  We consider an unknown quantum state shared between two parties, Alice and
Bob, and ask how much quantum communication is needed to transfer the full
state to Bob. This problem is known as state merging and was introduced in
[Horodecki et al., Nature, 436, 673 (2005)]. It has been shown that for free
classical communication the minimal number of quantum bits that need to be sent
from Alice to Bob is given by the conditional von Neumann entropy. However this
result only holds asymptotically (in the sense that Alice and Bob share
initially many identical copies of the state) and it was unclear how much
quantum communication is necessary to merge a single copy. We show that the
minimal amount of quantum communication needed to achieve this single-shot
state merging is given by minus the smooth conditional min-entropy of Alice
conditioned on the environment. This gives an operational meaning to the smooth
conditional min-entropy.
