Minimal orthonormal bases for pure quantum state estimation

Leonardo Zambrano1, Luciano Pereira2, and Aldo Delgado3

1ICFO - Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Barcelona, Spain
2Instituto de Física Fundamental IFF-CSIC, Calle Serrano 113b, Madrid 28006, Spain
3Instituto Milenio de Investigación en Óptica y Departamento de Física, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile

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We present an analytical method to estimate pure quantum states using a minimum of three measurement bases in any finite-dimensional Hilbert space. This is optimal as two bases are insufficient to construct an informationally complete positive operator-valued measurement (IC-POVM) for pure states. We demonstrate our method using a binary tree structure, providing an algorithmic path for implementation. The performance of the method is evaluated through numerical simulations, showcasing its effectiveness for quantum state estimation.

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