In this paper, we present the following quantum compression protocol ‘ $ \mathcal {P}$ ’: Let $\rho ,\sigma { \mathrm {S} \: \! \! { \left ({ \rho \big \| \sigma }\right )}} { \stackrel {\mathrm {def}}{=}} { \mathrm {Tr}} (\rho \log \rho - \rho \log \sigma )$ , the relative entropy between $\rho $ and $\sigma $ , is finite. Alice gets to know the eigendecomposition of $\rho $ . Bob gets to know the eigendecomposition of $\sigma { \mathrm {S} \: \! \! { \left ({ \rho \big \| \sigma }\right )}}$ and an error parameter $ {\varepsilon }$ . Alice and Bob use shared entanglement and after communication of $\mathcal {O}(( { \mathrm {S} \: \! \! { \left ({ \rho \big \| \sigma }\right )}}+1)/ {\varepsilon }^{4})$ bits from Alice to Bob, Bob ends up with a quantum state $\tilde {\rho }$ , such that $ \mathrm {F}(\rho , \tilde {\rho }) \geq 1 - 5 {\varepsilon }$ , where $ \mathrm {F}(\cdot )$ represents fidelity. This result can be considered as a non-commutative generalization of a result due to Braverman and Rao 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 $ \mathcal {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 et al.. We also present a variant of protocol $ \mathcal {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 analog of the widely used classical correlated-sampling protocol. For example, Holenstein used the classical correlated-sampling protocol in his proof of a parallel-repetition theorem for two-player one-round games.