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Five Open Problems in Quantum Information Theory

Paweł Horodecki, Łukasz Rudnicki, Karol Życzkowski

Published Mar 3, 2022• 95 citations
Publisher

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.

DOI
10.1103/prxquantum.3.010101

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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.
We identify five selected open problems in the theory of quantum information, which are rather simple to formulate, were 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, absolutely maximally entangled states for four subsystems with six levels each and bound entangled states with negative partial transpose. The fifth problem requires checking whether a certain state of a two-ququart system is 2-copy distillable. An award for solving each of them is announced.
This survey paper introduces several topics in quantum information theory, especially positive operator valued measures (POVMs) and entangled states. POVMs are needed to mathematically express measurements in the quantum world, and there are two special POVMs, named symmetric informationally complete POVMs and mutually unbiased bases. These two POVMs have very beautiful symmetry. We give the definitions and examples of these POVMs and discuss the existence or non-existence problem of these POVMs. Entanglement is some kind of relationship between two or more quanta. When several quanta have entanglement, the state of the quanta is called an entangled state. In this paper, quantum teleportation is introduced as an application using entangled states. Moreover, we investigate the problem of determining whether a given quantum state is an entangled state.
Recent theoretical developments are reviewed concentrating on the description of optimal generalised quantum measurement processes in physical systems with finite dimensional Hilbert spaces. Based on a self contained presentation of the required underlying quantum theoretical concepts this review particularly addresses recently explored basic theoretical properties and possible applications of informationally complete symmetric generalised measurements, a concept unifying the description of numerous so far largely separately explored measurement scenarios, such as measurements based on complete systems of mutually unbiased bases or symmetric informationally complete measurements. The recent results concerning the construction and existence of these informationally complete symmetric generalised measurements may also shed new light on still open questions concerning some of their special cases, such as the existence of complete sets of mutually unbiased bases in Hilbert spaces of non-prime power dimensions.
Determining when the multiparameter quantum Cramér--Rao bound (QCRB) is saturable with experimentally relevant single-copy measurements is a central open problem in quantum metrology. Here we establish an equivalence between QCRB saturation and the simultaneous hollowization of a set of traceless operators associated with the estimation model, i.e., the existence of complete (generally nonorthogonal) bases in which all corresponding diagonal matrix elements vanish. This formulation yields a geometric characterization: optimal rank-one measurement vectors are confined to a subspace orthogonal to a state-determined Hermitian span. This provides a direct criterion to construct optimal Positive Operator-Valued Measures(POVMs). We then identify conditions under which the partial commutativity condition proposed in [Phys. Rev. A 100, 032104(2019)] becomes necessary and sufficient for the saturation of the QCRB, demonstrate that this condition is not always sufficient, and prove the counter-intuitive uselessness of informationally-complete POVMs.
We present a brief review of discrete structures in a finite Hilbert space, relevant for the theory of quantum information. Unitary operator bases, mutually unbiased bases, Clifford group and stabilizer states, discrete Wigner function, symmetric informationally complete measurements, projective and unitary t--designs are discussed. Some recent results in the field are covered and several important open questions are formulated. We advocate a geometric approach to the subject and emphasize numerous links to various mathematical problems.
The aim of the present paper is twofold. First, to give the main ideas behind quantum computing and quantum information, a field based on quantum-mechanical phenomena. Therefore, a short review is devoted to (i) quantum bits or qubits (and more generally qudits), the analogues of the usual bits 0 and 1 of the classical information theory, and to (ii) two characteristics of quantum mechanics, namely, linearity, which manifests itself through the superposition of qubits and the action of unitary operators on qubits, and entanglement of certain multi-qubit states, a resource that is specific to quantum mechanics. A, second, focus is on some mathematical problems related to the so-called mutually unbiased bases used in quantum computing and quantum information processing. In this direction, the construction of mutually unbiased bases is presented via two distinct approaches: one based on the group SU(2) and the other on Galois fields and Galois rings.