Expected communication cost of distributed quantum tasks
Penghui Yao, University of Maryland, Baltimore
Data compression is a fundamental problem in quantum and classical information theory. A typical version of the problem is that the sender Alice receives a classical or quantum) state from some known ensemble and needs to transmit it to the receiver Bob with average error below some specified bound. We consider the case in which the message can have a variable length and goal is to minimise its expected length. For the classical case, this problem has a well-known solution given by the Huffman coding. In this scheme, the expected length of the message is equal to the Shannon entropy of the source (with a constant additive factor) and the protocol succeeds with zero error. For the quantum case, the asymptotic compression rate is given by the von-Neumann entropy. However, we show that there is no one-shot scheme which is able to match this rate, even if interactive communication is allowed. Our result has implications for direct sum theorems in quantum communication complexity and one-shot formulations of Quantum Reverse Shannon theorem.
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