A new proof for the existence of mutually unbiased bases

Abstract

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^m.

Revised version. To appear in the special issue of Algorithmica on Quantum Algorithms and Quantum Cryptography

References in corpus (5)

• Encoding a qubit in an oscillator
• Quantum Error Correction and Orthogonal Geometry
• Quantum cryptography with 3-state systems
• Quantum Cryptography using larger alphabets
• Optimal Encryption of Quantum Bits

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