Isoentangled Mutually Unbiased Bases, Symmetric Quantum Measurements, and Mixed-State Designs

Jakub Czartowski, Dardo Goyeneche, Markus Grassl, and Karol Życzkowski
Phys. Rev. Lett. 124, 090503 – Published 5 March 2020
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Abstract

Discrete structures in Hilbert space play a crucial role in finding optimal schemes for quantum measurements. We solve the problem of whether a complete set of five isoentangled mutually unbiased bases exists in dimension four, providing an explicit analytical construction. The reduced density matrices of these 20 pure states forming this generalized quantum measurement form a regular dodecahedron inscribed in a sphere of radius 3/20 located inside the Bloch ball of radius 1/2. Such a set forms a mixed-state 2-design—a discrete set of quantum states with the property that the mean value of any quadratic function of density matrices is equal to the integral over the entire set of mixed states with respect to the flat Hilbert-Schmidt measure. We establish necessary and sufficient conditions mixed-state designs need to satisfy and present general methods to construct them. Furthermore, it is shown that partial traces of a projective design in a composite Hilbert space form a mixed-state design, while decoherence of elements of a projective design yields a design in the classical probability simplex. We identify a distinguished two-qubit orthogonal basis such that four reduced states are evenly distributed inside the Bloch ball and form a mixed-state 2-design.

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  • Received 15 July 2019
  • Accepted 2 January 2020

DOI:https://doi.org/10.1103/PhysRevLett.124.090503

© 2020 American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & Technology

Authors & Affiliations

Jakub Czartowski1, Dardo Goyeneche2, Markus Grassl3,4, and Karol Życzkowski1,5

  • 1Institute of Physics, Jagiellonian University, ul. Łojasiewicza 11, 30-348 Kraków, Poland
  • 2Depto de Física, Fac. de Cs. Básicas, Universidad de Antofagasta, Casilla 170, Antofagasta, Chile
  • 3Max Planck Institute for the Science of Light, 91058 Erlangen, Germany
  • 4International Centre for Theory of Quantum Technologies, University of Gdańsk, 80-308 Gdańsk, Poland
  • 5Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warszawa, Poland

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Issue

Vol. 124, Iss. 9 — 6 March 2020

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