3,320 research outputs found
The ascending central series of nilpotent Lie algebras with complex structure
We obtain several restrictions on the terms of the ascending central series
of a nilpotent Lie algebra under the presence of a complex
structure . In particular, we find a bound for the dimension of the center
of when it does not contain any non-trivial -invariant ideal.
Thanks to these results, we provide a structural theorem describing the
ascending central series of 8-dimensional nilpotent Lie algebras
admitting this particular type of complex structures . Since our method is
constructive, it allows us to describe the complex structure equations that
parametrize all such pairs .Comment: 28 pages, 1 figure. To appear in Trans. Amer. Math. So
Systematic Analysis of Majorization in Quantum Algorithms
Motivated by the need to uncover some underlying mathematical structure of
optimal quantum computation, we carry out a systematic analysis of a wide
variety of quantum algorithms from the majorization theory point of view. We
conclude that step-by-step majorization is found in the known instances of fast
and efficient algorithms, namely in the quantum Fourier transform, in Grover's
algorithm, in the hidden affine function problem, in searching by quantum
adiabatic evolution and in deterministic quantum walks in continuous time
solving a classically hard problem. On the other hand, the optimal quantum
algorithm for parity determination, which does not provide any computational
speed-up, does not show step-by-step majorization. Lack of both speed-up and
step-by-step majorization is also a feature of the adiabatic quantum algorithm
solving the 2-SAT ``ring of agrees'' problem. Furthermore, the quantum
algorithm for the hidden affine function problem does not make use of any
entanglement while it does obey majorization. All the above results give
support to a step-by-step Majorization Principle necessary for optimal quantum
computation.Comment: 15 pages, 14 figures, final versio
Entanglement and Quantum Phase Transition Revisited
We show that, for an exactly solvable quantum spin model, a discontinuity in
the first derivative of the ground state concurrence appears in the absence of
quantum phase transition. It is opposed to the popular belief that the
non-analyticity property of entanglement (ground state concurrence) can be used
to determine quantum phase transitions. We further point out that the
analyticity property of the ground state concurrence in general can be more
intricate than that of the ground state energy. Thus there is no one-to-one
correspondence between quantum phase transitions and the non-analyticity
property of the concurrence. Moreover, we show that the von Neumann entropy, as
another measure of entanglement, can not reveal quantum phase transition in the
present model. Therefore, in order to link with quantum phase transitions, some
other measures of entanglement are needed.Comment: RevTeX 4, 4 pages, 1 EPS figures. some modifications in the text.
Submitted to Phys. Rev.
Optimal control of multiscale systems using reduced-order models
We study optimal control of diffusions with slow and fast variables and
address a question raised by practitioners: is it possible to first eliminate
the fast variables before solving the optimal control problem and then use the
optimal control computed from the reduced-order model to control the original,
high-dimensional system? The strategy "first reduce, then optimize"--rather
than "first optimize, then reduce"--is motivated by the fact that solving
optimal control problems for high-dimensional multiscale systems is numerically
challenging and often computationally prohibitive. We state sufficient and
necessary conditions, under which the "first reduce, then control" strategy can
be employed and discuss when it should be avoided. We further give numerical
examples that illustrate the "first reduce, then optmize" approach and discuss
possible pitfalls
Area law and vacuum reordering in harmonic networks
We review a number of ideas related to area law scaling of the geometric
entropy from the point of view of condensed matter, quantum field theory and
quantum information. An explicit computation in arbitrary dimensions of the
geometric entropy of the ground state of a discretized scalar free field theory
shows the expected area law result. In this case, area law scaling is a
manifestation of a deeper reordering of the vacuum produced by majorization
relations. Furthermore, the explicit control on all the eigenvalues of the
reduced density matrix allows for a verification of entropy loss along the
renormalization group trajectory driven by the mass term. A further result of
our computation shows that single-copy entanglement also obeys area law
scaling, majorization relations and decreases along renormalization group
flows.Comment: 15 pages, 6 figures; typos correcte
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