dorsal/arxiv
View SchemaMutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras
| Authors | P. Oscar Boykin, Meera Sitharam, Pham Huu Tiep, Pawel Wocjan |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0506089 |
| URL | https://arxiv.org/abs/quant-ph/0506089 |
Abstract
We establish a connection between the problem of constructing maximal collections of mutually unbiased bases (MUBs) and an open problem in the theory of Lie algebras. More precisely, we show that a collection of m MUBs in K^n gives rise to a collection of m Cartan subalgebras of the special linear Lie algebra sl_n(K) that are pairwise orthogonal with respect to the Killing form, where K=R or K=C. In particular, a complete collection of MUBs in C^n gives rise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse holds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under the adjoint operation. In this case, the Cartan subalgebras have unitary bases, and the above correspondence becomes equivalent to a result relating collections of MUBs to collections of maximal commuting classes of unitary error bases, i.e., orthogonal unitary matrices. It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a prime power. This corroborates further the general belief that a complete collection of MUBs can only exist in prime power dimensions. The connection to ODs of sl_n(C) potentially allows the application of known results on (partial) ODs of sl_n(C) to MUBs.
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"abstract": "We establish a connection between the problem of constructing maximal\ncollections of mutually unbiased bases (MUBs) and an open problem in the theory\nof Lie algebras. More precisely, we show that a collection of m MUBs in K^n\ngives rise to a collection of m Cartan subalgebras of the special linear Lie\nalgebra sl_n(K) that are pairwise orthogonal with respect to the Killing form,\nwhere K=R or K=C. In particular, a complete collection of MUBs in C^n gives\nrise to a so-called orthogonal decomposition (OD) of sl_n(C). The converse\nholds if the Cartan subalgebras in the OD are also *-closed, i.e., closed under\nthe adjoint operation. In this case, the Cartan subalgebras have unitary bases,\nand the above correspondence becomes equivalent to a result relating\ncollections of MUBs to collections of maximal commuting classes of unitary\nerror bases, i.e., orthogonal unitary matrices.\n It is a longstanding conjecture that ODs of sl_n(C) can only exist if n is a\nprime power. This corroborates further the general belief that a complete\ncollection of MUBs can only exist in prime power dimensions. The connection to\nODs of sl_n(C) potentially allows the application of known results on (partial)\nODs of sl_n(C) to MUBs.",
"arxiv_id": "quant-ph/0506089",
"authors": [
"P. Oscar Boykin",
"Meera Sitharam",
"Pham Huu Tiep",
"Pawel Wocjan"
],
"categories": [
"quant-ph"
],
"title": "Mutually Unbiased Bases and Orthogonal Decompositions of Lie Algebras",
"url": "https://arxiv.org/abs/quant-ph/0506089"
},
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