16 2 v 2 3 0 M ar 2 00 1 A new proof for the existence of mutually unbiased bases ∗
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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
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 ∗
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 ...
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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
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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...
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We work out the phase-space structure for a system of n qubits. We replace the field of real numbers that label the axes of the continuous phase space by the finite field GF(2n) and investigate the geometrical structures compatible with the notion of unbiasedness. These consist of bundles of discrete curves intersecting only at the origin and satisfying certain additional properties. We provide...
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تاریخ انتشار 2008