Static classical-quantum-entanglement trade-offs: an entropic converse and single-letter characterizations

The direct static capacity region of a bipartite quantum state describes its asymptotic conversion into classical communication, quantum communication, and entanglement, with all communication directed from Alice to Bob. This paper gives a unified entropic converse for this region. The proof treats generated and consumed resources simultaneously and derives all three information inequalities for one local instrument. Its main ingredients are no-signalling, the chain rule, strong subadditivity, and continuity of conditional entropy. A supporting-hyperplane characterization leads to the static capacity formula, which is helpful in determining single-letterization of the capacity region. For a pure state of entanglement entropy $h$ subjected to erasure with probability $p$, the complete region is the convex hull of $0$ and $(0,-ph,(1-p)h)$, plus the unit-resource cone. This statement holds for arbitrary collective instruments by using subset-entropy inequalities related to earlier methods used for the dynamic capacity region of the erasure channel. Complete characterizations also hold for symmetrically extendible states and locally flagged pure-state mixtures. For maximally correlated states, the optimization over local instruments admits an exact matrix formulation and an additive outer bound. An explicit qubit example shows that a nonorthogonal discarded quantum memory can outperform every efficient instrument and every instrument with conditionally commuting discarded states. For states obtained by qubit dephasing of Schmidt-aligned pure states, an exact threshold characterizes when the static capacity formula vanishes, at every blocklength and after regularization. Full single-letter characterizations for general Hadamard states and for the remaining trade-offs for dephased states remain open, with sufficient additivity conditions identified.