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.

Figures (8)

• Figure 1: (a) A cartoon of the pyramid structure of the $\mathrm{SU{(4)}}$ state space. Each layer of the pyramid is a square of size $(2 \ell + 1)$-by-$(2 \ell + 1)$ for the $(2 \ell + 1)$-many levels of each degree of freedom. (b) The two Bloch spheres with dipole length $\ell$ for the $\mathrm{SU{(2)}}\otimes\mathrm{SU{(2)}}$ subgroup of $\mathrm{SU{(4)}}$. Layer $\ell$ of the state space in (a) corresponds to many copies of this subgroup.
• Figure 2: (a) The state pyramid for $N=4$. (b) The layers of the state pyramid, where each $\ell$ corresponding to one of the irreducible representations of $\mathrm{SU{(2)}} \otimes \mathrm{SU{(2)}}$. Red arrows indicate an application of $\hat{J}_-$, blue arrows indicate an application of $\hat{K}_-$, and gray arrows indicate a Gram-Schmidt orthogonalization step. The $\ell$ label and the multiplicity $d^{\ell}_N$ are given to the right of each layer.
• Figure 3: The algebraic entanglement entropy for the collective internal states (corresponding to $J$) of the atoms for the Hamiltonian in Eq. \ref{['eq:H_E2']}. (a) shows results for $N = 4$ while (b) shows results for $N = 20$. We display the numerical results (red lines) and the analytical results from Eq. \ref{['eq:AnalyticSolution']} (blue lines). In both plots, we see that the entanglement entropy grows to the maximum possible value of $N \ln(2)$ (gray lines) given in Eq. \ref{['eq:S_E_max']} at $t = \pi / ( 2 \Omega)$. We find that the results from Algorithm \ref{['alg:EE']} for the algebraic entanglement entropy follows the analytical solutions exactly.
• Figure 4: The algebraic entanglement entropy for the collective internal states (corresponding to $J$) of the atoms for the BOAT Hamiltonian Eq. \ref{['eq:H_BOAT']}. (a) shows results for $N = 4$. Here, we can simulate the dynamics in the full single-particle basis and calculate the algebraic entanglement entropy (blue line), where we find it exactly matches the collective simulation (red line). (b) shows results for $N = 20$. Here, simulating the full $4^N$ dynamics is infeasible. However, the algorithm which runs on the $\mathcal{O}(N^3)$ state space still calculates the algebraic entanglement entropy efficiently.
• Figure 5: Results for the leaky BOAT model Eq. \ref{['eq:MasterEq']}. We display the algebraic entanglement entropy for $\hat{\rho}_J$ (blue) and $\hat{\rho}_K$ (red), the entropy for the total state $\hat{\rho}$ (black), and the coherent informations for $I(J \rangle K)$ (orange) and $I(K \rangle J)$ (purple). The parameters are (a) $\Gamma_c = 0.05 \chi$, (b) $\Gamma_c = 0.25 \chi$, and (c) $\Gamma_c = 0.5 \chi$. In all three, we see that both $I(J\rangle K)$ and $I(K \rangle J)$ are positive, indicating the presence of algebraic entanglement between $J$ and $K$. For larger decay rates, we see that the entanglement entropy of $\hat{\rho}_J$ and $\hat{\rho}_K$ differ due to entanglement with the environment, which is indicated by the growing entropy of $\hat{\rho}$.