20 problems

Let $\rho_{ABC}$ be a tripartite quantum state, and let $I_{\alpha}(A;B|C)_{\rho}$ denote a proposed Rényi generalization of the conditional quantum mutual information, with an ana…

Let $\rho_\beta$ be a reduced density matrix on a subsystem, and consider the cluster-expansion method applied to the logarithm of reduced density matrices in order to compute cond…

Let $A$, $B$, and $C$ be regions of a quantum system, and let $\rho_\beta$ be a quantum Gibbs state. Suppose … The conditional mutual information is … where $S_{\rho_\beta}(X)$ den…

Let $A$, $B$, and $C$ be regions of a quantum system $\rho_\beta$, with … where $\Lambda$ denotes the entire system. For a tripartite state, define the conditional mutual informati…

Extension conjecture. The conclusions established for the restricted class of Gibbs states and noise channels should hold in these more general cases as well.

Let a VC class have VC dimension $d$, and consider the randomized one-inclusion graph prediction rule together with a probability assignment for that randomized graph. eCMI conject…

Let a VC class have VC dimension $d$, and consider a proper learning algorithm for the class under the conditional mutual information (CMI) framework. Steinke's CMI conjecture. The…

Let $\rho$ be a matrix product density operator (MPDO), and let $I_\rho(A:C\mid B)$ denote its conditional mutual information for a tripartition into regions $A$, $B$, and $C$: … w…

Completely contractive data-processing conjecture for conditional mutual information. For any channel $\mathcal{E}:C\rightarrow C'$ with trivial correctable algebra, there exists a…

Generic MPDO conjecture. For a generic' MPDO, there exists a constant $c>0$ such that for any tripartition $ABC$ in which $B$ separates $A$ from $C$ by distance $\ell$,

Let $\mathcal{X}$ be a domain, let $\mathcal{W}$ be a class of functions $f:\mathcal{X}\to\{0,1\}$ with VC dimension $d\geq 1$, and let $n\geq 1$. For a dataset…

Let $\mathcal{X}$ be a domain, let $\mathcal{W}$ be a class of functions $f:\mathcal{X}\to\{0,1\}$ with VC dimension $d\geq 1$, and let $n\geq 1$. For a dataset…

Let $\mathcal{X}$ be a domain, let $\mathcal{W}$ be a class of functions $f:\mathcal{X}\to\{0,1\}$ with VC dimension $d\geq 1$, and let $n\geq 1$. For a dataset…

Sandwiched Rényi monotonicity conjecture. The inequalities

Rényi conditional mutual information monotonicity conjecture. The Rényi conditional quantum mutual information satisfies

Let $\rho_{ABE}$ be a tripartite quantum state, and let $I_{\alpha}(A;B|E)_{\rho}$ denote the Rényi conditional quantum mutual information for Rényi parameter…

Let $\rho_{ABC}\in\mathcal{S}(\mathcal{H}_{ABC})_{++}$, $\tau_{AC}\in\mathcal{S}(\mathcal{H}_{AC})_{++}$, $\theta_{BC}\in\mathcal{S}(\mathcal{H}_{BC})_{++}$, and…

Let $\rho_{ABC}$ be a tripartite state with reduced states $\rho_{AB}$, $\rho_{BC}$, and $\rho_B$, and let $U,V,W$ be unitaries over $\mathcal{H}_{ABC}$. Let $I(A:C|B)_\rho$ denote…

Let $\rho_{ABC}$ be a tripartite state, with reduced states $\rho_{AB}$, $\rho_{BC}$, and $\rho_B$. Define … and … Here $I(A:C|B)_\rho$ denotes the quantum conditional mutual infor…

Conditional-mutual-information stability conjecture. Small conditional mutual information implies that $\rho_{ABC}$ is a small perturbation of a quantum Markov chain.