dorsal/arxiv
View SchemaProjective Ring Line Encompassing Two-Qubits
| Authors | Metod Saniga, Michel Planat, Petr Pracna |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0611063 |
| URL | https://arxiv.org/abs/quant-ph/0611063 |
| DOI | 10.1007/s11232-008-0076-x |
| Journal | Theoretical and Mathematical Physics 155, 3 (2008) 905-913 |
Abstract
The projective line over the (non-commutative) ring of two-by-two matrices with coefficients in GF(2) is found to fully accommodate the algebra of 15 operators - generalized Pauli matrices - characterizing two-qubit systems. The relevant sub-configuration consists of 15 points each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified with the points in such a one-to-one manner that their commutation relations are exactly reproduced by the underlying geometry of the points, with the ring geometrical notions of neighbor/distant answering, respectively, to the operational ones of commuting/non-commuting. This remarkable configuration can be viewed in two principally different ways accounting, respectively, for the basic 9+6 and 10+5 factorizations of the algebra of the observables. First, as a disjoint union of the projective line over GF(2) x GF(2) (the "Mermin" part) and two lines over GF(4) passing through the two selected points, the latter omitted. Second, as the generalized quadrangle of order two, with its ovoids and/or spreads standing for (maximum) sets of five mutually non-commuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open up rather unexpected vistas for an algebraic geometrical modelling of finite-dimensional quantum systems and give their numerous applications a wholly new perspective.
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"abstract": "The projective line over the (non-commutative) ring of two-by-two matrices\nwith coefficients in GF(2) is found to fully accommodate the algebra of 15\noperators - generalized Pauli matrices - characterizing two-qubit systems. The\nrelevant sub-configuration consists of 15 points each of which is either\nsimultaneously distant or simultaneously neighbor to (any) two given distant\npoints of the line. The operators can be identified with the points in such a\none-to-one manner that their commutation relations are exactly reproduced by\nthe underlying geometry of the points, with the ring geometrical notions of\nneighbor/distant answering, respectively, to the operational ones of\ncommuting/non-commuting. This remarkable configuration can be viewed in two\nprincipally different ways accounting, respectively, for the basic 9+6 and 10+5\nfactorizations of the algebra of the observables. First, as a disjoint union of\nthe projective line over GF(2) x GF(2) (the \"Mermin\" part) and two lines over\nGF(4) passing through the two selected points, the latter omitted. Second, as\nthe generalized quadrangle of order two, with its ovoids and/or spreads\nstanding for (maximum) sets of five mutually non-commuting operators and/or\ngroups of five maximally commuting subsets of three operators each. These\nfindings open up rather unexpected vistas for an algebraic geometrical\nmodelling of finite-dimensional quantum systems and give their numerous\napplications a wholly new perspective.",
"arxiv_id": "quant-ph/0611063",
"authors": [
"Metod Saniga",
"Michel Planat",
"Petr Pracna"
],
"categories": [
"quant-ph"
],
"doi": "10.1007/s11232-008-0076-x",
"journal_ref": "Theoretical and Mathematical Physics 155, 3 (2008) 905-913",
"title": "Projective Ring Line Encompassing Two-Qubits",
"url": "https://arxiv.org/abs/quant-ph/0611063"
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