Abstract
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 them 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 the goal is to minimize its expected length.
For classical messages this problem has a wellknown solution given by 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 scheme succeeds with zero error. This is a singleshot result which implies the asymptotic result, viz. Shannon's source coding theorem, by encoding each state sequentially.
For the quantum case, the asymptotic compression rate is given by the vonNeumann entropy. However, we show that there is no oneshot scheme which is able to match this rate, even if interactive communication is allowed. This is a relatively rare case in quantum information theory when the cost of a quantum task is significantly different than the classical analogue. Our result has implications for direct sum theorems in quantum communication complexity and oneshot formulations of Quantum Reverse Shannon theorem.
BibTeX  Entry
@InProceedings{anshu_et_al:LIPIcs:2016:6684,
author = {Anurag Anshu and Ankit Garg and Aram W. Harrow and Penghui Yao},
title = {{Lower Bound on Expected Communication Cost of Quantum Huffman Coding}},
booktitle = {11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016)},
pages = {3:13:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770194},
ISSN = {18688969},
year = {2016},
volume = {61},
editor = {Anne Broadbent},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2016/6684},
URN = {urn:nbn:de:0030drops66843},
doi = {10.4230/LIPIcs.TQC.2016.3},
annote = {Keywords: Quantum information, quantum communication, expected communica tion cost, huffman coding}
}
Keywords: 

Quantum information, quantum communication, expected communica tion cost, huffman coding 
Seminar: 

11th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2016) 
Issue Date: 

2016 
Date of publication: 

13.09.2016 