Minimal Informationally Complete Measurements for Pure States
- 260 Downloads
We consider measurements, described by a positive-operator-valued measure (POVM), whose outcome probabilities determine an arbitrary pure state of a D-dimensional quantum system. We call such a measurement a pure-state informationally complete (PS I-complete) POVM. We show that a measurement with 2D−1 outcomes cannot be PS I-complete, and then we construct a POVM with 2D outcomes that suffices, thus showing that a minimal PS I-complete POVM has 2D outcomes. We also consider PS I-complete POVMs that have only rank-one POVM elements and construct an example with 3D−2 outcomes, which is a generalization of the tetrahedral measurement for a qubit. The question of the minimal number of elements in a rank-one PS I-complete POVM is left open.
KeywordsQuantum state tomography informationally complete measurement Positive Operator-Valued Measure
Unable to display preview. Download preview PDF.
- 1.Prugovečki, E. 1977Int J Theor Phys.16321Google Scholar
- 3.A. Peres, Quantum Theory: Concepts and Methods (Kluwer Academic, Dordrecht, The Netherlands, 1993). POVMs are discussed in Secs. 9-5 and 9-6, and PS I-complete measurements in Sec. 3–5.Google Scholar
- 7.Fuchs C.A., “On the quantumness of a Hilbert space,” arXiv.org e-print quant-ph/0404122.Google Scholar
- 9.W. Pauli, in Handbuch der Physik, Vol. XXIV, Pt. 1, edited by H. Geiger and K.~Scheel (Springer, Berlin, 1933), p. 98; reprinted in Encyclopedia of Physics, Vol. V, Part 1 (Springer, Berlin, 1958), p. 17.Google Scholar
- 15.Peres, A., Wootters, W.K. 1992Phys Rev A661119Google Scholar
- 16.Iyanaga, S.Kawada, Y. eds. 1980Encyclopedic Dictionary of MathematicsMIT PressCambridge, MA682Google Scholar