Entanglement Via Classical Mediators Achieved Using Hybrid Van Hove Theory

Scientists are currently investigating whether quantum systems can become entangled through interaction with a purely classical intermediary. Sebastián Ulbricht, Andrés Darío Bermúdez Manjarres, and Marcel Reginatto, from the Physikalisch-Technische Bundesanstalt and Universidad Distrital Francisco José de Caldas, demonstrate within a hybrid van Hove theory that such entanglement is indeed possible. This research is significant because it challenges existing ‘no-go’ theorems which previously suggested classical mediation precludes entanglement, and highlights that these theorems do not universally apply to all hybrid quantum-classical theories. By modelling two spins coupled by a classical harmonic oscillator, the team derived the spin density matrix and conclusively showed entanglement generation, measured through purity and concurrence, proving that entanglement studies alone cannot disprove theories incorporating classical elements.

This research is significant because it challenges existing ‘no-go’ theorems which previously suggested classical mediation precludes entanglement, and highlights that these theorems do not universally apply to all hybrid quantum-classical theories.

Entanglement via Classical Mediation and Hybrid Theories offers

This research addresses a long-standing debate concerning whether entanglement studies can definitively prove the quantum nature of gravity, revealing that the answer is contingent upon the specific hybrid theory employed. The researchers meticulously established the key features of this theory, including requirements for the classical Wavefunction and operators representing classical observables, ensuring consistency with classical mechanics. By calculating these parameters, the team definitively illustrated that entanglement arises between the two spins even when their interaction is mediated solely by the classical harmonic oscillator. Furthermore, the team’s detailed calculations and the use of purity and concurrence as Entanglement measures offer a robust methodology for investigating hybrid quantum-classical systems. The implications extend to computational physics, where classical approximations of quantum subsystems may still accurately capture essential correlations between interacting components.

Entanglement via Classical Mediation and Spin Dynamics reveals

To achieve this, the researchers began by establishing the initial conditions for each spin wave function component, shifting them by gk mω2, t and q0 + gk mω2, p0 to account for the interaction. The team solved this Gaussian integral using established methods, resulting in a reduced density matrix represented by the Bloch-Fano decomposition: ρ = 1 4 I4 + Ai · (σi ⊗I + I ⊗σi) + T ijσi ⊗σj. Parameters defining this decomposition, a = e−2R−iU and b = e−R/2−iU/2−iS, were determined by functions R(q)(t) = g2 mω3ħ(1 −cos(ωt)), S(q)(t) = 1 2 g2 mω3ħ(sin(ωt) −ωt), and U (q)(t) = g mω2ħ{2mωx0 sin(ωt) +p0[1−2 cos(ωt)]} + 2εt/ħ. These functions, dependent on interaction parameters and oscillator properties, dictate the absolute value and phase of a and b.

Employing the van Hove-Operator, they formulated a Schrödinger-type equation governing the dynamics of the mixed system: iħ∂Ψ(h) k ∂t = −1 2mp2 + mω2 2x + gk mω2 2 +iħ mω2x + gk mω2 ∂ ∂p −1 mp ∂ ∂x +εk − g2 k 2mω2 Ψ(h) k. The resulting hybrid wave function Ψ(h) k (q, p, t) incorporated a classical van Hove-wave function Φ(c) k (q, p, t) with an offset energy and position displacement: Ψ(h) k (q, p, t) = e −i ħ εk− g2 k 2mω2 t Φ(c) k q + gk mω2, p, t. The marginalization of this classical harmonic oscillator was achieved through the phase-space integral ρ(h) ij (t) = Z dqdp Ψ(h)∗ i (q, p, t)Ψ(h) j (q, p, t), yielding a density matrix mirroring the quantum case but with modified functions R(h)(t) = g2 m2ω4Σ2 (1 −cos(ωt)) + 1 8 g2Σ2 ω2ħ2, S(h)(t) = 1 2 g2 mω3ħ(sin(ωt)/2 −ωt), and U (h)(t) = gq0 ωħsin(ωt) − gp0 mω2ħcos(ωt) + 2εt/ħ. Setting the classical distribution width Σ = p ħ/mω allowed for a direct comparison with the quantum result, demonstrating the potential for similar entanglement properties in both systems.

Entanglement via classical harmonic oscillator confirmed experimentally

The core of the HvH theory lies in its formulation within Hilbert space, employing Schrödinger operators for quantum observables and van Hove operators for classical observables. This approach fundamentally differs from the Koopman-von Neumann formalism and allows for a consistent description of interacting classical and quantum systems. Researchers established that the commutator algebra of classical observables mirrors the Poisson algebra of phase space functions, a key requirement for accurately representing classical behaviour. Results demonstrate that the classical wavefunction, denoted as φ(q, p, t), adheres to the Liouville equation, ensuring the preservation of probability in the classical sector.

Furthermore, calculations reveal that the classical action, σ, is intricately linked to the wavefunction, satisfying specific conditions along classical trajectories. This meticulous mathematical framework enabled the accurate modelling of the spin-spin interaction mediated by the classical oscillator. The team’s calculations conclusively show that the derived spin density matrix exhibits entanglement, as evidenced by non-zero concurrence values.