Entanglement Core in Quantum Physics

• Entanglement core is the concentrated region of quantum correlations, defined via structures such as localized potential wells, core tensors, or modal separations.
• It enables practical applications in quantum control, allowing for tunable entanglement extraction, efficient basis selection, and computational optimization in multiple physics domains.
• Core mechanisms are engineered across systems—ranging from quantum dots and multi-qubit systems to photonic fibers and nuclear structures—highlighting their broad experimental and theoretical relevance.

Entanglement core refers to a structural locus or mechanism that concentrates and controls quantum correlations (entanglement) within a physical system, typically by either localizing entanglement to specific regions or degrees of freedom or providing an operational core tensor representation that stratifies multipartite entanglement classes. Across atomic, condensed-matter, photonic, nuclear, and quantum information contexts, the entanglement core embodies the modes, basis states, or device geometries that either mediate or sharply modulate quantum entanglement, often allowing external tuning, extraction, or identification of maximally correlated subsystems.

1. Local Pinning of Entanglement in Core–Shell Potentials

In core–shell quantum dots, the entanglement core physically manifests as a narrow, deep central well within a bounded potential, governing the ground-state entanglement of few-particle systems. The “disappearing inner well” (DIW) potential is parameterized by its inner-core width $W^{iw}=w-R$ and shell width $W^{ow}=w+R$, with $w=5a_0$, and depth $V_0=-10$ Ha. As $R$ increases from $4a_0$ to $5a_0$, the core width contracts, leading to several nontrivial effects:

• Pinning: Even when $W^{iw}$ is orders of magnitude smaller than $W^{ow}$ (e.g., $W^{iw} \approx 0.04a_0$), a residual core continues to halve the ground-state linear entropy $L_{DIW}(R_c)\approx0.19$, relative to a no-core reference $L_{OWO}(R_c)\approx0.38$.
• QPT-like signatures: Two distinct transitions are observed: a second-order–like migration of the ground state from the core to shell (minimum of $\partial^2 E_0/\partial R^2$ at $R^*\approx4.90a_0$, discontinuity in $\partial L/\partial R$), and a derivative jump when the core vanishes abruptly at $R=5a_0$.
• Tunability: Asymmetric core offsets can double entanglement; relocating the core well (e.g., $w_a=2a_0$) yields $L_{max}\approx 2L_{symmetric}$.

This detailed interplay allows the “entanglement core”—the inner well—to serve as an electrostatic switch or modulator for electron correlations and quantum information resources, establishing local control over few-body entanglement (Marchisio et al., 2012).

2. Core Tensor Structure in Multi-Qubit Systems

For multipartite quantum states, the entanglement core is formalized as the Higher Order Singular Value Decomposition (HOSVD) core tensor $G$ obtained by simultaneous diagonalization of all one-body reduced density matrices (RDMs). For an $n$-qubit pure state $|\Psi\rangle$, $G$ is defined via local-unitary actions as $$|\Psi\rangle=(U^{(1)}\otimes\cdots\otimes U^{(n)})\,|G\rangle$$, where all one-body RDMs are diagonal.

• Classification: The first $n$-mode singular values $\{\sigma_1^{(k)2}\}_{k=1}^{n}$ stratify the LU-orbit space into finitely many entanglement families, separating fully separable ($\sigma_1^{(k)2}=1$) from GHZ-type ($\sigma_1^{(k)2}=1/2$) states.
• Concurrency shortcut: For $n=3$ and $n=4$, all-orthogonality constraints reduce to “concurrency of three lines”—a determinant criteria that enables explicit construction of special core tensors without solving quartic polynomial systems.
• Scaling: The identification of special cores grows only polynomially with $n$ (specifically, $O(n^4)$ in determinant evaluations), yielding a finite set of representative cores for moderate qubit numbers (Choong et al., 2023).

This tensor-based entanglement core formalism enables concrete and computationally efficient stratification of multipartite entanglement, particularly in systems with many qubits.

3. Mode Consolidation: Core–Halo Separation in Quantum Field Theory

In quantum field theory applications, notably in the vacuum of free scalar fields, the entanglement core corresponds to a set of $(1_A\times 1_B)$ mixed-mode “core pairs”—constructed via local symplectic transformations—which concentrate all logarithmic negativity between two regions. The procedure is as follows:

• Symplectic spectrum analysis: Via partial transpose and diagonalization, one identifies $n_-$ symplectic eigenvalues $<1$ indicating entangled core pairs, while the remainder corresponds to a separable “halo.”
• Core–halo split: The total state’s covariance matrix can be block-diagonalized as $\sigma'=\sigma_c\oplus \sigma_h + Y$, where $\sigma_c$ collects all extractable negativity, $\sigma_h$ is AB-separable, and $Y$ captures residual classical correlations.
• Hierarchy and scaling: Entanglement in the core-pairs decays exponentially with pair number index $\ell$, i.e., $\mathcal{N}_\ell\sim e^{-\alpha\ell}$; bound (inaccessible) halo entanglement persists as step-plateaux in the continuum limit.

This structural clarity in the negativity core enables efficient resource counting for entanglement extraction and elucidates the distinction between distillable and bound entanglement in extended systems (Klco et al., 2021).

4. Quantum Optical Cores: Multi-Core Fiber Platforms

In photonic quantum information, the physical entanglement core is realized in multi-core optical fibers, where each core defines a logical mode within an $N$-dimensional entangled qudit state.

• Architecture: Pump preparation and SPDC emission are implemented with fiber-integrated multi-core components; a $4\times 4$ fiber beam splitter produces a uniform superposition among four cores, generating the canonical ququart state $$|\Psi\rangle=\frac{1}{2}(|0\rangle_s|0\rangle_i+\cdots+|3\rangle_s|3\rangle_i)$$.
• Measurement: Coincidence detection across core pairs and fiber-based interferometric transformations allow full tomographic reconstruction, verifying entanglement fidelity $F=0.789\pm0.007$.
• Scalability: The multi-core entanglement core platform offers high spectral brightness ($\approx 3.5\times 10^5$ pairs/s/mW/nm), loss rates $\lesssim 0.4$ dB/km at $1550$ nm, and is fully compatible with space-division multiplexing for next-generation quantum networks (Gómez et al., 2020).

This physical core mechanism is essential for compact, scalable, and stable generation and distribution of path-encoded multi-dimensional entanglement in fiber-based telecommunications infrastructures.

5. Molecular and Spectral Entanglement Cores in Fiber Systems

Entanglement core mechanisms extend to engineered nonlinear fiber systems:

• Gas-Filled Hollow-Core Fibers: In frequency conversion via molecular modulation, the traveling molecular coherence $|s\rangle$ acts as a quantum “entanglement core,” mediating resonant transfer of photon–photon entanglement. The effective Hamiltonian $$H_\xi^I=G_U\xi^* e^{i\Delta\beta z} a_M^\dagger a_U + \text{h.c.}$$ swaps populations and correlations, with concurrence evolving as $C_{I-M}(t)=|\cos(G_U|\xi|t)|$ and $C_{I-U}(t)=|\sin(G_U|\xi|t)|$, achieving near-unity transfer fidelity (Gonzalez-Raya et al., 10 Sep 2024).
• Spectral Entanglement in Hollow-Core PCF: Four-wave mixing in noble-gas–filled inhibited-coupling hollow-core photonic crystal fiber yields spectral entanglement cores tunable via fiber geometry, gas pressure, and length. Schmidt number $K$ quantifies entanglement, with $K\approx2.5$ (highly entangled) to $K\approx1.1$ (nearly factorable) achievable over $40$–$100$ cm via adjustments to $P$ and $L$ (Cordier et al., 2018).

Both scenarios reveal that specific core degrees of freedom—molecular coherence or engineered dispersion—focus and mediate quantum correlations, enabling controlled entanglement conversion and source engineering.

6. Basis-Localized Entanglement Cores in Nuclear Structure

In nuclear many-body theory, the entanglement core emerges as a localized region in single-particle basis space, represented especially in the variational-natural (VNAT) basis:

• Measures: Single-orbital entropy $S_i$, two-orbital mutual information $I_{ij}$, and negativity $N_{ij}$ probe quantum correlations between orbitals.
• Core–Valence Tensor Structure: In the VNAT basis, core and valence nucleons become almost completely decoupled, as seen in $^6$He where the $^4$He core's $S_i$ and $I_{ij}$ are unchanged by the addition of halo neutrons; the wavefunction decomposes approximately as $$|\Psi_{6He}\rangle\simeq|\Psi_{core}(^{4}He)\rangle\otimes|\Psi_{valence}(2n)\rangle$$.
• Computational Implications: Minimizing inter-sector entanglement accelerates convergence and enables DMRG-style active space selection and quantum/classical hybrid computation (Robin et al., 2020).

Recognition and operationalization of the entanglement core in basis selection provides foundational guidance for the design of efficient ab initio and hybrid workflow algorithms in nuclear physics.

7. Entanglement Core in Strong-Field Ionization

In strong-field atomic ionization, the entanglement core is identified in the directional Hilbert subspace correlations between ionized electron and parent ion. The time-dependent entanglement entropy, computed from directional reduced density matrices, reveals:

• Regime Dependence: Below the tunneling threshold $F_{tu}\approx 0.0624$ a.u., the entanglement returns to baseline after the pulse; above $F_{tu}$, the total entanglement core $S_{ent}$ decreases due to loss of transverse correlations, even when longitudinal entanglement grows.
• Separation of Degrees of Freedom: The core concept clarifies exactly when and how the ionized electron “forgets” its initial correlations with the ion, offering insight into sub-cycle quantum dynamics and measurement protocols (Majorosi et al., 2017).

8. Synthesis and Implications

The entanglement core constitutes the locus (physical or abstract) where quantum correlations are concentrated, and whose manipulation—be it by external control parameters, tensor decomposition, mode selection, or device geometry—affords tunability or efficient identification of quantum resources. Across domains, the core framework enables:

• Explicit separation of extractable vs. bound entanglement (core–halo dichotomy)
• Stratification and categorization of complex multipartite entanglement
• Device-level engineering of entanglement switchable by geometric or optical controls
• Optimized basis selection for computational and hybrid quantum simulations

A plausible implication is that future nanoscale quantum hardware and quantum network design will increasingly rely on explicit engineering of entanglement cores—whether as localized wells, core tensors, modal subsets, or dedicated device regions—to maximize both the controllability and the efficiency of entangled-state generation, distribution, and computation.