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Algorithmica

, Volume 34, Issue 4, pp 512–528 | Cite as

A New Proof for the Existence of Mutually Unbiased Bases

  •  Bandyopadhyay
  •  Boykin
  •  Roychowdhury
  •  Vatan

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 that are powers of primes 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 .

Key words. Quantum measurement, Mutually unbiased bases, Hilbert space, Quantum error-correcting codes. 

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Copyright information

© Springer-Verlag New York Inc. 2002

Authors and Affiliations

  •  Bandyopadhyay
    • 1
  •  Boykin
    • 1
  •  Roychowdhury
    • 1
  •  Vatan
    • 2
  1. 1.Electrical Engineering Department, UCLA, Los Angeles, CA 90095, USA. som@ee.ucla.edu, boykin@ee.ucla.edu, vwani@ee.ucla.edu.US
  2. 2.Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USA. Farrokh.Vatan@jpl.nasa.gov.US

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