  Entanglement purification protocols promise to play a critical role in the
future of quantum networks by distributing entanglement across noisy channels.
However, only the security of two-way purification protocols have been closely
studied. To address this, we propose a one-way entanglement purification
protocol which utilizes quantum sampling and prove its security against an
adversary restricted to single qubit Pauli gates. This is done through
leveraging the equivalence of one-way entanglement purification protocols with
error-correcting codes. To prove the security of this protocol, we first use
the quantum sampling framework introduced by Bouman and Fehr to estimate the
Hamming weight of the qubits which passed through the channel and then use the
estimated relative Hamming weight $\omega$ to determine the amount of
interference that Eve has subjected to the quantum channel. Since Eve is
restricted to single qubit Pauli gates, the number of applied gates can be
directly estimated using the Hamming weight. Estimating the number of
adversarial single qubit gates, allows us to perform error correction and
disentangle the logical qubit from Eve with probability
$1-\epsilon_{qu}^\delta$. Since this protocol allows communication only in one
direction, the distance of the code must be decided before transmission, and
therefore Bob will be forced to abort the protocol if he finds that Eve has
applied more gates than the code can correct. One-way protocols may find use
when communication is limited, or when we desire to decrease latency compared
to the multiple rounds of communication needed in two-way protocols. Further
research may investigate the security of this protocol against arbitrary single
or multi-qubit gates to obtain security guarantees against a more general
adversary.
