Section 5.3: relationships among MONO, MPT, SSA and JC

The following four logical statements hold: $(\mathcal M\Rightarrow\mathcal P)$, $(\mathcal P\Leftrightarrow\mathcal S)$, $(\mathcal S\Leftrightarrow\mathcal J)$, and $(\mathcal P\Rightarrow\mathcal M)$, where the properties are defined as follows. Write $S(X)=-\operatorname{Re}\operatorname{tr}[f(X)]$, with $f(x)=x\ln x$ applied by continuous functional calculus, and write $D_{\mathbb R}(X\Vert Y)$ for the real conversion of the extended-real quantity

Here $\log Y$ is defined by continuous functional calculus using the total real logarithm, whose value at zero is zero; conversion to a real number sends $+\infty$ to zero. Property $\mathcal M$ says that for every $n,m\in\mathbb N$, every channel $\Phi$ from $(n+1)\times(n+1)$ complex matrices to $(m+1)\times(m+1)$ complex matrices, and all positive semidefinite $X,Y$ with $\ker Y\subseteq\ker X$, one has both $\ker(\Phi Y)\subseteq\ker(\Phi X)$ and $D_{\mathbb R}(\Phi X\Vert\Phi Y)\le D_{\mathbb R}(X\Vert Y)$. A channel here means a map admitting a finite representation $\Phi(Z)=\sum_i E_iZE_i^\dagger$ with $\sum_iE_i^\dagger E_i=I$. Property $\mathcal P$ says that for every $n,m\in\mathbb N$ and all positive semidefinite matrices $X,Y$ on $\mathbb C^{n+1}\otimes\mathbb C^{m+1}$ with $\ker Y\subseteq\ker X$, tracing out the first factor preserves this kernel inclusion and does not increase $D_{\mathbb R}$. Property $\mathcal S$ says that for every $n,m,k\in\mathbb N$ and every positive semidefinite trace-one matrix $X_{ABC}$ on dimensions $n+1,m+1,k+1$,

where each subscript denotes the indicated partial trace. Property $\mathcal J$ says that for every $n,k\in\mathbb N$, nonnegative weights $w_0,\ldots,w_k$ summing to one, and positive semidefinite $(n+1)\times(n+1)$ matrices $X_j,Y_j$ satisfying $\ker Y_j\subseteq\ker X_j$, their weighted sums satisfy the same kernel inclusion and

All natural-number parameters include zero, so the register dimensions here include one but are never zero; $X,Y,X_j,Y_j$ need not have trace one and may be zero, and the weights may vanish.