New one shot quantum protocols with application to communication complexity

In this paper, we present the following quantum compression protocol ‘ $ \\mathcal {P}$ ’: Let $\\rho ,\\sigma $ be quantum states, such that $ { \\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 $ . Both Alice and Bob know $ { \\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.