Geometry of projective Hilbert space

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.

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

Quantum Reports ◽

10.3390/quantum3030029 ◽

2021 ◽

Vol 3 (3) ◽

pp. 444-457

Author(s):

Carlo Cafaro ◽

Paul M. Alsing

Keyword(s):

Hilbert Space ◽

Maximum Energy ◽

Quantum States ◽

Minimum Time ◽

Quantum Evolution ◽

Efficiency Measure ◽

Geometric Framework ◽

Constant Energy ◽

Projective Hilbert Space ◽

Arbitrary Quantum

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.

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The Projective Hilbert Space as a Classical Phase Space for Nonrelativistic Quantum Dynamics

International Journal of Theoretical Physics ◽

10.1007/s10773-005-8982-2 ◽

2005 ◽

Vol 44 (11) ◽

pp. 2041-2049 ◽

Cited By ~ 4

Author(s):

Igor Bjelaković ◽

Werner Stulpe

Keyword(s):

Hilbert Space ◽

Phase Space ◽

Quantum Dynamics ◽

Nonrelativistic Quantum ◽

Projective Hilbert Space ◽

Classical Phase Space

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Effect of Entanglement on Geometric Phase for Multi-Qubit States

Open Systems & Information Dynamics ◽

10.1142/s1230161209000244 ◽

2009 ◽

Vol 16 (02n03) ◽

pp. 305-323 ◽

Cited By ~ 2

Author(s):

Mark S. Williamson ◽

Vlatko Vedral

Keyword(s):

Hilbert Space ◽

Geometric Phase ◽

Correction Term ◽

Entangled State ◽

Cluster States ◽

W States ◽

Local Invariants ◽

Projective Hilbert Space ◽

Classical Correlations ◽

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.

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