Theory of Quantum Information with Memory

3 Probability measures and convex functions on 𝒮(ℍ)

The purpose of this chapter is to investigate probability measures on the set of quantum states $\mathcal{S}\mathrm{(}\mathbb{H}\mathrm{)}$ as a subset of the Banach space of trace-class operators $\mathfrak{T}\mathrm{(}\mathbb{H}\mathrm{)}$. Studies of probability measure and convex functions on $\mathcal{S}\mathrm{(}\mathbb{H}\mathrm{)}$ are essential in computation of Holevo χ-capacity (and hence of classical and quantum capacities) in an unconstrained and constrained quantum channel in later chapters.

Let $\mathbb{H}$ be a complex Hilbert space (finite- or infinite-dimensional) and let $\mathfrak{T}\mathrm{(}\mathbb{H}\mathrm{)}$ be the Banach space of trace-class operators on $\mathbb{H}$ under the trace norm $\Vert ·{\Vert }_1$ defined by $\Vert \mathbf{T}{\Vert }_1=\mathrm{tr}[|\mathbf{T}|]$ for $\mathbf{T}\in \mathfrak{T}\mathrm{(}\mathbb{H}\mathrm{)}$, where $|\mathbf{T}|=\sqrt{{\mathbf{T}}^{\ast }\mathbf{T}}$. Recall that ${\mathfrak{T}}_{+}\mathrm{(}\mathbb{H}\mathrm{)}=\{\mathit{\rho }\in \mathfrak{T}\mathrm{(}\mathbb{H}\mathrm{)}\mid \mathit{\rho }\ge 0\}$ is the positive cone in $\mathfrak{T}\mathrm{(}\mathbb{H}\mathrm{)}$ and $\mathcal{S}\mathrm{(}\mathbb{H}\mathrm{)}=\{\mathit{\rho }\in {\mathfrak{T}}_{+}\mathrm{(}\mathbb{H}\mathrm{)}\mid \mathrm{tr}[\mathit{\rho }]=1\}$ is the space ...