arXiv subjects

Ioannis Tsohantjis

Publications and source records attributed to Ioannis Tsohantjis.

8 recordsLinked to original sources

arXiv2025

Construction of Metaplectic Representations of $SL_2(\mathbb{Z}_{2^n})$ and Twisted Magnetic Translations

Unitary metaplectic representations of the group $SL_2(\mathbb{Z}_{2^n})$ are necessary to describe the evolution of $2^n$-dimensional quantum systems, such as systems involving $n$ qubits. It is shown that in order for the metaplectic property to be fulfilled, an increase in the dimensionality of the involved $n$-qubit Hilbert spaces, from $2^n$ to $2^{2n}$, is necessary. Thus we construct the general matrix form of such representations based on the magnetic translations of the diagonal subgroup $HW_{2^n} \otimes HW_{2^n}$. Comparisson with other approaches on this problem of the literature are discussed.

quant-ph↗

arXiv2009

Anyonic Quantum Walks

The one dimensional quantum walk of anyonic systems is presented. The anyonic walker performs braiding operations with stationary anyons of the same type ordered canonically on the line of the walk. Abelian as well as non-Abelian anyons are studied and it is shown that they have very different properties. Abelian anyonic walks demonstrate the expected quadratic quantum speedup. Non-Abelian anyonic walks are much more subtle. The exponential increase of the system's Hilbert space and the particular statistical evolution of non-Abelian anyons give a variety of new behaviors. The position distribution of the walker is related to Jones polynomials, topological invariants of the links created by the anyonic world-lines during the walk. Several examples such as the SU(2) level k and the quantum double models are considered that provide insight to the rich diffusion properties of anyons.

quant-ph↗

arXiv2005

Region Operators of Wigner Function: Transformations, Realizations and Bounds

An integral of the Wigner function of a wavefunction |psi >, over some region S in classical phase space is identified as a (quasi) probability measure (QPM) of S, and it can be expressed by the |psi > average of an operator referred to as the region operator (RO). Transformation theory is developed which provides the RO for various phase space regions such as point, line, segment, disk and rectangle, and where all those ROs are shown to be interconnected by completely positive trace increasing maps. The latter are realized by means of unitary operators in Fock space extended by 2D vector spaces, physically identified with finite dimensional systems. Bounds on QPMs for regions obtained by tiling with discs and rectangles are obtained by means of majorization theory.

quant-ph↗

arXiv2004

Pseudo Memory Effects, Majorization and Entropy in Quantum Random Walks

A quantum random walk on the integers exhibits pseudo memory effects, in that its probability distribution after N steps is determined by reshuffling the first N distributions that arise in a classical random walk with the same initial distribution. In a classical walk, entropy increase can be regarded as a consequence of the majorization ordering of successive distributions. The Lorenz curves of successive distributions for a symmetric quantum walk reveal no majorization ordering in general. Nevertheless, entropy can increase, and computer experiments show that it does so on average. Varying the stages at which the quantum coin system is traced out leads to new quantum walks, including a symmetric walk for which majorization ordering is valid but the spreading rate exceeds that of the usual symmetric quantum walk.

quant-ph↗

arXiv2002

Random Walk and Diffusion on a Smash Line Algebra

Working withing the framework of Hopf algebras, a random walk and the associated diffusion equation are constructed on a space that is algebraically described as the merging of the real line algebra with the anyonic line algebra. Technically this merged structure is a smash algebra, namely an algebra resulting by a braided tensoring of real with anyonic line algebras. The motivation of introducing the smashing results from the necessity of having non commuting increments in the random walk. Based on the observable-state duality provided by the underlying Hopf structure, the construction is cast into two dual forms: one using functionals determined by density probability functions and the other using the associated Markov transition operator. The ensuing diffusion equation is shown to possess triangular matrix realization. The study is completed by the incorporation of Hamiltonian dynamics in the above random walk model, and by the construction of the dynamical equation obeyed by statistical moments of the problem for generic entangled density functions.

quant-ph↗

arXiv2000

Brownian motion on a smash line

Brownian motion on a smash line algebra (a smash or braided version of the algebra resulting by tensoring the real line and the generalized paragrassmann line algebras), is constructed by means of its Hopf algebraic structure. Further, statistical moments, non stationary generalizations and its diffusion limit are also studied. The ensuing diffusion equation posseses triangular matrix realizations.

quant-ph↗

arXiv1996

Twisted Quantum Affine Superalgebra $U_q[sl(2|2)^{(2)}]$, $U_q[osp(2|2)]$ Invariant R-matrices and a New Integrable Electronic Model

We describe the twisted affine superalgebra $sl(2|2)^{(2)}$ and its quantized version $U_q[sl(2|2)^{(2)}]$. We investigate the tensor product representation of the 4-dimensional grade star representation for the fixed point subsuperalgebra $U_q[osp(2|2)]$. We work out the tensor product decomposition explicitly and find the decomposition is not completely reducible. Associated with this 4-dimensional grade star representation we derive two $U_q[osp(2|2)]$ invariant R-matrices: one of them corresponds to $U_q[sl(2|2)^{(2)}]$ and the other to $U_q[osp(2|2)^{(1)}]$. Using the R-matrix for $U_q[sl(2|2)^{(2)}]$, we construct a new $U_q[osp(2|2)]$ invariant strongly correlated electronic model, which is integrable in one dimension. Interestingly, this model reduces, in the $q=1$ limit, to the one proposed by Essler et al which has a larger, $sl(2|2)$, symmetry.

cond-mat.stat-mech↗

arXiv1996

Chord diagrams and BPHZ subtractions

The combinatorics of the BPHZ subtraction scheme for a class of ladder graphs for the three point vertex in $ϕ^3$ theory is transcribed into certain connectivity relations for marked chord diagrams (knots with transversal intersections). The resolution of the singular crossings using the equivalence relations in these examples provides confirmation of a proposed fundamental relationship between knot theory and renormalization in perturbative quantum field theory.

hep-th↗