1,398 research outputs found
Universal Quantum Information Compression
Suppose that a quantum source is known to have von Neumann entropy less than
or equal to S but is otherwise completely unspecified. We describe a method of
universal quantum data compression which will faithfully compress the quantum
information of any such source to S qubits per signal (in the limit of large
block lengths).Comment: RevTex 4 page
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
Identifying the Information Gain of a Quantum Measurement
We show that quantum-to-classical channels, i.e., quantum measurements, can
be asymptotically simulated by an amount of classical communication equal to
the quantum mutual information of the measurement, if sufficient shared
randomness is available. This result generalizes Winter's measurement
compression theorem for fixed independent and identically distributed inputs
[Winter, CMP 244 (157), 2004] to arbitrary inputs, and more importantly, it
identifies the quantum mutual information of a measurement as the information
gained by performing it, independent of the input state on which it is
performed. Our result is a generalization of the classical reverse Shannon
theorem to quantum-to-classical channels. In this sense, it can be seen as a
quantum reverse Shannon theorem for quantum-to-classical channels, but with the
entanglement assistance and quantum communication replaced by shared randomness
and classical communication, respectively. The proof is based on a novel
one-shot state merging protocol for "classically coherent states" as well as
the post-selection technique for quantum channels, and it uses techniques
developed for the quantum reverse Shannon theorem [Berta et al., CMP 306 (579),
2011].Comment: v2: new result about non-feedback measurement simulation, 45 pages, 4
figure
Universal quantum information compression and degrees of prior knowledge
We describe a universal information compression scheme that compresses any
pure quantum i.i.d. source asymptotically to its von Neumann entropy, with no
prior knowledge of the structure of the source. We introduce a diagonalisation
procedure that enables any classical compression algorithm to be utilised in a
quantum context. Our scheme is then based on the corresponding quantum
translation of the classical Lempel-Ziv algorithm. Our methods lead to a
conceptually simple way of estimating the entropy of a source in terms of the
measurement of an associated length parameter while maintaining high fidelity
for long blocks. As a by-product we also estimate the eigenbasis of the source.
Since our scheme is based on the Lempel-Ziv method, it can be applied also to
target sequences that are not i.i.d.Comment: 17 pages, no figures. A preliminary version of this work was
presented at EQIS '02, Tokyo, September 200
A quantum version of Sanov's theorem
We present a quantum extension of a version of Sanov's theorem focussing on a
hypothesis testing aspect of the theorem: There exists a sequence of typical
subspaces for a given set of stationary quantum product states
asymptotically separating them from another fixed stationary product state.
Analogously to the classical case, the exponential separating rate is equal to
the infimum of the quantum relative entropy with respect to the quantum
reference state over the set . However, while in the classical case the
separating subsets can be chosen universal, in the sense that they depend only
on the chosen set of i.i.d. processes, in the quantum case the choice of the
separating subspaces depends additionally on the reference state.Comment: 15 page
Entropy and Quantum Kolmogorov Complexity: A Quantum Brudno's Theorem
In classical information theory, entropy rate and Kolmogorov complexity per
symbol are related by a theorem of Brudno. In this paper, we prove a quantum
version of this theorem, connecting the von Neumann entropy rate and two
notions of quantum Kolmogorov complexity, both based on the shortest qubit
descriptions of qubit strings that, run by a universal quantum Turing machine,
reproduce them as outputs.Comment: 26 pages, no figures. Reference to publication added: published in
the Communications in Mathematical Physics
(http://www.springerlink.com/content/1432-0916/
Lossless quantum data compression and variable-length coding
In order to compress quantum messages without loss of information it is
necessary to allow the length of the encoded messages to vary. We develop a
general framework for variable-length quantum messages in close analogy to the
classical case and show that lossless compression is only possible if the
message to be compressed is known to the sender. The lossless compression of an
ensemble of messages is bounded from below by its von-Neumann entropy. We show
that it is possible to reduce the number of qbits passing through a quantum
channel even below the von-Neumann entropy by adding a classical side-channel.
We give an explicit communication protocol that realizes lossless and
instantaneous quantum data compression and apply it to a simple example. This
protocol can be used for both online quantum communication and storage of
quantum data.Comment: 16 pages, 5 figure
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