Entangled Bias Forces

• Entangled bias forces are effective interactions arising from nonlocal quantum and statistical correlations that alter subsystem dynamics beyond mean-field effects.
• They manifest in diverse settings—from atom–atom potentials and stochastic control to cosmological structure formation—with distinct scaling laws and experimental signatures.
• Their study enables enhanced modeling and control in complex systems, requiring precise entangled state preparation and sensitive force measurements to distinguish them from conventional forces.

Entangled bias forces are a class of effective interactions, corrections, or couplings that arise when entanglement or quantum/statistical correlations between subsystem degrees of freedom dynamically alter the form, magnitude, or direction of system-level forces. Manifesting in diverse settings—ranging from quantum atom–atom potentials, stochastic control of interacting particles, cosmological structure formation, to causal models of Bell nonlocality—these forces encode the influence of nonlocal or path-dependent correlations on the effective dynamics of subsystems. Unlike mean-field or independent-bias effects, entangled bias forces fundamentally depend on system-wide state or trajectory entanglement and typically cannot be decomposed into sums of local or separable components.

1. Quantum Field–Mediated Entanglement Forces Between Atoms

In the nonequilibrium quantum field theory approach, Behunin and Hu (Behunin et al., 2010) identified an “entanglement force” emergent between two neutral atoms, modeled as 3D harmonic oscillators. When the internal oscillator degrees of freedom of both atoms are initialized in an entangled, dipole-squeezed state with nonzero $\langle Q_1 Q_2 \rangle$, the field-mediated atom–atom influence functional acquires a novel cross-term at first order in $q_1 q_2$, distinct from London–van der Waals and Casimir–Polder forces. The force on atom 1 located at distance $z$ from atom 2 (assumed static at the origin) is

$$f_z^{\rm ent}(\tau) = -\frac{q_1 q_2}{4\pi} \left[z^3\left(\frac1z\frac d{dz}\right)^3 +5 z\left(\frac1z\frac d{dz}\right)^2\right] \frac1z\,g_{\rm ent}(\tau,\tau-z)\,,$$

where $g_{\rm ent}$ encodes the initial entangled correlator of $Q_1$ and $Q_2$. In the late-time, near-field regime ($\Omega z\ll 1$), the scaling becomes $f_z^{\rm ent} \propto 1/z^2$ (for isotropic squeezing) or $1/z^4$ (anisotropic case), much longer-ranged than ordinary $1/z^7$ (London) or $1/z^8$ (Casimir–Polder) scaling. Diagrammatically, this force results from a single exchange of a retarded photon, enabled by the pre-existing $\langle Q_1 Q_2 \rangle$, in contrast to the second-order, two-exchange nature of ordinary dispersion forces.

The entanglement force can be comparable to or exceed London dispersion interactions for suitably chosen squeeze parameters at nanometer separations with atomic transition frequencies in the optical regime. Detection would require preparation of stable entangled oscillator pairs with nonzero cross-correlators at nm-scale separation and force sensing at the sub-piconewton level in the presence of competing backgrounds.

2. Entangled Bias Forces in Stochastic Control—The Entangled Schrödinger Bridge

In the context of stochastic optimal control of multi-particle systems, the Entangled Schrödinger Bridge Matching (EntangledSBM) framework (Tang et al., 10 Nov 2025) defines the entangled bias forces as the optimal (in KL divergence) controls for large-scale stochastic dynamical systems, where each particle’s force at time $t$ is a function of the full joint configuration $(R_t, V_t)$,

$b^i_t = b^i_\theta(R_t, V_t)\,,$

entangling all particle velocities and positions. The controlled SDE,

$$dX_t = \left[f(X_t) + b_t(X_t)\right]\,dt + \Sigma\,dW_t\,,$$

steers the system from an initial distribution $\pi_{\mathcal A}$ to a target $\pi_{\mathcal B}$ by learning bias forces that dynamically encode inter-particle coupling. The bias is parameterized as

$$b^i_\theta = \alpha^i_\theta(R, V) \hat{s}_i + (I - \hat{s}_i \hat{s}_i^\top) h^i_\theta(R, V)\,,$$

where $$\hat{s}_i = \nabla_{r^i} \log \pi_{\mathcal B}/\|\nabla_{r^i} \log \pi_{\mathcal B}\|$$, ensuring both direct progress and flexible rotation in particle paths.

The associated convex cross-entropy loss over path space,

$$\mathcal{L}_{\rm CE}[b] = D_{\rm KL}(\mathbb{P}^\star \| \mathbb{P}^{b_\theta})\,,$$

possesses a unique minimizer $$b^\star = \arg\min_b D_{\rm KL}(\mathbb{P}^b\|\mathbb{P}^\star)$$, where $\mathbb{P}^\star$ is the Schrödinger bridge measure. Allowing “entangled” bias (not factorized over particles) strictly enlarges the control class, provably reducing the KL cost. Empirically, EntangledSBM achieves superior accuracy in both high-dimensional cell-trajectory and molecular transition-path sampling tasks, with velocity-conditioned entangled biases providing 2–4× reductions in pathwise MMD/Wasserstein error and outperforming SMD, PIPS, TPS-DPS on non-separable systems.

3. Primordial Entanglement–Induced Bias Forces in Cosmology

The phenomenon of entangled bias forces appears in cosmological structure formation as a quantum relic of inflationary dynamics (Tejerina-Pérez et al., 23 Mar 2024). During inflation, graviton–inflaton interaction generates Bell-like polarization entangled states of gravitons, which—with decoherence of one polarization mode—survives as a specific projection on the anisotropy of primordial tensor-scalar-scalar and tensor exchange correlators. This process seeds a correction to the Lagrangian halo bias,

$b_L(k) = b_G + \Delta b_{\rm ent}(k)\,,$

where

$$\Delta b_{\rm ent}(k) \approx (b_G-1)\,D_P\,A_P\,(k_*/k)^2\,,$$

with $D_P$ encoding the decoherence (entanglement) probability and $A_P$ the amplitude set by inflationary parameters. The correction yields a $k^{-2}$, polarization-dependent modulation in halo bias and can be recast as a nonlocal, anisotropic force kernel $F_{\rm ent}(r)$,

$$F_{\rm ent}(r) = \int \frac{d^3k}{(2\pi)^3} e^{i k\cdot r} \Delta b_{\rm ent}(k)\,,$$

resulting in $F_{\rm ent}(r) \propto r\,\cos2\theta/r^3$, analogous to the gradient of an anisotropic $1/r$ potential. This extra acceleration is a direct consequence of primordial entanglement projected along a surviving polarization mode. The same mechanism predicts a quadrupolar, $k^{-2}$ scaling in intrinsic galaxy alignment cross-correlation functions, serving as a specific observational signature of early quantum gravitational entanglement.

4. Causal Models and “Collider” Bias–Mediated Entangled Forces

A distinct formulation arises in foundational quantum theory. The “Connection across a Constrained Collider” (CCC) mechanism (Price et al., 7 Jun 2024) demonstrates how Bell-type, nonlocal correlations—interpretable as “entangled-bias forces”—can emerge in classical causal networks through conditioned collider variables. In both delayed-choice entanglement swapping (W-shaped diagrams) and standard Bell tests (V-shaped diagrams), imposing a boundary constraint (e.g., post-selecting a Bell state or pre-selecting a prepared entangled state) transforms a statistical collider bias into a robust, nonlocal correlation,

$$P(a,b\,|\,x,y,\,C{=}c^*) = P(a|x,c^*)\,P(b|y,c^*)\,,$$

where “locking” the collider $C$ generates quantum-like cosine-law statistics and can violate CHSH inequalities. While not a force in the dynamical sense, these correlations act as an emergent entangled-bias mechanism, propagating constraint-induced structure through otherwise local influences without requiring superluminal causation.

5. Entanglement and Bias-Driven Forces in Condensed Matter

In Bernal-stacked bilayer graphene, bias voltage terms in the tight-binding Hamiltonian act analogously to an “entanglement driver” (1705.01432). The applied voltage, mapped via an $SU(2)\otimes SU(2)$ (parity-spin) Dirac structure, spreads the lattice–layer entanglement (quantified by the quantum concurrence) away from Dirac points, reshaping the entangled component of the wavefunction across the Brillouin zone. The bias thus controls the spatial distribution and texture of quantum entanglement among lattice-layer degrees of freedom, manifesting as modulation of interlayer coupling and symmetry-breaking in the concurrence profile. Although not strictly a dynamical “force,” such voltage-driven entanglement redistribution parallels the broader theme of entangled bias terms altering subsystem-level physics via system-spanning correlations.

6. Comparative Scaling, Theoretical Properties, and Observational Prospects

The comparative features of entangled bias forces across contexts can be summarized as follows:

Context	Scaling/Structure	Distinctive Features
Atom–atom quantum	$1/z^2$ (isotropic)	First-order, field-mediated, prep-dependent
	$1/z^4$ (anisotropic)	Requires entangled initial state
Stochastic control	Path-dependent, global	Cross-particle dependence, KL-optimality
Cosmology	$k^{-2}$, $\cos2\phi$	Anisotropic, nonlocal, imprint of Bell state
Causal modeling	Non-factorizable	Collider constraint, (apparent) nonlocality
Bilayer graphene	Spread in $k$-space	Bias-driven delocalization of entanglement

A unifying property is the essential dependence on nonseparable initial or pathwise statistical structure—purely local or product-bias forces cannot replicate the described effects, as proven in the strict convexity and uniqueness results for both quantum field theory and optimal control (Behunin et al., 2010, Tang et al., 10 Nov 2025).

Experimental access or falsification generally requires: (i) precise preparation of entangled, non-product initial states; (ii) sensitive measurement or inference of force/response at scales and directions controlled by the entangled bias; (iii) elimination or control of classical or mean-field bias backgrounds. In many-body control and cosmology, entangled bias forces provide routes for more expressive modeling of system dynamics and potentially unique fingerprints of underlying quantum or statistical structure.

7. Conceptual Significance and Theoretical Implications

Entangled bias forces exemplify how global, system-level correlation structure—often originally quantum entanglement, Bell state preparation, or pathwise constraints—can generate distinctive, and sometimes dominant, corrections to subsystem dynamics. These corrections can outscale conventional (local, mean-field, or statistically independent) interactions, serve as unique signatures of underlying entanglement, and inform the foundations of nonlocality, optimal stochastic control, and condensed matter physics.

A plausible implication is that in any sufficiently complex interacting system—quantum, stochastic, or causal—the identification and exploitation of entangled bias or correlation-dependent force structures may be critical for accurate modeling, control, or interpretation of macroscopic observables, with applications spanning quantum technologies, molecular simulation, and cosmological inference.