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 $\mathfrak g$ under the presence of a complex structure $J$. In particular, we find a bound for the dimension of the center of $\mathfrak g$ when it does not contain any non-trivial $J$-invariant ideal. Thanks to these results, we provide a structural theorem describing the ascending central series of 8-dimensional nilpotent Lie algebras $\mathfrak g$ admitting this particular type of complex structures $J$. Since our method is constructive, it allows us to describe the complex structure equations that parametrize all such pairs $(\mathfrak g, J)$.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