In a topological space, a family of continuous mappings is called universal if its action, in at least one element of the space, is dense. If the mappings are unitary or trace-preserving completely positive, the notion of universality is closely related to the notion of controllability in either closed or open quantum systems. Quantum controllability in infinite dimensions is discussed in this ...

In this paper, we present a theoretical framework for understanding vector embedding, a fundamental building block of many deep learning models, especially in NLP. We discover a natural unitary-invariance in vector embeddings, which is required by the distributional hypothesis. This unitary-invariance states the fact that two embeddings are essentially equivalent if one can be obtained from the...

We combine the geometric realization of principal series representations of [28] with the Bott–Borel–Weil Theorem for direct limits of compact groups found in [22], obtaining limits of principal series representations for direct limits of real reductive Lie groups. We introduce the notion of weakly parabolic direct limits and relate it to the conditions that the limit representations are norm–p...

We generalize the notion of Lagrangian subspaces to self-orthogonal subspaces with respect to a (skew-)symmetric form, thus characterizing (skew-) self-adjoint and unitary operators by means of self-orthogonal subspaces. By orthogonality preserving mappings, these characterizations can be transferred to abstract boundary value spaces of (skew-)symmetric operators. Introducing the notion of boun...

We study the superalgebra of the M-theory on a fully supersymmetric pp-wave. We identify the algebra as the special unitary Lie superalgebra, su(2|4 ; 2, 0) or su(2|4 ; 2, 4), and analyze its root structure. We discuss the typical and atypical representations deriving the typicality condition explicitly in terms of the energy and other four quantum numbers. We classify the BPS multiplets preser...

Abstract. Area non-preserving transformations in the non-commutative plane are introduced with the aim to map the ν = 1 integer quantum Hall effect (IQHE) state on the fractional quantum Hall effect (FQHE) ν = 1 2p+1 FQHE states. Using the hydrodynamical description of the quantum Hall fluid, it is shown that these transformations are generated by vector fields satisfying the Gauss law in the i...

We introduce a new quantity for describing nonclassicality of an arbitrary optical two-mode Gaussian state which remains invariant under any global photon-number preserving unitary transformation of the covariance matrix of the state. The invariant naturally splits into an entanglement monotone and local-nonclassicality quantifiers applied to the reduced states. This shows how entanglement can ...

In this paper, we will obtain quantum work for a scale five dimensional Myers-Perry black hole. Unlike heat represented by Hawking radiation, the is unitary information preserving process, and becomes important holes only at small scales. It be observed that such short distances, corrected non-perturbative gravitational corrections. We use Jarzynski equality to modified These corrections also m...

We discuss the possibility of defining an emergent local temperature in extended quantum many-body systems evolving out equilibrium. For most simple case free-fermionic systems, we give explicit formula for effective of, not necessarily unitary, Gaussian preserving dynamics. In this framework, consider hopping fermions on a one-dimensional lattice submitted to randomly distributed projective me...

We study the superalgebra of the M-theory on a fully supersymmetric pp-wave. We identify the algebra as the special unitary Lie superalgebra, su(2|4 ; 2, 0) or su(2|4 ; 2, 4), and analyze its root structure. We discuss the typical and atypical representations deriving the typicality condition explicitly in terms of the energy and other four quantum numbers. We classify the BPS multiplets preser...