Corrigendum: Projective Hilbert space structures at exceptional points (2007 J. Phys. A: Math. Theor. 40 8815)

Abstract With an innovative idea of acceptability and usefulness of the non-Hermitian representations of Hamiltonians for the description of unitary quantum systems (dating back to the Dyson’s papers), the community of quantum physicists was offered a new and powerful tool for the building of models of quantum phase transitions. In this paper the mechanism of such transitions is discussed from the point of view of mathematics. The emergence of the direct access to the instant of transition (i.e., to the Kato’s exceptional point) is attributed to the underlying split of several roles played by the traditional single Hilbert space of states ℒ into a triplet (viz., in our notation, spaces K and ℋ besides the conventional ℒ ). Although this explains the abrupt, quantum-catastrophic nature of the change of phase (i.e., the loss of observability) caused by an infinitesimal change of parameters, the explicit description of the unitarity-preserving corridors of access to the phenomenologically relevant exceptional points remained unclear. In the paper some of the recent results in this direction are summarized and critically reviewed.

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

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887812500090 ◽

2012 ◽

Vol 09 (01) ◽

pp. 1250009 ◽

Cited By ~ 6

Author(s):

A. MAHDIFAR ◽

R. ROKNIZADEH ◽

M. H. NADERI

Keyword(s):

Hilbert Space ◽

Geometric Phase ◽

Coherent States ◽

Transition Probability ◽

Physical Space ◽

Spatial Curvature ◽

Curved Space ◽

Curvature Effects ◽

Projective Hilbert Space ◽

Physical Phenomena

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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Hausdorff–Besicovitch measure for random fractals of Chung's type

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004102005984 ◽

2002 ◽

Vol 133 (3) ◽

pp. 487-513

Author(s):

ALAIN LUCAS

Keyword(s):

Hilbert Space ◽

Unit Ball ◽

Wiener Process ◽

Bounded Variation ◽

Reproducing Kernel ◽

Reproducing Kernel Hilbert Space ◽

Exceptional Points ◽

Random Fractals ◽

Random Fractal ◽

Hilbert Norm

Let {W(t):t [ges ] 0} denote a Wiener process, and set [Sscr ] for the unit ball of the reproducing kernel Hilbert space pertaining to the restriction of W on [0,1], with Hilbert norm [mid ] · [mid ]H. Gorn and Lifshits [8] have shown that, whenever f ∈ [Sscr ] fulfills [mid ] f [mid ]H = 1 and has Lebesgue derivative of bounded variation, the rate of clustering of (2h log(1/h))−½(W(t + h·) − W(t)) to f is of the order O((log(1/h))−2/3. In this paper, we show that the set of exceptional points in [0,1] where this rate is reached constitutes a random fractal whose Hausdorff–Besicovitch measure is evaluated.

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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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Supersymmetry and Exceptional Points

Symmetry ◽

10.3390/sym12060892 ◽

2020 ◽

Vol 12 (6) ◽

pp. 892 ◽

Cited By ~ 1

Author(s):

Miloslav Znojil

Keyword(s):

Hilbert Space ◽

Quantum Theory ◽

Energy Levels ◽

Harmonic Oscillators ◽

Exceptional Point ◽

Basic Principles ◽

Exceptional Points ◽

Super Symmetry ◽

Physical Literature

The phenomenon of degeneracy of energy levels is often attributed either to an underlying (super)symmetry (SUSY), or to the presence of a Kato exceptional point (EP). In our paper a conceptual bridge between the two notions is proposed to be provided by the recent upgrade of the basic principles of quantum theory called, equivalently, PT − symmetric or three-Hilbert-space (3HS) or quasi-Hermitian formulation in the current physical literature. Although the original purpose of the 3HS approach laid in the mere simplification of technicalities, it is shown here to serve also as a natural theoretical link between the apparently remote concepts of EPs and SUSY. An explicit illustration of their close mutual interplay is provided by the description of infinitely many supersymmetric, mutually non-equivalent and EP-separated regularized spiked harmonic oscillators.

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Algebra and Hilbert space structures induced by quantum probes

Annals of Physics ◽

10.1016/j.aop.2019.168046 ◽

2020 ◽

Vol 412 ◽

pp. 168046

Author(s):

Go Kato ◽

Masaki Owari ◽

Koji Maruyama

Keyword(s):

Hilbert Space ◽

Space Structures

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Geometry of projective Hilbert space

Physical Review A ◽

10.1103/physreva.46.7292 ◽

1992 ◽

Vol 46 (11) ◽

pp. 7292-7294 ◽

Cited By ~ 6

Author(s):

A. N. Grigorenko

Keyword(s):

Hilbert Space ◽

Projective Hilbert 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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