Evaluating the Holevo Cramér-Rao bound for multiparameter quantum metrology

Only with the simultaneous estimation of multiple parameters are the quantum aspects of metrology fully revealed. This is due to the incompatibility of observables. The fundamental bound for multiparameter quantum estimation is the Holevo Cram´er-Rao bound (HCRB) whose evaluation has so far remained elusive. For finite-dimensional systems we recast its evaluation as a semidefinite program, with reduced size for rank-deficient states. We show that it also satisfies strong duality. We use this result to study phase and loss estimation in optical interferometry and three-dimensional magnetometry with noisy multiqubit systems. For the former, we show that, in some regimes, it is possible to attain the HCRB with the optimal (single-copy) measurement for phase estimation. For the latter, we show a nontrivial interplay between the HCRB and incompatibility and provide numerical evidence that projective single-copy measurements attain the HCRB in the noiseless 2-qubit case

Church-Rosser Systems, Codes with Bounded Synchronization Delay and Local Rees Extensions

What is the common link, if there is any, between Church-Rosser systems, prefix codes with bounded synchronization delay, and local Rees extensions? The first obvious answer is that each of these notions relates to topics of interest for WORDS: Church-Rosser systems are certain rewriting systems over words, codes are given by sets of words which form a basis of a free submonoid in the free monoid of all words (over a given alphabet) and local Rees extensions provide structural insight into regular languages over words. So, it seems to be a legitimate title for an extended abstract presented at the conference WORDS 2017. However, this work is more ambitious, it outlines some less obvious but much more interesting link between these topics. This link is based on a structure theory of finite monoids with varieties of groups and the concept of local divisors playing a prominent role. Parts of this work appeared in a similar form in conference proceedings where proofs and further material can be found.Comment: Extended abstract of an invited talk given at WORDS 201

Semigroup intersection problems in the Heisenberg groups

We consider two algorithmic problems concerning sub-semigroups of Heisenberg groups and, more generally, two-step nilpotent groups. The first problem is Intersection Emptiness, which asks whether a finite number of given finitely generated semigroups have empty intersection. This problem was first studied by Markov in the 1940s. We show that Intersection Emptiness is PTIME decidable in the Heisenberg groups $\operatorname{H}_{n}(\mathbb{K})$ over any algebraic number field $\mathbb{K}$, as well as in direct products of Heisenberg groups. We also extend our decidability result to arbitrary finitely generated 2-step nilpotent groups. The second problem is Orbit Intersection, which asks whether the orbits of two matrices under multiplication by two semigroups intersect with each other. This problem was first studied by Babai et al. (1996), who showed its decidability within commutative matrix groups. We show that Orbit Intersection is decidable within the Heisenberg group $\operatorname{H}_{3}(\mathbb{Q})$.Comment: 18 pages including appendix, 2 figure

Register automata with linear arithmetic

We propose a novel automata model over the alphabet of rational numbers, which we call register automata over the rationals (RA-Q). It reads a sequence of rational numbers and outputs another rational number. RA-Q is an extension of the well-known register automata (RA) over infinite alphabets, which are finite automata equipped with a finite number of registers/variables for storing values. Like in the standard RA, the RA-Q model allows both equality and ordering tests between values. It, moreover, allows to perform linear arithmetic between certain variables. The model is quite expressive: in addition to the standard RA, it also generalizes other well-known models such as affine programs and arithmetic circuits. The main feature of RA-Q is that despite the use of linear arithmetic, the so-called invariant problem---a generalization of the standard non-emptiness problem---is decidable. We also investigate other natural decision problems, namely, commutativity, equivalence, and reachability. For deterministic RA-Q, commutativity and equivalence are polynomial-time inter-reducible with the invariant problem

Algebraic Properties of Parikh Matrices of Words under an Extension of Thue Morphism

The Parikh matrix of a word $w$ over an alphabet $\{a_1, \cdots , a_k \}$ with an ordering $a_1 < a_2 < \cdots a_k,$ gives the number of occurrences of each factor of the word $a_1 \cdots a_k$ as a (scattered) subword of the word $w.$ Two words $u,v$ are said to be $M-$equivalent, if the Parikh matrices of $u$ and $v$ are the same. On the other hand properties of image words under different morphisms have been studied in the context of subwords and Parikh matrices. Here an extension to three letters, introduced by S$\acute{e}\acute{e}$bold (2003), of the well-known Thue morphism on two letters, is considered and properties of Parikh matrices of morphic images of words are investigated. The significance of the contribution is that various classes of binary words are obtained whose images are $M-$equivalent under this extended morphism

On the continuity of the commutative limit of the 4d N=4 non-commutative super Yang-Mills theory

We study the commutative limit of the non-commutative maximally supersymmetric Yang-Mills theory in four dimensions (N=4 SYM). The commutative limits of non-commutative spaces are important in particular in the applications of non-commutative spaces for regularisation of supersymmetric theories (such as the use of non-commutative spaces as alternatives to lattices for supersymmetric gauge theories and interpretations of some matrix models as regularised supermembrane or superstring theories), which in turn can play a prominent role in the study of quantum gravity via the gauge/gravity duality. In general, the commutative limits are known to be singular and non-smooth due to UV/IR mixing effects. We give a direct proof that UV effects do not break the continuity of the commutative limit of the non-commutative N=4 SYM to all order in perturbation theory, including non-planar contributions. This is achieved by establishing the uniform convergence (with respect to the non-commutative parameter) of momentum integrals associated with all Feynman diagrams appearing in the theory, using the same tools involved in the proof of finiteness of the commutative N=4 SYM.Comment: v1: 27 pages, 3 figures. v2: References update