THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

Minimum Time for the Evolution to a Nonorthogonal Quantum State and Upper Bound of the Geometric Efficiency of Quantum Evolutions

We present a simple proof of the fact that the minimum time TAB for quantum evolution between two arbitrary states A and B equals TAB=ℏcos−1A|B/ΔE with ΔE being the constant energy uncertainty of the system. This proof is performed in the absence of any geometrical arguments. Then, being in the geometric framework of quantum evolutions based upon the geometry of the projective Hilbert space, we discuss the roles played by either minimum-time or maximum-energy uncertainty concepts in defining a geometric efficiency measure ε of quantum evolutions between two arbitrary quantum states. Finally, we provide a quantitative justification of the validity of the inequality ε≤1 even when the system only passes through nonorthogonal quantum states.

DETECTION OF THE SPATIAL CURVATURE EFFECTS THROUGH PHYSICAL PHENOMENA: THE NONLINEAR COHERENT STATES APPROACH

In this paper, by using the nonlinear coherent states approach, we find a relation between the geometric structure of the physical space and the geometry of the corresponding projective Hilbert space. To illustrate the approach, we explore the quantum transition probability and the geometric phase in the curved space.

GAUGE THEORY AND TWO LEVEL SYSTEMS

We consider the time evolution of a two level system (a two level atom or a qubit) and show that it is characterized by a local (in time) gauge invariant evolution equation. The covariant derivative operator is constructed and related to the free energy. We show that the gauge invariant characterization of the time evolution of the two level system is analogous to the birefringence phenomenon in optics. The relation with Berry-like and Anandan–Aharonov phase is pointed out. Finally, we discuss entropy, environment effects and the distance in projective Hilbert space between two level states in their evolution.

On some hydrodynamical aspects of quantum mechanics

In this note we first set up an analogy between spin and vorticity of a perfect 2d-fluid flow, based on the complex polynomial (i.e. Borel-Weil) realization of the irreducible unitary representations of SU(2), and looking at the Madelung-Bohm velocity attached to the ensuing spin wave functions. We also show that, in the framework of finite dimensional geometric quantum mechanics, the Schrödinger velocity field on projective Hilbert space is divergence-free (being Killing with respect to the Fubini-Study metric) and fulfils the stationary Euler equation, with pressure proportional to the Hamiltonian uncertainty (squared). We explicitly determine the critical points of the pressure of this “Schrödinger fluid”, together with its vorticity, which turns out to depend on the spacings of the energy levels. These results follow from hydrodynamical properties of Killing vector fields valid in any (finite dimensional) Riemannian manifold, of possible independent interest.

Effect of Entanglement on Geometric Phase for Multi-Qubit States

When a multi-qubit state evolves under local unitaries it may obtain a geometric phase, a feature dependent on the geometry of the state projective Hilbert space. A correction term to this geometric phase, in addition to the local subsystem phases, may appear from correlations between the subsystems. We find that this correction term can be characterized completely either by the entanglement or by the classical correlations for several classes of entangled state. States belonging to the former set are W states and their mixtures, while members of the latter set are cluster states, GHZ states and two classes of bound entangled state. We probe the structures of these states more finely using local invariants and suggest that the cause of the entanglement correction is a recently introduced gauge field-like SL(2,ℂ)-invariant called twist.