Formalize Open Quantum Problem #23: SIC POVMs and Zauner's Conjecture

### What is the conjecture

This is problem [#23](https://oqp.iqoqi.oeaw.ac.at/sic-povms-and-zauners-conjecture) in [Reinhard F. Werner’s collection](https://arxiv.org/abs/quant-ph/0504166), later [collected by a community of quantum researchers](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems).

> **Problem (Open Quantum Problem #23: “SIC POVMs and Zauner’s Conjecture”).**
> We will give three variants of the problem, each being stronger than its predecessor.
>
> 1. Decide if SIC-POVMs exist in any dimension `d`.
> 2. Decide if group covariant SIC-POVMs exist in any dimension `d`.
> 3. Zauner’s conjecture: in any dimension `d`, a fiducial vector can be found among the eigenvectors of a distinguished Clifford operator `z`.

A standard formalization is in terms of **rank-1 POVMs / equiangular lines**:

* Fix a `d`-dimensional complex Hilbert space `H ≅ C^d`.
* A rank-1 POVM with `d^2` outcomes is a family
`Π_i = (1/d) |φ_i⟩⟨φ_i|`, `i = 1, …, d^2`,
where each `|φ_i⟩` is a unit vector and `∑_i Π_i = I_H`.
* It is **symmetric informationally complete** (a SIC-POVM) iff the pairwise overlaps are all equal:
`|⟨φ_i, φ_j⟩|^2 = 1/(d+1)` for all `i ≠ j`.
Equivalently,
`Tr(Π_i Π_j) = 1/(d^2(d+1))` for all `i ≠ j`.
* Equivalently, a SIC-POVM is the same as a set of `d^2` **equiangular complex lines** in `C^d`.
* The OQP page also notes that these operators are linearly independent, so the measurement statistics determine an arbitrary density operator on `H`.
* The OQP page also highlights the close relation with **spherical / complex projective `2`-designs**.

Then the three nested conjectural existence statements can be written as:

* **Existence of SIC-POVMs in all dimensions:** for every integer `d ≥ 2`, there exists a SIC-POVM in `C^d`.
* Fix a basis `{ |q⟩ }_{q=0}^{d-1}` and define
`X|q⟩ := |q+1 mod d⟩`,
`Z|q⟩ := ω^q |q⟩`,
where `ω = exp(2π i / d)`.
Let `w(p,q) = Z^p X^q` for `p,q ∈ Z_d` (up to the usual phase convention); these generate the **Heisenberg / Weyl–Heisenberg group**.
* A unit vector `|φ⟩` is a **fiducial vector** if the orbit
`{ w(p,q) |φ⟩⟨φ| w(p,q)^† }_{p,q ∈ Z_d}`
is a SIC-POVM.
* **Group-covariant SIC conjecture:** for every `d ≥ 2`, there exists such a fiducial vector.
* The **Clifford group** is the normalizer of the Heisenberg group in `U(d)`.
The OQP page singles out an element `z` satisfying
`z w(p,q) z^† = w(q-p, -p)`.
* **Zauner’s conjecture (the strongest version on the OQP page):** for every `d ≥ 2`, there exists a Heisenberg-covariant SIC fiducial vector that is an eigenvector of `z`.

So the problem can be viewed as a hierarchy of increasingly strong conjectures:
`Zauner-symmetric fiducial exists in every dimension`
`⇒ Heisenberg-covariant SIC exists in every dimension`
`⇒ SIC-POVM exists in every dimension`.

### Where to find the details / references

Primary sources:

* [Open Quantum Problems site (Problem #23)](https://oqp.iqoqi.oeaw.ac.at/sic-povms-and-zauners-conjecture)
* [Open Quantum Problems master list (for numbering/metadata)](https://oqp.iqoqi.oeaw.ac.at/open-quantum-problems)
* O. Krüger & R. F. Werner, “Some Open Problems in Quantum Information Theory” (Problem 23):
[https://arxiv.org/abs/quant-ph/0504166](https://arxiv.org/abs/quant-ph/0504166)
[https://arxiv.org/pdf/quant-ph/0504166](https://arxiv.org/pdf/quant-ph/0504166)

Key related references (as listed on the OQP page):

* J. M. Renes, R. Blume-Kohout, A. J. Scott, and C. M. Caves, “Symmetric Informationally Complete Quantum Measurements” (J. Math. Phys. 45, 2171 (2004); arXiv: quant-ph/0310075)
(introduces the modern SIC-POVM formulation; proves dimensions `2,3,4`; gives numerical evidence up to `45`)
* G. Zauner, “Quantendesigns — Grundzüge einer nichtkommutativen Designtheorie” (PhD thesis, University of Vienna, 1999)
[https://arnold-neumaier.at/papers/physpapers.html](https://arnold-neumaier.at/papers/physpapers.html)
(original source of the conjecture)
* G. Zauner, “Quantum Designs: Foundations of a Noncommutative Design Theory” (Int. J. Quantum Inf. 9(1), 445–507 (2011))
[https://doi.org/10.1142/S0219749911006776](https://doi.org/10.1142/S0219749911006776)
(English translation of the thesis)
* D. M. Appleby, “SIC-POVMs and the Extended Clifford Group” (J. Math. Phys. 46, 052107 (2005); arXiv: quant-ph/0412001)
(Clifford-group analysis; verifies compatibility of known numerical solutions with Zauner symmetry; exact fiducials in dimensions `7` and `19`)
* A. Klappenecker and M. Rötteler, “Mutually Unbiased Bases are Complex Projective 2-Designs” (2005 IEEE ISIT; arXiv: quant-ph/0502031)
(useful for the `2`-design viewpoint highlighted on the OQP page)
* M. Grassl, “On SIC-POVMs and MUBs in dimension 6” (arXiv: quant-ph/0406175)
(proves Zauner’s conjecture for `d = 6`)
* W. K. Wootters, “Quantum measurements and finite geometry” (arXiv: quant-ph/0406032)
* I. Bengtsson and Å. Ericsson, “Mutually Unbiased Bases and The Complementarity Polytope” (arXiv: quant-ph/0410120)

(If you want to include a brief “later progress” pointer section, these are especially useful:)

* A. J. Scott and M. Grassl, “SIC-POVMs: A new computer study” (J. Math. Phys. 51, 042203 (2010); arXiv: 0910.5784)
(numerical solutions up to `67`; new algebraic solutions in `24,35,48`)
* C. A. Fuchs, M. C. Hoang, and B. C. Stacey, “The SIC Question: History and State of Play” (Axioms 6(3):21 (2017); arXiv: 1703.07901)
(survey + extensive bibliography + later computational progress)
* M. Appleby, S. T. Flammia, and G. S. Kopp, “A Constructive Approach to Zauner’s Conjecture via the Stark Conjectures” (arXiv:2501.03970)
(modern number-theoretic approach; conditional all-d construction under Stark-type conjectures)

### Prerequisites needed

* Finite-dimensional complex Hilbert spaces `C^d`, unit vectors, rank-1 projectors, and POVMs
* Basic linear algebra / matrix theory over `C`: adjoints, trace, unitary matrices, Gram matrices
* Equiangular lines / frame-theoretic viewpoint; basic facts about informational completeness
* Finite group actions in linear algebra: Weyl–Heisenberg group, Clifford group, unitary conjugation
* (Helpful) complex projective `2`-designs / spherical `2`-designs
* (Optional, for later exact constructions) algebraic-number-theoretic and computational aspects of known SIC constructions

### [AMS categories](https://github.com/google-deepmind/formal-conjectures/labels?q=ams-)

* ams-81 (Quantum theory)
* ams-15 (Linear and multilinear algebra; matrix theory)
* ams-20 (Group theory and generalizations)
* ams-47 (Operator theory)
* ams-94 (Information and communication theory)

### Choose either option

* [ ] I plan on adding this conjecture to the repository
* [x] This issue is up for grabs: I would like to see this conjecture added by somebody else

---

Read the Open Quantum Problems page and the cited references to pin down the intended SIC-POVM and Zauner statements. The issue names no Lean files, entry points, or tests, so first inspect repository conventions for formalized conjectures. Done means adding a reviewed formal statement that matches the stated hierarchy and its required definitions.

Written by the indexing model from the issue text.