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Group theoretical construction of mutually unbiased bases in Hilbert spaces of prime dimensions. (English) Zbl 1134.81323

Summary: Mutually unbiased bases in Hilbert spaces of finite dimensions are closely related to the quantal notion of complementarity. An alternative proof of existence of a maximal collection of \((N + 1)\) mutually unbiased bases in Hilbert spaces of prime dimension \(N\) is given by exploiting the finite Heisenberg group (also called the Pauli group) and the action of \(SL(2,{\mathbb Z}_N) \) on finite phase space \({\mathbb Z}_N \times {\mathbb Z}_N \) implemented by unitary operators in the Hilbert space. Crucial for the proof is that, for prime \(N, {\mathbb Z}_N \) is also a finite field.

MSC:

81P68 Quantum computation
81R05 Finite-dimensional groups and algebras motivated by physics and their representations
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