Efficient Polynomial-Scaled Determination of Algebraic Entanglement Entropy Between Collective Degrees of Freedom

Abstract

In this work, we explore physical systems which support not only multipartite interparticle entanglement, but also intraparticle entanglement between different degrees of freedom of the constituent particles and entanglement between different degrees of freedom of different particles, i.e., algebraic entanglement. We derive a simple method for calculating the algebraic entanglement entropy between two of the particles' degrees of freedom from collective states of the whole ensemble. Our procedure makes use of underlying symmetries in these systems, in particular permutation symmetry of the particle indices, and shows a connection between the algebraic entanglement entropy in these systems and the irreducible representations of Lie groups which describe the particles' degrees of freedom. Namely, we use the direct sum over irreducible representations to diagonalize the reduced density matrices in a block-by-block manner, then utilize the multiplicity of these irreducible representations to reproduce the results from an exponentially-scaled Hilbert space in only polynomial complexity. We use this to explore a variety of systems where the constituent particles support two degrees of freedom each with two levels, such as atoms with two electronic states and two momentum states. Notably, these systems may be exactly simulated in a polynomial-scaled Hilbert space, yet they support an algebraic entanglement entropy that grows linearly with the particle number which typically requires an exponentially-scaled Hilbert space.