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Quantum isometries and group dual subgroups

Teodor Banica, Jyotishman Bhowmick, Kenny De Commer (2012)

Annales mathématiques Blaise Pascal

We study the discrete groups Λ whose duals embed into a given compact quantum group, Λ ^ ⊂ G . In the matrix case G ⊂ U n + the embedding condition is equivalent to having a quotient map Γ U → Λ , where F = { Γ U ∣ U ∈ U n } is a certain family of groups associated to G . We develop here a number of techniques for computing F , partly inspired from Bichon’s classification of group dual subgroups Λ ^ ⊂ S n + . These results are motivated by Goswami’s notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian...

Quantum Itô algebra and quantum martingale

Viacheslav Belavkin, Un Cig Ji (2007)

Banach Center Publications

In this paper, we study a representation of the quantum Itô algebra in Fock space and then by using a noncommutative Radon-Nikodym type theorem we study the density operators of output states as quantum martingales, where the output states are absolutely continuous with respect to an input (vacuum) state. Then by applying quantum martingale representation we prove that the density operators of regular, absolutely continuous output states belong to the commutant of the ⋆-algebra parameterizing the...

Quantum mechanics and nonabelian theta functions for the gauge group SU(2)

Răzvan Gelca, Alejandro Uribe (2015)

Fundamenta Mathematicae

We propose a direction of study of nonabelian theta functions by establishing an analogy between the Weyl quantization of a one-dimensional particle and the metaplectic representation on the one hand, and the quantization of the moduli space of flat connections on a surface and the representation of the mapping class group on the space of nonabelian theta functions on the other. We exemplify this with the cases of classical theta functions and of the nonabelian theta functions for the gauge group...

Quantum moment equations for a two-band k ⋅ p Hamiltonian

Luigi Barletti (2005)

Bollettino dell'Unione Matematica Italiana

The hydrodynamic moment equations for a quantum system described by a two-band k ⋅ p Hamiltonian are derived. In the case of pure states, it is proved that the order-0 and order-1 moment equations yield a closed system which is the two band analogue of Madelung's fluid equations.

Quantum random walk revisited

Kalyan B. Sinha (2006)

Banach Center Publications

In the framework of the symmetric Fock space over L²(ℝ₊), the details of the approximation of the four fundamental quantum stochastic increments by the four appropriate spin-matrices are studied. Then this result is used to prove the strong convergence of a quantum random walk as a map from an initial algebra 𝓐 into 𝓐 ⊗ ℬ (Fock(L²(ℝ₊))) to a *-homomorphic quantum stochastic flow.

Quantum scattering near the lowest Landau threshold for a Schrödinger operator with a constant magnetic field

Michael Melgaard (2003)

Open Mathematics

For fixed magnetic quantum number m results on spectral properties and scattering theory are given for the three-dimensional Schrödinger operator with a constant magnetic field and an axisymmetrical electric potential V. In various, mostly singular settings, asymptotic expansions for the resolvent of the Hamiltonian H m+Hom+V are deduced as the spectral parameter tends to the lowest Landau threshold. Furthermore, scattering theory for the pair (H m, H om) is established and asymptotic expansions...

Quantum states satisfying classical probability constraints

Elena R. Loubenets (2006)

Banach Center Publications

For linear combinations of quantum product averages in an arbitrary bipartite state, we derive new quantum Bell-form and CHSH-form inequalities with the right-hand sides expressed in terms of a bipartite state. This allows us to specify bipartite state properties sufficient for the validity of a classical CHSH-form inequality and the perfect correlation form of the original Bell inequality for any bounded quantum observables. We also introduce a new general condition on a bipartite state and quantum...

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