Particle systems admit a variety of tensor product structures (TPSs) depending on the complete system of commuting observables chosen for the analysis. Different notions of entanglement are associated with these different TPSs. Global symmetry transformations and dynamical transformations factor into products of local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs and corresponding measures of entanglement are defined for particle scattering systems.
This paper explores the connections between particle scattering and quantum information theory in the context of the non-relativistic, elastic scattering of two spin-1/2 particles. An untangled, pure, two-particle in-state is evolved by an S-matrix that respects certain symmetries and the entanglement of the pure out-state is measured. The analysis is phrased in terms of unitary, irreducible representations (UIRs) of the symmetry group in question, either the rotation group for the spin degrees of freedom or the Galilean group for non-relativistic particles. Entanglement may occurs when multiple UIRs appear in the direct sum decomposition of the direct product in-state, but it also depends of the scattering phase shifts.
Invariant operator-valued tensor fields on Lie groups are considered. These define classical tensor fields on Lie groups by evaluating them on a quantum state. This particular construction, applied on the local unitary group U(n) × U(n), may establish a method for the identification of entanglement monotone candidates by deriving invariant functions from tensors being by construction invariant under local unitary transformations. In particular, for n = 2, we recover the purity and a concurrence related function (Wootters 1998) as a sum of inner products of symmetric and anti-symmetric parts of the considered tensor fields. Moreover, we identify a distinguished entanglement monotone candidate by using a non-linear realization of the Lie algebra of SU(2) × SU(2). The functional dependence between the latter quantity and the concurrence is illustrated for a subclass of mixed states parametrized by two variables.
A model of quantum-mechanical scattering is given in terms of the Abelian subalgebras of observables which are associated with the initial preparation and final measurement procedures. The scattering is described by an operator, the
S matrix, which defines an isomorphism between these two subalgebras. By using the concept of presymmetry for Galilean-invariant systems, these algebras are shown to be generated by the generalized momenta \ensuremath{'}P which are related to the dynamical situation by $H{P}_{\mathrm{ac}}=\frac{{|{\mathrm{P}}^{\ensuremath{'}}|}^{2}}{2\ensuremath{\mu}}$. Here
H is the total Hamiltonian effective at the time of measurement,
Pac is the projection onto its absolutely continuous spectrum, and $\ensuremath{\mu}$ is the mass of the particle. The
S matrix which is defined in this way gives the same formal expressions for the differential cross sections as the usual one, and it is automatically unitary. Furthermore, since the differential cross section is defined as a relative probability in terms of the results of the two measurement procedures, it can be finite even when the classical cross section is infinite. Scattering can therefore be defined in this model for spherically symmetric potentials which decrease as slowly as ${r}^{\ensuremath{-}\ensuremath{\epsilon}}$, for any $\ensuremath{\epsilon}>0$, provided that the potential is repulsive at large separations between the particles.
Using the quantum two-body system as a familiar model, this talk will describe how entanglement can be used to select preferred observables for interrogating a physical system. The symmetries and dynamics of the quantum two-body system provide a backdrop for testing the relativity of entanglement with respect to observable-induced tensor product structures. We believe this exploration leads us to a general statement: the physically-meaningful observable subalgebras are the ones that minimize entanglement in typical states.
The group invariance of entanglement is obtained within a very general and simple setup of the latter, given by a recently introduced considerably extended concept of tensor products. This general approach to entanglement - unlike the usual one given in the particular setup of tensor products of vector spaces - turns out not to need any specific algebraic structure. The resulting advantage is that, entanglement being in fact defined by a negation, its presence in a general setup increases the chances of its manifestations, thus also its availability as a resource.