4,385 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
Simulation of many-qubit quantum computation with matrix product states
Matrix product states provide a natural entanglement basis to represent a
quantum register and operate quantum gates on it. This scheme can be
materialized to simulate a quantum adiabatic algorithm solving hard instances
of a NP-Complete problem. Errors inherent to truncations of the exact action of
interacting gates are controlled by the size of the matrices in the
representation. The property of finding the right solution for an instance and
the expected value of the energy are found to be remarkably robust against
these errors. As a symbolic example, we simulate the algorithm solving a
100-qubit hard instance, that is, finding the correct product state out of ~
10^30 possibilities. Accumulated statistics for up to 60 qubits point at a slow
growth of the average minimum time to solve hard instances with
highly-truncated simulations of adiabatic quantum evolution.Comment: 5 pages, 4 figures, final versio
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 generalized quantum measurements for arbitrary spin systems
Positive operator valued measurements on a finite number of N identically
prepared systems of arbitrary spin J are discussed. Pure states are
characterized in terms of Bloch-like vectors restricted by a SU(2 J+1)
covariant constraint. This representation allows for a simple description of
the equations to be fulfilled by optimal measurements. We explicitly find the
minimal POVM for the N=2 case, a rigorous bound for N=3 and set up the analysis
for arbitrary N.Comment: LateX, 12 page
Fine-grained entanglement loss along renormalization group flows
We explore entanglement loss along renormalization group trajectories as a
basic quantum information property underlying their irreversibility. This
analysis is carried out for the quantum Ising chain as a transverse magnetic
field is changed. We consider the ground-state entanglement between a large
block of spins and the rest of the chain. Entanglement loss is seen to follow
from a rigid reordering, satisfying the majorization relation, of the
eigenvalues of the reduced density matrix for the spin block. More generally,
our results indicate that it may be possible to prove the irreversibility along
RG trajectories from the properties of the vacuum only, without need to study
the whole hamiltonian.Comment: 5 pages, 3 figures; minor change
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