  The classical-input quantum-output (cq) wiretap channel is a communication
model involving a classical sender $X$, a legitimate quantum receiver $B$, and
a quantum eavesdropper $E$. The goal of a private communication protocol that
uses such a channel is for the sender $X$ to transmit a message in such a way
that the legitimate receiver $B$ can decode it reliably, while the eavesdropper
$E$ learns essentially nothing about which message was transmitted. The
$\varepsilon $-one-shot private capacity of a cq wiretap channel is equal to
the maximum number of bits that can be transmitted over the channel, such that
the privacy error is no larger than $\varepsilon\in(0,1)$. The present paper
provides a lower bound on the $\varepsilon$-one-shot private classical
capacity, by exploiting the recently developed techniques of Anshu,
Devabathini, Jain, and Warsi, called position-based coding and convex
splitting. The lower bound is equal to a difference of the hypothesis testing
mutual information between $X$ and $B$ and the "alternate" smooth
max-information between $X$ and $E$. The one-shot lower bound then leads to a
non-trivial lower bound on the second-order coding rate for private classical
communication over a memoryless cq wiretap channel.
