On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States
<|reference_start|>On Approximately Symmetric Informationally Complete Positive Operator-Valued Measures and Related Systems of Quantum States: We address the problem of constructing positive operator-valued measures (POVMs) in finite dimension $n$ consisting of $n^2$ operators of rank one which have an inner product close to uniform. This is motivated by the related question of constructing symmetric informationally complete POVMs (SIC-POVMs) for which the inner products are perfectly uniform. However, SIC-POVMs are notoriously hard to construct and despite some success of constructing them numerically, there is no analytic construction known. We present two constructions of approximate versions of SIC-POVMs, where a small deviation from uniformity of the inner products is allowed. The first construction is based on selecting vectors from a maximal collection of mutually unbiased bases and works whenever the dimension of the system is a prime power. The second construction is based on perturbing the matrix elements of a subset of mutually unbiased bases.   Moreover, we construct vector systems in $\C^n$ which are almost orthogonal and which might turn out to be useful for quantum computation. Our constructions are based on results of analytic number theory.<|reference_end|>
@article{klappenecker2005on,
    title={On Approximately Symmetric Informationally Complete Positive
  Operator-Valued Measures and Related Systems of Quantum States},
    author={Andreas Klappenecker, Martin Roetteler, Igor Shparlinski, Arne
  Winterhof},
    journal={Journal of Mathematical Physics, 46:082104, 2005},
    year={2005},
    doi={10.1063/1.1998831},
    archivePrefix={arXiv},
    eprint={quant-ph/0503239},
    primaryClass={quant-ph cs.ET}
}