Proton-Neutron Entanglement Entropy

• Proton-neutron entanglement entropy is defined as the von Neumann entropy of reduced density matrices that quantifies the quantum correlations between proton and neutron subspaces.
• Studies reveal that entanglement peaks at N=Z due to shell structure and pairing effects, while spin-orbit and tensor forces significantly modulate its magnitude.
• Computational methods like Lanczos diagonalization and IMSRG demonstrate how reduced proton-neutron entanglement can optimize resource demands in quantum simulations.

Proton-neutron entanglement entropy quantifies the quantum correlations between proton and neutron degrees of freedom in nuclear many-body systems. Defined mainly as the von Neumann entropy of reduced density matrices following a partition into proton and neutron subspaces, it provides insight into the interplay between shell structure, pairing, collective excitations, and short-range nuclear forces. This metric is now routinely computed in shell-model, ab initio, and pair-condensate frameworks, as well as in the analysis of few-nucleon scattering, and is employed to characterize mode entanglement, tensor-force effects, cross-shell mixing, and computational tractability in quantum simulations.

1. Formal Definition and Computation

In the shell-model or ab initio framework, the full many-body wavefunction $|\Psi\rangle$ is expressed as a superposition of proton and neutron Slater determinants,

$$|\Psi\rangle = \sum_{\mu,\sigma} c_{\mu\sigma} | \mu_\pi \rangle \otimes | \sigma_\nu \rangle,$$

where the $c_{\mu\sigma}$ are configuration-interaction amplitudes. The reduced density matrix for the proton sector is constructed by tracing over neutron states: $$(\rho_p)_{\mu\mu'} = \sum_\sigma c_{\mu\sigma} c^*_{\mu'\sigma}.$$ Diagonalizing $\rho_p$ yields eigenvalues $\lambda_i \in [0,1]$, constrained by $\sum_i \lambda_i = 1$. The proton-neutron entanglement entropy is the von Neumann entropy of $\rho_p$ (or, equivalently, $\rho_n$), given by

$$S_{pn} = -\mathrm{Tr}(\rho_p \ln \rho_p) = -\sum_i \lambda_i \ln \lambda_i,$$

where the logarithm is usually base 2 for results in bits (Pérez-Obiol et al., 2023, Johnson et al., 2022, Shinde et al., 10 Jan 2026).

Operationally, computation involves explicit construction of the many-body basis, solution of the eigenvalue problem via Lanczos diagonalization (shell model), or IMSRG transformation and occupation-number representation (VS-IMSRG), followed by tracing out subspace degrees of freedom and entropy evaluation.

2. Systematics in Nuclear Shells and Isotopic Chains

Extensive numerical studies in the $p$-, $sd$-, and $pf$-shells reveal consistent patterns:

• Peak at $N=Z$: In Ne, Mg, Si chains, $S_{pn}$ sharply peaks at self-conjugate nuclei (e.g., $^{24}$Mg, $^{28}$Si), typically reaching $S_{pn} \sim 1.2$–$1.8$ bits for ground states (central interaction), then falls off by $0.1$–$0.2$ bits per neutron added or removed (Shinde et al., 24 Jun 2025).
• Shell Structure Effects: The shell gap and its monopole contributions set the available valence space, determining entanglement: closed-shell configurations exhibit suppressed $S_{pn}$ ($^{20}$Ne, $^{28}$Si), while mid-shell nuclei (e.g., $^{24}$Mg, $^{24}$Ne) are more entangled (Pérez-Obiol et al., 2023, Johnson et al., 2022).
• Role of Spin-Orbit and Tensor Forces: Inclusion of spin-orbit splits reduces $S_{pn}$ by $\sim20\%$; tensor interactions partially restore entanglement by attracting $\pi j_>$ and $\nu j_<$ orbitals (Shinde et al., 24 Jun 2025). Table summarizing central, spin-orbit, and tensor decompositions for $N=Z$ nuclei:

Nucleus	$S^{\text{central}}$	$S^{\text{cent+LS}}$	$S^{\text{full}}$
$^{20}$Ne	1.30	1.02	1.12
$^{24}$Mg	1.55	1.22	1.37
$^{28}$Si	1.80	1.48	1.65
$^{32}$S	2.00	1.68	1.88

• Suppression at $N>Z$: For isotopes with neutron excess (e.g., $^{10}$Be, $^{26}$Ne), $S_{pn}$ drops by $\sim50\%$ compared to $N=Z$ analogs; this suppression is absent for generic random two-body interactions, indicating its origin in realistic nuclear forces, especially isospin-dependent monopoles (Johnson et al., 2022).

3. Interpretation: Pairing, Collectivity, and Shell Evolution

The low proton–neutron entanglement in most nuclei reflects the dominance of like-particle (proton–proton, neutron–neutron) pairing and the approximate SU(2) isospin symmetry. Enhanced $S_{pn}$ correlates with collective modes—particularly cross-shell excitations and intruder configurations in the “island of inversion” (IoI) region (Shinde et al., 10 Jan 2026):

• IoI Behavior: For even-A Ne, Mg, Si isotopes near $N=20$, $S_{pn}$ drops approaching shell closure, then rises inside the IoI, tracking the onset of strong cross-shell mixing: $S_{pn}(^{32}\text{Mg}, 0^+)\approx 1.0$ bits, increasing to $1.4$ bits for $2^+$ states. In Si, $S_{pn}(^{34}\text{Si}, 0^+)\approx 0.4$, consistent with isolation from IoI.
• Pair Condensates: Variational optimization in $N=Z$ nuclei yields low Shannon entropies ($S\lesssim 0.1$), signifying near-Hartree-Fock (uncorrelated) limits. Like-nucleon pair condensates (NN or PP) in semi-magic chains present much larger entropies ($S \sim 0.2$–$0.8$), characteristic of strongly entangled superfluid phases (Liang et al., 2024).
• External Tuning and Phase Transition: By varying artificial pairing strengths, a rapid transition in $S$ is observed: $S$ increases to $\sim 0.25$ in T=1–dominated paired phase, then sharply to $\sim 0.5$ for pure T=0 (spin-triplet) pairing, resembling an order-parameter jump in Landau theory (Liang et al., 2024).

4. Entanglement in Few-Nucleon Scattering and Short-Range Correlations

In two-body scattering, proton–neutron spin entanglement (entanglement power) is evaluated via the averaged von Neumann entropy of reduced spin density matrices:

$$\epsilon_1(k) = \frac{1}{6}\sin^2[2(\delta_0(k)-\delta_1(k))]$$

where $\delta_0(k)$ and $\delta_1(k)$ are $s$-wave singlet/triplet phase shifts (Kirchner et al., 2023). At threshold, $\epsilon_1$ is suppressed due to the near-equality of singlet/triplet scattering lengths (Wigner SU(4) symmetry); maxima correspond to optimal mixing of channels, but absolute values remain small ($\sim$0.03).

For short-range correlations (SRCs), the orbital entanglement entropy $S_{ij}^{\rm SRC}$ relates directly to nuclear contacts $c_{ij}^\alpha$ through the reduced occupation probability

$$S_{ij}^{\rm SRC} = -\bigl[c_{ij}^\alpha\ln c_{ij}^\alpha + (1-c_{ij}^\alpha)\ln(1-c_{ij}^\alpha)\bigr].$$

Empirically, in symmetric nuclei, the ratio $S_{pp}/S_{np} \simeq c_{pp}/c_{np} \simeq 2.0$, holding across a wide range of $A$ values (Kou et al., 2023). This scaling reflects the relative abundances of PP and NP pairs in dominant $s$-wave and deuteron channels.

5. Mutual Information, Relative Entropy, and Complementary Measures

Beyond the standard von Neumann entropy, mutual information $I(A:B)$ and quantum relative entropy $S(\rho \| \sigma)$ provide additional quantification of both quantum and classical correlations:

• Mutual information: $I(A:B) = S(A) + S(B) - S(AB)$ captures total correlations between partitions, sensitive to pairing and deformation (Pérez-Obiol et al., 2023, Shinde et al., 10 Jan 2026).
• Quantum relative entropy: $$S(\rho \| \sigma) = \mathrm{Tr}[\,\rho\,(\ln \rho - \ln \sigma)\,]$$ measures state distinguishability; its symmetrized Jensen–Shannon divergence defines a metric and is bounded.

These metrics enhance the resolution of collective phenomena, cross-shell mixing, and quantum phase transitions in excited states, with $S_{pn}$ serving as a robust subset (Shinde et al., 10 Jan 2026).

6. Computational and Quantum Simulation Implications

The consistent finding—across shell-model, ab initio, and variational pair-condensate approaches—is that the proton–neutron partition yields the minimal entanglement, often substantially lower than other equipartitions (e.g., those splitting angular momentum quantum numbers). This suggests protons and neutrons may be efficiently treated in separate quantum registers. For quantum algorithms in the noisy intermediate-scale quantum (NISQ) era, this insularity limits required cross-register entanglement to $\lesssim 1$ bit, reducing variational circuit depth and measurement overhead (Pérez-Obiol et al., 2023, Johnson et al., 2022).

The small number of significant Schmidt components in the proton–neutron cut motivates tensor network truncations, low-rank factorization, and density-matrix renormalization-group (DMRG)–like schemes, particularly for neutron-rich heavy nuclei. This provides an algorithmic rationale for partitioning Hilbert space based on actual entanglement structure, not solely shell-model quantum numbers.

7. Physical Insights and Universality

Proton–neutron entanglement entropy traces emergent nuclear structure from a quantum-information standpoint. Its suppression reflects robust isospin symmetry and like-nucleon pairing, while deviations (such as in IoI and collective modes) serve as direct probes of shell-gap evolution and cross-species correlations. Scaling laws in short-range entanglement, as evidenced in $S_{pp}/S_{np} \approx 2.0$, connect to universal features of nuclear contacts. Exceptionally, artificially tuned pair condensates allow for high-entropy, strongly entangled phases rarely encountered in atomic nuclei but relevant for cold-atom analogs.

Entanglement entropy thus provides not only a technical metric for computational efficiency but also a quantitative probe of fundamental nuclear phenomena such as pairing, shell evolution, collectivity, phase transitions, and short-range interaction universality.