dorsal/arxiv
View SchemaA Note on the correspondence between Qubit Quantum Operations and Special Relativity
| Authors | Pablo Arrighi, Christophe Patricot |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0212135 |
| URL | https://arxiv.org/abs/quant-ph/0212135 |
| DOI | 10.1088/0305-4470/36/20/101 |
| Journal | J. Phys. A: Math. Gen. 36 (2003) L287-L296 |
Abstract
We exploit a well-known isomorphism between complex hermitian $2\times 2$ matrices and $\mathbb{R}^4$, 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 $\mathbb{E}^{1,3}$, 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.
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"abstract": "We exploit a well-known isomorphism between complex hermitian $2\\times 2$\nmatrices and $\\mathbb{R}^4$, which yields a convenient real vector\nrepresentation of qubit states. Because these do not need to be normalized we\nfind that they map onto a Minkowskian future cone in $\\mathbb{E}^{1,3}$, whose\nvertical cross-sections are nothing but Bloch spheres. Pure states are\nrepresented by light-like vectors, unitary operations correspond to special\northogonal transforms about the axis of the cone, positive operations\ncorrespond to pure Lorentz boosts. We formalize the equivalence between the\ngeneralized measurement formalism on qubit states and the Lorentz\ntransformations of special relativity, or more precisely elements of the\nrestricted Lorentz group together with future-directed null boosts. The note\nends with a discussion of the equivalence and some of its possible\nconsequences.",
"arxiv_id": "quant-ph/0212135",
"authors": [
"Pablo Arrighi",
"Christophe Patricot"
],
"categories": [
"quant-ph"
],
"doi": "10.1088/0305-4470/36/20/101",
"journal_ref": "J. Phys. A: Math. Gen. 36 (2003) L287-L296",
"title": "A Note on the correspondence between Qubit Quantum Operations and Special Relativity",
"url": "https://arxiv.org/abs/quant-ph/0212135"
},
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