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in Dimensions Two to Five

STEPHEN D. BRIERLEY, Stefan Weigert, Ingemar Bengtsson

Published Jan 1, 2010• 0 citations
OpenAlex

Abstract

All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.

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Mutually unbiased bases and Hadamard matrices of order six

Ingemar Bengtsson, Wojciech Bruzda, Åsa Ericsson +3 more

We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.

arXiv:quant-ph/0610161•Journal of Mathematical Physics•May 1, 2007•130 citations
All complex Hadamard matrices in dimensions two to five are known. We use this fact to derive all inequivalent sets of mutually unbiased (MU) bases in low dimensions. We find a three-parameter family of triples of MU bases in dimension four and two inequivalent classes of MU triples in dimension five. We confirm that the complete sets of (d+1) MU bases are unique (up to equivalence) in dimensions below six, using only elementary arguments for d less than five.
The density matrix of a qudit may be reconstructed with optimal efficiency if the expectation values of a specific set of observables are known. The required observables only exist if it is possible to identify d mutually unbiased (MU) complex (d×d) Hadamard matrices, defining a complete set of d+1 MU bases. This thesis is an exploration of sets of r≤d+1 MU bases in low dimensions. We derive all inequivalent sets of MU bases in dimensions two to five confirming that in these dimensions, the complete sets of (d+1) MU bases are unique. In dimension six, we prescribe a first Hadamard matrix and construct all others mutually unbiased to it, using algebraic computations performed by a computer program. We repeat this calculation many times, sampling all known complex Hadamard matrices, and never find more than two that are mutually unbiased. We also study subsets of a complete set of MU bases by considering sets of pure states which satisfy the desired properties. We use this concept to provide the strongest numerical evidence so far that no seven MU bases exist in dimension six. In the final part of the thesis, we introduce a new quantum key distribution protocol that uses d-level quantum systems to encode an alphabet with c letters. It has the property that the error rate introduced by an intercept-and-resend attack is higher than the BB84 or six-state protocols when the legitimate parties use a complete set of MU bases.
We present a new method for constructing affine families of complex Hadamard matrices in every even dimension. This method has an intersection with Diţă’s construction and generalizes Szöllősi's method. We extend some known families and present new ones existing in even dimensions. In particular, we find more than 13 millon inequivalent affine families in dimension 32. We also find analytical restrictions for any set of four mutually unbiased bases existing in dimension six and for any family of complex Hadamard matrices existing in every odd dimension.
We report on a search for mutually unbiased bases (MUBs) in six dimensions. We find only triplets of MUBs, and thus do not come close to the theoretical upper bound 7. However, we point out that the natural habitat for sets of MUBs is the set of all complex Hadamard matrices of the given order, and we introduce a natural notion of distance between bases in Hilbert space. This allows us to draw a detailed map of where in the landscape the MUB triplets are situated. We use available tools, such as the theory of the discrete Fourier transform, to organize our results. Finally, we present some evidence for the conjecture that there exists a four dimensional family of complex Hadamard matrices of order 6. If this conjecture is true the landscape in which one may search for MUBs is much larger than previously thought.