Session 6
Finite Quantum Systems and Combinatorial Physics
Chairs:
Allan Solomon (Open University, UK)
Apostol Vourdas (University of Bradford, UK)
Titles and Abstracts
Dariusz Chruscinski (Nicolaus Copernicus University, Poland)
Title: Markovianity criteria for quantum evolution
Abstract: We characterize a class of Markovian dynamics using the concept of
divisible dynamical map. Moreover we provide a family of criteria which can
distinguish Markovian and non-Markovian dynamics. These "Markovianity criteria"
are based on a simple observation that Markovian dynamics implies monotonic
behavior of several well known quantities like distinguishability of states, fidelity,
relative entropy and genuine entanglement measures. Finally, we compare the notion
of Markovianity based on divisibility with that based on distinguishability of quantum
states. The talk is illustrated by simple examples of qubit dynamics. It is shown how
the symmetry properties of the corresponding time-dependent generator influences the
dynamics of the quantum system.
Bob Coecke (University of Oxford, UK)
Title: Graphical quantum physics, complementarity and non-locality
Abstract: We describe a graphical language which enables one to reason abstractly
and at the same time very intuitively about quantum phenomena. The language is
universal for full qubit quantum theory and complete for the stabilizer fragment (all
equations are graphically provable). E.g. see “quantum picturalism” (arXiv:0908.1787)
and also “strong complementarity and non-locality in categorical quantum
mechanics” (arXiv:1203.4988). The later makes extensive use of group theory in
classification results, and also identifies non-locality in group theoretic terms.
Nicolae Cotfas (University of Bucharest, Romania)
Title: On a phase space approach to finite quantum systems
Abstract: The quantum observables used in the case of quantum systems with
finite-dimensional Hilbert space are defined either algebraically in terms of an
orthonormal basis and discrete Fourier transformation or by using a continuous
system of coherent states. We present an alternative approach to these important
quantum systems based on the finite frame quantization. Finite systems of coherent
states, usually called finite tight frames, can be defined in a natural way in the case of
finite quantum systems. The quantum observables used in our approach are obtained
by starting from certain classical observables described by functions defined on the
discrete phase space corresponding to the system. They are obtained by using a
finite frame and a Klauder-Berezin-Toeplitz type quantization. The finite frames used
in our approach are defined by starting from the finite Gaussian used by Mehta in its
paper on the eigenvectors of the finite Fourier transform. In the three-dimensional
case, the nine projectors corresponding to our frame are linearly independent and can
be used in order to define an alternative description for qutrits. We present a more
general class of finite Gaussians and compute explicitly the corresponding Fourier
transforms and Wigner functions.
David Finkelstein (Georgia Institute of Technology, USA)
Title: Covariant finite simplicial quantum field theory
Abstract: Fermion field theory is de-contracted to a finite quantum complex. The
statevector space is a higher-order Clifford algebra, which is relativistically covariant
and finite-dimensional, unlike Hilbert space. This extends the Yang 1947
de-contraction of the Poincare'-Heisenberg Lie algebra. It weakens the Heisenberg
uncertainty relation small distances and momenta, as required for binding. The
statistics of fermion pairs is now exactly Palev, which contracts to Bose. The invariant
action of least degree is now cubic in the group generators, a vertex with 6 fermion
lines, contracting to 4. A similar de-contraction of gauge field theory is under study.
Jean Pierre Gazeau (Universite Paris Diderot Paris 7, France)
Title: Generalized binomial distributions (joint work with H. Bergeron, E.M.F.
Curado, and Ligia M.C.S. Rodrigues)
Abstract: In most of the realistic models in Physics one must take correlations into
account; events, which are usually presented as independent, like in a binomial
Bernoulli process, are actually submitted to correlative perturbations. These
perturbations lead to deformations of the mathematical independent laws. In
accordance with this statement, we present a generalization of the binomial
distribution associated with a sequence of positive numbers. It involves asymmetric
and symmetric expressions of probabilities for ("win-loss") sequences of trials. Our
approach is based on generating functions and produces, in the symmetric case,
polynomials of the binomial type. Poisson-like limits, Leibniz triangle rules and
related entropy(ies) are considered. Our generalizations are illustrated by various
analytical and numerical examples.
[1] E. M. F. Curado, J. P. Gazeau, and L. M. C. S. Rodrigues, Non-linear coherent
states for optimizing Quantum Information, Phys. Scr.82 038108--1-9 (2010).
[2] E.M.F. Curado, J.P. Gazeau, L. M. C. S. Rodrigues, On a Generalization of the
Binomial Distribution and Its Poisson-like Limit, J. Stat. Phys. 146 264-280 (2012);
arXiv:1105.3889 math-ph
[3] H. Bergeron, E.M.F. Curado, J.P. Gazeau, L. M. C. S. Rodrigues, Generating
functions for generalized binomial distributions, submitted (2012); arXiv:1203.3936
math-ph
Hubert de Guise (Lakehead University, Canada)
Title: Different angles on phase operators
Abstract: I will contrast two possible approaches – based on polar decomposition and
on generalized Fourier transforms - to the construction of phase operators in systems
with SU(n) symmetries, focusing largely on the SU(3) case. Neither approach is fully
satisfactory, although both yield interesting results in the limit of large
representations.
Andrzej Horzela (H. Niewodniczanski Institute of Nuclear Physics, Poland)
Title: A measure free approach to coherent states
Abstract: In the still running process of generalizing coherent states the situation
when the measure - customarily incorporated in their definition - is indeterminate
becomes unavoidable. A more dramatic situation may happen if there is no measure
which makes the reproducing kernel Hilbert space, involved in the construction of
coherent states, isometrically included in an ${\cal L}^{2}$ space. Therefore a need
appears to redefine coherent states making their definition measure free. Starting out
with the reproducing kernel property we ensure the basic feature of coherent states resolution of the identity - to be maintaned. The only investment in the whole
undertaking is an orthonormal sequence $(\varPhi_{n}(x))_{n=0}^{d}$ satisfying the
condition $\sum_{n}|\varPhi_{n}(x)|^2 < \infty$ for $x\in X$. The rest, including the
aforesaid resolution of the identity, is a consequence of our choice. The approach is
supported by examples which make clear the circumstances under which the sequence
$(\varPhi_{n})_{n=0}^{d}$ appears - complex Hermite polynomials are one of them.
This is a joint work with Franciszek Hugon Szafraniec.
Richard Kerner (University Pierre et Marie Curie (Paris-VI), France)
Title: Z2, Z3 and the Lorentz Group: from discrete to continuous symmetries.
Abstract: We argue that the discrete symmetries of quantum physics combined with
the superposition principle lead to the continuous symmetries in the macroscopic
world of classical physics. In particular, we show how to derive the Lorentz group
from discrete symmetries imposed on fermion and quarks states. A tentative
generalization of Dirac equation for the Z3-graded Pauli principle is also introduced.
Maurice Kibler (Lyon University and CNRS/IN2P3, France)
Title: Construction of phase states and coherent states for generalized WeylHeisenberg algebras
Abstract: This work presents two facets.
The first one deals with the construction of phase operators and (vector) phase states
for two one-parameter generalized Weyl-Heisenberg algebras Aϰ(1) and Aϰ(2). The
Aϰ(1) algebra covers the su(1,1), su(2) and h(4) cases while the Aϰ(2) algebra covers
the su(2,1), su(3) and h(4)xh(4) cases according to as the parameter ϰ is > 0, < 0 and 0,
respectively (h(4) is the ordinary Weyl-Heisenberg algebra). Finite- and
infinite-dimensional representations of these algebras are derived for ϰ < 0 and ϰ > 0
or = 0, respectively. Phase operators associated with Aϰ(1) and Aϰ(2) are defined and
temporally stable (vector) phase states are constructed from the eigenstates of the
phase operators.
The second facet of this work concerns the construction of coherent states à la
Perelomov and à la Barut-Girardello for a polynomial Weyl-Heisenberg algebra. This
algebra, noted A{ϰ}, depends on r real parameters and is an extension of the
one-parameter algebra Aϰ(1). For r arbitrary, the generalized Weyl-Heisenberg algebra
admits finite- or infinite-dimensional representations depending on the values of the
parameters. For finite-dimensional representations of A{ϰ} and A{ϰ},s, where A{ϰ},s is a
truncation of order s of A{ϰ} in the sense of Pegg and Barnett, a connection is
established with k-fermionic algebras (or quon algebras). Coherent states of the
Perelomov type are derived in finite and infinite dimensions through a
Fock-Bargmann approach based on the use of complex (or bosonic) variables. The
same approach is applied for deriving coherent states of the Barut-Girardello type in
infinite dimension. In contrast, the construction of coherent states à la
Barut-Girardello in finite dimension can be achieved solely at the price to replace
complex variables by generalized Grassmann (or k-fermionic) variables.
As applications, (i) we discuss a relation between quantized phase states and a
quadratic discrete Fourier transform and show how to use these states for constructing
mutually unbiased bases useful in quantum information and (ii) some preliminary
developments are given for the study of Bargmann functions associated with some of
the coherent states obtained in this work.
Work in collaboration with M. Daoud. For references, see: Maurice R. Kibler and
Mohammed Daoud, “Generalized Coherent States for Polynomial Weyl-Heisenberg
Algebras”, posted on arXiv arXiv:1201.1811v1 [quant-ph].
Miroslav Korbelar (Masaryk University, Czech Republic)
Title: Symmetries of finite Heisenberg groups and their applications (joint work with
J. Tolar)
Abstract: Finite Heisenberg groups are basic operator structures of finite-dimensional
quantum mechanics. Their symmetries are described for single as well as arbitrary
composite systems. Of their numerous applications they can be applied for an
alternative proof of existence of mutually unbiased bases in prime power dimensions.
Karol Andrzej Penson (University Pierre et Marie Curie (Paris-VI), France)
Title: Fractional Fuss-Catalan Numbers and their Moments Representation
Abstract: We consider fractional Fuss-Catalan numbers C(k,l,n), n=0,1,… . for
fractional values of parameter in the form k/l , with k end l positive integers. We
furnish exact solution of the Hausdorff moment problem, with C(k,l,n) as nth moment
of a positive function W(k,l,x) which can be expressed explicitly as a finite sum of
hypergeometric functions. We cite instances when W(k,l,x) are expressible via
elementary functions. The functions W(k,l,x) are studied analytically and graphically.
These results are discussed in the perspective of the free probabilities, where the
functions W(k,l,x) are shown to be equal to free fractional powers of the standard
Marchenko-Pastur distribution. (Joint work with W. Mlotkowski (Wroclaw) and K.
Zyczkowski (Cracow) )
Michel Planat (Institut FEMTO-ST/CNRS, France)
Title: Quantum states arising from the Pauli groups: symmetries and paradoxes
Abstract: We investigate single qudit and multiple qudit Pauli groups [1] and the
quantum states/rays arising from their maximal bases. Remarkably, for the multiple
qubit Pauli group Pd, d = 2n, the real rays are carried by a Barnes-Wall lattice Bn. We
focus on the smallest subsets of rays allowing a proof of the Bell-Kochen-Specker
theorem (BKS). BKS theorem rules out realistic non-contextual theories by resorting
to impossible assignments of rays among a selected set of maximal orthogonal bases.
We investigate the geometrical structure of small BKS-proofs v − l involving v rays
and l 2n-dimensional bases of n-qubits (1 < n < 5) [2]. Specifically, we look at the
parity proofs 18 − 9 with two qubits (A. Cabello, 1996 [3]), the parity proofs 36–11
with three qubits (M. Kernaghan & A. Peres, 1995 [4]) and a newly discovered
nonparity proof 80-21 with four qubits (that improves a work at P. K Aravind’s group
in 2008 [5]). One finds universal signatures of the distances among the bases, that
carry various symmetries in their graphs.
Bibliography
[1] Planat M. 2011 Pauli graphs when the Hilbert space dimension contains a square:
why the
Dedekind psi function? J. Phys. A: Math. Theor. 44, 045301 (16pp).
[2] Planat M 2012 On small proofs of Bell-Kochen-Specker theorem for two, three
and four qubits, in preparation.
[3] Cabello A., Estebaranz J. M. and Alcaine G. G. 1996 Bell-Kochen-Specker
theorem: a proof with 18 vectors Phys. Lett. A 212, 183-187.
[4] Kernaghan M. and Peres A. 1965 Kochen-Specker theorem for 8-dimensional
space Phys. Lett. A198, 1.
[5] Harvey C. and Chryssanthacopoulos 2008 BKS theorem and Bell’s theorem in 16
dimensions,Worcester Polytechnic Institute, project number: PH-PKA-JC08 (63pp).
Luis Lorenzo Sanchez Soto (Universidad Complutense de Madrid, Spain)
Title: A simple toolbox for picturing qubits in phase space
Abstract: I will explain the basic techniques to deal with a system of n qubits in
phase space, which turns out to be a discrete 2n × 2n phase space. The phase space is
based on the finite field having 2n elements, and its geometric structure leads
naturally to the construction of a complete set of 2n + 1 mutually conjugate bases.
Sophie Schirmer (Swansea University, UK)
Title: Controlling entanglement dissipation
Abstract: We consider the problem of entanglement dissipation and how to
counteract it using control. If the environment is Markovian it will be shown that
open-loop control alone cannot counteract the environmental decoherence, and
feedback control is required to achieve stabilization of entangled states (in most cases).
Conversely, we also illustrate how environmental effects can be harnessed to create
entanglement via control in different settings.
Joris Van der Jeugt (Ghent University, Belgium)
Title: Finite oscillator models described by the Lie superalgebra sl(2|1)
Abstract: We investigate new models for a finite quantum oscillator based upon the
Lie superalgebra sl(2|1), where the position and momentum operators are represented
as odd elements of the algebra. We discuss interesting properties of the spectrum of
the position operator, and of the (discrete) position wave functions.
Stefan Weigert (University of York, UK)
Title: Mutually unbiased bases in composite dimensions
Abstract: Two orthonormal bases of a d-dimensional Hilbert space are called
mutually unbiased (MU) if the d^2 transition probabilities from any state of one basis
to any state of the other basis coincide (cf. [1] for a recent review). This property
expresses the notion of complementarity for discrete variables, by analogy with the
eigenbases of the canonical position and momentum operators of a quantum particle.
It is always possible to construct three MU bases in the state space of a finite quantum
system with d orthogonal states. If the dimension d is a prime number or a prime
power, even (d+1) pairwise MU bases can be found. Pairs as well as larger sets of MU
bases have a number of useful applications. For example, they provide optimal means
to reconstruct unknown quantum states, and they allow one to establish secret keys for
cryptographic ends. A quantum system consisting of two or more unequal parts lives
in a state space of `composite' dimension, d=6,10,12,... The allowed maximum of
(d+1) MU bases has never been found in these case, in spite of considerable efforts.
In this presentation, I will summarize what is known about MU bases for quantum
systems with six levels. Using numerical, computer-algebraic and analytic methods,
various partial results have been obtained all of which are compatible with the
conjecture that no more than three MU bases exist. I will emphasise the case of MU
bases consisting of product states only for which strong analytic results have been
obtained recently [2,3].
From an abstract point of view, the existence problem of MU bases in composite
dimensions is interesting because it represents another instance of the close ties
between quantum mechanics and number theory. On a technical level, this relation
manifests itself in the absence of certain group-theoretical concepts in composite
dimensions which can be taken for granted in prime and prime power dimensions.
[1] T. Durt, B-G. Englert, I. Bengtsson and K. Zyczkowski: On Mutually Unbiased
Bases; Int. J. Quant. Inf. 8 (2010) 535
[2] D. McNulty and S. Weigert: The limited role of mutually unbiases product bases
in dimension 6; J. Phys. A 45 (2012) 102001
[3] D. McNulty and S. Weigert: On the Impossibility to Extend Triples of Mutually
Unbiased Product Bases in Dimension Six; J. Quant. Inf. Comp. (in print),
arXiv:1203.6887
Kurt Bernardo Wolf (Universidad Nacional Autonoma de Mexico, Mexico)
Title: From finite unitarity to classical canonicity
Abstract: Canonical transformations in quantum mechanics produce unitary
transformations of the Hilbert space of wavefunctions that respect the basic
Heisenberg commutation relations. In classical systems on the other hand, canonical
transformations must preserve the basic Poisson brackets that characterize its
Hamiltonian structure. In the D=1-dimensional case, the quadratic classical, quantum
and discrete systems can be covariant under SU(2) or SU(1,1) (harmonic or repulsive
oscillator), or ISO(2)} (free system). All these algebras possess a compact generator
whose discrete spectrum we identify as the discrete position observable. Since
measuring apparata can only register a finite number of values, we should note that
N-point finite signals (states) in discrete systems can be unitarily transformed by the
much larger group U(N). Here we examine how, as N grows to infinity, U(N)
becomes the group of all classical canonical transformations -linear (paraxial) and
nonlinear (aberrations) - of phase space.
Joshua Zak (Technion, Israel)
Title: Inversion operators in finite phase plane
Abstract：There has been much interest in recent years in quantum mechanics in
finite phase plane. In this talk, we use inversion operators on a lattice in finite phase
plane for building a complete set of mutually orthogonal Hermitian operators. The
lattice is given by tc in the x direction and by
in the p direction; c is an
arbitrary length constant and M is the dimension of the space; s and t assume the
values from 0 to M-1. For M odd the M2 inversion operators on the lattice form a
complete set of mutually orthogonal operators. For M even we assign a sum of 4
inversion operators (a quartet) to each site of the lattice (t,s). We prove that these
quartets for t,s=0,1,..., M-1 form a mutually orthogonal set of M2 Hermitian operators.
Karol Zyczkowski (Jagiellonian University, Poland)
Title: Almost Hadamard matrices and incomplete block designes (in collaboration
with Teodor Banica and Ion Nechita)
Abstract: We analyze "almost Hadamard matrices"- orthogonal matrices of a given
order N with modulus of all elements distributed as uniform as possible. Formally an
Almost Hadamard matrix is an orthogonal matrix, for which the 1-norm on O(N)
achieves a local maximum. Our study includes a discussion the two-entry case
closely linked to 'balanced incomplete block designs' (BIBD). A another
generalization of real Hadamard matrices is obtained if one considers complex unitary
matrices with entries of the same modulus. A brief review of problems related to
complex Hadamard matrices and Mutually Unbiased Bases (MUB) is also presented.