nt - p h / 01 03 16 2 v 1 2 9 M ar 2 00 1 A new proof for the existence of mutually unbiased bases ∗
نویسندگان
چکیده
We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a constructive proof of the existence of mutually unbiased bases for dimensions which are power of a prime is presented. It is also proved that in any dimension d the number of mutually unbiased bases is at most d + 1. An explicit representation of mutually unbiased observables in terms of Pauli matrices are provided for d = 2.
منابع مشابه
16 2 v 2 3 0 M ar 2 00 1 A new proof for the existence of mutually unbiased bases ∗
We develop a strong connection between maximally commuting bases of orthogonal unitary matrices and mutually unbiased bases. A necessary condition of the existence of mutually unbiased bases for any finite dimension is obtained. Then a constructive proof of the existence of mutually unbiased bases for dimensions which are power of a prime is presented. It is also proved that in any dimension d ...
متن کاملar X iv : q ua nt - p h / 04 09 18 4 v 2 2 5 N ov 2 00 4 Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes ?
This note is a short conceptual elaboration of the conjecture of Saniga et al (J. Opt. B: Quantum Semiclass. 6 (2004) L19-L20) by regarding a set of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space as an analogue of an arc in a (finite) projective plane of order d. Complete sets of MUBs thus correspond to (d+1)-arcs, i.e., ovals. In the Desarguesian case, the existence of two pri...
متن کاملar X iv : q ua nt - p h / 04 09 18 4 v 1 2 7 Se p 20 04 Sets of Mutually Unbiased Bases as Arcs in Finite Projective Planes ?
This note is a short elaboration of the conjecture of Saniga et al (J. Opt. B: Quantum Semiclass. 6 (2004) L19-L20) by regarding a set of mutually unbiased bases (MUBs) in a d-dimensional Hilbert space, d being a power of a prime, as an analogue of an arc in a (Desarguesian) projective plane of order d. Complete sets of MUBs thus correspond to (d+1)-arcs, i.e., ovals. The existence of two princ...
متن کاملar X iv : m at h - ph / 0 50 60 57 v 2 2 6 Ju l 2 00 5 Hjelmslev Geometry of Mutually Unbiased Bases
The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space Hq, q = p with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p and rank r. The q vectors of a basis of Hq corr...
متن کاملar X iv : m at h - ph / 0 50 60 57 v 3 8 N ov 2 00 5 Hjelmslev Geometry of Mutually Unbiased Bases
The basic combinatorial properties of a complete set of mutually unbiased bases (MUBs) of a q-dimensional Hilbert space Hq, q = p with p being a prime and r a positive integer, are shown to be qualitatively mimicked by the configuration of points lying on a proper conic in a projective Hjelmslev plane defined over a Galois ring of characteristic p and rank r. The q vectors of a basis of Hq corr...
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