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Quantum state transfer in a q-deformed chain

Cosmo Lupo, Stefano Mancini

Published Jan 1, 2008• 4 citations
OpenAlex

Abstract

We investigate the quantum state transfer in a chain of particles satisfying q-deformed oscillators algebra. This allows us to interpolate between a spin chain and a boson chain. We find conditions for perfect state transfer depending on the number of sites and excitations on the chain. They are formulated by means of irreducible representations of a quantum algebra realized through Jordan-Schwinger maps. PACS: 03.67.Hk (Quantum communication); 03.65.Fd (Algebraic methods); 02.20.Uw (Quantum groups) 1

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We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.

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We investigate the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. We consider first the simplest possible spin chain, where the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are constant throughout the chain. The time evolution of a single spin state is determined, and this time evolution is illustrated by means of an animation. Some years ago it was discovered that when the spin chain data are of a special form so-called perfect state transfer takes place. These special spin chain data can be linked to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials. We discuss here the case related to Krawtchouk polynomials, and illustrate the possibility of perfect state transfer by an animation showing the time evolution of the spin chain from an initial single spin state. Very recently, these ideas were extended to discrete orthogonal polynomials of q -hypergeometric type. Here, a remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q -Krawtchouk polynomials. This case is discussed here, and again illustrated by means of an animation.

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Chang-Pu Sun, Xu‐Feng Liu, Jing-fa Lu +1 more

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Journal of Physics A Mathematical and General•Jan 21, 1992•3 citations
We investigate the quantum state transfer in a chain of particles satisfying q-deformed oscillators algebra. This allows us to interpolate between a spin chain and a boson chain. We find conditions for perfect state transfer depending on the number of sites and excitations on the chain. They are formulated by means of irreducible representations of a quantum algebra realized through Jordan-Schwinger maps. PACS: 03.67.Hk (Quantum communication); 03.65.Fd (Algebraic methods); 02.20.Uw (Quantum groups) 1
We investigate the quantum state transfer in a chain of particles satisfying q-deformed oscillators algebra. This general algebraic setting includes the spin chain and the bosonic chain as limiting cases. We study conditions for perfect state transfer depending on the number of sites and excitations on the chain. They are formulated by means of irreducible representations of a quantum algebra realized through Jordan-Schwinger maps. Playing with deformation parameters, we can study the effects of nonlinear perturbations or interpolate between the spin and bosonic chain.
We study the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. Some years ago it was discovered that when the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are related to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials, so-called perfect state transfer takes place. The extension of these ideas to other types of discrete orthogonal polynomials did not lead to new models with perfect state transfer, but did allow more insight in the general computation of the correlation function. In the present paper, we extend the study to discrete orthogonal polynomials of q-hypergeometric type. A remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q-Krawtchouk polynomials. The other cases studied here (affine q-Krawtchouk polynomials, quantum q-Krawtchouk polynomials, dual q-Krawtchouk polynomials, q-Hahn polynomials, dual q-Hahn polynomials and q-Racah polynomials) do not give rise to models with perfect state transfer. However, the computation of the correlation function itself is quite interesting, leading to advanced q-series manipulations.
All classical Lie algebras can be realized a la Schwinger (1965) in terms of fermionic oscillators. We show that the same can be done for their q-deformed counterparts by simply replacing the fermionic oscillators with anyonic ones defined on a two-dimensional lattice. The deformation parameter q is a phase related to the anyonic statistical parameter. A crucial role in this construction is played by a sort of bosonization formula which gives the generators of the quantum algebras in terms of the undeformed ones. The entire procedure works even on one-dimensional chains; in such a case q can also be real.
We discuss the connection between anyons (particles with fractional statistics) and deformed Lie algebras (quantum groups). After a brief review of the main properties of anyons, we present the details of the anyonic realization of all deformed classical Lie algebras in terms of anyonic oscillators. The deformation parameter of the quantum groups is directly related to the statistics parameter of the anyons. Such a realization is a direct generalization of the Schwinger construction in terms of fermions and is based on a sort of bosonization formula which yields the generators of the deformed algebra in terms of the undeformed ones. The entire procedure is well defined on two-dimensional lattices, but it can be consistently reduced also to one-dimensional chains.
We investigate the time evolution of a single spin excitation state in certain linear spin chains, as a model for quantum communication. We consider first the simplest possible spin chain, where the spin chain data (the nearest neighbour interaction strengths and the magnetic field strengths) are constant throughout the chain. The time evolution of a single spin state is determined, and this time evolution is illustrated by means of an animation. Some years ago it was discovered that when the spin chain data are of a special form so-called perfect state transfer takes place. These special spin chain data can be linked to the Jacobi matrix entries of Krawtchouk polynomials or dual Hahn polynomials. We discuss here the case related to Krawtchouk polynomials, and illustrate the possibility of perfect state transfer by an animation showing the time evolution of the spin chain from an initial single spin state. Very recently, these ideas were extended to discrete orthogonal polynomials of q -hypergeometric type. Here, a remarkable result is a new analytic model where perfect state transfer is achieved: this is when the spin chain data are related to the Jacobi matrix of q -Krawtchouk polynomials. This case is discussed here, and again illustrated by means of an animation.