GEOMETRY OF MIXED STATES AND DEGENERACY STRUCTURE OF GEOMETRIC PHASES FOR MULTI-LEVEL QUANTUM SYSTEMS: A UNITARY GROUP APPROACH (original) (raw)
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The manifold of pure quantum states can be regarded as a complex projective space endowed with the unitary-invariant Riemannian geometry of Fubini and Study. According to the principles of geometric quantum mechanics, the detailed physical characteristics of a given quantum system can be represented by specific geometrical features that are selected and preferentially identified in this complex manifold. In particular, any specific feature of projective geometry gives rise to a physically realisable characteristic in quantum mechanics. Here we construct a number of examples of such geometrical features as they arise in the state spaces for spin-1 2 , spin-1, and spin-3 2 systems, and for pairs of spin-1 2 systems. A study is undertaken on the geometry of entangled states, and a natural measure is assigned to the degree of entanglement of a given state for a general multi-particle system. The properties of this measure are analysed in detail for the entangled states of a pair of spin-1 2 particles, thus enabling us to determine the structure of the space of maximally entangled states. With the specification of a quantum Hamiltonian, the resulting Schrödinger trajectories induce an isometry of the Fubini-Study manifold. For a generic quantum evolution, the corresponding Killing trajectory is quasiergodic on a toroidal subspace of the energy surface. When the dynamical trajectory is lifted orthogonally to Hilbert space, it induces a geometric phase shift on the wave function. The uncertainty of an observable in a given state is the length of the gradient vector of the level surface of the expectation of the observable in that state, a fact that allows us to calculate higher order corrections to the Heisenberg relations. A general mixed state is determined by a probability density function on the state space, for which the associated first moment is the density matrix. The advantage of the idea of a general state is in its applicability in various attempts to go beyond the standard quantum theory, some of which admit a natural phase-space characterisation.