Quantum Entanglement and the Lorentz Group
Résumé
The Lorentz metric represented by the diagonal matrix G = diag(1,-1,-1,-1) acts on Minkowski space-time quadrivectors. In the language of Quantum Information the operator G can be viewed as an entangling gate this because it acts in a similar way as the Controlled-Z gate on the computational basis of a 2-qubit separable quantum vector. The entangling power corresponds to the fact that the resulting vector, considered as a 2-qubit vector, cannot be put into a Kronecker product of two 1-qubit vectors.
One can represent the generators of the Lorentz group by 4x4 matrices. An example is given by the Lorentz rotation considered as a controlled 2-qubit gate structure similar to the Control-NOT logical quantum gate.
The Minkowski metric, according to the positive energy theorem in General Relativity, represents the ground state of the metric field, its maximally entangled character analogous to the singlet Bell state suggests that entanglement and hyperbolic space are intimately interconnected.
Quantum Information associated with the logic linear algebraic structures, as proposed in Eigenlogic, could provide a new tool to revisit the geometric structures (rotational and hyperbolic) of the Lorentz group linking Quantum Mechanics and Relativity Theory.
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