[Paper Review] Teleportation seen from space-time

Marek Czachor|ArXiv.org|Mar 24, 2008

Quantum Mechanics and Applications8 references4 citations

TL;DR

This paper reveals deep structural analogies between quantum teleportation protocols and 2-spinor calculus in relativistic space-time, showing that Minkowski tetrads mirror Bell bases and that null tetrads enable qubits resistant to relativistic noise. By defining qubits via projections on principal null directions (PNDs), the paper demonstrates a relativistic error correction mechanism that preserves polarization under Lorentz transformations, offering a geometric framework for robust quantum information processing across inertial frames.

ABSTRACT

Formal similarity between Minkowski tetrads and Bell bases allows to think of metric tensors in terms of quantum teleportation protocols. The role of null tetrads for quantum information processing is different. They define qubits resistant to a special kind of noise that occurs if coding and decoding of quantum information is performed in different reference frames. These examples show that mutual links between quantum information and the 2-spinor calculus may be nontrivial and worthy of further studies.

Motivation & Objective

• To explore structural parallels between quantum teleportation and 2-spinor calculus in Minkowski space-time.
• To address the problem of relativistic noise in quantum information caused by momentum-dependent Wigner rotations in different reference frames.
• To propose PND-projected qubits as a solution to frame-dependent depolarization in quantum communication.
• To advocate for new experimental approaches in quantum optics that align with geometric, relativistic principles rather than standard polarizer-based methods.
• To stimulate interdisciplinary research at the intersection of quantum information and relativistic space-time geometry.

Proposed method

• Using Penrose’s abstract index notation to formalize the isomorphism between 2-spinor structures and quantum information protocols.
• Mapping the teleportation protocol’s four steps to the decomposition of a Lorentzian metric into Minkowski tetrads, which formally resemble Bell basis states.
• Analyzing the transformation properties of qubits under SL(2,C) and SU(2) representations, showing that momentum-dependent Wigner phases cause depolarization in linearly polarized states.
• Introducing PND qubits via eigenstates of the Pauli-Lubanski vector projected on null directions, which remain invariant under Lorentz boosts.
• Deriving the transformation matrix Λ(p) for PND qubits and showing it is p-independent, thus preserving polarization under boosts.
• Defining ω-spinors as the physical realization of qubits in momentum space, ensuring unitary evolution under relativistic symmetries.

Experimental results

Research questions

• RQ1 Can the structure of quantum teleportation be formally mapped onto geometric objects in 2-spinor calculus, such as tetrads in Minkowski space-time?
• RQ2 Why do standard qubit encoding schemes (e.g., linear polarization) become depolarized under Lorentz transformations, and what is the origin of this relativistic noise?
• RQ3 How can qubits be defined such that their polarization remains invariant across different inertial reference frames?
• RQ4 What is the role of principal null directions (PNDs) of SL(2,C) transformations in constructing noise-resistant quantum information carriers?
• RQ5 Can the use of PND-projected qubits serve as a form of relativistic error correction in quantum communication?

Key findings

• The formal structure of the teleportation protocol corresponds to the decomposition of a metric tensor into Minkowski tetrads, which are isomorphic to Bell basis states.
• Momentum-dependent Wigner rotations cause depolarization in linearly polarized qubits, introducing fundamental relativistic noise in quantum communication.
• Qubits defined as eigenstates of the Pauli-Lubanski vector projected on PNDs remain invariant under Lorentz transformations, as their rotation angle φ is independent of momentum.
• The transformation matrix Λ(p) for PND qubits becomes p-independent, specifically Λ(p) = diag(e^{-iφ}, e^{iφ}), ensuring consistent polarization evolution.
• The entropy of the spin-reduced density matrix for PND qubits is independent of the Lorentz boost, confirming their robustness against relativistic noise.
• ω-spinors provide a physically consistent representation of qubits in momentum space that transform unitarily under SU(2), resolving the conflict between unitary quantum mechanics and non-unitary position-space spinors.