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

Abstract

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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References

  1. 1.

    Appleby, D.M., Ericsson, A., Fuchs, C.A.: Properties of qbist state spaces. http://arxiv.org/abs/0910.2750

  2. 2.

    Bengtsson, I., Życzkowski, K.: Geometry of Quantum States. Cambridge University Press, Cambridge (2006)

  3. 3.

    Fuchs, C.A.: Quantum mechanics as quantum information (and only a little more). http://arxiv.org/abs/quant-ph/0205039 (2002)

  4. 4.

    Fuchs, C.A., Schack, R.: A quantum-Bayesian route to quantum-state space. Foundations of Physics (2010)

  5. 5.

    Keyl, M.: Fundamentals of quantum information theory. Phys. Rep. 369, 431–548 (2002)

  6. 6.

    Kimura, G.: The Bloch vector for n-level systems. Phys. Lett. A 314, 339–349 (2003)

  7. 7.

    Kimura, G., Kossakowski, A.: The Bloch-vector space for n-level systems: the spherical-coordinate point of view. Open Syst. Inf. Dyn. 12, 207–229 (2005)

  8. 8.

    Renes, J.M., Blume-Kohout, R., Scott, A.J., Caves, C.M.: Symmetric informationally complete quantum measurements. J. Math. Phys. 45, 2171–2180 (2004)

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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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Keywords

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