[Paper Review] Five open problems in quantum information

Paweł Horodecki, Łukasz Rudnicki+1 more|arXiv (Cornell University)|Feb 8, 2020

Quantum Information and Cryptography92 references14 citations

TL;DR

This paper identifies five profound open problems in quantum information theory—existence of symmetric informationally complete generalized measurements in infinite dimensions, mutually unbiased bases in dimension six, absolutely maximally entangled states for four ququarts, bound entangled states with negative partial transpose, and 2-copy distillability of a specific two-ququart state—each with deep mathematical roots and high potential for breakthroughs. Solving any earns the Golden KCIK Award of 2021 EUR, with extensions and increases for unsolved problems.

ABSTRACT

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.

Motivation & Objective

• To identify and highlight five technically challenging, well-studied open problems in quantum information theory with broad mathematical and foundational significance.
• To stimulate foundational research by offering a financial incentive (Golden KCIK Award) for solving each problem, encouraging innovation beyond current theoretical toolkits.
• To emphasize problems with universal relevance—connected to group theory, number theory, tensor product structures, and matrix analysis—rather than platform-specific or protocol-bound challenges.
• To promote progress in quantum combinatorics, entanglement theory, and quantum information processing by focusing on existence and structural questions of fundamental quantum objects.
• To inspire new mathematical techniques that could impact not only quantum information but also linear algebra, quantum metrology, and secure quantum communication.

Proposed method

• Formulate each problem as a precise mathematical existence question rooted in quantum information structures: SIC-POVMs, MUBs, AME states, NPT bound entanglement, and 2-copy distillability.
• Use the Kronecker sum of matrices, $ A igoplus B = A igotimes \mathbf{1} + \mathbf{1} \bigotimes B $, as a key algebraic tool to analyze entanglement and state properties.
• Leverage symmetry and geometric constraints in Hilbert space to analyze SIC-POVMs and MUBs, particularly in dimension six where the existence remains unproven.
• Apply tensor product structures to study the existence of absolutely maximally entangled states across four subsystems of dimension six.
• Analyze the 2-copy distillability of a specific two-ququart state using criteria based on partial transposition and entanglement distillation protocols.
• Connect each problem to broader mathematical frameworks—such as finite geometry, combinatorics, and matrix analysis—to guide potential solution strategies.

Experimental results

Research questions

• RQ1 Do symmetric informationally complete generalized measurements (SIC-POVMs) exist in an infinite sequence of dimensions?
• RQ2 Do mutually unbiased bases (MUBs) exist in dimension six?
• RQ3 Do absolutely maximally entangled (AME) states exist for four subsystems, each of dimension six?
• RQ4 Do bound entangled states with negative partial transpose (NPT) exist?
• RQ5 Is the given two-ququart state 2-copy distillable?

Key findings

• The existence of SIC-POVMs in all dimensions remains unproven, with the case of infinite-dimensional sequences posing a deep open problem.
• The existence of mutually unbiased bases in dimension six is still unconfirmed, despite extensive study and numerical evidence suggesting non-existence.
• No known construction of absolutely maximally entangled states for four ququarts (d=6) has been found, despite their importance in quantum networks and error correction.
• The existence of bound entangled states with negative partial transpose (NPT) remains an open question, with implications for the distillability of quantum entanglement.
• The 2-copy distillability of a specific two-ququart state is not yet determined, though it is considered a tractable case due to symmetry and structure.
• Solving any of these problems is expected to yield new insights into tensor product structures, matrix algebra, and the foundations of quantum information, with potential applications in quantum cryptography and secure communication.