3,253 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
Matrix Product States Algorithms and Continuous Systems
A generic method to investigate many-body continuous-variable systems is
pedagogically presented. It is based on the notion of matrix product states
(so-called MPS) and the algorithms thereof. The method is quite versatile and
can be applied to a wide variety of situations. As a first test, we show how it
provides reliable results in the computation of fundamental properties of a
chain of quantum harmonic oscillators achieving off-critical and critical
relative errors of the order of 10^(-8) and 10^(-4) respectively. Next, we use
it to study the ground state properties of the quantum rotor model in one
spatial dimension, a model that can be mapped to the Mott insulator limit of
the 1-dimensional Bose-Hubbard model. At the quantum critical point, the
central charge associated to the underlying conformal field theory can be
computed with good accuracy by measuring the finite-size corrections of the
ground state energy. Examples of MPS-computations both in the finite-size
regime and in the thermodynamic limit are given. The precision of our results
are found to be comparable to those previously encountered in the MPS studies
of, for instance, quantum spin chains. Finally, we present a spin-off
application: an iterative technique to efficiently get numerical solutions of
partial differential equations of many variables. We illustrate this technique
by solving Poisson-like equations with precisions of the order of 10^(-7).Comment: 22 pages, 14 figures, final versio
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
General entanglement scaling laws from time evolution
We establish a general scaling law for the entanglement of a large class of
ground states and dynamically evolving states of quantum spin chains: we show
that the geometric entropy of a distinguished block saturates, and hence
follows an entanglement-boundary law. These results apply to any ground state
of a gapped model resulting from dynamics generated by a local hamiltonian, as
well as, dually, to states that are generated via a sudden quench of an
interaction as recently studied in the case of dynamics of quantum phase
transitions. We achieve these results by exploiting ideas from quantum
information theory and making use of the powerful tools provided by
Lieb-Robinson bounds. We also show that there exist noncritical fermionic
systems and equivalent spin chains with rapidly decaying interactions whose
geometric entropy scales logarithmically with block length. Implications for
the classical simulatability are outlined.Comment: 4 pages, 1 figure (see also related work by S. Bravyi, M. Hastings,
and F. Verstraete, quant-ph/0603121); replaced with final versio
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
Half the entanglement in critical systems is distillable from a single specimen
We establish that the leading critical scaling of the single-copy
entanglement is exactly one half of the entropy of entanglement of a block in
critical infinite spin chains in a general setting, using methods of conformal
field theory. Conformal symmetry imposes that the single-copy entanglement for
critical many-body systems scales as E_1(\rho_L)=(c/6) \log L- (c/6)
(\pi^2/\log L) + O(1/L), where L is the number of constituents in a block of an
infinite chain and c corresponds to the central charge. This proves that from a
single specimen of a critical chain, already half the entanglement can be
distilled compared to the rate that is asymptotically available. The result is
substantiated by a quantitative analysis for all translationally invariant
quantum spin chains corresponding to general isotropic quasi-free fermionic
models. An analytic example of the XY model shows that away from criticality
the above simple relation is only maintained near the quantum phase transition
point.Comment: 4 pages RevTeX, 1 figure, final versio
Optimal minimal measurements of mixed states
The optimal and minimal measuring strategy is obtained for a two-state system
prepared in a mixed state with a probability given by any isotropic a priori
distribution. We explicitly construct the specific optimal and minimal
generalized measurements, which turn out to be independent of the a priori
probability distribution, obtaining the best guesses for the unknown state as
well as a closed expression for the maximal mean averaged fidelity. We do this
for up to three copies of the unknown state in a way which leads to the
generalization to any number of copies, which we then present and prove.Comment: 20 pages, no figure
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