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Five Open Problems in Quantum Information Theory

Paweł Horodecki, Łukasz Rudnicki, and Karol Życzkowski
PRX Quantum 3, 010101 – Published 3 March 2022

Abstract

We identify five selected open problems in the theory of quantum information, which are rather simple to formulate, are well studied in the literature, but are technically not easy. As these problems enjoy diverse mathematical connections, they offer a huge breakthrough potential. The first four concern existence of certain objects relevant for quantum information, namely a family of symmetric informationally complete generalized measurements in an infinite sequence of dimensions, mutually unbiased bases in dimension six, measurements saturating multiparameter Cramér-Rao bound and bound entangled states with negative partial transpose. The fifth problem requires checking whether a certain state of a two-ququart system is two-copy distillable.

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  • Received 21 December 2020
  • Revised 1 December 2021

DOI:https://doi.org/10.1103/PRXQuantum.3.010101

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Quantum Information, Science & TechnologyGeneral Physics

Authors & Affiliations

Paweł Horodecki1,2,3, Łukasz Rudnicki1,3,4,*, and Karol Życzkowski3,4,5

  • 1International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk, Gdańsk 80-308, Poland
  • 2Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, Gdańsk 80-952, Poland
  • 3National Quantum Information Centre (KCIK), University of Gdańsk, Poland
  • 4Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, Warsaw 02-668, Poland
  • 5Institute of Theoretical Physics, Jagiellonian University, Kraków 30-348, Poland

  • *lukasz.rudnicki@ug.edu.pl

Popular Summary

Quantum information gradually enters the era, in which seminal theoretical and experimental research is being turned into quantum technologies. The current aim of the field mainly lays in taking well-known theoretical concepts, such as quantum cryptography, and involving them in operational devices. Such devices, while based on `standard' technologies developed so far, shall possess essential functionalities solely operating on quantum principles such as quantum entanglement.

Just looking at the case of quantum entanglement, we can therefore track the development relevant for its understanding, departing from a famous `spooky action at a distance' by Einstein, through entanglement's essential role in a theoretical proposal by Shor, providing an algorithm about how to factorize (very large) numbers, arriving at dozens of quite-well controllable entangled qubits to-be-accessible nowadays.

Being well aware of this turning moment for the community, with our contribution we aim to endorse areas of research within theoretical quantum information, which while deeply rooted in the good old quantum information, still nurture a significant potential for a wider development of the whole field. To this end, we select five well-known and very difficult open problems pertaining to existence of symmetric structures in quantum theory, quantum metrology and distillability of entanglement. We provide a thorough review of the subfields each problem belongs to, propose an extended motivation behind it, as well as sketch potential approaches to find the solution.

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Vol. 3, Iss. 1 — March - May 2022

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It is not necessary to obtain permission to reuse this article or its components as it is available under the terms of the Creative Commons Attribution 4.0 International license. This license permits unrestricted use, distribution, and reproduction in any medium, provided attribution to the author(s) and the published article's title, journal citation, and DOI are maintained. Please note that some figures may have been included with permission from other third parties. It is your responsibility to obtain the proper permission from the rights holder directly for these figures.

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