55 research outputs found
Energy Requirements for Quantum Data Compression and 1-1 Coding
By looking at quantum data compression in the second quantisation, we present
a new model for the efficient generation and use of variable length codes. In
this picture lossless data compression can be seen as the {\em minimum energy}
required to faithfully represent or transmit classical information contained
within a quantum state.
In order to represent information we create quanta in some predefined modes
(i.e. frequencies) prepared in one of two possible internal states (the
information carrying degrees of freedom). Data compression is now seen as the
selective annihilation of these quanta, the energy of whom is effectively
dissipated into the environment. As any increase in the energy of the
environment is intricately linked to any information loss and is subject to
Landauer's erasure principle, we use this principle to distinguish lossless and
lossy schemes and to suggest bounds on the efficiency of our lossless
compression protocol.
In line with the work of Bostr\"{o}m and Felbinger \cite{bostroem}, we also
show that when using variable length codes the classical notions of prefix or
uniquely decipherable codes are unnecessarily restrictive given the structure
of quantum mechanics and that a 1-1 mapping is sufficient. In the absence of
this restraint we translate existing classical results on 1-1 coding to the
quantum domain to derive a new upper bound on the compression of quantum
information. Finally we present a simple quantum circuit to implement our
scheme.Comment: 10 pages, 5 figure
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Parallel data compression
Data compression schemes remove data redundancy in communicated and stored data and increase the effective capacities of communication and storage devices. Parallel algorithms and implementations for textual data compression are surveyed. Related concepts from parallel computation and information theory are briefly discussed. Static and dynamic methods for codeword construction and transmission on various models of parallel computation are described. Included are parallel methods which boost system speed by coding data concurrently, and approaches which employ multiple compression techniques to improve compression ratios. Theoretical and empirical comparisons are reported and areas for future research are suggested
The Degree of a Finite Set of Words
We generalize the notions of the degree and composition from uniquely decipherable codes to arbitrary finite sets of words. We prove that if X = Y?Z is a composition of finite sets of words with Y complete, then d(X) = d(Y) ? d(Z), where d(T) is the degree of T. We also show that a finite set is synchronizing if and only if its degree equals one.
This is done by considering, for an arbitrary finite set X of words, the transition monoid of an automaton recognizing X^* with multiplicities. We prove a number of results for such monoids, which generalize corresponding results for unambiguous monoids of relations
On the Construction of Prefix-Free and Fix-Free Codes with Specified Codeword Compositions
We investigate the construction of prefix-free and fix-free codes with
specified codeword compositions. We present a polynomial time algorithm which
constructs a fix-free code with the same codeword compositions as a given code
for a special class of codes called distinct codes. We consider the
construction of optimal fix-free codes which minimizes the average codeword
cost for general letter costs with uniform distribution of the codewords and
present an approximation algorithm to find a near optimal fix-free code with a
given constant cost
Testing decipherability of directed figure codes with domino graphs
Various kinds of decipherability of codes, weaker than unique decipherability, have been studied since mid-1980s. We consider decipherability
of directed gure codes, where directed gures are de ned as labelled polyomi-
noes with designated start and end points, equipped with catenation operation
that may use a merging function to resolve possible con
icts. This setting ex-
tends decipherability questions from words to 2D structures. In the present
paper we develop a (variant of) domino graph that will allow us to decide some
of the decipherability kinds by searching the graph for speci c paths. Thus the
main result characterizes directed gure decipherability by graph properties
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Heisenberg Codes and Channels
We construct a classical code, called a Heisenberg code, which is not uniquely decipherable in order to mimic the quantum behavior of uncertainty. We classify this code according to two properties and determine the possible codeword lengths for a Heisenberg code. We suggest a possible example of a physical system which utilizes Heisenberg codes. We define a channel for Heisenberg codes, called a Heisenberg channel, which is a composite of a sender state and a receiver state which are matrices of probability amplitudes. We demonstrate that Heisenberg channels have partial trace properties similar to density matrices for quantum states. Next, we show that certain Heisenberg channels can be associated to the correlations between different partite systems of a quantum states, and define Heisenberg states and Heisenberg density matrices which are sender states and Heisenberg channels with complex entries, respectively. We prove that a Heisenberg state exists for any quantum state and that a Heisenberg density matrix relating to an n-qubit quantum state is itself a density matrix for a (2n − 1)-qubit quantum state
Note on islands in path-length sequences of binary trees
An earlier characterization of topologically ordered (lexicographic)
path-length sequences of binary trees is reformulated in terms of an
integrality condition on a scaled Kraft sum of certain subsequences (full
segments, or islands). The scaled Kraft sum is seen to count the set of
ancestors at a certain level of a set of topologically consecutive leaves is a
binary tree.Comment: 4 page
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