39 problems

Cover's conjecture. The feedback capacity is at most as large as the nonfeedback capacity under twice the power:

Let $\mathcal{K}(r,V)$ be the minimum of $\mathbb{E}[\Phi(\Pi)]$ over probability distributions $P_\Pi$ on $\mathbb{R}$ with mean $r$ and variance $V$, where $\Phi$ is the standard…

Constrained minimum output entropy conjecture. The output entropy satisfies

MMSE conjecture. If

Tightness conjecture. The upper bound on the linearithmic capacity of deterministic identification over additive white Gaussian noise channels should be tight, and the use of typic…

Multimode constrained minimum output entropy conjecture. For every $m$-mode state $\rho$,

Butman's feedback-capacity conjecture. The quantity $C^{L,B}(\kappa)$ is the feedback capacity of the channel.

Let $C_\infty$ denote the asymptotic feedback capacity of a Gaussian channel with memory. An input process is stationary when its finite-dimensional distributions are invariant und…

Yang–Kavčić–Tatikonda's conjecture. $C_{\textit{FB}}$ can be characterized as the solution of a maximization problem whose size depends only on the order of the noise process.

Let $h>0$ be finite, and let $\lambda_h$ solve $h=L(\lambda_h)-\lambda_h L'(\lambda_h)$, with $\mathsf{d}_h=1/\sqrt{-L'(\lambda_h)}$. Denote by…

For the Gaussian channel with signal-to-noise ratio $gamma$, let the channel input be an infinite standard normal vector and use the squared Euclidean distance between the transmit…

Consider a Gaussian channel with noiseless feedback and an estimator used for sensing. The identification rate may be measured using any scaling, including double-exponential and t…

Detectability redundancy conjecture. The detectability condition in Theorem is redundant: if $(\Gamma,\Pi,\hat{\Sigma})$ solves the optimization in, then…

Support-evolution conjecture. For a sufficiently small increment in SNR, at most a single mass point is added to the support, and this mass point arises at

Support-size order conjecture. The order of this lower bound is tight; equivalently, the support size of the capacity-achieving input grows with the same order in $\mathsf{A}$ as…

Let a linear bosonic channel be a discrete convex combination of conjugations by different displacement operators. Non-Gaussian dilatability conjecture. All such linear bosonic cha…

Let $\mathcal{E}_\lambda$ be the quantum-limited attenuator, let $n\geq 1$, and let $\hat{\rho}$ be an $n$-mode input state. Let $S$ denote von Neumann entropy and let $g^{-1}$ be…

Gaussian optimizer conjecture. The Holevo information of $\Phi$ is achieved by such a Gaussian ensemble of pure Gaussian states. This is an instance of the principle that Gaussian…

Let $\Phi$ be a gauge-covariant Gaussian quantum channel, and let the vacuum state be used as input or, more generally, let an arbitrary state be used as input. Gaussian majorizati…

Let $\mathcal{H}$ be a separable Hilbert space, and let $\mathfrak{T}_p$ be the Schatten class for $1\leq p<\infty$, with $\mathfrak{T}_\infty$ the compact operators equipped with…

Let a quantum Gaussian channel be given, and let the input quantum states have a fixed von Neumann entropy. Constrained minimum output entropy conjecture. Gaussian input states min…

Let a Gaussian channel act from a Schatten space $\mathcal S^p$ to a Schatten space $\mathcal S^q$. A state is Gaussian when it is a Gaussian quantum state. Gaussian optimizer conj…

The $p\to q$ norm conjecture. For any $\kappa\ge1$ and any $p,q\ge1$, every quantum state $\hat\rho$ satisfies

Gaussian-input optimality conjecture. The Gaussian input is optimal for sufficiently large interference-to-noise ratio $\mathsf{I}$.

Let a phase-covariant quantum Gaussian channel act on an input quantum state, and let $S$ denote the von Neumann entropy. The vacuum is the pure state with zero entropy. Gaussian m…