arXiv subjects
Publications and source records attributed to Stefan Weigert.
A new construction of complex Hadamard matrices of composite order d=pq, with primes p,q, is presented which is based on pairs of mutually unbiased bases containing only product states. For product dimensions d < 100, we illustrate the method by deriving many previously unknown complex Hadamard matrices. We obtain at least 12 new isolated matrices of Butson type, with orders ranging from 9 to 91.
An analytic proof is given which shows that it is impossible to extend any triple of mutually unbiased (MU) product bases in dimension six by a single MU vector. Furthermore, the 16 states obtained by removing two orthogonal states from any MU product triple cannot figure in a (hypothetical) complete set of seven MU bases. These results follow from exploiting the structure of MU product bases in a novel fashion, and they are among the strongest ones obtained for MU bases in dimension six without recourse to computer algebra.
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.
We show that a complete set of seven mutually unbiased bases in dimension six, if it exists, cannot contain more than one product basis.
The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. In dimension six, the required observables only exist if it is possible to identify six mutually unbiased complex 6x6 Hadamard matrices. Prescribing a first Hadamard matrix, we construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and we never find more than two that are mutually unbiased. This result adds considerable support to the conjecture that no seven mutually unbiased bases exist in dimension six.
All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.
It has been conjectured that a complete set of mutually unbiased bases in a space of dimension d exists if and only if there is an affine plane of order d. We introduce affine constellations and compare their existence properties with those of mutually unbiased constellations, mostly in dimension six. The observed discrepancies make a deeper relation between the two existence problems unlikely.
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.
The concept of mutually unbiased bases is studied for N pairs of continuous variables. To find mutually unbiased bases reduces, for specific states related to the Heisenberg-Weyl group, to a problem of symplectic geometry. Given a single pair of continuous variables, three mutually unbiased bases are identified while five such bases are exhibited for two pairs of continuous variables. For N = 2, the golden ratio occurs in the definition of these mutually unbiased bases suggesting the relevance of number theory not only in the finite-dimensional setting.
We study sets of pure states in a Hilbert space of dimension d which are mutually unbiased (MU), that is, the squares of the moduli of their scalar products are equal to zero, one, or 1/d. These sets will be called a MU constellation, and if four MU bases were to exist for d=6, they would give rise to 35 different MU constellations. Using a numerical minimisation procedure, we are able to identify only 18 of them in spite of extensive searches. The missing MU constellations provide the strongest numerical evidence so far that no seven MU bases exist in dimension six.
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.
Simple minimal but informationally complete positive operator-valued measures are constructed out of the expectation-value representation for qudits. Upon suitable modification, the procedure transforms any set of d^2 linearly independent hermitean operators into such an observable. Minor changes in the construction lead to closed-form expressions for informationally complete positive measures in d-dimensional Hilbert spaces.
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.
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.
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.
A new approach to play games quantum mechanically is proposed. We consider two players who perform measurements in an EPR-type setting. The payoff relations are defined as functions of *correlations*, i.e. without reference to classical or quantum mechanics. Classical bi-matrix games are reproduced if the input states are classical and perfectly anti-correlated, that is, for a classical correlation game. However, for a quantum correlation game, with an entangled singlet state as input, qualitatively different solutions are obtained. For example, the Prisoners' Dilemma acquires a Nash equilibrium if the players both apply a mixed strategy. It appears to be conceptually impossible to reproduce the properties of quantum correlation games within the framework of classical games.
The eigenstates of a diagonalizable PT-symmetric Hamiltonian satisfy unconventional completeness and orthonormality relations. These relations reflect the properties of a pair of bi-orthonormal bases associated with non-hermitean diagonalizable operators. In a similar vein, such a dual pair of bases is shown to possess, in the presence of PT symmetry, a Gram matrix of a particular structure: its inverse is obtained by simply swapping the signs of some its matrix elements.
Lueders theorem states that two observables commute if measuring one of them does not disturb the measurement outcomes of the other. We study measurements which are described by continuous positive operator-valued measurements (or POVMs) associated with coherent states on a Lie group. In general, operators turn out to be invariant under the Lueders map if their P- and Q-symbols coincide. For a spin corresponding to SU(2), the identity is shown to be the only operator with this property. For a particle, a countable family of linearly independent operators is identified which are invariant under the Lueders map generated by the coherent states of the Heisenberg-Weyl group, H_3. The Lueders map is also shown to implement the anti-normal ordering of creation and annihilation operators of a particle.