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Geometric Origin of Quantum Entanglement

Marco Zaopo

Abstract

We investigate massless representations related to the extension of Poincarè group constructed in [1]. These representations differ from Wigner's ones of standard Poincarè group because the stabilizer of lightlike orbits has extra degrees of freedom. The unitary irreducible representations (UIRs) of massless particles in this extension must decompose as a direct sum of a massless forward (positive zeroth component momentum) and massless backward (negative zeroth component momentum) Wigner's representations linked by internal two valued degree of freedom. We prove that these representations are unitarily equivalent to entangled states of two qubits. This provides a geometric origin of quantum entanglement for photons in the framework of quantum field theory: photons appear as superpositions of backward and forward propagating electromagnetic waves depending on a two valued parameter and this dependency gives rise to correlations between the values of local observables identical to those experienced with an entangled state of two qubits. Finally we describe an experiment capable of distinguishing the two different values of the parameter that links backward and forward massless representations providing experimental falsification of the theory.

Geometric Origin of Quantum Entanglement

Abstract

We investigate massless representations related to the extension of Poincarè group constructed in [1]. These representations differ from Wigner's ones of standard Poincarè group because the stabilizer of lightlike orbits has extra degrees of freedom. The unitary irreducible representations (UIRs) of massless particles in this extension must decompose as a direct sum of a massless forward (positive zeroth component momentum) and massless backward (negative zeroth component momentum) Wigner's representations linked by internal two valued degree of freedom. We prove that these representations are unitarily equivalent to entangled states of two qubits. This provides a geometric origin of quantum entanglement for photons in the framework of quantum field theory: photons appear as superpositions of backward and forward propagating electromagnetic waves depending on a two valued parameter and this dependency gives rise to correlations between the values of local observables identical to those experienced with an entangled state of two qubits. Finally we describe an experiment capable of distinguishing the two different values of the parameter that links backward and forward massless representations providing experimental falsification of the theory.

Paper Structure

This paper contains 12 sections, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. UIRs of Extended Poincaré Group
  3. Non Massless Representations
  4. Massless Representations
  5. Equivalence of Massless UIRs with Entangled Two--Qubits States
  6. Experimental Proposal: Tomography of $\varepsilon$ Eigenvalue
  7. Specialisation to a fixed direction and single--mode photons
  8. Preparation stage
  9. Measurement stage and dependence on $\varepsilon$
  10. Interpretation and falsifiability
  11. Relation to existing single--particle entanglement experiments
  12. Conclusion

Figures (1)

  • Figure 1: Schematic of the proposed single--photon experiment. BS$_1$ prepares a superposition of counterpropagating modes $|+\rangle,|-\rangle$ (forward/backward sectors). The elements $H,V$ encode polarisation $|H\rangle,|V\rangle$ on each arm, implementing the state $\tfrac{1}{\sqrt2}(|+\rangle\otimes|H\rangle+\varepsilon|-\rangle\otimes|V\rangle)$. BS$_2$ measures $\sigma_x$ on the direction qubit; in each output, a HWP (Half Wave Plate) + PBS (Polarization Beam Splitter) measures $\sigma_x$ on polarisation. Joint detector clicks ($D_i$) estimate $\langle\sigma_x^{(\text{dir})}\otimes\sigma_x^{(\text{pol})}\rangle=\varepsilon$.