939 problems

For an integer $d\geq 2$, let $\mathbb C^d$ be the $d$-dimensional complex vector space. For $x,\omega\in\mathbb Z_d$, define the cyclic translation and modulation operators by … w…

Positive–PPT composition conjecture. The composition

The classical entropy for a quantum-mechanical density matrix on $L^2(\mathbb{R})$ is defined through the associated Husimi function, obtained by evaluating the density matrix on G…

Let the noise modes c_i_{i=1}^n be in a product state … of $n$ vacuum states, and assume that $H(A^n)_\rho=n g(N_A)$. Minimum output-entropy conjecture. Then … This conjecture is…

Additivity conjecture. Quantum entanglement does not enhance the channel capacity, and the classical capacity of a quantum channel is additive.

Let $\mathcal{H}_A$ and $\mathcal{H}_B$ be finite-dimensional Hilbert spaces, and let $\mathcal{T}_A$ and $\mathcal{T}_B$ be super-operators on $\mathcal{H}_A$ and $\mathcal{H}_B$,…

Let $\rho_A$ be the reduced density matrix obtained by partially tracing one factor of a random Gaussian vector in $\mathbb{C}^D\otimes\mathbb{C}^D$, with the normalization convent…

De Palma–Trevisan conjecture. The divergences defined this way are genuine metrics on quantum state spaces. The conjecture has recently been justified under certain additional assu…

A topological order is described through ground states of Hamiltonians with local interactions; the area law concerns the entanglement properties of such ground states, and project…

Let $\chi(\mathcal{N})$ denote the Holevo information of a quantum channel $\mathcal{N}$. Holevo-information additivity conjecture. In the general case, $\chi$ is additive for the…

Triangle inequality conjecture. The quantity $D_{\Delta}$ satisfies the triangle inequality.

Constrained minimum output entropy conjecture. The output entropy satisfies

Let the axioms ${\bf A0}$ and ${\bf A1}$ hold approximately on an intermediate length scale, with sufficiently small violations. Coarse-graining conjecture. Coarse-graining the lat…

Rhodes's conjecture. The rank of $t$ is at most $2n-1$.

Let $\mathcal{H}=\mathcal{H}_A\otimes\mathcal{H}_B$ be a finite-dimensional Hilbert space of dimension $D=d_A d_B$, and let $W$ be a normalized entanglement witness, so that…

Let $M=N$ and consider the special case $\beta=2$ for the Wishart ensemble. Denote the minimum eigenvalue by $\lambda_{\rm min}$ and expectation by $\langle\cdot\rangle$. Znidaric'…

Let $p$ satisfy $1\leq p\leq2$, and let $\mathcal{N}_1$ and $\mathcal{N}_2$ be quantum channels. Their maximal $p$-norms are denoted by $\nu_p(\mathcal{N}_1)$ and…

Georgiou–Pavon conjecture. Given a positivity-improving completely positive trace-preserving map $\mathcal{E}$ and two density matrices $\rho$ and $\omega$, there exist invertible…

Capacity-region equality conjecture. The capacity regions are identical in all cases, including the remaining open regime:

Let $n$ be a positive integer, let $sigma$ be a density operator, and let $M^c$ denote the complement of $M$ in $\{1,2,\cdots,n\}$. For a subset $M$, write $\sigma^{\otimes M^c}$ f…

Let $\mathcal{X}$ be an input alphabet, let $\mathcal{B}$ be an output quantum system, and let $x\mapsto\rho_{\mathcal{B}}^x$ be a classical–quantum channel. For an input distribut…

For $d\in\mathbb{N}$, consider arrangements of distinct points in $\mathbb{CP}^{d-1}$, with distance measured by the Fubini--Study metric. Zauner's conjecture. For every…

Let a quantum measurement be a POVM on a finite-dimensional quantum system. A measurement uses a single auxiliary qubit if its implementation requires only one ancillary two-level…

Let $\mathcal{P}_1\in U_1\otimes\cdots\otimes U_n$ and $\mathcal{P}_2\in V_1\otimes\cdots\otimes V_n$ be persistent tensors, where $U_1,\ldots,U_n$ and $V_1,\ldots,V_n$ are vector…

Let $\rho_\beta$ be an arbitrary quantum Gibbs state on a system partitioned as $\Lambda=A\sqcup B\sqcup C$, and let $d_{A,C}$ denote the distance between $A$ and $C$. Write…