3,372 research outputs found
Diagonal unitary entangling gates and contradiagonal quantum states
Nonlocal properties of an ensemble of diagonal random unitary matrices of
order are investigated. The average Schmidt strength of such a bipartite
diagonal quantum gate is shown to scale as , in contrast to the behavior characteristic to random unitary gates. Entangling power of a
diagonal gate is related to the von Neumann entropy of an auxiliary quantum
state , where the square matrix is obtained by
reshaping the vector of diagonal elements of of length into a square
matrix of order . This fact provides a motivation to study the ensemble of
non-hermitian unimodular matrices , with all entries of the same modulus and
random phases and the ensemble of quantum states , such that all their
diagonal entries are equal to . Such a state is contradiagonal with
respect to the computational basis, in sense that among all unitary equivalent
states it maximizes the entropy copied to the environment due to the coarse
graining process. The first four moments of the squared singular values of the
unimodular ensemble are derived, based on which we conjecture a connection to a
recently studied combinatorial object called the "Borel triangle". This allows
us to find exactly the mean von Neumann entropy for random phase density
matrices and the average entanglement for the corresponding ensemble of
bipartite pure states.Comment: 14 pages, 6 figure
Denominator Bounds and Polynomial Solutions for Systems of q-Recurrences over K(t) for Constant K
We consider systems A_\ell(t) y(q^\ell t) + ... + A_0(t) y(t) = b(t) of
higher order q-recurrence equations with rational coefficients. We extend a
method for finding a bound on the maximal power of t in the denominator of
arbitrary rational solutions y(t) as well as a method for bounding the degree
of polynomial solutions from the scalar case to the systems case. The approach
is direct and does not rely on uncoupling or reduction to a first order system.
Unlike in the scalar case this usually requires an initial transformation of
the system.Comment: 8 page
Pseudo-Unitary Operators and Pseudo-Unitary Quantum Dynamics
We consider pseudo-unitary quantum systems and discuss various properties of
pseudo-unitary operators. In particular we prove a characterization theorem for
block-diagonalizable pseudo-unitary operators with finite-dimensional diagonal
blocks. Furthermore, we show that every pseudo-unitary matrix is the
exponential of times a pseudo-Hermitian matrix, and determine the
structure of the Lie groups consisting of pseudo-unitary matrices. In
particular, we present a thorough treatment of pseudo-unitary
matrices and discuss an example of a quantum system with a
pseudo-unitary dynamical group. As other applications of our general results we
give a proof of the spectral theorem for symplectic transformations of
classical mechanics, demonstrate the coincidence of the symplectic group
with the real subgroup of a matrix group that is isomorphic to the
pseudo-unitary group U(n,n), and elaborate on an approach to second
quantization that makes use of the underlying pseudo-unitary dynamical groups.Comment: Revised and expanded version, includes an application to symplectic
transformations and groups, accepted for publication in J. Math. Phy
Applications of the group SU(1,1) for quantum computation and tomography
This paper collects miscellaneous results about the group SU(1,1) that are
helpful in applications in quantum optics. Moreover, we derive two new results,
the first is about the approximability of SU(1,1) elements by a finite set of
elementary gates, and the second is about the regularization of group
identities for tomographic purposes.Comment: 11 pages, no figure
Harmonic equiangular tight frames comprised of regular simplices
An equiangular tight frame (ETF) is a sequence of unit-norm vectors in a
Euclidean space whose coherence achieves equality in the Welch bound, and thus
yields an optimal packing in a projective space. A regular simplex is a simple
type of ETF in which the number of vectors is one more than the dimension of
the underlying space. More sophisticated examples include harmonic ETFs which
equate to difference sets in finite abelian groups. Recently, it was shown that
some harmonic ETFs are comprised of regular simplices. In this paper, we
continue the investigation into these special harmonic ETFs. We begin by
characterizing when the subspaces that are spanned by the ETF's regular
simplices form an equi-isoclinic tight fusion frame (EITFF), which is a type of
optimal packing in a Grassmannian space. We shall see that every difference set
that produces an EITFF in this way also yields a complex circulant conference
matrix. Next, we consider a subclass of these difference sets that can be
factored in terms of a smaller difference set and a relative difference set. It
turns out that these relative difference sets lend themselves to a second,
related and yet distinct, construction of complex circulant conference
matrices. Finally, we provide explicit infinite families of ETFs to which this
theory applies
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