41 problems

Let $M(n)$ denote the maximum number of pairwise mutually unbiased bases in $\mathbb{C}^n$. For a prime power $n=p^r$, one has $M(p^r)=p^r+1$; the unresolved case concerns dimensio…

Let $N(d)$ denote the maximum number of mutually unbiased bases in dimension $d$. Dimension-six MUB conjecture. The maximum number of mutually unbiased bases in dimension six is ……

Prime-power conjecture for orthogonal decompositions. Orthogonal decompositions of $sl_d(\mathbb{C})$ can exist only if $d$ is a prime power.

SPR conjecture. If $d$ differs from a power of a prime number, then the question of whether a $d$-dimensional Hilbert space has a set of $d+1$ MUBs is equivalent to the question of…

The projective-plane conjecture. Non-existence of a finite projective plane of order $d$ implies that there are fewer than $d+1$ mutually unbiased bases in $\mathcal{H}^d$, and vic…

WH-MUB bound conjecture. The quantity $M_\alpha(|\psi\rangle)$ is bounded by the bound with $\Delta=0$, and equality is attained uniquely by the WH-MUB fiducial states.

WH-MUB optimality conjecture. WH-MUBs saturate the optimal bound for general $n$-qubit systems, except when $n=1$ or $n=3$.

Szöllősi-family non-extendability conjecture. Any matrix $S\in X(\alpha)$ in the Szöllősi family cannot be part of an MUB-quadruplet.

Hadamard-cube vanishing conjecture. For every $\pi\in S_6$ and every $j=1,2,3$,

Let a 2-design in dimension $d$ be a quantum state 2-design, and let a SIC-POVM denote a symmetric informationally complete positive-operator-valued measure and a maximal set of MU…

A classical design is an object in the domain of the functor $Q$, and $Q$ maps such designs to objects that may be interpreted as sets of mutually unbiased bases, denoted by…

Conjecture on MUB inner products. This property holds when $N$ is any higher power of $2$. The conjecture predicts that the observed phase restrictions for cross-inner products bet…

Let $\mathbb{C}^6$ be the six-dimensional complex Hilbert space. Two orthonormal bases $\{\phi_1,\dots,\phi_6\}$ and $\{\psi_1,\dots,\psi_6\}$ are mutually unbiased when … for all…

A collection of orthonormal bases in $\mathbb{C}^d$ is mutually unbiased when vectors from distinct bases satisfy $\lvert\langle\psi_i^k\mid\psi_j^l\rangle\rvert=\frac{1}{\sqrt d}$…

Difference-matrix conjecture. Difference matrices may also be applied to construct mutually unbiased bases in multipartite systems.

Let $U(d)$ be the unitary group, let $\nu$ be normalized Haar measure on $U(d)$, let $e$ be its identity, and let $\mu_{[\bm H]}$ be the smoothed measure associated with the equiva…

Let $U(d)$ be the unitary group, let $H(d)$ be the set of complex Hadamard matrices in $U(d)$, and let $B(d)$ be the optimal positive-definite-functions upper bound for mutually un…

Let $U(d)$ be the unitary group, let $H(d)$ denote the set of complex Hadamard matrices in $U(d)$, and let $B_{\leq k}(d)$ be the infimum of $h(\bm I)$ over real-valued positive de…

Nonexistence conjecture for four MUBs in dimension six. Four mutually unbiased bases should not exist in $\bbC^6$.

Let $N$ be a power of $2$, and let $\{{\mathbf{U}}_1,\ldots,{\mathbf{U}}_N\}$ be the $N$ mutually unbiased orthonormal bases of $\mathbb{C}^N$ constructed for powers of $2$. Thus,…

Dimension-six MUB conjecture. It was conjectured that

Let an optimistic linked system of symmetric designs have parameters $(v,k,\lambda;w)$, where $v$ is the number of points in each design and $w$ is the number of linked designs. Th…

A set of mutually unbiased bases (MUBs) in $\mathbb{C}^{2^m}$ is strongly $\mathbb{C}$-unextendible if no additional basis can be added even after allowing the bases to be changed…

Unextendible-set/EUR conjecture. The connection between the existence of unextendible sets of mutually unbiased bases and the tightness of the corresponding entropic uncertainty re…

Let $d$ be an odd prime-power dimension satisfying $d\equiv3\pmod 4$, and let MUB-balanced states be the states balanced with respect to the complete set of mutually unbiased bases…