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We prove a family of $n$-th order identities for quantum $R$-matrices of Baxter-Belavin type in fundamental representation. The set of identities includes the unitarity condition as the simplest one ($n=2$). Our study is inspired by the fact that the third order identity provides commutativity of the Knizhnik-Zamolodchikov-Bernard connections. On the other hand the same identity gives rise to $R$-matrix valued Lax pairs for the classical integrable systems of Calogero type. The latter construction uses interpretation of quantum $R$-matrix as matrix generalization of the Kronecker function. We present a proof of the higher order scalar identities for the Kronecker functions which is then naturally generalized to the $R$-matrix identities.

Quantum relative entropy $D(\rho\|\sigma)\defeq\Tr \rho (\log \rho- \log \sigma)$ plays an important role in quantum information and related fields. However, there are many quantum analogues of relative entropy. In this paper, we characterize these analogues from information geometrical viewpoint. We also consider the naturalness of quantum relative entropy among these analogues.

Since Renes et al. [J. Math. Phys. 45, 2171 (2004)], there has been much effort in the quantum information community to prove (or disprove) the existence of symmetric informationally complete (SIC) sets of quantum states in arbitrary finite dimension. This paper strengthens the urgency of this question by showing that if SIC-sets exist: 1) by a natural measure of orthonormality, they are as close to being an orthonormal basis for the space of density operators as possible, and 2) in prime dimensions, the standard construction for complete sets of mutually unbiased bases and Weyl-Heisenberg covariant SIC-sets are intimately related: The latter represent minimum uncertainty states for the former in the sense of Wootters and Sussman. Finally, we contribute to the question of existence by conjecturing a quadratic redundancy in the equations for Weyl-Heisenberg SIC-sets.

We formulate and prove a free quantum analogue of the first fundamental theorems of invariant theory. More precisely, the polynomial functions algebras are replaced by free algebras, while the universal cosovereign Hopf algebras play the role of the general linear group.

We study quantum Schubert varieties from the point of view of regularity conditions. More precisely, we show that these rings are domains which are maximal orders and are AS-Cohen-Macaulay and we determine which of them are AS-Gorenstein. One key fact that enables us to prove these results is that quantum Schubert varieties are quantum graded algebras with a straightening law that have a unique minimal element in the defining poset. We prove a general result showing when such quantum graded algebras are maximal orders. Finally, we exploit these results to show that quantum determinantal rings are maximal orders.

For non-Hermitian equilateral q-pointed star-shaped quantum graphs of paper I [Can. J. Phys. 90, 1287 (2012), arXiv 1205.5211] we show that due to certain dynamical aspects of the model as controlled by the external, rotation-symmetric complex Robin boundary conditions, the spectrum is obtainable in a closed asymptotic-expansion form, in principle at least. Explicit formulae up to the second order are derived for illustration, and a few comments on their consequences are added.

A new class of stochastic variables, governed by a specifice set of rules, is introduced. These rules force them to loose some properties usually assumed for this kind of variables. We demonstrate that stochastic processes driven by these random sources must be described using an probability amplitude formalism in a close resemblance to Quantum Theory. This fact shows, for the first time, that probability amplitudes are a general concept and is not exclusive to the formalism of Quantum Theory. Application of these rules to a noisy, one-dimensional motion, leds to a probability structure homomorphic to Quantum Mechanics.

We obtain off-critical (elliptic) Boltzmann weights for lattice models whose continuum limits correspond to massive, $N=2$ supersymmetric, quantum integrable field theories. We also compute the free energies of these models and show that they are analytic in the region of parameter space where we believe that the supersymmetry is unbroken. While the supersymmetry is not directly realized on the lattice, there is still a very close connection between the models described here and topological lattice models. A simple example is discussed in detail and some corner transfer matrix computations are also presented.

We obtain off-critical (elliptic) Boltzmann weights for lattice models whose continuum limits correspond to massive, $N=2$ supersymmetric, quantum integrable field theories. We also compute the free energies of these models and show that they are analytic in the region of parameter space where we believe that the supersymmetry is unbroken. While the supersymmetry is not directly realized on the lattice, there is still a very close connection between the models described here and topological lattice models. A simple example is discussed in detail and some corner transfer matrix computations are also presented.

The work is divided in three parts. We devote the first part to the study of analog Hawking radiation in Bose-Einstein condensates. We study numerically the birth of a sonic black hole in an outcoupled Bose-Einstein condensate after relaxing the confinement provided by an optical lattice. We also study possible signatures of spontaneous Hawking radiation. We propose that the violation of CS inequalities is a smoking gun of the presence of the Hawking effect. We compare this criterion with the presence of entaglement, finding that both are equivalent under usual assumptions. Finally, we study a different gravitational analogue: the so-called black-hole laser. The most interesting result is the appearance of a regime of continuous and periodic emission of solitons, providing the most strong analogue with optical lasers. In the second part, we analyze the effect of the introduction of a short Bragg pulse in a thermal cloud. We show that the induced periodic density pattern decays to the equilibrium profile. However, instead of the usual collisional relaxation, the mechanism responsible for the decay is the thermal disorder of the particles, with a characteristic time that only depends on the temperature. We find a very good agreement with actual experimental data. In the last part, we switch to a very different system: the $\nu=0$ quantum Hall state of bilayer graphene. After re-deriving the corresponding mean-field phase diagram, we compute the collective modes within the zero Landau level. Among the most remarkable results, we have found that at the boundary between the FLP and the F phases a gapless mode appears resulting from an accidental symmetry that can be regarded as a remanent of a broken $SO(5)$ symmetry. On the other hand, the CAF and PLP phases can present dynamical instabilities. We straightforwardly extend the previous results to monolayer graphene.

We treat quantum counterparts of testing problems whose optimal tests are given by chi-square, t and F tests. These quantum counterparts are formulated as quantum hypothesis testing problems concerning quantum Gaussian states families, and contain disturbance parameters, which have group symmetry. Quantum Hunt-Stein Theorem removes a part of these disturbance parameters, but other types of difficulty still remain. In order to remove them, combining quantum Hunt-Stein theorem and other reduction methods, we establish a general reduction theorem that reduces a complicated quantum hypothesis testing problem to a fundamental quantum hypothesis testing problem. Using these methods, we derive quantum counterparts of chi-square, t and F tests as optimal tests in the respective settings.

Hardy's nonlocality is a "nonlocality proof without inequalities": it exemplifies that quantum correlations can be qualitatively stronger than classical correlations. This paper introduces variants of Hardy's nonlocality in the CHSH scenario which are realized by the PR-box, but not by quantum correlations. Hence this new kind of Hardy-type nonlocality is a proof without inequalities showing that superquantum correlations can be qualitatively stronger than quantum correlations.

Conceptual analogies among statistical mechanics and classical (or quantum) mechanics often appeared in the literature. For classical two-body mean field models, an analogy develops into a proper identification between the free energy of Curie-Weiss type magnetic models and the Hamilton-Jacobi action for a one dimensional mechanical system. Similarly, the partition function plays the role of the wave function in quantum mechanics and satisfies the heat equation that plays, in this context, the role of the Schrodinger equation in quantum mechanics. We show that this identification can be remarkably extended to include a wide family of magnetic models classified by normal forms of suitable real algebraic dispersion curves. In all these cases, the model turns out to be completely solvable as the free energy as well as the order parameter are obtained as solutions of an integrable nonlinear PDE of Hamilton-Jacobi type. We observe that the mechanical analog of these models can be viewed as the relativistic analog of the Curie-Weiss model and this helps to clarify the connection between generalised self-averaging and in statistical thermodynamics and the semi-classical dynamics of viscous conservation laws.

In the present paper, we are interested in natural quantum analogues of Richardson varieties in the type A grassmannians. To be more precise, the objects that we investigate are quantum analogues of the homogeneous coordinate rings of Richardson varieties which appear naturally in the theory of quantum groups. Our point of view, here, is geometric: we are interested in the regularity properties of these "non-commutative varieties", such as their irreducibility, normality, Cohen-Macaulayness... in the spirit of non-commutative algebraic geometry. A major step in our approach is to show that these algebras have the structure of an Algebra with a Straightening Law. From this, it follows that they degenerate to some quantum analogues of toric varieties.

We obtain lattice models whose continuum limits correspond to $N=2$ superconformal coset models. This is done by taking the well known vertex model whose continuum limit is the $G \times G/G$ conformal field theory, and twisting the transfer matrix and modifying the quantum group truncation. We find that the natural order parameters of the new models are precisely the chiral primary fields. The integrable perturbations of the conformal field theory limit also have natural counterparts in the lattice formulation, and these can be incorporated into an affine quantum group structure. The topological, twisted $N=2$ superconformal models also have lattice analogues, and these emerge as an intermediate part of our analysis.

In this paper we analyze the structure of some subalgebras of quantized enveloping algebras corresponding to unipotent and solvable subgroups of a simple Lie group G. These algebras have the non--commutative structure of iterated algebras of twisted polynomials with a derivation, an object which has often appeared in the general theory of non-commutative rings. In particular, we find maximal dimensions of their irreducible representations. Our results confirm the validity of the general philosophy that the representation theory is intimately connected to the Poisson geometry.

The first step in quantum information theory is the identification of entanglement as a valuable resource. The next step is learning how to exploit this resource efficiently. We learn how to exploit entanglement efficiently by applying analogues of thermodynamical concepts. These concepts include reversibility, entropy, and the distinction between intensive and extensive quantities. We discuss some of these analogues and show how they lead to a measure of entanglement for pure states. We also ask whether these analogues are more than analogues, and note that, l o c a l l y, entropy of entanglement is thermodynamical entropy.

The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense "q-commutative", the latter one is related to its noncommutative counterpart. We introduce also a two parameter deformation of the invariant integral on a sphere.

The q-Legendre polynomials can be treated as some special "functions in the quantum double cosets $U(1)\setminus SU_q(2)/U(1)$". They form a family (depending on a parameter $q$) of polynomials in one variable. We get their further generalization by introducing a two parameter family of polynomials. If the former family arises from an algebra which is in a sense "q-commutative", the latter one is related to its noncommutative counterpart. We introduce also a two parameter deformation of the invariant integral on a sphere.

Chirped-pulse interferometry (CPI) captures the metrological advantages of quantum Hong-Ou-Mandel (HOM) interferometry in a completely classical system. Modified HOM interferometers are the basis for a number of seminal quantum-interference effects. Here, the corresponding modifications to CPI allow for the first observation of classical analogues to the HOM peak and quantum beating. They also allow a new classical technique for generating phase super-resolution exhibiting a coherence length dramatically longer than that of the laser light, analogous to increased two-photon coherence lengths in entangled states.

We pick up a solvable ${\cal PT}-$symmetric quantum square well on an interval of $x \in := (-L,L)\mathbb{G}^{(2)}$ (with an $\alpha-$dependent non-Hermiticity given by Robin boundary conditions) and generalize it. In essence, we just replace the support interval $\mathbb{G}^{(2)}$ (reinterpreted as an equilateral two-pointed star graph with the Kirchhoff matching at the vertex $x=0$) by a $q-$pointed equilateral star graph $\mathbb{G}^{(q)}$ endowed with the simplest complex-rotation-symmetric external $\alpha-$dependent Robin boundary conditions. The remarkably compact form of the secular determinant is then deduced. Its analysis reveals that (1) at any integer $q=2,3,...$, there exists the same, $q-$independent and infinite subfamily of the real energies, and (2) at any special $q=2,6,10,...$, there exists another, additional and $q-$dependent infinite subfamily of the real energies. In the spirit of the recently proposed dynamical construction of the Hilbert space of a quantum system, the physical bound-state interpretation of these eigenvalues is finally proposed.

We recently considered the task of summoning an unknown quantum state and proved necessary and sufficient conditions for Alice to be able to guarantee to complete the task when there may be several possible calls, of which she need only respond to one. We showed that these are strictly stronger conditions than those previously established by Hayden and May for the case where Alice knows there will only be one call. We introduced the concept of a {\it quantum paradox of choice} to summarize the implications of these results: Alice is given more options to complete our version of the task, yet one can easily construct examples where our version is impossible and the apparently simpler version considered by Hayden-May is possible. Finkelstein has argued that one can identify analogous classical paradoxes of choice in a relativistic setting. We examine Finkelstein's proposed classical tasks and explain why they seem to us disanalogous.

In this paper we explore the quantum behaviour of a SQUID ring which has a significant Josephson coupling energy. We show that that the eigenfunctions of the Hamiltonian for the ring can be used to create macroscopic quantum superposition states of the ring. We also show that the ring potential may be utilised to squeeze coherent states. With the SQUID ring as a strong contender as a device for manipulating quantum information, such properties may be of great utility in the future. However, as with all candidate systems for quantum technologies, decoherence is a fundamental problem. In this paper we apply an open systems approach to model the effect of coupling a quantum mechanical SQUID ring to a thermal bath. We use this model to demonstrate the manner in which decoherence affects the quantum states of the ring.

In a multidimensional model with several scalar fields and an m-form we deal with classical spherically symmetric solutions with one (electric or magnetic) p-brane and Ricci-flat internal spaces and the corresponding solutions to the Wheeler--DeWitt (WDW) equation. Classical black holes are considered and their quantum analogues (e.g. for M2 and M5 extremal solutions in D =11 supergravity, electric and magnetic charges in D=4 gravity) are suggested when the curvature coupling in the WDW equation is zero.

We discuss two quantum analogues of Fisher information, symmetric logarithmic derivative (SLD) Fisher information and Kubo-Mori-Bogoljubov (KMB) Fisher information from a large deviation viewpoint of quantum estimation and prove that the former gives the true bound and the latter gives the bound of consistent superefficient estimators. In another comparison, it is shown that the difference between them is characterized by the change of the order of limits.

We describe Nuclear Magnetic Resonance (NMR) demonstrations of the quantum Zeno effect, and discuss briefly how these are related to similar phenomena in more conventional NMR experiments.

A Schroedinger type equation on the superspace R^{D|2n} is studied, which involves a potential inversely proportional to the negative of the osp(D|2n) invariant "distance" away from the origin. An osp(2,D+1|2n) dynamical supersymmetry for the system is explicitly constructed, and the bound states of the system are shown to form an irreducible highest weight module for this superalgebra. A thorough understanding of the structure of the irreducible module is obtained. This in particular enables the determination of the energy eigenvalues and the corresponding eigenspaces as well as their respective dimensions.