A complete set of Lorentz-invariant wave packets and modified uncertainty relation

We define a set of fully Lorentz-invariant wave packets and show that it spans the corresponding one-particle Hilbert subspace, and hence the whole Fock space as well, with a manifestly Lorentz-invariant completeness relation (resolution of identity). The position–momentum uncertainty relation for this Lorentz-invariant wave packet deviates from the ordinary Heisenberg uncertainty principle, and reduces to it in the non-relativistic limit.

Universal quantum simulation of single-qubit nonunitary operators using duality quantum algorithm

Quantum information processing enhances human’s power to simulate nature in quantum level and solve complex problem efficiently. During the process, a series of operators is performed to evolve the system or undertake a computing task. In recent year, research interest in non-Hermitian quantum systems, dissipative-quantum systems and new quantum algorithms has greatly increased, which nonunitary operators take an important role in. In this work, we utilize the linear combination of unitaries technique for nonunitary dynamics on a single qubit to give explicit decompositions of the necessary unitaries, and simulate arbitrary time-dependent single-qubit nonunitary operator F(t) using duality quantum algorithm. We find that the successful probability is not only decided by F(t) and the initial state, but also is inversely proportional to the dimensions of the used ancillary Hilbert subspace. In a general case, the simulation can be achieved in both eight- and six-dimensional Hilbert spaces. In phase matching conditions, F(t) can be simulated by only two qubits. We illustrate our method by simulating typical non-Hermitian systems and single-qubit measurements. Our method can be extended to high-dimensional case, such as Abrams–Lloyd’s two-qubit gate. By discussing the practicability, we expect applications and experimental implementations in the near future.

Amenability properties of unitary co-representations of locally compact quantum groups

For locally compact quantum groups [Formula: see text], we initiate an investigation of stable states with respect to unitary co-representations [Formula: see text] of [Formula: see text] on Hilbert spaces [Formula: see text]; in particular, we study the subject on the multiplicative unitary operator [Formula: see text] of [Formula: see text] with some examples on locally compact quantum groups arising from discrete groups and compact groups. As the main result, we consider the one co-dimensional Hilbert subspace of [Formula: see text] associated to a suitable vector [Formula: see text], to present an operator theoretic characterization of stable states with respect to a related unitary co-representation [Formula: see text]. This provides a quantum version of an interesting result on unitary representations of locally compact groups given by Lau and Paterson in 1991.

Reproducing kernel Hilbert spaces of CR functions for the Euclidean motion group

We study the geometric structure of the reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the group of Euclidean motions of the plane SE(2). A natural Hilbert norm for functions on the group is constructed that makes the wavelet transform an isometry, but since the considered representations are not square integrable, the resulting Hilbert space will not coincide with L2( SE (2)). The reproducing kernel Hilbert subspace generated by the wavelet transform, for the case of a minimal uncertainty mother wavelet, can be characterized in terms of the complex regularity defined by the natural CR structure of the group. Relations with the Bargmann transform are presented.

A version of the Hörmander–Malliavin theorem in 2-smooth Banach spaces

We consider a stochastic evolution equation in a 2-smooth Banach space with a densely and continuously embedded Hilbert subspace. We prove that under Hörmander's bracket condition, the image measure of the solution law under any finite-rank bounded linear operator is absolutely continuous with respect to the Lebesgue measure. To obtain this result, we apply methods of the Malliavin calculus.

Average entanglement of spin 1 and 1/2 pair

We study the entanglement features of the ground state of a system composed of spin 1 and 1/2 parts. In the light of the ground state degeneracy, the notion of average entanglement is used to measure the entanglement of the Hilbert subspace. The entanglement properties of both a general superposition as well as the mixture of the degenerate ground states are discussed by means of average entanglement and the negativity respectively.

Generalized Spin Bases for Quantum Chemistry and Quantum Information

Symmetry-adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3) or its double group (i.e., spinor group) SU(2). We exploit this fact for generating spin bases of relevance for quantum systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is based on the use of generalized Pauli matrices arising from a polar decomposition of SU(2). This approach leads to a complete solution for the construction of mutually unbiased bases in the case where the dimension d of the considered Hilbert subspace is a prime number. We also give the starting point for studying the case where d is the power of a prime number. A connection of this work to the unitary group U(d) and the Pauli group is briefly underlined.

MODULAR LOCALIZATION AND WIGNER PARTICLES

We propose a framework for the free field construction of algebras of local observables which uses as an input the Bisognano–Wichmann relations and a representation of the Poincaré group on the one-particle Hilbert space. The abstract real Hilbert subspace version of the Tomita–Takesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the Reeh–Schlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and of de Sitter spacetime.

HIGHER-DERIVATIVE TWO-DIMENSIONAL MASSIVE FERMION THEORIES

We consider the canonical quantization of a generalized two-dimensional massive fermion theory containing higher odd-order derivatives. The requirements of Lorentz invariance, hermiticity of the Hamiltonian and absence of tachyon excitations suffice to fix the mass term, which contains a derivative coupling. We show that the basic quantum excitations of a higher-derivative theory of order 2N+1 consist of a physical usual massive fermion, quantized with positive metric, plus 2N unphysical massless fermions, quantized with opposite metrics. The positive-metric Hilbert subspace, which is isomorphic to the space of states of a massive free fermion theory, is selected by a subsidiary-like condition. Employing the standard bosonization scheme, the equivalent boson theory is derived. The results obtained are used as a guideline to discuss the solution of a theory including a current–current interaction.