Zauner's Conjecture (SIC-POVM)

Zauner's Conjecture (SIC-POVM)

Definitions and mathematical statement

In ${\mathbb{C}}^d$ with its standard Hermitian inner product, seek $d^2$ unit vectors satisfying \[ |\langle v_i,v_j\rangle|^2=\frac1{d+1}\quad(i\ne j) \] for every integer $d\ge2$. Their lines form a symmetric informationally complete configuration. Writing $P_i=v_iv_i^*$, the same configuration has $\sum_iP_i=dI_d$; hence the positive operators $M_i=P_i/d$ sum to $I_d$ and form a POVM, with $\operatorname{Tr}(M_iM_j)=1/[d^2(d+1)]$ for $i\ne j$. The assertion asks existence only. Additional group covariance, an order-three symmetry, algebraic coordinates, or a uniform construction algorithm are not required.

Formulation and concept references: [S1].

Short English statement

Does every complex dimension admit the maximal symmetric family of equiangular lines needed for a symmetric informationally complete quantum measurement?

Research attempt

Current frontier.

The formulation source relates SICs to tight frames and complex projective designs, and distinguishes numerical evidence from an all-dimensional existence theorem. [S1]. A recent adjacent result of Cobucci and coauthors, revised in February 2026 and published in Physical Review Letters, shows that optimizing a natural measure of non-projectivity does not in general select SICs beyond qubits. That is a caution against changing the objective, not a nonexistence theorem for SICs. [E1]. No all-dimensional existence or nonexistence result was verified in this source check. The present target imposes no Weyl--Heisenberg covariance; failure of a particular covariance ansatz would not disprove it.

Attack route.

The first route minimizes a frame potential on a compact space. It needs a proof that its lower bound is attained, not merely a proof of existence of a minimizer. A second places a regular simplex in the real space of traceless Hermitian matrices; it needs the vertices to lie on the rank-one-projector submanifold. A third tensors known small SICs, which preserves tightness but might not preserve equiangularity. We combine these routes to derive an exact gap identity and construct positive-gap stationary configurations. The aim is to test whether variational or dimension-building arguments can close the existence gap without adding a covariance restriction.

Attempt.

1. An exact nonnegative gap functional. Fix $d\geq2$, set $N=d^2$, and let $v_1,\ldots,v_N$ be arbitrary unit vectors. Put $P_i=v_iv_i^*$, $s_{ij}=\operatorname{Tr}(P_iP_j)$ and \[ S=\sum_iP_i,\qquad \Phi_1=\sum_{i,j}s_{ij}=\operatorname{Tr}(S^2),\qquad \Phi_2=\sum_{i,j}s_{ij}^2. \] Since $\operatorname{Tr}S=N$, the sum-of-squares inequality on its $d$ eigenvalues gives \begin{equation}\label{wsref-0-31-33-30-35-3a-74-69-67-68-74} \Phi_1\geq N^2/d=d^3, \qquad \Phi_1=d^3\Longleftrightarrow S=dI_d. \end{equation} Write $c=(d+1)^{-1}$. Expanding the off-diagonal squares and using $s_{ii}=1$ and $N=d^2$ gives the exact identity \begin{equation}\label{wsref-0-31-33-30-35-3a-67-61-70} \Phi_2-\frac{2d^3}{d+1} =\sum_{i\ne j}(s_{ij}-c)^2 +\frac2{d+1}\left(\Phi_1-d^3\right). \end{equation} For clarity, the constant in $\sum_{i\ne j}(s_{ij}-c)^2=\Phi_2-2c\Phi_1$ vanishes because $-N+2cN+c^2N(N-1)=0$. Both terms on the right of (1305:gap) are nonnegative. Equality holds exactly when all off-diagonal overlaps have the target value and $S=dI_d$. Indeed, the overlaps alone also imply tightness: then $\operatorname{Tr}(S^2)=N+N(N-1)c=d^3$. Thus attaining this bound is exactly the unrestricted existence problem, not a relaxation to a different quantum measurement.

The same lower bound follows by considering $A=\sum_i|v_i\otimes v_i\rangle\langle v_i\otimes v_i|$ on the symmetric square, whose dimension is $d(d+1)/2$: $\operatorname{Tr}A=N$ and $\operatorname{Tr}A^2=\Phi_2$. This is the projective-design interpretation used in [S1]. The domain $(S^{2d-1})^{d^2}$ is compact, so $\Phi_2$ has a global minimum. Compactness does not prove that the minimum equals its Cauchy--Schwarz lower bound. That unproved equality is exactly what a variational solution still needs.

2. The regular-simplex construction has an extra quadratic constraint. Pass to $Q_i=P_i-I_d/d$ in the $D=d^2-1$ dimensional real inner-product space of traceless Hermitian matrices, with inner product $\operatorname{Tr}(AB)$. A SIC would give \begin{equation}\label{wsref-0-31-33-30-35-3a-73-69-6d-70-6c-65-78} \operatorname{Tr}(Q_i^2)=\frac{d-1}{d},\qquad \operatorname{Tr}(Q_iQ_j)=-\frac1{d(d+1)}\quad(i\ne j),\qquad \sum_iQ_i=0. \end{equation} A regular simplex with those inner products exists abstractly in every such real vector space. However, being the translate of a rank-one projector also requires \begin{equation}\label{wsref-0-31-33-30-35-3a-72-61-6e-6b-6f-6e-65} Q_i^2=\left(1-\frac2d\right)Q_i+\frac{d-1}{d^2}I_d. \end{equation} Conversely, a traceless Hermitian $Q$ satisfying (1305:rankone) has eigenvalues $(d-1)/d$ and $-1/d$. The trace forces multiplicity one for the former. Hence $Q+I_d/d$ is indeed a positive rank-one projector. This proves that the omitted constraint is both necessary and sufficient, not a cosmetic positivity convention.

An orthogonal rotation of the real space preserves (1305:simplex), but need not preserve (1305:rankone). For example, apply the orthogonal map $Q\mapsto-Q$ to any actual SIC in dimension $d>2$. The reconstructed matrix is \[ P'=\frac{2}{d}I_d-P, \] which has eigenvalue $2/d-1<0$ along the original vector. Thus the reflected simplex satisfies all the proposed Gram conditions but is not a family of positive projectors. The dimension-three example below makes this an explicit test independent of any unproved large-dimensional existence. The real-simplex construction cannot simply discard the matrix quadratic equations.

3. Exact seeds in dimensions two and three. Let $\omega=e^{2\pi i/3}$. In dimension two take \[ (1,0),\qquad \left(\frac1{\sqrt3},\sqrt{\frac23}\,\omega^k\right) \quad(k=0,1,2). \] For two different nonfirst vectors, the squared overlap is $|1+2\omega^{k-\ell}|^2/9=1/3$; overlap with the first vector also has square $1/3$. In dimension three take the nine vectors consisting of the cyclic coordinate permutations of \[ \frac1{\sqrt2}(0,1,-\omega^k),\qquad k=0,1,2. \] Two vectors with the same zero coordinate have squared overlap $|1+\omega^{k-\ell}|^2/4=1/4$ when distinct. With different zero coordinates they have just one overlapping nonzero coordinate, so again the square is $1/4$. These exact elementary seeds are well-known small SICs, not claimed as new constructions. The companion calculation checks their Gram matrices and tight-frame sums in exact algebraic arithmetic.

4. Tensoring gives a stationary obstruction, not induction. Suppose SICs are available in dimensions $a,b\geq2$. Their Cartesian tensor products give $a^2b^2$ unit vectors in dimension $d=ab$ and a tight frame, since \[ \sum_{i,j}P_i\otimes R_j=aI_a\otimes bI_b=dI_d. \] But overlaps between distinct vectors can be $1/(a+1)$, $1/(b+1)$, or $1/[(a+1)(b+1)]$, depending on which factors agree. They are not all $1/(ab+1)$. More quantitatively, factorization of the full ordered frame-potential sum gives \begin{align} \Phi_2^{\rm tensor} &=\frac{2a^3}{a+1}\frac{2b^3}{b+1} =\frac{4d^3}{(a+1)(b+1)},\notag\\ \Phi_2^{\rm tensor}-\frac{2d^3}{d+1} &=\frac{2d^3(a-1)(b-1)}{(a+1)(b+1)(ab+1)}>0. \label{wsref-0-31-33-30-35-3a-74-65-6e-73-6f-72-67-61-70} \end{align} Thus the most immediate dimension-building construction fails by a specified positive amount.

It has an additional useful feature: it is stationary for $\Phi_2$ on the full product of unit spheres, not just under tensor-product variations. The first variation with respect to a vector $v_i$ is proportional to \[ \operatorname{Re}\left\langle\delta v_i, \left(\sum_j s_{ij}P_j\right)v_i\right\rangle. \] For a SIC in dimension $a$, \[ \sum_j s_{ij}P_j =P_i+\frac1{a+1}(aI_a-P_i) =\frac{a}{a+1}(I_a+P_i). \] It therefore sends $v_i$ to $2a/(a+1)$ times $v_i$. For the tensor family the corresponding operator factorizes, sending $v_i\otimes w_k$ to $4ab/[(a+1)(b+1)]$ times that same vector. Every real tangent variation to the unit sphere is orthogonal in real inner product to its vector, so the first variation vanishes, even for variations that are not tensor-product variations.

Using the exact dimensions $2$ and $3$ gives an explicit stationary configuration in dimension $6$ with \begin{equation}\label{wsref-0-31-33-30-35-3a-73-69-78} \Phi_2=72,\qquad \frac{2d^3}{d+1}=\frac{432}{7},\qquad \text{gap}=\frac{72}{7}. \end{equation} This refutes the proposed shortcut that all stationary configurations must be SICs. No assertion that this configuration is a local minimum is made: first-order stationarity does not determine its Hessian. A descent proof would need a mechanism to escape such positive-gap critical configurations and would still need to identify the global minimum.

Stress test.

The objective in (1305:gap) is exactly tied to the prescribed overlap condition. It is not the different non-projectivity objective studied in [E1]. The regular-simplex test retains all its linear and Gram constraints; its failure occurs precisely at the projector constraint. Tensor tightness and the count $d^2$ are checked, but neither substitutes for equal pairwise overlaps.

All finite examples and the value (1305:six) are exact algebraic calculations, not small numerical residuals promoted to existence in unbounded dimension. Conversely, a successful numerical search in every dimension tested would not cover every integer $d$. No order-three symmetry, group covariance, or algorithmic construction requirement has been added to the conjecture. A failure confined to any such ansatz would remain a failure of that method.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The attempt derives an exact sum-of-squares/tightness gap identity, identifies the quadratic constraint omitted by the abstract-simplex route, and constructs an explicit positive-gap stationary configuration by tensoring exact small SICs. These are mathematical diagnostics of the proposed existence methods, with no claim of novelty priority for their elementary frame-theoretic ingredients.

What remains is a proof that the global minimum in (1305:gap) is zero in every dimension, or an obstruction showing a positive minimum in at least one dimension. Neither conclusion is obtained. The stationary tensor example is not a SIC nonexistence example.

Sources

• S1 Joseph M. Renes, Robin Blume-Kohout, A. J. Scott, Carlton M. Caves, Symmetric Informationally Complete Quantum Measurements, arXivquant-ph/0310075v1,13October2003,8pages. https://arxiv.org/pdf/quant-ph/0310075v1. Catalogue formulation source.
• E1 G. Cobucci, R. Brinster, S. Khandelwal, H. Kampermann, D. Bru

  \ss

  , N. Wyderka and A. Tavakoli, Maximally non-projective measurements are not always symmetric informationally complete, Physical Review Letters 136 $2026$, 060201; arXiv:2508.03652v2, 9 February 2026, Abstract and the non-projectivity optimization framework. https://arxiv.org/abs/2508.03652v2. Consulted 27 September 2026; used only to distinguish objectives, not as an all-dimensional SIC existence or nonexistence theorem.