Single-shot Quantum State Merging

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.