1,372 research outputs found
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
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 ‘‘deterministic randomness’’ 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 ×N) 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
How to determine a quantum state by measurements: The Pauli problem for a particle with arbitrary potential
The problem of reconstructing a pure quantum state ¿¿> from measurable quantities is considered for a particle moving in a one-dimensional potential V(x). Suppose that the position probability distribution ¿¿(x,t)¿2 has been measured at time t, and let it have M nodes. It is shown that after measuring the time evolved distribution at a short-time interval ¿t later, ¿¿(x,t+¿t)¿2, the set of wave functions compatible with these distributions is given by a smooth manifold M in Hilbert space. The manifold M is isomorphic to an M-dimensional torus, TM. Finally, M additional expectation values of appropriately chosen nonlocal operators fix the quantum state uniquely. The method used here is the analog of an approach that has been applied successfully to the corresponding problem for a spin system
Reconstruction of the spin state
System of 1/2 spin particles is observed repeatedly using Stern-Gerlach
apparatuses with rotated orientations. Synthesis of such non-commuting
observables is analyzed using maximum likelihood estimation as an example of
quantum state reconstruction. Repeated incompatible observations represent a
new generalized measurement. This idealized scheme will serve for analysis of
future experiments in neutron and quantum optics.Comment: 4 pages, 1 figur
Discrete Moyal-type representations for a spin
In Moyal’s formulation of quantum mechanics, a quantum spin s is described in terms of continuous symbols, i.e., by smooth functions on a two-dimensional sphere. Such prescriptions to associate operators with Wigner functions, P or Q symbols, are conveniently expressed in terms of operator kernels satisfying the Stratonovich-Weyl postulates. In analogy to this approach, a discrete Moyal formalism is defined on the basis of a modified set of postulates. It is shown that appropriately modified postulates single out a well-defined set of kernels that give rise to discrete symbols. Now operators are represented by functions taking values on (2s+1)2 points of the sphere. The discrete symbols contain no redundant information, contrary to the continuous ones. The properties of the resulting discrete Moyal formalism for a quantum spin are worked out in detail and compared to the continuous formalism
Resolution in Linguistic Propositional Logic based on Linear Symmetrical Hedge Algebra
The paper introduces a propositional linguistic logic that serves as the
basis for automated uncertain reasoning with linguistic information. First, we
build a linguistic logic system with truth value domain based on a linear
symmetrical hedge algebra. Then, we consider G\"{o}del's t-norm and t-conorm to
define the logical connectives for our logic. Next, we present a resolution
inference rule, in which two clauses having contradictory linguistic truth
values can be resolved. We also give the concept of reliability in order to
capture the approximative nature of the resolution inference rule. Finally, we
propose a resolution procedure with the maximal reliability.Comment: KSE 2013 conferenc
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