Representation of Quantum States as Points in a Probability Simplex Associated to a SIC-POVM


The quantum state of a d-dimensional system can be represented by a probability distribution over the d 2 outcomes of a Symmetric Informationally Complete Positive Operator Valued Measure (SIC-POVM), and then this probability distribution can be represented by a vector of \(\mathbb {R}^{d^{2}-1}\) in a (d 2−1)-dimensional simplex, we will call this set of vectors \(\mathcal{Q}\). Other way of represent a d-dimensional system is by the corresponding Bloch vector also in \(\mathbb {R}^{d^{2}-1}\), we will call this set of vectors \(\mathcal{B}\). In this paper it is proved that with the adequate scaling \(\mathcal{B}=\mathcal{Q}\). Also we indicate some features of the shape of \(\mathcal{Q}\).

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Correspondence to José Ignacio Rosado.

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Rosado, J.I. Representation of Quantum States as Points in a Probability Simplex Associated to a SIC-POVM. Found Phys 41, 1200–1213 (2011) doi:10.1007/s10701-011-9540-9

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  • Bloch vectors
  • Probability simplex
  • SIC-POVM’s
  • Quantum state space