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Physical Review C

Entanglement and seniority

András T. Kruppa, J. Kovács, P Salamon, Örs Legeza, Gergely Zaránd

Published Aug 3, 2022• 35 citations
Publisher

Abstract

Background: In quantum mechanics, entanglement is the most striking phenomenon which has no counterpart in classical systems. Although different approaches have already been developed to study correlations in the case of indistinguishable particles, the exploration of the so-called mode entanglement is still in its initial stage in nuclear physics.Purpose: We aim to study mode entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority $\ensuremath{\nu}=0,\phantom{\rule{0.16em}{0ex}}1,\phantom{\rule{0.16em}{0ex}}2$, and 3, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. In addition, we compare the predictions of the seniority model with the results of the shell model to gain insight into the structure and correlations of the ground and lowest yrast states.Methods: In the seniority model the analytic results are calculated using the quasispin formalism. The numerical shell model calculations are carried out in the standard shell model framework using an inert core. The applied realistic effective interactions are derived earlier using the $G$-matrix formalism. The density matrix renormalization group method is also applied in order to directly calculate the mode entropies.Results: In the seniority model simple analytical expressions are given for the mode entropies. The peculiar behavior of the half-filled shells and the seniority-zero states is revealed. Numerical results are presented for the lightest stable calcium isotopes and for $^{94}\mathrm{Ru}$ nucleus.Conclusions: For $^{94}\mathrm{Ru}$, the seniority model accounts for the $0{g}_{9/2}$ mode entropies, but seniority mixing is important for certain yrast states. Interaction induced quantum fluctuations decrease the occupation of the $0{f}_{5/2}$, $1{p}_{3/2}$, and $1{p}_{1/2}$ shells, and result in non-negligible mode entropies on these shells, too, clearly outside the scope of the simple ${(0{g}_{9/2})}^{4}$ seniority model. The $0{f}_{7/2}$ shell based seniority model is more accurate for the $\mathrm{Ca}$ isotopes, but seniority mixing is substantial for some $^{44}\mathrm{Ca}$ yrast states, too. Mode and one-body entanglement entropies are useful tools to investigate the structure of quantum correlations in nuclei.

DOI
10.1103/physrevc.106.024303

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•Jan 1, 2021•21 citations
Background: In quantum mechanics, entanglement is the most striking phenomenon which has no counterpart in classical systems. Although different approaches have already been developed to study correlations in the case of indistinguishable particles, the exploration of the so-called mode entanglement is still in its initial stage in nuclear physics.Purpose: We aim to study mode entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority \ensuremathν=0,1,2\ensuremath{\nu}=0,\phantom{\rule{0.16em}{0ex}}1,\phantom{\rule{0.16em}{0ex}}2\ensuremathν=0,1,2, and 3, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. In addition, we compare the predictions of the seniority model with the results of the shell model to gain insight into the structure and correlations of the ground and lowest yrast states.Methods: In the seniority model the analytic results are calculated using the quasispin formalism. The numerical shell model calculations are carried out in the standard shell model framework using an inert core. The applied realistic effective interactions are derived earlier using the GGG-matrix formalism. The density matrix renormalization group method is also applied in order to directly calculate the mode entropies.Results: In the seniority model simple analytical expressions are given for the mode entropies. The peculiar behavior of the half-filled shells and the seniority-zero states is revealed. Numerical results are presented for the lightest stable calcium isotopes and for 94Ru^{94}\mathrm{Ru}94Ru nucleus.Conclusions: For 94Ru^{94}\mathrm{Ru}94Ru, the seniority model accounts for the 0g9/20{g}_{9/2}0g9/2​ mode entropies, but seniority mixing is important for certain yrast states. Interaction induced quantum fluctuations decrease the occupation of the 0f5/20{f}_{5/2}0f5/2​, 1p3/21{p}_{3/2}1p3/2​, and 1p1/21{p}_{1/2}1p1/2​ shells, and result in non-negligible mode entropies on these shells, too, clearly outside the scope of the simple (0g9/2)4{(0{g}_{9/2})}^{4}(0g9/2​)4 seniority model. The 0f7/20{f}_{7/2}0f7/2​ shell based seniority model is more accurate for the Ca\mathrm{Ca}Ca isotopes, but seniority mixing is substantial for some 44Ca^{44}\mathrm{Ca}44Ca yrast states, too. Mode and one-body entanglement entropies are useful tools to investigate the structure of quantum correlations in nuclei.
We study mode-entanglement in the seniority model, derive analytic formulas for the one-body reduced density matrix of states with seniority ν=0,  1,  2ν= 0,\;1,\;2ν=0,1,2, and ν=3ν=3ν=3, and also determine the particle number dependence of the one-body reduced density matrix for arbitrary seniority. We carry out numerical calculations for the lightest calcium isotopes and for 94Ru^{94}{\rm Ru}94Ru nucleus, and investigate the structure of their ground and low energy yrast states. We investigate the fulfillment of several predictions of the seniority model regarding the behavior of one-mode entropies, which we compare with the results of configuration interaction (CI) and density matrix renormalization group (DMRG) computations. For 94Ru^{94}{\rm Ru}94Ru, the seniority model accounts for the 0g9/20g_{9/2}0g9/2​ mode entropies, but seniority mixing is important for certain yrast states. Interaction induced quantum fluctuations decrease the occupation of the 0f5/20f_{5/2}0f5/2​, 1p3/21p_{3/2}1p3/2​ and 1p1/21p_{1/2}1p1/2​ shells, and amount in finite mode entropies on these shells, too, clearly outside the scope of the simple (0g9/2)4(0g_{9/2})^4(0g9/2​)4 seniority model. The 0f7/20f_{7/2}0f7/2​ shell based seniority model is more accurate for the light Ca{\rm Ca}Ca isotopes, but seniority mixing is substantial for some 44Ca^{44}{\rm Ca}44Ca yrast states, too.
Abstract We examine mode entanglement and correlation of two fermionic particles analytically and numerically. We study the one- and two-mode entropies and a global characteristic, the one-body entanglement entropy considering angular momentum coupled state with a single configuration. We show that with rearrangement of the single-particle orbitals the Slater decomposition can be obtained which can also be applied for configuration mixing subject to special restriction. With the help of the Slater decomposition, we derive analytical expressions for the entanglement measures, which becomes very simple for zero total angular momentum. The Slater decomposition allows us to define associated modes, and it turns out that they have identical one- and two-mode entropies. Furthermore, we show that specific single configurations describe maximally entangled states when the total angular momentum is zero. The numerical shell model study of two valence neutrons in the sd shell is revealed that the one-body entanglement entropy of the ground state is close to the maximal value, and the associated modes have the largest mutual information.
A discussion of the seniority quantum number in many-body systems is presented. The analysis is carried out for bosons and fermions simultaneously but is restricted to identical particles occupying a single shell. The emphasis of the paper is on the possibility of partial conservation of seniority which turns out to be a peculiar property of spin-9/2 fermions but prevalent in systems of interacting bosons of any spin. Partial conservation of seniority is at the basis of the existence of seniority isomers, frequently observed in semi-magic nuclei, and also gives rise to peculiar selection rules in one-nucleon transfer reactions. Key words: quantum mechanics, many-body systems, seniority, nuclear shell model, interacting bosons
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 4He^{4}\mathrm{He}4He and 6He^{6}\mathrm{He}6He. 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 4He^{4}\mathrm{He}4He and 6He^{6}\mathrm{He}6He 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 6He^{6}\mathrm{He}6He. 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.
Abstract In this study, we explore the entanglement and correlation in two-nucleon systems using isospin formalism. With the help of Slater decomposition, we derive analytical expressions for various entanglement measures. Specifically, we analyse the one- and two-mode entropies, mutual informations, and a basis-independent characteristic known as the one-body entanglement entropy. To understand the impact of pairing, we consider interactions involving isovector and isoscalar L = 0 pairing terms. Our findings show that certain pairing interactions can maximize one-body entanglement entropy of ground states when both total angular momentum and total isospin have zero projections. We provide numerical examples for the sd shell and explore the mutual informations in LS coupled and jj coupled single-particle bases. We find that the shell structure and angular momentum coupling significantly impact the measures of entanglement. We outline the implications of conserving angular momentum and isospin on one-mode entropies, irrespective of particle number.
We examine the mode entanglement and correlation of two fermionic particles. We study the one- and two-mode entropy and a global characteristic, the one-body entanglement entropy. We consider not only angular momentum coupled states with single configuration but use the configuration interaction method. With the help of the Slater decomposition, we derive analytical expressions for the entanglement measures. We show that when the total angular momentum is zero specific single configurations describe maximally entangled states. It turns out that for a finite number of associated modes the one- and two-mode entropies have identical values. In the shell model framework, we numerically study two valence neutrons in the sdsdsd shell. The one-body entanglement entropy of the ground state is close to the maximal value and the associated modes have the largest mutual information.