Entanglement rearrangement in self-consistent nuclear structure calculations

Background: Entanglement plays a central role in a diverse array of increasingly important research areas, including quantum computation, simulation, measurement, sensing, and communication. Extensive suites of investigations have been performed to better understand entanglement in atomic and molecular quantum many-body systems, while the exploration of entanglement in the structure of nuclei and their reactions is presently in its infancy.

Purpose: The goal of this work is to begin investigating the entanglement properties of nuclei from first-principles nuclear many-body calculations. We attempt to identify common features and emergent structures of entanglement that could ultimately lead to new and natural many-body schemes. With an eye toward quantum accelerators in future hybrid-supercomputers, criteria for partitioning nuclear many-body calculations into quantum and classical components may provide advantages in future large-scale computations. Along the way we look for explanations of the relative success of phenomenological models such as the nuclear shell model, and for better ways to match to low-energy nuclear effective field theories and lattice QCD calculations to nuclear many-body techniques that are based upon entanglement.

Method: We explore the entanglement between single-particle states in ${}^4\mathrm{He}$ and ${}^6\mathrm{He}$. The patterns of entanglement emerging from different single-particle bases are compared, and possible links with the convergence of observables are explored, in particular, ground-state energies. The nuclear wave functions are obtained by performing active-space no-core configuration-interaction calculations using a two-body nucleon-nucleon interaction derived from chiral effective field theory. Entanglement measures within single-particle bases exhibiting different degrees of complexity are determined, in particular, harmonic oscillator (HO), Hartree-Fock (HF), natural (NAT) and variational natural (VNAT) bases. Specifically, single-orbital entanglement entropy, two-orbital mutual information, and negativity are studied.

Results: The entanglement structures in ${}^4\mathrm{He}$ and ${}^6\mathrm{He}$ are found to be more localized within NAT and VNAT bases than within a HO basis for the optimal HO parameters we have worked with. In particular a core-valence structure clearly emerges from the full no-core calculation of ${}^6\mathrm{He}$. The two-nucleon mutual information shows that the VNAT basis, which typically exhibits good convergence properties, effectively decouples the active and inactive spaces.

Conclusions: Measures of one- and two-nucleon entanglement are found to be useful in analyzing the structure of nuclear wave functions, in particular the efficacy of basis states, and may provide useful metrics toward developing more efficient schemes for ab initio computations of the structure and reactions of nuclei, and quantum many-body systems more generally.

Figure 1

Single-orbital entanglement entropy $S_i^{\left(1\right)}$ of the HO, HF, NAT, and VNAT single-neutron states $i=(n_i,l_i,j_i,m_i,{\tau }_i=-1/2)$ in ${}^4\mathrm{He}$, obtained with a model space of three shells. The left panels shows $S^{\left(1\right)}$ ordered by the states in the HO basis, while the right panels correspond to ordering by decreasing values of $S^{\left(1\right)}$, which coincides with the ordering of the calculation (by occupation number) for the NAT and VNAT bases.

Figure 2

Single-orbital entanglement entropy $S_{nljm}^{\left(1\right)}=S_{nlj}^{\left(1\right)}$ of the HO, HF, NAT, and VNAT single-neutron states in ${}^4\mathrm{He}$, for different sizes of the active model space.

Figure 3

The mutual information within the neutron-neutron orbitals of ${}^4\mathrm{He}$ using five-shell active spaces in the HO (upper panel) and VNAT (lower panel) bases. The states are ordered by quantum numbers, and the vertical scales are the same.

Figure 4

Mutual information between two HO (left) and VNAT (right) states, obtained for a model space of $N_{\mathrm{tot}}=5$ shells. Each pixel corresponds to the MI between states $i=(n_i,l_i,j_i,m_i,{\tau }_i)$. The states are ordered by quantum numbers $N_i=1n_i+l_i,j_i$ and AM projection $m_i=+1/2,-1/2,+3/2,-3/2$, etc. The top panels show neutron-neutron MI and the bottom panels show the proton-neutron MI. In the bottom panels, the proton (neutron) states are on the $y$ ($x$) axis.

Figure 5

Neutron-neutron (left panel) and proton-neutron (right panel) MI obtained using VNAT orbitals ordered with occupation numbers, for $N_{\mathrm{tot}}=5$ shells. In the right panel, the proton (neutron) states are shown on the $y$ ($x$) axis.

Figure 6

The negativity within the neutron-neutron orbitals of ${}^4\mathrm{He}$ using five-shell active spaces in the HO (upper panel) and VNAT (lower panel) bases. The vertical scales are the same in both panels.

Figure 7

The entanglement entropy of a single-neutron (top) and single-proton (bottom) HO and VNAT state, in ${}^6\mathrm{He}$ obtained with $N_{\mathrm{tot}}=4$ shells.

Figure 8

The mutual information within two neutron orbitals in ${}^6\mathrm{He}$ with the HO (upper panel) and VNAT (lower panel) bases using $N_{\mathrm{tot}}=4$ active shells.

Figure 9

Mutual information of HO (left) and VNAT (right) single-particle states in ${}^6\mathrm{He}$ obtained for an active space of $N_{\mathrm{tot}}=4$ shells. The top panels show the MI between neutron orbitals, the middle panels show the MI between proton orbitals, and the bottom ones show the proton-neutron MI. (The $f$ shells are not shown as their MI is not visible). In the bottom panels the proton (neutron) states are on the $y$ ($x$) axis.

Figure 10

Neutron-neutron MI for NAT (upper) and VNAT (lower) single-particle states in ${}^6\mathrm{He}$ obtained with $N_{\mathrm{tot}}=3$ and $N_{\mathrm{tot}}^{\prime }=5$. The dashed lines show the initial active space of $N_{\mathrm{tot}}=3$ shells.