Galilean and Dynamical Invariance of Entanglement in Particle Scattering

Abstract

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...

Entanglement for all quantum states

“…The only possibility to preserve the factorization is when the degrees of freedom are not mixed, that is, the transformations are of the type A = A(X 1 ) and B = B(X 2 ). Therefore, the unitary transformations in the Hilbert space H 1 ⊗ H 2 that leave the TPS invariant are of the type U A ⊗ U B , for instance, the time evolution of internal and external degrees of freedom [2]. A generalization of this to TPS involving any number of factors is evident.…”

“…The fact that factorizability and entanglement are not preserved in a change of the degrees of freedom used to describe the system has been analysed by experts, specially those involved in quantum computation research [1,2], but this important feature of quantum mechanics is ignored in textbooks, even advanced ones. In this work we present simple calculations that emphasize this remarkable feature and provide thereby a didactic complement for a modern quantum mechanics course.…”

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

“…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].…”

“…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.…”

Decoherence: A Closed-System Approach

“…By following Harshman and Wickramasekara ( [53]), we will use the expression "tensor product structure" (TPS) to call any factorization In general, a quantum system U admits a variety of TPSs, that is, of decompositions into S A and S B . Among all these possible decompositions, there may be a particular TPS that remains dynamically invariant (see [53]). This is the case when there is no interaction between S A and S B , H AB = 0, and, then,…”