754 research outputs found
Occam's Quantum Strop: Synchronizing and Compressing Classical Cryptic Processes via a Quantum Channel
A stochastic process's statistical complexity stands out as a fundamental
property: the minimum information required to synchronize one process generator
to another. How much information is required, though, when synchronizing over a
quantum channel? Recent work demonstrated that representing causal similarity
as quantum state-indistinguishability provides a quantum advantage. We
generalize this to synchronization and offer a sequence of constructions that
exploit extended causal structures, finding substantial increase of the quantum
advantage. We demonstrate that maximum compression is determined by the
process's cryptic order---a classical, topological property closely allied to
Markov order, itself a measure of historical dependence. We introduce an
efficient algorithm that computes the quantum advantage and close noting that
the advantage comes at a cost---one trades off prediction for generation
complexity.Comment: 10 pages, 6 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/oqs.ht
Optimizing Quantum Models of Classical Channels: The reverse Holevo problem
Given a classical channel---a stochastic map from inputs to outputs---the
input can often be transformed to an intermediate variable that is
informationally smaller than the input. The new channel accurately simulates
the original but at a smaller transmission rate. Here, we examine this
procedure when the intermediate variable is a quantum state. We determine when
and how well quantum simulations of classical channels may improve upon the
minimal rates of classical simulation. This inverts Holevo's original question
of quantifying the capacity of quantum channels with classical resources. We
also show that this problem is equivalent to another, involving the local
generation of a distribution from common entanglement.Comment: 13 pages, 6 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/qfact.htm; substantially updated
from v
Reductions of Hidden Information Sources
In all but special circumstances, measurements of time-dependent processes
reflect internal structures and correlations only indirectly. Building
predictive models of such hidden information sources requires discovering, in
some way, the internal states and mechanisms. Unfortunately, there are often
many possible models that are observationally equivalent. Here we show that the
situation is not as arbitrary as one would think. We show that generators of
hidden stochastic processes can be reduced to a minimal form and compare this
reduced representation to that provided by computational mechanics--the
epsilon-machine. On the way to developing deeper, measure-theoretic foundations
for the latter, we introduce a new two-step reduction process. The first step
(internal-event reduction) produces the smallest observationally equivalent
sigma-algebra and the second (internal-state reduction) removes sigma-algebra
components that are redundant for optimal prediction. For several classes of
stochastic dynamical systems these reductions produce representations that are
equivalent to epsilon-machines.Comment: 12 pages, 4 figures; 30 citations; Updates at
http://www.santafe.edu/~cm
Information Accessibility and Cryptic Processes: Linear Combinations of Causal States
We show in detail how to determine the time-reversed representation of a
stationary hidden stochastic process from linear combinations of its
forward-time -machine causal states. This also gives a check for the
-cryptic expansion recently introduced to explore the temporal range over
which internal state information is spread.Comment: 6 pages, 9 figures, 2 tables;
http://users.cse.ucdavis.edu/~cmg/compmech/pubs/iacplcocs.ht
Prediction, Retrodiction, and The Amount of Information Stored in the Present
We introduce an ambidextrous view of stochastic dynamical systems, comparing
their forward-time and reverse-time representations and then integrating them
into a single time-symmetric representation. The perspective is useful
theoretically, computationally, and conceptually. Mathematically, we prove that
the excess entropy--a familiar measure of organization in complex systems--is
the mutual information not only between the past and future, but also between
the predictive and retrodictive causal states. Practically, we exploit the
connection between prediction and retrodiction to directly calculate the excess
entropy. Conceptually, these lead one to discover new system invariants for
stochastic dynamical systems: crypticity (information accessibility) and causal
irreversibility. Ultimately, we introduce a time-symmetric representation that
unifies all these quantities, compressing the two directional representations
into one. The resulting compression offers a new conception of the amount of
information stored in the present.Comment: 17 pages, 7 figures, 1 table;
http://users.cse.ucdavis.edu/~cmg/compmech/pubs/pratisp.ht
Anatomy of a Spin: The Information-Theoretic Structure of Classical Spin Systems
Collective organization in matter plays a significant role in its expressed
physical properties. Typically, it is detected via an order parameter,
appropriately defined for each given system's observed emergent patterns.
Recent developments in information theory, however, suggest quantifying
collective organization in a system- and phenomenon-agnostic way: decompose the
system's thermodynamic entropy density into a localized entropy, that solely
contained in the dynamics at a single location, and a bound entropy, that
stored in space as domains, clusters, excitations, or other emergent
structures. We compute this decomposition and related quantities explicitly for
the nearest-neighbor Ising model on the 1D chain, the Bethe lattice with
coordination number k=3, and the 2D square lattice, illustrating its generality
and the functional insights it gives near and away from phase transitions. In
particular, we consider the roles that different spin motifs play (in cluster
bulk, cluster edges, and the like) and how these affect the dependencies
between spins.Comment: 12 pages, 8 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/ising_bmu.ht
Extreme Quantum Advantage for Rare-Event Sampling
We introduce a quantum algorithm for efficient biased sampling of the rare
events generated by classical memoryful stochastic processes. We show that this
quantum algorithm gives an extreme advantage over known classical biased
sampling algorithms in terms of the memory resources required. The quantum
memory advantage ranges from polynomial to exponential and when sampling the
rare equilibrium configurations of spin systems the quantum advantage diverges.Comment: 11 pages, 9 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/eqafbs.ht
The Computational Complexity of Symbolic Dynamics at the Onset of Chaos
In a variety of studies of dynamical systems, the edge of order and chaos has
been singled out as a region of complexity. It was suggested by Wolfram, on the
basis of qualitative behaviour of cellular automata, that the computational
basis for modelling this region is the Universal Turing Machine. In this paper,
following a suggestion of Crutchfield, we try to show that the Turing machine
model may often be too powerful as a computational model to describe the
boundary of order and chaos. In particular we study the region of the first
accumulation of period doubling in unimodal and bimodal maps of the interval,
from the point of view of language theory. We show that in relation to the
``extended'' Chomsky hierarchy, the relevant computational model in the
unimodal case is the nested stack automaton or the related indexed languages,
while the bimodal case is modeled by the linear bounded automaton or the
related context-sensitive languages.Comment: 1 reference corrected, 1 reference added, minor changes in body of
manuscrip
Prediction and Generation of Binary Markov Processes: Can a Finite-State Fox Catch a Markov Mouse?
Understanding the generative mechanism of a natural system is a vital
component of the scientific method. Here, we investigate one of the fundamental
steps toward this goal by presenting the minimal generator of an arbitrary
binary Markov process. This is a class of processes whose predictive model is
well known. Surprisingly, the generative model requires three distinct
topologies for different regions of parameter space. We show that a previously
proposed generator for a particular set of binary Markov processes is, in fact,
not minimal. Our results shed the first quantitative light on the relative
(minimal) costs of prediction and generation. We find, for instance, that the
difference between prediction and generation is maximized when the process is
approximately independently, identically distributed.Comment: 12 pages, 12 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/gmc.ht
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