  Convex splitting is a powerful technique in quantum information theory used
in proving the achievability of numerous information-processing protocols such
as quantum state redistribution and quantum network channel coding. In this
work, we establish a one-shot error exponent and a one-shot strong converse for
convex splitting with trace distance as an error criterion. Our results show
that the derived error exponent (strong converse exponent) is positive if and
only if the rate is in (outside) the achievable region. This leads to new
one-shot exponent results in various tasks such as communication over quantum
wiretap channels, secret key distillation, one-way quantum message compression,
quantum measurement simulation, and quantum channel coding with side
information at the transmitter. We also establish a near-optimal one-shot
characterization of the sample complexity for convex splitting, which yields
matched second-order asymptotics. This then leads to stronger one-shot analysis
in many quantum information-theoretic tasks.
