Absolutely Maximally Entangled states, combinatorial designs and multi-unitary matrices

Absolutely Maximally Entangled (AME) states are those multipartite quantum states that carry absolute maximum entanglement in all possible partitions. AME states are known to play a relevant role in multipartite teleportation, in quantum secret sharing and they provide the basis novel tensor networks related to holography. We present alternative constructions of AME states and show their link with combinatorial designs. We also analyze a key property of AME, namely their relation to tensors that can be understood as unitary transformations in every of its bi-partitions. We call this property multi-unitarity.Comment: 18 pages, 2 figures. Comments are very welcom

Parity of transversals of Latin squares

We introduce a notion of parity for transversals, and use it to show that in Latin squares of order $2 \bmod 4$, the number of transversals is a multiple of 4. We also demonstrate a number of relationships (mostly congruences modulo 4) involving $E_1,\dots, E_n$, where $E_i$ is the number of diagonals of a given Latin square that contain exactly $i$ different symbols. Let $A(i\mid j)$ denote the matrix obtained by deleting row $i$ and column $j$ from a parent matrix $A$. Define $t_{ij}$ to be the number of transversals in $L(i\mid j)$, for some fixed Latin square $L$. We show that $t_{ab}\equiv t_{cd}\bmod2$ for all $a,b,c,d$ and $L$. Also, if $L$ has odd order then the number of transversals of $L$ equals $t_{ab}$ mod 2. We conjecture that $t_{ac} + t_{bc} + t_{ad} + t_{bd} \equiv 0 \bmod 4$ for all $a,b,c,d$. In the course of our investigations we prove several results that could be of interest in other contexts. For example, we show that the number of perfect matchings in a $k$-regular bipartite graph on $2n$ vertices is divisible by $4$ when $n$ is odd and $k\equiv0\bmod 4$. We also show that $${\rm per}\, A(a \mid c)+{\rm per}\, A(b \mid c)+{\rm per}\, A(a \mid d)+{\rm per}\, A(b \mid d) \equiv 0 \bmod 4$$ for all $a,b,c,d$, when $A$ is an integer matrix of odd order with all row and columns sums equal to $k\equiv2\bmod4$

Implementing Brouwer's database of strongly regular graphs

Andries Brouwer maintains a public database of existence results for strongly regular graphs on $n\leq 1300$ vertices. We implemented most of the infinite families of graphs listed there in the open-source software Sagemath, as well as provided constructions of the "sporadic" cases, to obtain a graph for each set of parameters with known examples. Besides providing a convenient way to verify these existence results from the actual graphs, it also extends the database to higher values of $n$.Comment: 18 pages, LaTe

Multipartite entanglement and quantum algorithms

[eng] Quantum information science has grown from being a very small subfield in the 70s until being one of the most dynamic fields in physics, both in fundamentals and applications. In the theoretical section, perhaps the feature that has attracted most interest is the notion of entanglement, the ghostly relation between particles that dazzled Einstein and has provided fabulous challenges to build a coherent interpretation of quantum mechanics. While not completely solved, we have today learned enough to feel less uneasy with this fundamental problem, and the focus has shifted towards its potential powerful applications. Entanglement is now being studied from different perspectives as a resource for performing information processing tasks. With bipartite entanglement being largely understood nowadays, many questions remain unanswered in the multipartite case. The first part of this thesis deals with multipartite entanglement in different contexts. In the first chapters it is studied within the whole corresponding Hilbert space, and we investigate several entanglement measures searching for states that maximize them, including violations of Bell inequalities. Later, focus is shifted towards hamiltonians that have entangled ground states, and we investigate entanglement as a way to establish a distance between theories and we study frustration and methods to efficiently solve hamiltonians that exhibit it. In the practical section, the most promised upcoming technological advance is the advent of quantum computers. In the 90s some quantum algorithms improving the performance of all known classical algorithms for certain problems started to appear, while in the 2000s the first universal computers of few atoms began to be built, allowing implementation of those algorithms in small scales. The D-Wave machine already performs quantum annealing in thousands of qubits, although some controversy over the true quantumness of its internal workings surrounds it. Many countries in the planet are devoting large amounts of money to this field, with the recent European flagship and the involvement of the largest US technological companies giving reasons for optimism. The second part of this thesis deals with some aspects of quantum computation, starting with the creation of the field of cloud quantum computation with the appearance of the first computer available to the general public through internet, which we have used and analysed extensively. Also small incursions in quantum adiabatic computation and quantum thermodynamics are present in this second part.[cat] La informació quàntica ha crescut des d'un petit subcamp als anys setanta fins a esdevenir un dels camps més dinàmics de la física actualment, tant en aspectes fonamentals com en les seves aplicacions. En la secció teòrica, potser la propietat que ha atret més interès és la noció d'entrellaçament, la relació fantasmagòrica entre partícules que va deixar estupefacte Einstein i que ha suposat un enorme desafiament per a construir una interpretació coherent de la mecànica quàntica. Sense estar totalment solucionat, hem après prou per sentir-nos menys incòmodes amb aquest problema fonamental i el focus s'ha desplaçat a les seves aplicacions potencials. L'entrellaçament s'estudia avui en dia des de diferents perspectives com a recurs per realitzar tasques de processament de la informació. L'entrellaçament bipartit està ja molt ben comprès, però en el cas multipartit queden moltes qüestions obertes. La primera part d'aquesta tesi tracta de l'entrellaçament multipartit en diferents contextos. Estudiem l'hiperdeterminant com a mesura d'entrellaçament el cas de 4 qubits, analitzem l'existència i les propietats matemàtiques dels estats absolutament màximament entrellaçats, trobem noves desigualtats de Bell, estudiem l'espectre d'entrellaçament com a mesura de distància entre teories i estudiem xarxes tensorials per tractar eficientment sistemes frustrats. En l'apartat pràctic, el més prometedor avenç tecnològic del camp és l'adveniment dels ordinadors quàntics. La segona part de la tesi tracta d'alguns aspectes de computació quàntica, començant per la creació del camp de la computació quàntica al núvol, amb l'aparició del primer ordinador disponible per al públic general, que hem usat extensament. També fem petites incursions a la computació quàntica adiabàtica i a la termodinàmica quàntica en aquesta segona par

Optimum experimental designs for models with a skewed error distribution: with an application to stochastic frontier models

In this thesis, optimum experimental designs for a statistical model possessing a skewed error distribution are considered, with particular interest in investigating possible parameter dependence of the optimum designs. The skewness in the distribution of the error arises from its assumed structure. The error consists of two components (i) random error, say V, which is symmetrically distributed with zero expectation, and (ii) some type of systematic error, say U, which is asymmetrically distributed with nonzero expectation. Error of this type is sometimes called 'composed' error. A stochastic frontier model is an example of a model that possesses such an error structure. The systematic error, U, in a stochastic frontier model represents the economic efficiency of an organisation. Three methods for approximating information matrices are presented. An approximation is required since the information matrix contains complicated expressions, which are difficult to evaluate. However, only one method, 'Method 1', is recommended because it guarantees nonnegative definiteness of the information matrix. It is suggested that the optimum design is likely to be sensitive to the approximation. For models that are linearly dependent on the model parameters, the information matrix is independent of the model parameters but depends on the variance parameters of the random and systematic error components. Consequently, the optimum design is independent of the model parameters but may depend on the variance parameters. Thus, designs for linear models with skewed error may be parameter dependent. For nonlinear models, the optimum design may be parameter dependent in respect of both the variance and model parameters. The information matrix is rank deficient. As a result, only subsets or linear combinations of the parameters are estimable. The rank of the partitioned information matrix is such that designs are only admissible for optimal estimation of the model parameters, excluding any intercept term, plus one linear combination of the variance parameters and the intercept. The linear model is shown to be equivalent to the usual linear regression model, but with a shifted intercept. This suggests that the admissible designs should be optimal for estimation of the slope parameters plus the shifted intercept. The shifted intercept can be viewed as a transformation of the intercept in the usual linear regression model. Since D_A-optimum designs are invariant to linear transformations of the parameters, the D_A-optimum design for the asymmetrically distributed linear model is just the linear, parameter independent, D_A-optimum design for the usual linear regression model with nonzero intercept. C-optimum designs are not invariant to linear transformations. However, if interest is in optimally estimating the slope parameters, the linear transformation of the intercept to the shifted intercept is no longer a consideration and the C-optimum design is just the linear, parameter independent, C-optimum design for the usual linear regression model with nonzero intercept. If interest is in estimating the slope parameters, and the shifted intercept, the C-optimum design will depend on (i) the design region; (ii) the distributional assumption on U; (iii) the matrix used to define admissible linear combinations of parameters; (iv) the variance parameters of U and V; (v) the method used to approximate the information matrix. Some numerical examples of designs for a cross-sectional log-linear Cobb-Douglas stochastic production frontier model are presented to demonstrate the nonlinearity of designs for models with a skewed error distribution. Torsney's (1977) multiplicative algorithm was implemented in finding the optimum designs