  In this paper we present the following quantum compression protocol:
  P : Let $\rho,\sigma$ be quantum states such that $S(\rho || \sigma) =
\text{Tr} (\rho \log \rho - \rho \log \sigma)$, the relative entropy between
$\rho$ and $\sigma$, is finite. Alice gets to know the eigen-decomposition of
$\rho$. Bob gets to know the eigen-decomposition of $\sigma$. Both Alice and
Bob know $S(\rho || \sigma)$ and an error parameter $\epsilon$. Alice and Bob
use shared entanglement and after communication of $\mathcal{O}((S(\rho ||
\sigma)+1)/\epsilon^4)$ bits from Alice to Bob, Bob ends up with a quantum
state $\tilde{\rho}$ such that $F(\rho, \tilde{\rho}) \geq 1 - 5\epsilon$,
where $F(\cdot)$ represents fidelity.
  This result can be considered as a non-commutative generalization of a result
due to Braverman and Rao [2011] where they considered the special case when
$\rho$ and $\sigma$ are classical probability distributions (or commute with
each other) and use shared randomness instead of shared entanglement. We use P
to obtain an alternate proof of a direct-sum result for entanglement assisted
quantum one-way communication complexity for all relations, which was first
shown by Jain, Radhakrishnan and Sen [2005,2008]. We also present a variant of
protocol P in which Bob has some side information about the state with Alice.
We show that in such a case, the amount of communication can be further
reduced, based on the side information that Bob has.
  Our second result provides a quantum analogue of the widely used classical
correlated-sampling protocol. For example, Holenstein [2007] used the classical
correlated-sampling protocol in his proof of a parallel-repetition theorem for
two-player one-round games.
