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Communications in Theoretical Physics

Coordinate Eigenstate Representation in Noncommutative Space as a New Entangled State in Quantum Mechanics

Fan Hong‐Yi, Xubing Tang, Tongtong Wang

Published Oct 1, 2007• 3 citations
Publisher

Abstract

For the first time we construct the eigenstate |τ⟩ of noncommutative coordinate. It turns out that |τ⟩ is an entangled state in the ordinary space. Remarkably, its Schmidt decomposition has definite expression in the coordinate representation and the momentum representation. The ⟨τ| representation can simplify some calculations for obtaining energy level of two-dimensional oscillator in noncommutative space.

DOI
10.1088/0253-6102/48/4/012

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For the first time we construct the eigenstate |τ⟩ of noncommutative coordinate. It turns out that |τ⟩ is an entangled state in the ordinary space. Remarkably, its Schmidt decomposition has definite expression in the coordinate representation and the momentum representation. The ⟨τ| representation can simplify some calculations for obtaining energy level of two-dimensional oscillator in noncommutative space.
We introduce new representations to formulate quantum mechanics on noncommutative coordinate space, which explicitly display entanglement properties between degrees of freedom of different coordinate components and hence could be called entangled state representations. Furthermore, we derive unitary transformations between the new representations and the ordinary one used in noncommutative quantum mechanics (NCQM) and obtain eigenfunctions of some basic operators in these representations. To show the potential applications of the entangled state representations, a two-dimensional harmonic oscillator on the noncommutative plane with both coordinate-coordinate and momentum-momentum couplings is exactly solved.
The eigenstate existing in the problem of EPR paradox is introduced.The commutation of related operators is analyzed and the wave-function in coordinate representation of the eigenstate is verified and its physical signification is discussed.The eigenstate is expended in the direct product space of the moment eigenstates of a two-particle system,subsequently the wave-function in momentum representation is obtained.The relation of the form of the wave-function in momentum representation and the selection of the origin of coordinates,including its physical meaning was analyzed.Results showed that the eigenstate was a entangled state of the coordinate and momentum.
In this paper, we investigate the quantum entanglement induced by phase-space noncommutativity. Both the position–position and momentum–momentum noncommutativity are incorporated to study the entanglement properties of coordinate and momentum degrees of freedom under the shade of oscillators in noncommutative space. Exact solutions for the systems are obtained after the model is re-expressed in terms of canonical variables, by performing a particular Bopp’s shift to the noncommuting degrees of freedom. It is shown that the bipartite Gaussian state for an isotropic oscillator is always separable. To extend our study for the time-dependent system, we allow arbitrary time dependency on parameters. The time-dependent isotropic oscillator is solved with the Lewis–Riesenfeld invariant method. It turns out that even for arbitrary time-dependent scenarios, the separability property does not alter. We extend our study to the anisotropic oscillator, which provides an entangled state even for time-independent parameters. The Wigner quasi-probability distribution is constructed for a bipartite Gaussian state. The noise matrix (covariance matrix) is explicitly studied with the help of Wigner distribution. Simon’s separability criterion (generalized Peres–Horodecki criterion) has been employed to find the unique function of the (mass and frequency) parameters, for which the bipartite states are separable. In particular, we show that the mere inclusion of non-commutativity of phase-space is not sufficient to generate the entanglement, rather anisotropy is important at the same footing. We explore the experimental viability of our result through the computation of extractable work for the current situation.
The structure of the state-vector space of identical bosons in noncommutative spaces is investigated. To maintain Bose–Einstein statistics the commutation relations of phase space variables should simultaneously include coordinate-coordinate non-commutativity and momentum-momentum non-commutativity, which leads to a kind of deformed Heisenberg-Weyl algebra. Although there is no ordinary number representation in this state-vector space, several set of orthogonal and complete state-vectors can be derived which are common eigenvectors of corresponding pairs of commuting Hermitian operators. As a simple application of this state-vector space, an explicit form of two-dimensional canonical coherent state is constructed and its properties are discussed.
Entanglement is defined for each vector subspace of the tensor product of two finite-dimensional Hilbert spaces, by applying the notion of operator entanglement to the projection operator onto that subspace. The operator Schmidt decomposition of the projection operator defines a string of Schmidt coefficients for each subspace, and this string is assumed to characterize its entanglement, so that a first subspace is more entangled than a second, if the Schmidt string of the second majorizes the Schmidt string of the first. The idea is applied to the antisymmetric and symmetric tensor products of a finite-dimensional Hilbert space with itself, and also to the tensor product of an angular momentum jjj with a spin 1∕21∕21∕2. When adapted to the subspaces of states of the nonrelativistic hydrogen atom with definite total angular momentum (orbital plus spin), within the space of bound states with a given total energy, this leads to a complete ordering of those subspaces by their Schmidt strings.