The hydrogen atom as an entangled electron–proton system

Tommasini, Paolo; Timmermans, E.A.H.; Piza, A. F. R. de Toledo

doi:10.1119/1.18977

Cited by 38 publications

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“…In the case of a central interaction and spin-orbit coupling, the reduced S-matrix is just exp(2iδ(W ) l,s )δ ll ′ δ ss ′ , where δ(W ) l,s are called scattering phase shifts. Clearly, any Galilean invariant S-matrix factors into local unitaries on the IE TPS (10). Therefore, the amount of IE entanglement in an in-state will be invariant under any scattering dynamics that respects Galilean symmetry.…”

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“…When solving the bound state problem, one typically assumes that there is no IE entanglement and so the wave function for the net motion can be factored out from the internal wave function. As understood in the context of the hydrogen atom [10], having zero internal-external entanglement certainly does not imply that there is no interparticle entanglement, i.e., entanglement with respect to the TPS (4). We will discuss this further in the next section.…”

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Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Harshman

Wickramasekara

2007

Phys. Rev. Lett.

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Particle systems admit a variety of tensor product structures (TPSs) depending on the algebra of observables chosen for analysis. Global symmetry transformations and dynamical transformations may be resolved into local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs are defined and various notions of entanglement are considered for scattering states.PACS numbers: 03.67. Mn, 03.65.Nk, The interaction of particle systems via scattering is a fundamental theoretical and experimental paradigm. The quantum information theory of particle scattering is, however, still in its infancy. Results, theoretical and computational, exist for the entanglement between the momenta [1] or the angular momenta [2] of two particles generated in scattering, but many problems remain open. The challenges are partly technical due to the greater complexity of dealing with entanglement in continuous variable systems [3] and partly conceptual as in defining a measure of entanglement that has meaningful properties under space-time symmetry transformations. See, for example, the literature on spin-entanglement of relativistic particles [4,5] where different types of entanglement (between two particles, between two particles' spins, and between a single particle's spin and momentum) have been discussed and occasionally confused.In this letter, we examine how some of these difficulties may be resolved by combining two approaches: (1) the generalized tensor product structures (TPSs) and observable-dependent entanglement developed by Zanardi and others [6], and (2) the representation theory of space-time symmetry groups, which has a long and fruitful history in quantum mechanics. Using these methods, TPSs for single particle and multi-particle systems are explored. These methods allow one to distinguish between TPSs that are symmetry invariant and/or dynamically invariant and TPSs that are not, and, in the latter case, to obtain quantitative expressions for the change of entanglement. The reason why certain TPSs have entanglement measures which are symmetry or dynamically invariant is that the space-time symmetries or the time evolution operator, respectively, act as a product of local unitaries with respect to these TPSs.As an application of these general concepts and methods, we will study non-relativistic elastic scattering of two particles. In this context, several interesting results emerge. First, there are single particle TPSs that are invariant under transformations between inertial reference frames, and these TPSs allow one to define intraparticle entanglement between momentum and spin degrees of freedom in a Galilean invariant manner. Second, there are multiple, inequivalent two particle TPSs that are symmetry invariant. In particular, these TPSs can be used to define Galilean invariant entanglement between the internal and external degrees of freedom of the two particle system. Finally, this internal-external entanglement is also dynamically invariant, i.e., i...

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Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Harshman

Wickramasekara

2007

Phys. Rev. Lett.

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“…Certain observables can be preferred because of the form of interactions, the sources of error and decoherence, the physical accessibility of measurement and control, or for other reasons. For example, in bound states of a proton and an electron, energy eigenstates are unentangled with respect to the tensor product induced by the center-of-mass/relative observables, but they are highly entangled with respect to the particle observables [12]. By turning on external fields, one can induce entanglement between the center-of-mass and relative observables, i.e.…”

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Observables can be tailored to change the entanglement of any pure state

Harshman

Ranade

2011

Phys. Rev. A

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We show that for a finite-dimensional Hilbert space, there exist observables that induce a tensor product structure such that the entanglement properties of any pure state can be tailored. In particular, we provide an explicit, finite method for constructing observables in an unstructured d-dimensional system so that an arbitrary known pure state has any Schmidt decomposition with respect to an induced bipartite tensor product structure. In effect, this article demonstrates that in a finite-dimensional Hilbert space, entanglement properties can always be shifted from the state to the observables and all pure states are equivalent as entanglement resources in the ideal case of complete control of observables.PACS numbers: 03.65.Aa, 03.65.UdThe entanglement of a quantum state is only defined with respect to a tensor product structure within the Hilbert space that represents the quantum system. In turn, a tensor product structure of the Hilbert space is induced by the algebra of observables. Zanardi and colleagues [1,2] have provided criteria for the algebra of observables of a finite-dimensional system to induce a tensor product structure. The algebra of observables must be partitioned into subalgebras that satisfy two mathematical requirements, the subalgebras must be independent and complete (see Corollary 3 for a precise formulation of Zanardi's Theorem), and one physical requirement, the subalgebras must be locally accessible. Such observableinduced partitions of the Hilbert space have been referred to as virtual subsystems and can be thought of as a generalization from entanglement between subsystems to entanglement between degrees of freedom (see also [3,4]). This mathematical framework has found applications to studies of multi-level encoding [5], decoherence [6], operator quantum error correction [7], entanglement in fermionic systems [8], single-particle entanglement [9,10], and entanglement in scattering systems [11].In this Letter, we extend this mathematical framework and prove what we call the Tailored Observables Theorem (Theorem 6): observables can be constructed such that any pure state in a finite-dimensional Hilbert space H = C d has any amount of entanglement possible for any given factorization of the dimension d of H. This means all pure states are equivalent as entanglement resources in the ideal case of complete control of observables. To establish the framework, we provide a brief, relatively self-contained introduction to Zanardi's Theorem and obtain some necessary preliminary results about observable algebras in finite dimensions. We then prove Theorem 6, which applies to bipartite tensor product structures, and present an illustrative example. We will also provide a corollary of the theorem (Corollary 7) applied to multipartite tensor product structures. Before delving into the technical details, we present a more intuitive discussion of this result. this Hilbert space could represent states of a quantum system composed from N subsystems each represented by Hilbert spaces H i = C ki . Fo...

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“…So, the Schrödinger dynamics for the hydrogen atom as a whole simultaneously preserves separability of states for CM + R and induces entanglement for e + p structure. A dynamical proof of the presence of entanglement for the e + p structure can be found in the literature [16].…”

Section: Entanglement In the Hydrogen Atommentioning

confidence: 99%

Quantum Structures of the Hydrogen Atom

Jeknić-Dugić¹,

Dugić²,

Francom³

et al. 2014

OALib

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How to cite this paper: Jeknić-Dugić, J., Dugić, M., Francom, A. and Arsenijević, M. (2014) Abstract Modern quantum theory introduces quantum structures (decompositions into subsystems) as a new discourse that is not fully comparable with the classical-physics counterpart. To this end, socalled Entanglement Relativity appears as a corollary of the universally valid quantum mechanicsthat can provide for a deeper and more elaborate description of the composite quantum systems. In this paper we employ this new concept to describe the hydrogen atom. We offer a consistent picture of the hydrogen atom as an open quantum system that naturally answers the following important questions: 1) how do the so called "quantum jumps" in atomic excitation and de-excitation occur? and 2) why does the classically and seemingly artificial "center-of-mass + relative degrees of freedom" structure appear as the primarily operable form in most of the experimental reality of atoms?

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The hydrogen atom as an entangled electron–proton system

Cited by 38 publications

References 9 publications

Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Observables can be tailored to change the entanglement of any pure state

Quantum Structures of the Hydrogen Atom

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