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.Received 21 December 2020Revised 1 December 2021DOI:https://doi.org/10.1103/PRXQuantum.3.010101Published 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 SocietyPhysics Subject Headings (PhySH)Research AreasQuantum correlations in quantum informationQuantum measurementsQuantum InformationGeneral Physics
Highlights
Roman Stanisław Ingarden, one of the founding fathers of the field, wrote in 1975: “The aim of the present paper was only to give a general formulation of the quantum-information theory of the Shannon type
While we have just declared that the number of proposed problems shall be lower than six, in the Appendix we describe the sixth problem, extending the discussion of the current section to cover the third constellation, perhaps better known in classical considerations, namely the Latin squares (LSs)
The goal of this perspective and the competition announced is to stimulate further research on interesting mathematical problems directly related to quantuminformation applications
Summary
Roman Stanisław Ingarden, one of the founding fathers of the field, wrote in 1975: “The aim of the present paper was only to give a general formulation of the quantum-information theory of the Shannon type. [2,3]) with its cornerstones of pioneering discoveries of quantum money [4], quantum cryptography [5,6], quantum dense coding [7], quantum teleportation [8], quantuminformation compression [9], and quantum computing [10–12] has visibly matured recently, more and more often we hear and read about quantum technologies The latter aim at turning famous theoretical concepts such as quantum cryptography into fully operational devices. Being aware of the currently relevant, particular challenges of theoretical quantum information, we ask ourselves whether there is still room for ground-breaking, though not completely unexpected, developments. To let this question have an affirmative answer, we identify open problems with such a breakthrough potential [16]. While the above justification gives as good a reason as any other reason, the first two problems described below are associated with symmetric configurations in discrete Hilbert spaces, and the second one is to some extent concerned with this special dimension
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