Einstein velocity addition endows the unit ball of Rn with the structure of a gyrocommutative gyrogroup together with the hyperbolic Hilbert-Klein Riemannian metric. First, we propose a self-contained derivation of these properties by rewriting Einstein addition in the Clifford algebra of the Euclidean space R^n , in which gyroautomorphisms are given by spinors. Then, our main result shows that the Einstein gyrogroup arises naturally in the context of quantum measurements in real spin factors. By embedding the state space of the Jordan algebra R ⊕ R^n into a multipartite rebit system realized as a tensor product of the algebra of 2 × 2 real symmetric matrices, we prove that Lüders state-update operations induce a gyrogroup morphism from the unit ball endowed with Einstein addition to a gyrogroup of density matrices. This extends known results in the low-dimensional case for the rebit and the qubit, and confirms that relativistic velocities and quantum state-updates in real spin factor share a common algebraic structure in arbitrary dimensions.Einstein velocity addition endows the unit ball of Rn with the structure of a gyrocommutative gyrogroup together with the hyperbolic Hilbert-Klein Riemannian metric. First, we propose a self-contained derivation of these properties by rewriting Einstein addition in the Clifford algebra of the Euclidean space R^n , in which gyroautomorphisms are given by spinors. Then, our main result shows that the Einstein gyrogroup arises naturally in the context of quantum measurements in real spin factors. By embedding the state space of the Jordan algebra R ⊕ R^n into a multipartite rebit system realized as a tensor product of the algebra of 2 × 2 real symmetric matrices, we prove that Lüders state-update operations induce a gyrogroup morphism from the unit ball endowed with Einstein addition to a gyrogroup of density matrices. This extends known results in the low-dimensional case for the rebit and the qubit, and confirms that relativistic velocities and quantum state-updates in real spin factor share a common algebraic structure in arbitrary dimensions.

Mar 20, 20260 citations

Metrics of unitary matrices and their applications in quantum information theory

Kai-yan Lee

Jan 1, 20111 citation

Fidelity of Mobius matrices related with Lorentz boosts

Sejong Kim

In this article we consider the extended version of a real counterpart of qubit density matrices, called the Möbius matrix, and we see that it is a normalized Lorentz boost. Using the isomorphic gyrogroup structures between the set $\PP$ of all Lorentz boosts and the Einstein gyrogroup on the open unit ball $\B$ of $\R^n$ we give an explicit formula of the fidelity for Möbius matrices in terms of Lorentz gamma factors.In this article we consider the extended version of a real counterpart of qubit density matrices, called the Möbius matrix, and we see that it is a normalized Lorentz boost. Using the isomorphic gyrogroup structures between the set $\PP$ of all Lorentz boosts and the Einstein gyrogroup on the open unit ball $\B$ of $\R^n$ we give an explicit formula of the fidelity for Möbius matrices in terms of Lorentz gamma factors.

arXiv:1310.1981Oct 8, 20131 citation

Metricity of separable quantum optimal transport with the Hilbert-Schmidt cost

Tomasz Miller

We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.We prove that the square root of the separable quantum optimal transport cost associated with the orthogonal projection onto the antisymmetric subspace defines a genuine distance between density matrices. Equivalently, this establishes the triangle inequality for the order-two Beatty-França quantum optimal transport construction induced by the Hilbert-Schmidt distance between pure states. The result also proves metricity of the corresponding distance derived from separable SWAP fidelity. The proof replaces the unavailable gluing argument by convex-roof duality and a dimension-independent interpolation result for Hermitian operators.

quant-pharXiv:2609.03769Sep 4, 20260 citations

On the Riemannian metric on the space of density matrices

Jochen Dittmann

Reports on Mathematical PhysicsOct 1, 199532 citations

\R^n

we give an explicit formula of the fidelity for Möbius matrices in terms of Lorentz gamma factors.