IOPcollections

We give a separability criterion for three qubit states in terms of diagonal and anti-diagonal entries. This gives us a complete characterization of separability when all the entries are zero except for diagonal and anti-diagonals. The criterion is expressed in terms of a norm arising from anti-diagonal entries. We compute this norm in several cases, so that we get criteria with which we can decide the separability by routine computations.

We investigate a non-Abelian generalization of the Kuramoto model proposed by Lohe and given by N quantum oscillators (‘nodes’) connected by a quantum network where the wavefunction at each node is distributed over quantum channels to all other connected nodes. It leads to a system of Schrödinger equations coupled by nonlinear self-interacting potentials given by their correlations. We give a complete picture of synchronization results, given on the relative size of the natural frequency and the coupling constant, for two non-identical oscillators and show complete phase synchronization for arbitrary identical oscillators. Our results are mainly based on the analysis of the ODE system satisfied by the correlations and on the introduction of a quantum order parameter, which is analogous to the one defined by Kuramoto in the classical model. As a consequence of the previous results, we obtain the synchronization of the probability and the current densities defined via the Madelung transformations.

We derive a closed formula for the Baker–Campbell–Hausdorff series expansion in the case of complex matrices. For arbitrary matrices A and B, and a matrix Z such that , our result expresses Z as a linear combination of A and B, their commutator , and the identity matrix I. The coefficients in this linear combination are functions of the traces and determinants of A and B, and the trace of their product. The derivation proceeds purely via algebraic manipulations of the given matrices and their products, making use of relations developed here, based on the Cayley–Hamilton theorem, as well as a characterization of the consequences of and/or its determinant being zero or otherwise. As a corollary of our main result we also derive a closed formula for the Zassenhaus expansion. We apply our results to several special cases, most notably the parametrization of the product of two matrices and a verification of the recent result of Van-Brunt and Visser (2015 J. Phys. A: Math. Theor. 48 225207) for complex matrices, in this latter case deriving also the related Zassenhaus formula which turns out to be quite simple. We then show that this simple formula should be valid for all matrices and operators.

The lambda model is a one parameter deformation of the principal chiral model that arises when regularizing the non-compactness of a non-abelian T dual in string theory. It is a current–current deformation of a WZW model that is known to be integrable at the classical and quantum level. The standard techniques of the quantum inverse scattering method cannot be applied because the Poisson bracket is non ultra-local. Inspired by an approach of Faddeev and Reshetikhin, we show that in this class of models, there is a way to deform the symplectic structure of the theory leading to a much simpler theory that is ultra-local and can be quantized on the lattice whilst preserving integrability. This lattice theory takes the form of a generalized spin chain that can be solved by standard algebraic Bethe Ansatz techniques. We then argue that the IR limit of the lattice theory lies in the universality class of the lambda model implying that the spin chain provides a way to apply the quantum inverse scattering method to this non ultra-local theory. This points to a way of applying the same ideas to other lambda models and potentially the string world-sheet theory in the gauge-gravity correspondence.

We study the partition function of the six-vertex model in the rational limit on arbitrary Baxter lattices with reflecting boundary. Every such lattice is interpreted as an invariant of the twisted Yangian. This identification allows us to relate the partition function of the vertex model to the Bethe wave function of an open spin chain. We obtain the partition function in terms of creation operators on a reference state from the algebraic Bethe ansatz and as a sum of permutations and reflections from the coordinate Bethe ansatz.

We describe a procedure that creates an explicit complex-valued polynomial function of three-dimensional space, whose nodal lines are the three-twist knot 5 ₂. The construction generalizes a similar approach for lemniscate knots: a braid representation is engineered from finite Fourier series and then considered as the nodal set of a certain complex polynomial which depends on an additional parameter. For sufficiently small values of this parameter, the nodal lines form the three-twist knot. Further mathematical properties of this map are explored, including the relationship of the phase critical points with the Morse–Novikov number, which is nonzero as this knot is not fibred. We also find analogous functions for other simple knots and links. The particular function we find, and the general procedure, should be useful for designing knotted fields of particular knot types in various physical systems.

The Segal–Bargmann transform is a unitary map between the Schrödinger and Fock space, which is used, for example, to show the integrability of quantum Rabi models. Slice monogenic functions provide the framework in which functional calculus for quaternionic quantum mechanics can be developed. In this paper, a generalisation of the Segal–Bargmann transform, to the context of slice monogenic functions, is constructed and studied in detail. It is shown to interact appropriately with the recently constructed slice Fourier transform. This leads furthermore to a construction of a slice Fock space, which is shown to be a reproducing kernel space.

We prove that the energy gap of the model proposed by Zhang et al (2016 arXiv:1606.07795) is exponentially small in the square of the system size. In Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA) a class of exactly solvable quantum spin chain models was proposed that have integer spins ( s), with a nearest neighbors Hamiltonian, and a unique ground state. The ground state can be seen as a uniform superposition of all s-colored Motzkin walks. The half-chain entanglement entropy provably violates the area law by a square root factor in the system’s size (∼ ) for s  >  1. For s  =  1, the violation is logarithmic (Bravyi et al 2012 Phys. Rev. Lett. 109 207202). Moreover in Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA) it was proved that the gap vanishes polynomially and is O( n ^{− c} ) with .

Recently, a deformation of Movassagh and Shor (2016 Proc. Natl Acad. Sci. USA), which we call ‘weighted Motzkin quantum spin chain’ was proposed Zhang et al (2016 arXiv:1606.07795). This model has a unique ground state that is a superposition of the s-colored Motzkin walks weighted by with t  >  1. The most surprising feature of this model is that it violates the area law by a factor of n. Here we prove that the gap of this model is upper bounded by for t  >  1 and s  >  1.

We address the problem of estimating the mass of a quantum particle in a gravitational field and seek the ultimate bounds to precision of quantum-limited detection schemes. In particular, we study the effect of the field on the achievable sensitivity and address the question of whether quantumness of the probe state may provide a precision enhancement. The ultimate bounds to precision are quantified in terms of the corresponding quantum Fisher information. Our results show that states with no classical limit perform better than semiclassical ones and that a non-trivial interplay exists between the external field and the statistical model. More intense fields generally lead to a better precision, with the exception of position measurements in the case of freely-falling systems.

In this note we propose an approach for a fast analytic determination of all possible sets of Bethe roots corresponding to eigenstates of rational integrable spin chains of given not too large length, in terms of Baxter Q-functions. We observe that all exceptional solutions, if any, are automatically correctly accounted.

The key intuition behind the approach is that the equations on the Q-functions are determined solely by the Young diagram, and not by the choice of the rank of the symmetry. Hence we can choose arbitrary and that accommodate the desired representation. Then we consider all distinguished Q-functions at once, not only those following a certain Kac–Dynkin path.