SICUP Matrix: Topological and SIC-POVM Perspectives

• SICUP matrix is a symmetric, integral, circulant, unimodular, and positive-definite object used in topology for studying cyclic branched covers and equivariant surgery.
• In quantum information, the closely related SIC-UP projector is derived from SIC-POVMs, leading to constructions such as Hadamard matrices and equiangular tight frames.
• The two variants of SICUP matrices encapsulate deep symmetry properties that influence both instanton Floer theory in topology and Weyl–Heisenberg covariance in quantum measurements.

“SICUP matrix” has two distinct uses in current arXiv literature. In low-dimensional topology and gauge theory, a $d\times d$ SICUP matrix is a symmetric, integral, circulant, unimodular, positive-definite matrix used in the study of cyclic branched covers and irreducible $SU(2)$-representations (Ghosh et al., 27 Aug 2025). In the SIC-POVM literature, the closely related label “SIC-UP” denotes the projector $Q=\frac{d+1}{2d}(G\circ G)$ canonically obtained from a symmetric informationally complete measurement, where $G$ is a Hermitian Gram matrix and $\circ$ is the Hadamard product (Appleby et al., 2019). These usages belong to different research settings: the former to equivariant surgery and instanton Floer homology, the latter to Weyl–Heisenberg-covariant SICs, Hadamard matrices, equiangular tight frames, and tight fusion frames (0910.5784).

1. Terminological scope

The acronymic overlap is exact enough to cause ambiguity, but the underlying objects are different.

Usage	Defining object	Research setting
SICUP matrix	$A\in M_{d\times d}(\mathbb Z)$ with properties (S), (I), (C), (U), (P)	Branched covers, tangles, instanton Floer theory
SIC-UP projector	$Q=\frac{d+1}{2d}(G\circ G)$	SIC-POVMs, Hadamard matrices, ETFs

In the topological usage, the five defining letters are explicit: symmetry, integrality, circulancy, unimodularity, and positive-definiteness. In the quantum-information usage, “SIC-UP” abbreviates a lifted matrix associated to a SIC-POVM. This suggests treating “SICUP matrix” as an ambiguous term unless the ambient subject area is specified.

2. The $d\times d$ SICUP matrix in branched-cover theory

Let $d\ge 2$ be an odd integer and write $r=(d-1)/2$. A matrix

$SU(2)$0

is called a SICUP matrix if it satisfies the following five properties: $SU(2)$1, every entry is integral, the matrix is circulant, $SU(2)$2, and all eigenvalues are $SU(2)$3 (Ghosh et al., 27 Aug 2025). The circulant condition is written with indices read modulo $SU(2)$4: there exist integers $SU(2)$5 such that

$SU(2)$6

Because $SU(2)$7 is circulant and symmetric, the all-ones vector is an eigenvector with eigenvalue

$SU(2)$8

and unimodularity forces $SU(2)$9 (Ghosh et al., 27 Aug 2025). The remaining eigenvalues come in complex-conjugate pairs of multiplicity two. This spectral structure is a basic constraint on the class.

The smallest explicit classification result supplied in the paper is that the only $Q=\frac{d+1}{2d}(G\circ G)$0 SICUP matrix is the identity,

$Q=\frac{d+1}{2d}(G\circ G)$1

The same paper points to higher-order arithmetic structure by mentioning Pell-equation connections in the $Q=\frac{d+1}{2d}(G\circ G)$2 case (Ghosh et al., 27 Aug 2025). A plausible implication is that classification rapidly becomes diophantine rather than purely linear-algebraic.

3. Adapted tangles and equivariant surgery

A SICUP matrix enters the topology of cyclic branched covers through the notion of an adapted tangle. Let

$Q=\frac{d+1}{2d}(G\circ G)$3

be a $Q=\frac{d+1}{2d}(G\circ G)$4 SICUP matrix. A $Q=\frac{d+1}{2d}(G\circ G)$5-strand tangle $Q=\frac{d+1}{2d}(G\circ G)$6 is adapted to $Q=\frac{d+1}{2d}(G\circ G)$7 if three conditions hold (Ghosh et al., 27 Aug 2025).

First condition: the usual closure $Q=\frac{d+1}{2d}(G\circ G)$8 is the unknot.

Second condition: the $Q=\frac{d+1}{2d}(G\circ G)$9-fold concatenation $G$0 closes to a $G$1-component link

$G$2

whose linking matrix is exactly $G$3, so that

$G$4

Third condition: a Floer-homological constraint is imposed on the first component $G$5 via the Baldwin–Sivek invariant $G$6. The paper requires either $G$7 and $G$8, or $G$9 with $\circ$0 “$\circ$1-shaped” and $\circ$2, or $\circ$3 with $\circ$4 “$\circ$5-shaped” and $\circ$6 (Ghosh et al., 27 Aug 2025).

The basic example occurs for $\circ$7. Since the only $\circ$8 SICUP matrix is the identity, one may adapt the 3-strand tangle

$\circ$9

to $A\in M_{d\times d}(\mathbb Z)$0. Its closure is the unknot, $A\in M_{d\times d}(\mathbb Z)$1 closes to a 3-component link of pairwise linking zero with each self-linking $A\in M_{d\times d}(\mathbb Z)$2, and $A\in M_{d\times d}(\mathbb Z)$3, so the condition on $A\in M_{d\times d}(\mathbb Z)$4 is satisfied (Ghosh et al., 27 Aug 2025).

The same framework is tied to equivariant surgery. If a knot $A\in M_{d\times d}(\mathbb Z)$5 can be unknotted by a single twist along an unlink $A\in M_{d\times d}(\mathbb Z)$6 with $A\in M_{d\times d}(\mathbb Z)$7, then the $A\in M_{d\times d}(\mathbb Z)$8-fold branched cover $A\in M_{d\times d}(\mathbb Z)$9 is obtained by an equivariant, $Q=\frac{d+1}{2d}(G\circ G)$0-periodic surgery on $Q=\frac{d+1}{2d}(G\circ G)$1 (Ghosh et al., 27 Aug 2025). The lifts of $Q=\frac{d+1}{2d}(G\circ G)$2 form a link $Q=\frac{d+1}{2d}(G\circ G)$3, and the resulting linking matrix is block-circulant, symmetric, and integral. In the single-component case it is directly a circulant symmetric matrix of size $Q=\frac{d+1}{2d}(G\circ G)$4. Equivariance under the covering transformation forces the circulant pattern.

4. Instanton Floer input and irreducible $Q=\frac{d+1}{2d}(G\circ G)$5-representations

The topological significance of SICUP matrices is mediated by instanton Floer theory. The paper isolates three ingredients (Ghosh et al., 27 Aug 2025).

Trace-cobordism vanishing: for a knot $Q=\frac{d+1}{2d}(G\circ G)$6, the trace-cobordism map

$Q=\frac{d+1}{2d}(G\circ G)$7

vanishes if and only if $Q=\frac{d+1}{2d}(G\circ G)$8.

Equivariant comparison: if a 4-dimensional surgery cobordism $Q=\frac{d+1}{2d}(G\circ G)$9 from $d\times d$0 to $d\times d$1 has negative-definite intersection form $d\times d$2, then

$d\times d$3

Representation-theoretic consequence: if $d\times d$4, then $d\times d$5 cannot be all reducible, so there must be an irreducible $d\times d$6-representation.

Combining these statements, the paper derives a criterion for the existence of irreducible $d\times d$7-representations when the surgery description is controlled by a suitable unimodular matrix and the relevant $d\times d$8-inequality holds (Ghosh et al., 27 Aug 2025). The resulting main theorem states: if $d\times d$9 is a prime knot and $d\ge 2$0 is such that $d\ge 2$1 is an integer homology 3-sphere, then $d\ge 2$2 admits an irreducible $d\ge 2$3-representation whenever either $d\ge 2$4 is 2-periodic or $d\ge 2$5 is the closure of a tangle adapted to a $d\ge 2$6 SICUP matrix (Ghosh et al., 27 Aug 2025).

Within this theorem, the SICUP matrix functions as a rigid equivariant-surgery datum: it packages the cyclic symmetry, the framing/linking information, and the unimodularity needed to place the problem inside the instanton-theoretic vanishing mechanism.

5. SIC-POVM background: Gram matrices, Weyl–Heisenberg covariance, and existence

The second usage of the term arises from symmetric informationally complete measurements. Let $d\ge 2$7 be a set of $d\ge 2$8 unit vectors in $d\ge 2$9 defining a rank-one POVM

$r=(d-1)/2$0

The SIC condition is

$r=(d-1)/2$1

equivalently

$r=(d-1)/2$2

The associated real Gram matrix

$r=(d-1)/2$3

has ones on the diagonal and constant off-diagonal value $r=(d-1)/2$4 (0910.5784).

A standard construction uses a single fiducial vector and the $r=(d-1)/2$5-element Weyl–Heisenberg displacement group. In a fixed orthonormal basis one sets

$r=(d-1)/2$6

with

$r=(d-1)/2$7

and defines

$r=(d-1)/2$8

The orbit

$r=(d-1)/2$9

is a SIC precisely when the fiducial obeys

$SU(2)$00

This reduces the full equiangularity problem to a single orbit equation (0910.5784).

The existence problem is central. A 2009 computer study reported numerical solutions in all dimensions $SU(2)$01, with $SU(2)$02 remaining slightly inconclusive, and a putatively complete list of Weyl–Heisenberg covariant solutions for $SU(2)$03 (0910.5784). Exact algebraic fiducials were listed in dimensions $SU(2)$04. The guiding symmetry is the order-three Zauner unitary $SU(2)$05, a special Clifford-group operator. Choosing a fiducial in an eigenspace of $SU(2)$06 reduces the number of real variables from $SU(2)$07 to roughly $SU(2)$08, which makes the defining polynomial systems more tractable (0910.5784). The broader conjecture is that maximal complex projective codes of this type exist in all finite dimensions.

6. The SIC-UP projector $SU(2)$09 and associated matrix constructions

Starting from a SIC in $SU(2)$10, one may instead use the Hermitian Gram matrix

$SU(2)$11

and form its entrywise square

$SU(2)$12

The SIC-UP projector is then

$SU(2)$13

a $SU(2)$14 Hermitian matrix (Appleby et al., 2019).

Its fundamental property is idempotence: $SU(2)$15 Hence $SU(2)$16 is a projector of rank

$SU(2)$17

with spectrum

$SU(2)$18

If the underlying SIC is Weyl–Heisenberg covariant, then $SU(2)$19 is covariant in the sense that

$SU(2)$20

equivalently

$SU(2)$21

(Appleby et al., 2019).

From $SU(2)$22 one obtains a Hermitian Hadamard-type involution

$SU(2)$23

This satisfies

$SU(2)$24

and has constant-modulus entries: $SU(2)$25 After the rescaling $SU(2)$26, one gets a unitary matrix all of whose entries have unit modulus. Equivalently, $SU(2)$27 is a Hermitian complex Hadamard matrix up to the overall scale $SU(2)$28, and

$SU(2)$29

gives a one-to-one correspondence between the projector and the Hadamard (Appleby et al., 2019).

The same construction yields two equiangular tight-frame Gram matrices,

$SU(2)$30

in dimensions $SU(2)$31 and $SU(2)$32. In the odd-$SU(2)$33, Weyl–Heisenberg-covariant case, one also gets two symmetric tight-fusion-frame projectors of ranks $SU(2)$34 in dimension $SU(2)$35 (Appleby et al., 2019).

A notable feature is deformation. Whereas SICs themselves appear isolated for $SU(2)$36, the associated $SU(2)$37 and $SU(2)$38 lie on nontrivial continuous families in the lowest nontrivial cases $SU(2)$39 (Appleby et al., 2019). Restricted-defect calculations through $SU(2)$40 are strictly positive for every known SIC with $SU(2)$41, implying that each corresponding $SU(2)$42 lies on at least a one-parameter manifold of solutions. In $SU(2)$43, the paper gives an explicit one-parameter family $SU(2)$44 that remains a rank-10 projection for all $SU(2)$45.

7. Distinctions, conceptual links, and open directions

The two meanings of “SICUP matrix” are structurally different. The topological SICUP matrix is a $SU(2)$46 integral matrix constrained by symmetry, circulancy, determinant $SU(2)$47, and positive-definiteness, and it serves as a linking matrix in an equivariant surgery construction (Ghosh et al., 27 Aug 2025). The SIC-UP matrix $SU(2)$48 is instead a $SU(2)$49 Hermitian projector derived from a SIC Gram matrix and used to produce Hadamard matrices, ETFs, and tight fusion frames (Appleby et al., 2019). The papers therefore treat “SICUP” and “SIC-UP” as labels for separate constructions rather than as variants of a single object.

Their shared conceptual feature is that both compress highly structured symmetry into a matrix formalism. In the branched-cover setting, circulancy reflects the action of the covering transformation on lifted surgery components. In the SIC setting, Weyl–Heisenberg covariance and Zauner symmetry organize the fiducial equations and the induced projector structure (0910.5784).

Open problems also diverge. On the topology side, the paper identifies three directions: extending the “commuting trick” beyond 2-periodicity to other symmetries such as invertibility and amphichirality, studying $SU(2)$50 more finely in relation to negative definite block-circulant matrices, and classifying SICUP matrices in higher sizes, including the Pell-equation phenomena seen in the $SU(2)$51 case (Ghosh et al., 27 Aug 2025). On the SIC-POVM side, the longstanding issue is the general existence of $SU(2)$52 equiangular lines in $SU(2)$53, conjectured in all finite dimensions, with numerical solutions known through $SU(2)$54 and exact algebraic solutions known only in selected dimensions (0910.5784). The deformation theory of the SIC-UP projector adds a second layer of questions, since the lifted objects appear to admit continuous families even when the originating SICs do not (Appleby et al., 2019).