57 research outputs found
Beyond the Spectral Theorem: Spectrally Decomposing Arbitrary Functions of Nondiagonalizable Operators
Nonlinearities in finite dimensions can be linearized by projecting them into
infinite dimensions. Unfortunately, often the linear operator techniques that
one would then use simply fail since the operators cannot be diagonalized. This
curse is well known. It also occurs for finite-dimensional linear operators. We
circumvent it by developing a meromorphic functional calculus that can
decompose arbitrary functions of nondiagonalizable linear operators in terms of
their eigenvalues and projection operators. It extends the spectral theorem of
normal operators to a much wider class, including circumstances in which poles
and zeros of the function coincide with the operator spectrum. By allowing the
direct manipulation of individual eigenspaces of nonnormal and
nondiagonalizable operators, the new theory avoids spurious divergences. As
such, it yields novel insights and closed-form expressions across several areas
of physics in which nondiagonalizable dynamics are relevant, including
memoryful stochastic processes, open non unitary quantum systems, and
far-from-equilibrium thermodynamics.
The technical contributions include the first full treatment of arbitrary
powers of an operator. In particular, we show that the Drazin inverse,
previously only defined axiomatically, can be derived as the negative-one power
of singular operators within the meromorphic functional calculus and we give a
general method to construct it. We provide new formulae for constructing
projection operators and delineate the relations between projection operators,
eigenvectors, and generalized eigenvectors.
By way of illustrating its application, we explore several, rather distinct
examples.Comment: 29 pages, 4 figures, expanded historical citations;
http://csc.ucdavis.edu/~cmg/compmech/pubs/bst.ht
Ultimate limit on learning non-Markovian behavior: Fisher information rate and excess information
We address the fundamental limits of learning unknown parameters of any
stochastic process from time-series data, and discover exact closed-form
expressions for how optimal inference scales with observation length. Given a
parametrized class of candidate models, the Fisher information of observed
sequence probabilities lower-bounds the variance in model estimation from
finite data. As sequence-length increases, the minimal variance scales as the
square inverse of the length -- with constant coefficient given by the
information rate. We discover a simple closed-form expression for this
information rate, even in the case of infinite Markov order. We furthermore
obtain the exact analytic lower bound on model variance from the
observation-induced metadynamic among belief states. We discover ephemeral,
exponential, and more general modes of convergence to the asymptotic
information rate. Surprisingly, this myopic information rate converges to the
asymptotic Fisher information rate with exactly the same relaxation timescales
that appear in the myopic entropy rate as it converges to the Shannon entropy
rate for the process. We illustrate these results with a sequence of examples
that highlight qualitatively distinct features of stochastic processes that
shape optimal learning
The impossibility of Landauer's bound for almost every quantum state
The thermodynamic cost of resetting an arbitrary initial state to a
particular desired state is lower bounded by Landauer's bound. However, here we
demonstrate that this lower bound is necessarily unachievable for nearly every
initial state, for any reliable reset mechanism. Since local heating threatens
rapid decoherence, this issue is of substantial importance beyond mere energy
efficiency. For the case of qubit reset, we find the minimally dissipative
state analytically for any reliable reset protocol, in terms of the
entropy-flow vector introduced here. This allows us to verify a recent theorem
about initial-state dependence of entropy production for any finite-time
transformation, as it pertains to quantum state preparation.Comment: 9 pages plus 3 pages of appendices, 3 figure
Initial-State Dependence of Thermodynamic Dissipation for any Quantum Process
New exact results about the nonequilibrium thermodynamics of open quantum
systems at arbitrary timescales are obtained by considering all possible
variations of initial conditions of a system, its environment, and correlations
between them. First we obtain a new quantum-information theoretic equality for
entropy production, valid for an arbitrary initial joint state of system and
environment. For any finite-time process with a fixed initial environment, we
then show that the contraction of the system's distinction -- relative to the
minimally dissipative state -- exactly quantifies its thermodynamic
dissipation. The quantum component of this dissipation is the change in
coherence relative to the minimally dissipative state. Implications for quantum
state preparation and local control are explored. For nonunitary processes --
like the preparation of any particular quantum state -- we find that mismatched
expectations lead to divergent dissipation as the actual initial state becomes
orthogonal to the anticipated one.Comment: 6 pages plus 14 pages of appendices, 1 figur
A Closed-Form Shave from Occam's Quantum Razor: Exact Results for Quantum Compression
The causal structure of a stochastic process can be more efficiently
transmitted via a quantum channel than a classical one, an advantage that
increases with codeword length. While previously difficult to compute, we
express the quantum advantage in closed form using spectral decomposition,
leading to direct computation of the quantum communication cost at all encoding
lengths, including infinite. This makes clear how finite-codeword compression
is controlled by the classical process' cryptic order and allows us to analyze
structure within the length-asymptotic regime of infinite-cryptic order (and
infinite Markov order) processes.Comment: 21 pages, 13 figures;
http://csc.ucdavis.edu/~cmg/compmech/pubs/eqc.ht
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
Spectral Simplicity of Apparent Complexity, Part I: The Nondiagonalizable Metadynamics of Prediction
Virtually all questions that one can ask about the behavioral and structural
complexity of a stochastic process reduce to a linear algebraic framing of a
time evolution governed by an appropriate hidden-Markov process generator. Each
type of question---correlation, predictability, predictive cost, observer
synchronization, and the like---induces a distinct generator class. Answers are
then functions of the class-appropriate transition dynamic. Unfortunately,
these dynamics are generically nonnormal, nondiagonalizable, singular, and so
on. Tractably analyzing these dynamics relies on adapting the recently
introduced meromorphic functional calculus, which specifies the spectral
decomposition of functions of nondiagonalizable linear operators, even when the
function poles and zeros coincide with the operator's spectrum. Along the way,
we establish special properties of the projection operators that demonstrate
how they capture the organization of subprocesses within a complex system.
Circumventing the spurious infinities of alternative calculi, this leads in the
sequel, Part II, to the first closed-form expressions for complexity measures,
couched either in terms of the Drazin inverse (negative-one power of a singular
operator) or the eigenvalues and projection operators of the appropriate
transition dynamic.Comment: 24 pages, 3 figures, 4 tables; current version always at
http://csc.ucdavis.edu/~cmg/compmech/pubs/sdscpt1.ht
Resolved Neutral Outflow from a Lensed Dusty Star Forming Galaxy at z=2.09
We report the detection of a massive neutral gas outflow in the z=2.09
gravitationally lensed Dusty Star-Forming Galaxy HATLASJ085358.9+015537
(G09v1.40), seen in absorption with the OH+(1_1-1_0) transition using spatially
resolved (0.5"x0.4") Atacama Large Millimeter/submillimeter Array (ALMA)
observations. The blueshifted OH+ line is observed simultaneously with the
CO(9-8) emission line and underlying dust continuum. These data are
complemented by high angular resolution (0.17"x0.13") ALMA observations of
CH+(1-0) and underlying dust continuum, and Keck 2.2 micron imaging tracing the
stellar emission. The neutral outflow, dust, dense molecular gas and stars all
show spatial offsets from each other. The total atomic gas mass of the observed
outflow is 6.7x10^9 M_sun, >25% as massive as the gas mass of the galaxy. We
find that a conical outflow geometry best describes the OH+ kinematics and
morphology and derive deprojected outflow properties as functions of possible
inclination (0.38 deg-64 deg). The neutral gas mass outflow rate is between
83-25400 M_sun/yr, exceeding the star formation rate (788+/-300 M_sun/yr) if
the inclination is >3.6 deg (mass-loading factor = 0.3-4.7). Kinetic energy and
momentum fluxes span 4.4-290x10^9 L_sun and 0.1-3.7x10^37 dyne, respectively
(energy-loading factor = 0.013-16), indicating that the feedback mechanisms
required to drive the outflow depend on the inclination assumed. We derive a
gas depletion time between 29 and 1 Myr, but find that the neutral outflow is
likely to remain bound to the galaxy, unless the inclination is small, and may
be re-accreted if additional feedback processes do not occur.Comment: 33 pages, 20 figure
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