1,988 research outputs found
Spectral Simplicity of Apparent Complexity, Part II: Exact Complexities and Complexity Spectra
The meromorphic functional calculus developed in Part I overcomes the
nondiagonalizability of linear operators that arises often in the temporal
evolution of complex systems and is generic to the metadynamics of predicting
their behavior. Using the resulting spectral decomposition, we derive
closed-form expressions for correlation functions, finite-length Shannon
entropy-rate approximates, asymptotic entropy rate, excess entropy, transient
information, transient and asymptotic state uncertainty, and synchronization
information of stochastic processes generated by finite-state hidden Markov
models. This introduces analytical tractability to investigating information
processing in discrete-event stochastic processes, symbolic dynamics, and
chaotic dynamical systems. Comparisons reveal mathematical similarities between
complexity measures originally thought to capture distinct informational and
computational properties. We also introduce a new kind of spectral analysis via
coronal spectrograms and the frequency-dependent spectra of past-future mutual
information. We analyze a number of examples to illustrate the methods,
emphasizing processes with multivariate dependencies beyond pairwise
correlation. An appendix presents spectral decomposition calculations for one
example in full detail.Comment: 27 pages, 12 figures, 2 tables; most recent version at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt2.ht
Reversible transformations from pure to mixed states, and the unique measure of information
Transformations from pure to mixed states are usually associated with
information loss and irreversibility. Here, a protocol is demonstrated allowing
one to make these transformations reversible. The pure states are diluted with
a random noise source. Using this protocol one can study optimal
transformations between states, and from this derive the unique measure of
information. This is compared to irreversible transformations where one does
not have access to noise. The ideas presented here shed some light on attempts
to understand entanglement manipulations and the inevitable irreversibility
encountered there where one finds that mixed states can contain "bound
entanglement".Comment: 10 pages, no figures, revtex4, table added, to appear in Phys. Rev.
Notes on the integration of numerical relativity waveforms
A primary goal of numerical relativity is to provide estimates of the wave
strain, , from strong gravitational wave sources, to be used in detector
templates. The simulations, however, typically measure waves in terms of the
Weyl curvature component, . Assuming Bondi gauge, transforming to the
strain reduces to integration of twice in time. Integrations
performed in either the time or frequency domain, however, lead to secular
non-linear drifts in the resulting strain . These non-linear drifts are not
explained by the two unknown integration constants which can at most result in
linear drifts. We identify a number of fundamental difficulties which can arise
from integrating finite length, discretely sampled and noisy data streams.
These issues are an artifact of post-processing data. They are independent of
the characteristics of the original simulation, such as gauge or numerical
method used. We suggest, however, a simple procedure for integrating numerical
waveforms in the frequency domain, which is effective at strongly reducing
spurious secular non-linear drifts in the resulting strain.Comment: 23 pages, 10 figures, matches final published versio
Unambiguous comparison of the states of multiple quantum systems
We consider N quantum systems initially prepared in pure states and address
the problem of unambiguously comparing them. One may ask whether or not all
systems are in the same state. Alternatively, one may ask whether or not the
states of all N systems are different. We investigate the possibility of
unambiguously obtaining this kind of information. It is found that some
unambiguous comparison tasks are possible only when certain linear independence
conditions are satisfied. We also obtain measurement strategies for certain
comparison tasks which are optimal under a broad range of circumstances, in
particular when the states are completely unknown. Such strategies, which we
call universal comparison strategies, are found to have intriguing connections
with the problem of quantifying the distinguishability of a set of quantum
states and also with unresolved conjectures in linear algebra. We finally
investigate a potential generalisation of unambiguous state comparison, which
we term unambiguous overlap filtering.Comment: 20 pages, no figure
Quantum information can be negative
Given an unknown quantum state distributed over two systems, we determine how
much quantum communication is needed to transfer the full state to one system.
This communication measures the "partial information" one system needs
conditioned on it's prior information. It turns out to be given by an extremely
simple formula, the conditional entropy. In the classical case, partial
information must always be positive, but we find that in the quantum world this
physical quantity can be negative. If the partial information is positive, its
sender needs to communicate this number of quantum bits to the receiver; if it
is negative, the sender and receiver instead gain the corresponding potential
for future quantum communication. We introduce a primitive "quantum state
merging" which optimally transfers partial information. We show how it enables
a systematic understanding of quantum network theory, and discuss several
important applications including distributed compression, multiple access
channels and multipartite assisted entanglement distillation (localizable
entanglement). Negative channel capacities also receive a natural
interpretation
Signal velocity, causality, and quantum noise in superluminal light pulse propagation
We consider pulse propagation in a linear anomalously dispersive medium where
the group velocity exceeds the speed of light in vacuum (c) or even becomes
negative. A signal velocity is defined operationally based on the optical
signal-to-noise ratio, and is computed for cases appropriate to the recent
experiment where such a negative group velocity was observed. It is found that
quantum fluctuations limit the signal velocity to values less than c.Comment: 4 Journal pages, 3 figure
The Resource Theory of Quantum States Out of Thermal Equilibrium
The ideas of thermodynamics have proved fruitful in the setting of quantum
information theory, in particular the notion that when the allowed
transformations of a system are restricted, certain states of the system become
useful resources with which one can prepare previously inaccessible states. The
theory of entanglement is perhaps the best-known and most well-understood
resource theory in this sense. Here we return to the basic questions of
thermodynamics using the formalism of resource theories developed in quantum
information theory and show that the free energy of thermodynamics emerges
naturally from the resource theory of energy-preserving transformations.
Specifically, the free energy quantifies the amount of useful work which can be
extracted from asymptotically-many copies of a quantum system when using only
reversible energy-preserving transformations and a thermal bath at fixed
temperature. The free energy also quantifies the rate at which resource states
can be reversibly interconverted asymptotically, provided that a sublinear
amount of coherent superposition over energy levels is available, a situation
analogous to the sublinear amount of classical communication required for
entanglement dilution.Comment: 4.5 pages main text, 12 pages appendix; v3: improvements to
presentation of the main resul
All-Versus-Nothing Proof of Einstein-Podolsky-Rosen Steering
Einstein-Podolsky-Rosen steering is a form of quantum nonlocality
intermediate between entanglement and Bell nonlocality. Although Schr\"odinger
already mooted the idea in 1935, steering still defies a complete
understanding. In analogy to "all-versus-nothing" proofs of Bell nonlocality,
here we present a proof of steering without inequalities rendering the
detection of correlations leading to a violation of steering inequalities
unnecessary. We show that, given any two-qubit entangled state, the existence
of certain projective measurement by Alice so that Bob's normalized conditional
states can be regarded as two different pure states provides a criterion for
Alice-to-Bob steerability. A steering inequality equivalent to the
all-versus-nothing proof is also obtained. Our result clearly demonstrates that
there exist many quantum states which do not violate any previously known
steering inequality but are indeed steerable. Our method offers advantages over
the existing methods for experimentally testing steerability, and sheds new
light on the asymmetric steering problem.Comment: 7 pages, 2 figures. Accepted in Sci. Re
Signal Processing
Contains research objectives and reports on two research projects.Joint Services Electronics Programs (U. S. Army, U. S. Navy, and U. S. Air Force) under Contract DA 28-043-AMC-02536(E
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