Quantum B-splines

Quantum splines are piecewise polynomials whose quantum derivatives (i.e. certain discrete derivatives or equivalently certain divided differences) agree up to some order at the joins. Just like classical splines, quantum splines admit a canonical basis with compact support: the quantum B-splines. These quantum B-splines are the q-analogues of classical B-splines. Here quantum B-spline bases and quantum B-spline curves are investigated, using a new variant of the blossom: the q (quantum)-blossom. The q-blossom of a degree d polynomial is the unique symmetric, multiaffine function in d variables that reduces to the polynomial along the q-diagonal. By applying the q-blossom, algorithms and identities for quantum B-spline bases and quantum B-spline curves are developed, including quantum variants of the de Boor algorithms for recursive evaluation and quantum differentiation, knot insertion procedures for converting from quantum B-spline to piecewise quantum Bézier form, and a quantum variant of Marsden’s identity.     

Quantum Pfaffians and hyper-Pfaffians

The concept of the quantum Pfaffian is rigorously examined and refurbished using the new method of quantum exterior algebras. We derive a complete family of Plücker relations for the quantum linear transformations, and then use them to give an optimal set of relations required for the quantum Pfaffian. We then give the formula between the quantum determinant and the quantum Pfaffian and prove that any quantum determinant can be expressed as a quantum Pfaffian. Finally the quantum hyper-Pfaffian is introduced, and we prove a similar result of expressing quantum determinants in terms of quantum hyper-Pfaffians at modular cases.     

A state-space approach to quantum permutations

In this exposition of quantum permutation groups, an alternative to the ‘Gelfand picture’ of compact quantum groups is proposed. This point of view is inspired by algebraic quantum mechanics and interprets the states of an algebra of continuous functions on a quantum permutation group as quantum permutations. This interpretation allows talk of an element of a quantum permutation group, and allows a clear understanding of the difference between deterministic, random, and quantum permutations. The interpretation is illustrated by various quantum permutation group phenomena.     

The quantum euler class and the quantum cohomology of the Grassmannians

The Poincaré duality of classical cohomology and the extension of this duality to quantum cohomology endows these rings with the structure of a Frobenius algebra. Any such algebra possesses a canonical “characteristic element;” in the classical case this is the Euler class, and in the quantum case this is a deformation of the classical Euler class which we call the “quantum Euler class.” We prove that the characteristic element of a Frobenius algebraA is a unit if and only ifA is semisimple, and then apply this result to the cases of the quantum cohomology of the finite complex Grassmannians, and to the quantum cohomology of hypersurfaces. In addition we show that, in the case of the Grassmannians, the [quantum] Euler class equals, as [quantum] cohomology element and up to sign, the determinant of the Hessian of the [quantum] Landau-Ginzbug potential.     

Quantum B-algebras

The concept of quantale was created in 1984 to develop a framework for non-commutative spaces and quantum mechanics with a view toward non-commutative logic. The logic of quantales and its algebraic semantics manifests itself in a class of partially ordered algebras with a pair of implicational operations recently introduced as quantum B-algebras. Implicational algebras like pseudo-effect algebras, generalized BL- or MV-algebras, partially ordered groups, pseudo-BCK algebras, residuated posets, cone algebras, etc., are quantum B-algebras, and every quantum B-algebra can be recovered from its spectrum which is a quantale. By a two-fold application of the functor “spectrum”, it is shown that quantum B-algebras have a completion which is again a quantale. Every quantale Q is a quantum B-algebra, and its spectrum is a bigger quantale which repairs the deficiency of the inverse residuals of Q. The connected components of a quantum B-algebra are shown to be a group, a fact that applies to normal quantum B-algebras arising in algebraic number theory, as well as to pseudo-BCI algebras and quantum BL-algebras. The logic of quantum B-algebras is shown to be complete.     

Clocks and fisher information

In a broad sense, any parametric family of quantum states can be viewed as a quantum clock. The time, which is the parameter, is encoded in the corresponding quantum states. The quality of such a clock depends on how precisely we can distinguish the states or, equivalently, estimate the parameter. In view of the quantum Cramér—Rao inequalities, the quality of quantum clocks can be characterized by the quantum Fisher information. We address the issue of quantum clock synchronization in terms of quantum Fisher information and demonstrate its fundamental difference from the classical paradigm. The key point is the superadditivity of Fisher information, which always holds in the classical case but can be violated in quantum mechanics. The violation can occur for both pure and mixed states. Nevertheless, we establish the superadditivity of quantum Fisher information for any classical-quantum state. We also demonstrate an alternative form of superadditivity and propose a weak form of superadditivity. The violation of superadditivity can be exploited to enhance quantum clock synchronization.     

Quantum stochastic convolution cocycles III

Every Markov-regular quantum Lévy process on a multiplier C ^*-bialgebra is shown to be equivalent to one governed by a quantum stochastic differential equation, and the generating functionals of norm-continuous convolution semigroups on a multiplier C ^*-bialgebra are then completely characterised. These results are achieved by extending the theory of quantum Lévy processes on a compact quantum group, and more generally quantum stochastic convolution cocycles on a C ^*-bialgebra, to locally compact quantum groups and multiplier C ^*-bialgebras. Strict extension results obtained by Kustermans, together with automatic strictness properties developed here, are exploited to obtain existence and uniqueness for coalgebraic quantum stochastic differential equations in this setting. Then, working in the universal enveloping von Neumann bialgebra, we characterise the stochastic generators of Markov-regular, *-homomorphic (respectively completely positive and contractive), quantum stochastic convolution cocycles.     

Quantum Counting: Algorithm and Error Distribution

Counting is one of the most basic procedures in mathematics and statistics. In statistics literature it is usually done via the proportion estimation method. In this article we manifest a radically different counting procedure first proposed in the late 1990’s based on the techniques of quantum computation. It combines two major tools in quantum computation, quantum Fourier transform and quantum amplitude amplification, and shares similar structure to the quantum part of the celebrated Shor’s factoring algorithm. We present complete details of this quantum counting algorithm and the analysis of its error distribution. Comparing it with the conventional proportion estimation method, we demonstrate that this quantum approach achieves much faster convergence rate than the classical approach.     

Sur les matrices quantiques

We show here that row-reducing a quantum matrix produces another quantum matrix satisfying the same relations as those of the original quantum matrix ring M_q(n). As a corollary, the image of the quantum determinant in the abelianization of the total ring of quotients of M_q(n ), equals the Dieudonné determinant of the quantum matrix.     

Global well-posedness and the classical limit of the solution for the quantum Zakharov system

In this paper, we study the quantum Zakharov system, which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The global well-posedness result of this system in the energy and above energy spaces is obtained in the case d = 1, 2, 3. Moreover, the classical limit behavior of the quantum Zakharov system is also investigated as the quantum parameter tends to zero.     

I-factorial quantum torsors and Heisenberg algebras of quantized universal enveloping type

We introduce a notion of I-factorial quantum torsor, which consists of an integrable ergodic action of a locally compact quantum group on a type I-factor such that also the crossed product is a type I-factor. We show that any such I-factorial quantum torsor is at the same time a I-factorial quantum torsor for the dual locally compact quantum group, in such a way that the construction is involutive. As a motivating example, we show that quantized compact semisimple Lie groups, when amplified via a crossed product construction with the function algebra on the associated weight lattice, admit I-factorial quantum torsors, and give an explicit realization of the dual quantum torsor in terms of a deformed Heisenberg algebra for the Borel part of a quantized universal enveloping algebra.     

A twisted quantum toroidal algebra

As an analog of the quantum TKK algebra, a twisted quantum toroidal algebra of type A ₁ is introduced. Explicit realization of the new quantum TKK algebra is constructed with the help of twisted quantum vertex operators over a Fock space.     

Quantum exotic PDE’s

Following the previous works on the Prástaro’s formulation of algebraic topology of quantum (super) PDE’s, it is proved that a canonical Heyting algebra (integral Heyting algebra) can be associated to any quantum PDE. This is directly related to the structure of its global solutions. This allows us to recognize a new inside in the concept of quantum logic for microworlds. Furthermore, the Prástaro’s geometric theory of quantum PDE’s is applied to the new category of quantum hypercomplex manifolds, related to the well-known Cayley–Dickson construction for algebras. Theorems of existence for local and global solutions are obtained for (singular) PDE’s in this new category of noncommutative manifolds. Finally, the extension of the concept of exotic PDE’s, recently introduced by Prástaro, has been extended to quantum PDE’s. Then a smooth quantum version of the quantum (generalized) Poincaré conjecture is given too. These results extend ones for quantum (generalized) Poincaré conjecture, previously given by Prástaro.     

矩阵代数收敛至球面与非交换度量几何

介绍了Rieffel定义的紧致量子度量空间与量子Gromov-Hausdorff距离和近来Latrémolière定义的量子Gromov-Hausdorff邻距,分别讨论了矩阵代数如何在这两种量子距离下收敛至球面.     

A representation theorem for quantum systems

In this note representations of quantum systems are investigated. We propose a unital bipolar theorem for unital quantum cones, which plays a key role in proving a representation theorem for quantum systems. It turns out that each quantum system is identified with a certain quantum L^∞-system up to a quantum order isomorphism.     

A new approach to induction and imprimitivity results

Given a closed quantum subgroup of a locally compact quantum group, we study induction of unitary corepresentations of the quantum subgroup to the ambient quantum group. More generally, we study induction given a coaction of the quantum subgroup on a C^*-algebra. We prove imprimitivity theorems that unify the existing theorems for actions and coactions of groups. This means that we define quantum homogeneous spaces as C^*-algebras and that we prove Morita equivalence of crossed products and homogeneous spaces. We essentially use von Neumann algebraic techniques to prove these Morita equivalences between C^*-algebras.     

(Co)bordism groups in quantum super PDE's. III: Quantum super Yang–Mills equations

In this third part of a series of three papers devoted to the study of geometry of quantum super PDE's [A. Prástaro, (Co)bordism groups in quantum super PDE's. I: quantum supermanifolds, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.007; (Co)bordism groups in quantum super PDE's. II: quantum super PDE's, Nonlinear Anal. Real World Appl., in press, doi:10.1016/j.nonrwa.2005.12.008], we apply our theory, developed in the first two parts, to quantum super Yang–Mills equations and quantum supergravity equations. For such equations we determine their integral bordism groups, and by using some surgery techniques, we obtain theorems of existence of global solutions, also with nontrivial topology, for Cauchy problems and boundary value problems. Quantum tunnelling effects are described in this context. Furthermore, for quantum supergravity equations we prove existence of solutions of the type quantum black holes evaporation processes just by using an extension to quantum super PDEs of our theory of integral (co)bordism groups. Our proof is constructive, i.e., we give geometric methods to build such solutions. In particular a criterion to recognize quantum global (smooth) solutions with mass-gap, for the quantum super Yang–Mills equation, is given. Finally it is proved that quantum super PDE's contain also solutions that come from Dirac quantization of their superclassical counterparts. This proves that quantum super PDE's are (nonlinear) generalizations of Dirac quantized superclassical PDE's. Applications of this result to free quantum super Yang–Mills equations are given.     

On cocycle twisting of compact quantum groups

We provide a class of examples of compact quantum groups and unitary 2-cocycles on them, such that the twisted quantum groups are non-compact, but still locally compact quantum groups (in the sense of Kustermans and Vaes). This also gives examples of cocycle twists where the underlying C^∗-algebra of the quantum group changes.     

On projective representations for compact quantum groups

We study actions of compact quantum groups on type I-factors, which may be interpreted as projective representations of compact quantum groups. We generalize to this setting some of Woronowicz?s results concerning Peter-Weyl theory for compact quantum groups. The main new phenomenon is that for general compact quantum groups (more precisely, those which are not of Kac type), not all irreducible projective representations have to be finite-dimensional. As applications, we consider the theory of projective representations for the compact quantum groups associated with group von Neumann algebras of discrete groups, and consider a certain non-trivial projective representation for quantum SU(2).