The SIC Question: History and State of Play (1703.07901v3)

Abstract: Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott's code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.

• The paper details a comprehensive exploration of SICs, transitioning from early theoretical models to high-precision, computer-assisted verifications.
• It highlights Zauner's conjecture as a pivotal strategy to simplify the dimensional search for symmetric informationally complete measurements.
• The study integrates algebraic number theory with advanced computation to link SIC constructions to optimal quantum-state tomography and broader applications.

An Overview of "The SIC Question: History and State of Play"

The paper "The SIC Question: History and State of Play" provides a comprehensive examination of symmetric informationally complete quantum measurements (SICs), a topic that intersects several domains such as algebraic number theory, quantum information theory, and Lie algebras. It explores both historical achievements and recent advancements in constructing maximal sets of complex equiangular lines. The text describes various computational efforts and mathematical conjectures, reinforcing the importance of SICs within the broader scientific community.

Symmetric Informationally Complete quantum measurements, or SICs, are crucial constructs in quantum mechanics and mathematical analysis. These are sets of vectors in $\mathbb{C}^d$, a complex vector space, that satisfy stringent equiangular conditions. The core equation describing a SIC in a $d$-dimensional space is:

$$\left|\braket{\psi_j}{\psi_k}\right|^2 = \frac{1}{d+1} \quad \text{for} \quad j \neq k.$$

Central to SIC research is the challenge of proving their existence across all possible dimensions. Although advancements in both computational and algebraic approaches have expanded knowledge up to dimension 151, and sporadically beyond, the problem remains unsolved in general despite many pragmatic solutions. These findings are linked to a range of mathematical structures such as higher-dimensional sphere packing, Lie and Jordan algebras, and finite groups.

The extensive chronology detailed in the paper indicates the incredibly interdisciplinary effort engaged in SIC research. Beginning with early explorations in the 1990s by Gerhard Zauner, further developments have seen progress through numerical solutions up to dimension 67, with computer-assisted searches now successful in providing results in all dimensions up to 151 and select higher dimensions. The work of Andrew Scott, who extended these findings to $d = 121$ with his numerical algorithms, is particularly noteworthy.

An important conjecture driving much of the work is that posed by Zauner. It suggests that a Weyl--Heisenberg covariant SIC fiducial exists as an eigenvector of a certain symmetrizing unitary operator. Such a hypothesis effectively reduces the dimensionality of the problem, simplifying the search for SICs and coinciding with various observed symmetrical properties of known SICs.

Computationally, SIC research has evolved from manual calculations to leveraging robust algorithms run on powerful computing clusters such as the Chimera supercomputer at UMass Boston. These efforts have achieved high-precision results verified against theoretical bounds, although distinct numerical performance still varies unexpectedly across dimensions due to unsolved aspects like differing numbers of solutions in given dimensions. This variability reflects the need for exhaustive exploration, especially when considering the broader implications for fields like quantum state tomography and communications.

The synthesis of these efforts is the generation of exhaustive SIC catalogues and the theoretical underpinning connecting SIC constructions to fundamental mathematical questions, such as those related to algebraic number theory. The paper highlights how algebraic approaches provide insights into the SIC phases' meaning, relating them to units in ray class fields, and intriguingly tying them to Hilbert's twelfth problem.

SICs have also been identified as optimal measurements for quantum-state tomography, underscoring their applied significance. Moreover, they continue to play a profound role in contexts outside traditional quantum mechanics, including radar technology and beyond.

In conclusion, "The SIC Question: History and State of Play" articulates both the breadth and depth of SIC research, setting the stage for future inquiries. The paper hints that untapped dimensional spaces up to $d = 2048$ and further theoretical insights remain as frontiers for researchers, offering a tantalizing potential for discoveries that may both answer enduring theoretical questions and enhance applied quantum technologies.