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Andrzej M. Frydryszak

Publications and source records attributed to Andrzej M. Frydryszak.

8 recordsLinked to original sources
arXiv2018

Probing the geometry of two-qubit state space by evolution

We derive an explicit expressions for geometric description of state manifold obtained from evolution governed by a three parameter family of Hamiltonians covering most cases related to real interacting two-qubit systems. We discuss types of evolution in terms of the defining parameters and obtain relevant explicit description of the pure state spaces and their Remannian geometry with the Fubini-Study metric . In particular, there is given an analysis of the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. For families of states resulting from the unitary evolution, we characterize a degree of entanglement using the squared concurrence as its measure.

quant-ph↗
arXiv2015

Geometric measure of entanglement for pure states and mean value of spin

We derive an explicit expression for geometric measure of entanglement for spin and other quantum system. A relation of entanglement in pure state with the mean value of spin is given, thus, at the experimental level the local measurement of spin may allow to find the value of entanglement. The obtained form of the measure is applied to the explicit characterization of bipartite entanglement for $n$-qubit systems in the Werner state, Dicke state, GHZ state and trigonometric states. In particular for Werner-like states the rule of sums is found and it is shown that deviations from the symmetricity of such states diminishes the amount of entanglement. For Dicke states the maximal value of bipartite entanglement is achieved when number of excitations is half of the total number of qubits in these states. For trigonometric states the bipartite entanglement is maximal and does not depend on the number of qubits. We also consider entanglement of discrete-continuous systems on the example of entanglement of spin with continuous variables of electron. The relation of entanglement with the mean value of spin is very useful for calculation of entanglement. With the help of this relation we find in explicit form the entanglement of the one spin with the rest in a spin chain during the evolution determined by the Ising Hamiltonian.

quant-ph↗
arXiv2014

$Q$-representations and unitary representations of the super-Heisenberg group and harmonic superanalysis

We juxtapose two approaches to the representations of the super-Heisenberg group. Physical one, sometimes called concrete approach, based on the super-wave functions depending on the anti-commuting variables, yielding the harmonic superanalysis and recently developed strict theory of unitary representations of the nilpotent super Lie groups covering the unitary representations of the super-Heisenberg group.

math-ph↗
arXiv2012

Qubit, superqubit and squbit

We analyze recently proposed formalisms which use nilpotent variables to describe and/or generalize qubits and notion of entanglement. There are two types of them distinguished by the commutativity and or anti-commutativity of basics variables. While nilpotent commuting variables suit the new description of the qubits and entanglement, the anticommuting do not, but they can be used to describe generalized objects - superqubits. A squbit, in the present context, is a version of superqubit introduced to cure some of problematic properties of the superqubit.

math-ph↗
arXiv2008

Nilpotent quantum mechanics, qubits, and flavors of entanglement

We address the question of description of qubit system in a formalism based on the nilpotent commuting variables. In this formalism qubits exhibit properties of composite objects being subject of the Pauli exclusion principle, but otherwise behaving boson-like. They are not fundamental particles. In such an approach the classical limit yields the nilpotent mechanics. Using the space of $η$-wavefunctions, generalized Schrödinger equation etc. we study properties of pure qubit systems and also properties of some composed, hybrid models: fermion-qubit, boson-qubit. The fermion-qubit system can be truly supersymmetric, with both SUSY partners having identical spectra. It is new and very interesting that SUSY transformations relate here only nilpotent object. The $η$-eigenfunctions for the qubit-qubit system give the set of Bloch vectors as a natural basis. Then the $η$-formalism is applied to the description of the pure state entanglement. Nilpotent commuting variables were firstly used in this context in [A. Mandilara, et. al., Phys. Rev. A \textbf{74}, 022331 (2006)], we generalize and extend approach presented there. Our main tool for study the entanglement or separability of states are Wronskians of $η$-functions. The known invariants and entanglement monotones for systems of $n=2,3,4$ qubits are expressed in terms of the Wronskians. This approach gives criteria for separability of states and insight into the flavor of entanglement of the system and simplifies description.

quant-ph↗
arXiv2006

Nilpotent classical mechanics: s-geometry

We introduce specific type of hyperbolic spaces. It is not a general linear covariant object, but of use in constructing nilpotent systems. In the present work necessary definitions and relevant properties of configuration and phase spaces are indicated. As a working example we use a D=2 isotropic harmonic oscillator.

math-ph↗