1,747 research outputs found
An Algorithmic Test for Diagonalizability of Finite-Dimensional PT-Invariant Systems
A non-Hermitean operator does not necessarily have a complete set of
eigenstates, contrary to a Hermitean one. An algorithm is presented which
allows one to decide whether the eigenstates of a given PT-invariant operator
on a finite-dimensional space are complete or not. In other words, the
algorithm checks whether a given PT-symmetric matrix is diagonalizable. The
procedure neither requires to calculate any single eigenvalue nor any numerical
approximation.Comment: 13 pages, 1 figur
A quantum search for zeros of polynomials
A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved
PT-symmetry and its spontaneous breakdown explained by anti-linearity
The impact of an anti-unitary symmetry on the spectrum of non-Hermitian operators is studied. Wigner's normal form of an anti-unitary operator accounts for the spectral properties of non-Hermitian, PE-symmetric Harniltonians. The occurrence of either single real or complex conjugate pairs of eigenvalues follows from this theory. The corresponding energy eigenstates span either one- or two-dimensional irreducible representations of the symmetry PE. In this framework, the concept of a spontaneously broken PE-symmetry is not needed
Integrability and level crossing manifolds in a quantum Hamiltonian system
We consider a two-spin model, represented classically by a nonlinear
autonomous Hamiltonian system with two degrees of freedom and a nontrivial
integrability condition, and quantum mechanically by a real symmetric
Hamiltonian matrix with blocks of dimensionalities K=l(l+1)/2, l=1,2,... In the
six-dimensional (6D) parameter space of this model, classical integrability is
satisfied on a 5D hypersurface, and level crossings occur on 4D manifolds that
are completely embedded in the integrability hypersurface except for some
lower-D sub-manifolds. Under mild assumptions, the classical integrability
condition can be reconstructed from a purely quantum mechanical study of level
degeneracies in finite-dimensional invariant blocks of the Hamiltonian matrix.
Our conclusions are based on rigorous results for K=3 and on numerical results
for K=6,10.Comment: 8 pages, 3 figure
How to Test for Diagonalizability: The Discretized PT-Invariant Square-Well Potential
Given a non-hermitean matrix M, the structure of its minimal polynomial
encodes whether M is diagonalizable or not. This note will explain how to
determine the minimal polynomial of a matrix without going through its
characteristic polynomial. The approach is applied to a quantum mechanical
particle moving in a square well under the influence of a piece-wise constant
PT-symmetric potential. Upon discretizing the configuration space, the system
is decribed by a matrix of dimension three. It turns out not to be
diagonalizable for a critical strength of the interaction, also indicated by
the transition of two real into a pair of complex energy eigenvalues. The
systems develops a three-fold degenerate eigenvalue, and two of the three
eigenfunctions disappear at this exceptional point, giving a difference between
the algebraic and geometric multiplicity of the eigenvalue equal to two.Comment: 5 page
Signatures of quantum integrability and nonintegrability in the spectral properties of finite Hamiltonian matrices
For a two-spin model which is (classically) integrable on a five-dimensional
hypersurface in six-dimensional parameter space and for which level
degeneracies occur exclusively (with one known exception) on four-dimensional
manifolds embedded in the integrability hypersurface, we investigate the
relations between symmetry, integrability, and the assignment of quantum
numbers to eigenstates. We calculate quantum invariants in the form of
expectation values for selected operators and monitor their dependence on the
Hamiltonian parameters along loops within, without, and across the
integrability hypersurface in parameter space. We find clear-cut signatures of
integrability and nonintegrability in the observed traces of quantum invariants
evaluated in finite-dimensional invariant Hilbert subspaces, The results
support the notion that quantum integrability depends on the existence of
action operators as constituent elements of the Hamiltonian.Comment: 11 page
Chaos and quantum-nondemolition measurements
The problem of chaotic behavior in quantum mechanics is investigated against the background of the theory of quantum-nondemolition (QND) measurements. The analysis is based on two relevant features: The outcomes of a sequence of QND measurements are unambiguously predictable, and these measurements actually can be performed on one single system without perturbing its time evolution. Consequently, QND measurements represent an appropriate framework to analyze the conditions for the occurrence of ââdeterministic randomnessââ in quantum systems. The general arguments are illustrated by a discussion of a quantum system with a time evolution that possesses nonvanishing algorithmic complexity
Quantum diagonalization of Hermitean matrices
To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource: Hermitean (N ĂN) matrices can be diagonalized, in principle, by performing appropriate quantum mechanical measurements. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As the underlying working principle is the `collapse of the wavefunction' associated with a measurement, the procedure is neither a digital nor an analogue calculation - it defines thus a new example of a quantum mechanical method of computation
Minimal Informationally Complete Measurements for Pure States
We consider measurements, described by a positive-operator-valued measure
(POVM), whose outcome probabilities determine an arbitrary pure state of a
D-dimensional quantum system. We call such a measurement a pure-state
informationally complete (PSI-complete) POVM. We show that a measurement with
2D-1 outcomes cannot be PSI-complete, and then we construct a POVM with 2D
outcomes that suffices, thus showing that a minimal PSI-complete POVM has 2D
outcomes. We also consider PSI-complete POVMs that have only rank-one POVM
elements and construct an example with 3D-2 outcomes, which is a generalization
of the tetrahedral measurement for a qubit. The question of the minimal number
of elements in a rank-one PSI-complete POVM is left open.Comment: 2 figures, submitted for the Asher Peres festschrif
Small denominators, frequency operators, and Lie transforms for nearly integrable quantum spin systems
Based on the previously proposed notions of action operators and of quantum integrability, frequency operators are introduced in a fully quantum-mechanical setting. They are conceptually useful because another formulation can be given to unitary perturbation theory. When worked out for quantum spin systems, this variant is found to be formally equivalent to canonical perturbation theory applied to nearly integrable systems consisting of classical spins. In particular, it becomes possible to locate the quantum-mechanical operator-valued equivalent of the frequency denominators that may cause divergence of the classical perturbation series. The results that are established here link the concept of quantum-mechanical integrability to a technical question, namely, the behavior of specific perturbation series
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