dorsal/arxiv
View SchemaMutually Unbiased Bases, Generalized Spin Matrices and Separability
| Authors | Arthur O. Pittenger, Morton H. Rubin |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0308142 |
| URL | https://arxiv.org/abs/quant-ph/0308142 |
| Journal | Linear Alg. Appl. 390, 255 (2004) |
Abstract
A collection of orthonormal bases for a complex dXd Hilbert space is called mutually unbiased (MUB) if for any two vectors v and w from different bases the square of the inner product equals 1/d: |<v,w>| ^{2}=1/d. The MUB problem is to prove or disprove the the existence of a maximal set of d+1 bases. It has been shown in [W. K. Wootters, B. D. Fields, Annals of Physics, 191, no. 2, 363-381, (1989)] that such a collection exists if d is a power of a prime number p. We revisit this problem and use dX d generalizations of the Pauli spin matrices to give a constructive proof of this result. Specifically we give explicit representations of commuting families of unitary matrices whose eigenvectors solve the MUB problem. Additionally we give formulas from which the orthogonal bases can be readily computed. We show how the techniques developed here provide a natural way to analyze the separability of the bases. The techniques used require properties of algebraic field extensions, and the relevant part of that theory is included in an Appendix.
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"abstract": "A collection of orthonormal bases for a complex dXd Hilbert space is called\nmutually unbiased (MUB) if for any two vectors v and w from different bases the\nsquare of the inner product equals 1/d: |\u003cv,w\u003e| ^{2}=1/d. The MUB problem is to\nprove or disprove the the existence of a maximal set of d+1 bases. It has been\nshown in [W. K. Wootters, B. D. Fields, Annals of Physics, 191, no. 2, 363-381,\n(1989)] that such a collection exists if d is a power of a prime number p. We\nrevisit this problem and use dX d generalizations of the Pauli spin matrices to\ngive a constructive proof of this result. Specifically we give explicit\nrepresentations of commuting families of unitary matrices whose eigenvectors\nsolve the MUB problem. Additionally we give formulas from which the orthogonal\nbases can be readily computed. We show how the techniques developed here\nprovide a natural way to analyze the separability of the bases. The techniques\nused require properties of algebraic field extensions, and the relevant part of\nthat theory is included in an Appendix.",
"arxiv_id": "quant-ph/0308142",
"authors": [
"Arthur O. Pittenger",
"Morton H. Rubin"
],
"categories": [
"quant-ph"
],
"journal_ref": "Linear Alg. Appl. 390, 255 (2004)",
"title": "Mutually Unbiased Bases, Generalized Spin Matrices and Separability",
"url": "https://arxiv.org/abs/quant-ph/0308142"
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