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In parameter estimation, nuisance parameters refer to parameters that are not of interest but nevertheless affect the precision of estimating other parameters of interest. For instance, the strength of noises in a probe can be regarded as a nuisance parameter. Despite its long history in classical statistics, the nuisance parameter problem in quantum estimation remains largely unexplored. The goal of this article is to provide a systematic review of quantum estimation in the presence of nuisance parameters, and to supply those who work in quantum tomography and quantum metrology with tools to tackle relevant problems. After an introduction to the nuisance parameter and quantum estimation theory, we explicitly formulate the problem of quantum state estimation with nuisance parameters. We extend quantum Cramér–Rao bounds to the nuisance parameter case and provide a parameter orthogonalization tool to separate the nuisance parameters from the parameters of interest. In particular, we put more focus on the case of one-parameter estimation in the presence of nuisance parameters, as it is most frequently encountered in practice.

A review of Boundary and defect conformal field theory: open problems and applications, following a workshop held at Chicheley Hall, Buckinghamshire, UK, 7–8 Sept. 2017. We attempt to provide a broad, bird's-eye view of the latest progress in boundary and defect conformal field theory in various sub-fields of theoretical physics, including the renormalization group, integrability, conformal bootstrap, topological field theory, supersymmetry, holographic duality, and more. We also discuss open questions and promising research directions in each of these sub-fields, and combinations thereof.

We propose a new approach to studying electrical networks interpreting the Ohm law as the operator which solves certain local Yang–Baxter equation. Using this operator and the medial graph of the electrical network we define a vertex integrable statistical model and its boundary partition function. This gives an equivalent description of electrical networks. We show that, in the important case of an electrical network on the standard graph introduced in [Curtis E B et al 1998 Linear Algebr. Appl.283 115–50], the response matrix of an electrical network, its most important feature, and the boundary partition function of our statistical model can be recovered from each other. Defining the electrical varieties in the usual way we compare them to the theory of the Lusztig varieties developed in [Berenstein A et al 1996 Adv. Math.122 49–149]. In our picture the former turns out to be a deformation of the later. Our results should be compared to the earlier work started in [Lam T and Pylyavskyy P 2015 Algebr. Number Theory9 1401–18] on the connection between the Lusztig varieties and the electrical varieties. There the authors introduced a one-parameter family of Lie groups which are deformations of the unipotent group. For the value of the parameter equal to 1 the group in the family acts on the set of response matrices and is related to the symplectic group. Using the data of electrical networks we construct a representation of the group in this family which corresponds to the value of the parameter −1 in the symplectic group and show that our boundary partition functions belong to it. Remarkably this representation has been studied before in the work on six vertex statistical models and the representations of the Temperley–Lieb algebra.

We propose a natural ${\mathbb{Z}}_{2}{\times}{\mathbb{Z}}_{2}$-graded generalisation of d = 2, $\mathcal{N}=\left(1,1\right)$ supersymmetry and construct a ${\mathbb{Z}}_{2}^{2}$-space realisation thereof. Due to the grading, the supercharges close with respect to, in the classical language, a commutator rather than an anticommutator. This is then used to build classical (linear and non-linear) sigma models that exhibit this novel supersymmetry via mimicking standard superspace methods. The fields in our models are bosons, right-handed and left-handed Majorana–Weyl spinors, and exotic bosons. The bosons commute with all the fields, the spinors belong to different sectors that cross commute rather than anticommute, while the exotic boson anticommute with the spinors. As a particular example of one of the models, we present a 'double-graded' version of supersymmetric sine-Gordon theory.

Using the algebraic approach Lie symmetries of time dependent Schrödinger equations for charged particles interacting with superpositions of scalar and vector potentials are classified. Namely, all the inequivalent equations admitting symmetry transformations with respect to continuous groups of transformations are presented. This classification is completed and includes the specification of symmetries and admissible equivalence relations for such equations. In particular, a simple mapping between the free Schrödinger equation and the repulsive oscillator is found which has a clear group-theoretical sense.

The notion of an odd quasi-connection on a supermanifold, which is loosely an affine connection that carries non-zero Grassmann parity, is examined. Their torsion and curvature are defined, however, in general, they are not tensors. A special class of such generalised connections, referred to as odd connections in this paper, have torsion and curvature tensors. Part of the structure is an odd involution of the tangent bundle of the supermanifold and this puts drastic restrictions on the supermanifolds that admit odd connections. In particular, they must have equal number of even and odd dimensions. Amongst other results, we show that an odd connection is defined, up to an odd tensor field of type (1, 2), by an affine connection and an odd endomorphism of the tangent bundle. Thus, the theory of odd connections and affine connections are not completely separate theories. As an example relevant to physics, it is shown that $\mathcal{N}=1$ super-Minkowski spacetime admits a natural odd connection.

Quantized topological invariants characterizing topological phases in quasi-1D materials are usually considered only on the basis of spatial inversion parity eigenvalues. However, symmetry of quasi-1D systems is far more complex and their complete topological characterisation can be obtained only on the basis of elementary band representations (EBRs) for the relevant symmetry groups. We derive complete sets of inequivalent EBRs for line groups (LG), the symmetry groups of all quasi-1D systems with either translational or helical periodicity. Besides, we determine also EBRs for double-LGs, accounting for spin degree of freedom. In order to illustrate applicability of the results obtained, we analyze electronic-band topology of a chiral single-wall carbon nanotube, using EBRs for relevant (double)-LG and discuss Su-Schrieffer–Heeger model from EBR-perspective.

We prove that, under some natural conditions, Hamiltonian systems on a contact manifold C can be split into a Reeb dynamics on an open subset of C and a Liouville dynamics on a submanifold of C of codimension 1. For the Reeb dynamics we find an invariant measure. Moreover, we show that, under certain completeness conditions, the existence of an invariant measure for the Liouville dynamics can be characterized using the notion of a symplectic sandwich with contact bread.

We study the chiral current of non-interacting bosons in a three-leg lattice subjected to a uniform magnetic flux. The model is equivalent to a spin-1 bosonic lattice with three internal degrees of freedom or a one-dimensional lattice with a three-site synthetic dimension. By manipulating a bias field between the legs which acts as a quadratic Zeeman shift, we can effectively reverse the chiral current on the lattice. The current can also be reversed by increasing the magnetic flux or the inter-leg hopping coefficients, provided that the quadratic field is applied. Three types of current reversal, either related or unrelated to the Meissner-vortex phase transition, are revealed.

It is shown that the enhanced (nonlinear) realignment criterion is equivalent to the family of linear criteria based on correlation tensor. These criteria generalize the original (linear) realignment criterium and give rise to the family of entanglement witnesses. An appropriate limiting procedure is proposed which leads to a novel class of witnesses which are as powerful as the enhanced realignment criterion.

Following a generic approach that leads to Bogomolny–Prasad–Sommerfield (BPS) soliton solutions by imposing self-duality, we investigate three different types of non-Hermitian field theories. We consider a complex version of a logarithmic potential that possess BPS super-exponential kink and antikink solutions and two different types of complex generalizations of systems of coupled sine-Gordon models with kink and antikink solution of complex versions of arctan type. Despite the fact that all soliton solutions obtained in this manner are complex in the non-Hermitian theories we show that they possess real energies. For the complex extended sine-Gordon model we establish explicitly that the energies are the same as those in an equivalent pair of a non-Hermitian and Hermitian theory obtained from a pseudo-Hermitian approach by means of a Dyson map. We argue that the reality of the energy is due to the topological properties of the complex BPS solutions. These properties result in general from modified versions of antilinear CPT symmetries that relate self-dual and an anti-self-dual theories.

We present trapped solitary wave solutions of a coupled nonlinear Schrödinger (NLS) system in 1 + 1 dimensions in the presence of an external, supersymmetric and complex $\mathcal{PT}$-symmetric potential. The Schrödinger system this work focuses on possesses exact solutions whose existence, stability, and spatio-temporal dynamics are investigated by means of analytical and numerical methods. Two different variational approximations are considered where the stability and dynamics of the solitary waves are explored in terms of eight and twelve time-dependent collective coordinates (CCs). We find regions of stability for specific potential choices as well as analytic expressions for the small oscillation frequencies in the CC approximation. Our findings are further supported by performing systematic numerical simulations of the NLS system.