262 research outputs found
The Physical Interpretation of PT-invariant Potentials
A purely imaginary potential can provide a phenomenological description of
creation and absorption of quantum mechanical particles. PT-invariance of such
a potential ensures that the non-unitary phenomena occur in a balanced manner.
In spite of wells and sinks which locally violate the conservation of quantum
probability, there is no net get loss or gain of particles. This, in turn, is
intuitively consistent with real energy eigenvalues.Comment: 4 page
Detecting broken PT-symmetry
A fundamental problem in the theory of PT-invariant quantum systems is to determine whether a given system 'respects' this symmetry or not. If not, the system usually develops non-real eigenvalues. It is shown in this contribution how to algorithmically detect the existence of complex eigenvalues for a given PT-symmetric matrix. The procedure uses classical results from stability theory which qualitatively locate the zeros of real polynomials in the complex plane. The interest and value of the present approach lies in the fact that it avoids diagonalization of the Hamiltonian at hand
Mutually Unbiased Bases and Semi-definite Programming
A complex Hilbert space of dimension six supports at least three but not more
than seven mutually unbiased bases. Two computer-aided analytical methods to
tighten these bounds are reviewed, based on a discretization of parameter space
and on Grobner bases. A third algorithmic approach is presented: the
non-existence of more than three mutually unbiased bases in composite
dimensions can be decided by a global optimization method known as semidefinite
programming. The method is used to confirm that the spectral matrix cannot be
part of a complete set of seven mutually unbiased bases in dimension six.Comment: 11 pages
All Mutually Unbiased Product Bases in Dimension Six
All mutually unbiased bases in dimension six consisting of product states
only are constructed. Several continuous families of pairs and two triples of
mutually unbiased product bases are found to exist but no quadruple. The
exhaustive classification leads to a proof that a complete set of seven
mutually unbiased bases, if it exists, cannot contain a triple of mutually
unbiased product bases.Comment: 32 pages, 3 figures, identical to published versio
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