  We present a security proof for establishing private entanglement by means of
recurrence-type entanglement distillation protocols over noisy quantum
channels. We consider protocols where the local devices are imperfect, and show
that nonetheless a confidential quantum channel can be established, and used to
e.g. perform distributed quantum computation in a secure manner. While our
results are not fully device independent (which we argue to be unachievable in
settings with quantum outputs), our proof holds for arbitrary channel noise and
noisy local operations, and even in the case where the eavesdropper learns the
noise. Our approach relies on non-trivial properties of distillation protocols
which are used in conjunction with de-Finetti and post-selection-type
techniques to reduce a general quantum attack in a non-asymptotic scenario to
an i.i.d. setting. As a side result, we also provide entanglement distillation
protocols for non-i.i.d. input states.

Quantum Physics