1+1$ dimensions.Comment: 18 pages, no figure

Finite-Function-Encoding Quantum States

We investigate the encoding of higher-dimensional logic into quantum states. To that end we introduce finite-function-encoding (FFE) states which encode arbitrary $d$-valued logic functions and investigate their structure as an algebra over the ring of integers modulo $d$. We point out that the polynomiality of the function is the deciding property for associating hypergraphs to states. Given a polynomial, we map it to a tensor-edge hypergraph, where each edge of the hypergraph is associated with a tensor. We observe how these states generalize the previously defined qudit hypergraph states, especially through the study of a group of finite-function-encoding Pauli stabilizers. Finally, we investigate the structure of FFE states under local unitary operations, with a focus on the bipartite scenario and its connections to the theory of complex Hadamard matrices.Comment: Comments welcom

Mutually unbiased bases and related structures

A set of bases of a d dimensional complex vector space, each pair of which is unbiased, is a set of mutually unbiased bases (MUBs). MUBs have applications in quantum physics and quantum information theory. Although the motivation to study MUBs comes from physical properties, MUBs are a mathematical structure. This is a mathematical investigation. There are many open problems in the theory of MUBS, some with conjectured solutions. For example: What is the maximum number of MUBs in a d dimensional vector space? Do complete sets of MUBs exist in all dimensions? One such conjectured solution states that a complete set of MUBs exists in a d dimensional complex vector space if and only if a complete set of mutually orthogonal Latin squares (MOLS) of order d exists (Saniga et. al., Journal of Optics B, 6: L19-20, 2004). The aim of this research was to find evidence for or against this conjecture. Inspired by constructions of MUBs that use sets of MOLS, complete sets of MOLS were constructed from two complete sets of MUBs. It is interesting to note that the MOLS structure emerges not from the vectors, but from the inner products of the vectors. Analogous properties between Hjelmslev planes and MUBs, and gaps in this knowledge motivated investigation of Hjelmslev planes. The substructures of a Hjelmslev plane over a Galois ring, and a combinatorial algorithm for generating Hjelmslev planes were developed. It was shown that the analogous properties of Hjelmslev planes and MUBs occur only for odd prime powers, making a strong connection between MUBs and Hjelmslev planes unlikely. A construction of MUBs that uses planar functions was generalised by using an automorphism on the additive group of a Galois field. It is still unclear whether this generalisation is equivalent to the original construction. Relation algebras were constructed from the structure of MUBs which do not share any similarities with relation algebras constructed from MOLS. It is possible that further investigation may yield relation algebras that are similar. It was shown that a set of Wooters and Fields type MUBs, when represented as elements of a group ring, forms a commutative monoid, whereas a set of Alltop type MUBs when similarly represented does not form a closed algebraic structure. It is known that WF and Alltop MUBs are equivalent, thus the lack of a closed structure in the Alltop MUBs suggests that the monoid is not a property of MUBs in general. Complete sets of MOLS and complete sets of MUBs are `similar in spirit', but perhaps this is not an inherent feature of MUBs and MOLS. Since all the known constructions of MUBs rely on algebraic structures which exist only in prime power dimensions, the connection may not be with MOLS, but with algebraic structures which generate both MOLS and MUBs

Machine learning methods for discriminating natural targets in seabed imagery

The research in this thesis concerns feature-based machine learning processes and methods for discriminating qualitative natural targets in seabed imagery. The applications considered, typically involve time-consuming manual processing stages in an industrial setting. An aim of the research is to facilitate a means of assisting human analysts by expediting the tedious interpretative tasks, using machine methods. Some novel approaches are devised and investigated for solving the application problems. These investigations are compartmentalised in four coherent case studies linked by common underlying technical themes and methods. The first study addresses pockmark discrimination in a digital bathymetry model. Manual identification and mapping of even a relatively small number of these landform objects is an expensive process. A novel, supervised machine learning approach to automating the task is presented. The process maps the boundaries of ≈ 2000 pockmarks in seconds - a task that would take days for a human analyst to complete. The second case study investigates different feature creation methods for automatically discriminating sidescan sonar image textures characteristic of Sabellaria spinulosa colonisation. Results from a comparison of several textural feature creation methods on sonar waterfall imagery show that Gabor filter banks yield some of the best results. A further empirical investigation into the filter bank features created on sonar mosaic imagery leads to the identification of a useful configuration and filter parameter ranges for discriminating the target textures in the imagery. Feature saliency estimation is a vital stage in the machine process. Case study three concerns distance measures for the evaluation and ranking of features on sonar imagery. Two novel consensus methods for creating a more robust ranking are proposed. Experimental results show that the consensus methods can improve robustness over a range of feature parameterisations and various seabed texture classification tasks. The final case study is more qualitative in nature and brings together a number of ideas, applied to the classification of target regions in real-world sonar mosaic imagery. A number of technical challenges arose and these were surmounted by devising a novel, hybrid unsupervised method. This fully automated machine approach was compared with a supervised approach in an application to the problem of image-based sediment type discrimination. The hybrid unsupervised method produces a plausible class map in a few minutes of processing time. It is concluded that the versatile, novel process should be generalisable to the discrimination of other subjective natural targets in real-world seabed imagery, such as Sabellaria textures and pockmarks (with appropriate features and feature tuning.) Further, the full automation of pockmark and Sabellaria discrimination is feasible within this framework