2002/12/31 by Pablo Arrighi, Christophe Patricot
Physics and Astronomy · #quant-ph
paper · pdf · doi:10.1088/0305-4470/36/20/101
published as J. Phys. A: Math. Gen. 36 (2003) L287-L296 · 6 pages, revtex, v3: revised discussion
arxiv created 2003/05/10 · arxiv updated 2009/12/01
We exploit a well-known isomorphism between complex hermitian 2× 2 matrices and ℝ4, which yields a convenient real vector representation of qubit states. Because these do not need to be normalized we find that they map onto a Minkowskian future cone in 𝔼1,3, whose vertical cross-sections are nothing but Bloch spheres. Pure states are represented by light-like vectors, unitary operations correspond to special orthogonal transforms about the axis of the cone, positive operations correspond to pure Lorentz boosts. We formalize the equivalence between the generalized measurement formalism on qubit states and the Lorentz transformations of special relativity, or more precisely elements of the restricted Lorentz group together with future-directed null boosts. The note ends with a discussion of the equivalence and some of its possible consequences.