A New Proof for the Existence of Mutually Unbiased Bases

By Somshubhro Bandyopadhyay, P. Oscar Boykin, Vwani P. Roychowdhury, Farrokh Vatan. Algorithmica, 2001

564 citations recorded by the source corpus.

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=2m .

Subfields: Artificial Intelligence, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/8273efab5c7bdb0f1557cf85789e9c6bc198c2a6

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Citations in the corpus

Geometry of Quantum States: Index (2007, background): “Turning this into an ‘if and only if’ statement is not very hard [88].”
Constructions of Mutually Unbiased Bases (2003, background): “Any collection of pairwise mutually unbiased bases of C has cardinality d + 1 or less, see [3, 11, 13, 15, 22].”
Quantum-Bayesian Coherence (2009, background): “ually be constructed in dimensions dthat are power of a prime number (say d= 7 or d= 36 or d= 832, etc.) In these dimensions, complete sets of mutually unbiased bases (MUBs) are known to always exist [108, 109], and when they do, they form a quantum two-design [110]. What we mean by this is that the”
Tight informationally complete quantum measurements (2006, background): “Although constructions are known for prime-power dimensions [25, 26] (see also [71, 72, 73]), a complete set of MUBs is unlikely to exist in all dimensions.”
Entropic uncertainty relations—a survey (2009, methodology): “If d is a prime, it is known that the d + 1 MUBs constructed ﬁrst by Wootters and Fields (Wootters and Fields, 1989) can also be obtained as the eigenvectors of the matrices Bandy-opadhyay et al. , 2002).”
On SIC-POVMs and MUBs in Dimension 6 (2004, background): “., [4]), the link between SIC-POVMs and ﬁnite aﬃne planes is on a more abstract level. Both ﬁnite aﬃne planes and maximal sets of MUBs can be constructed if the dimension is a prime power (see, e.g., [1, 5]). In neither case constructions for non-primepowers are known, but there is numerical evidenc”
Quantum random number generators (2016, methodology): “The certification method in (Vallone et al., 2014) is designed to ensure privacy without full tomography by switching between two mutually unbiased bases (Bandyopadhyay et al., 2002; Durt et al., 2010).”