Entanglement monotones

In the context of quantifying entanglement we study those functions of a multipartite state which do not increase under the set of local transformations. A mathematical characterization of these monotone magnitudes is presented. They are then related to optimal strategies of conversion of shared states. More detailed results are presented for pure states of bipartite systems. It is show that more than one measure are required simultaneously in order to quantify completely the non-local resources contained in a bipartite pure state, while examining how this fact does not hold in the so-called asymptotic limit. Finally, monotonicity under local transformations is proposed as the only natural requirement for measures of entanglement.Comment: Revtex, 13 pages, no figures. Previous title: "On the characterization of entanglement". Major changes in notation and structure. Some new results, comments and references have been adde

Operational criterion and constructive checks for the separability of low rank density matrices

We consider low rank density operators $\varrho$ supported on a $M\times N$ Hilbert space for arbitrary $M$ and $N$ ($M\leq N$) and with a positive partial transpose (PPT) $\varrho^{T_A}\ge 0$. For rank $r(\varrho) \leq N$ we prove that having a PPT is necessary and sufficient for $\varrho$ to be separable; in this case we also provide its minimal decomposition in terms of pure product states. It follows from this result that there is no rank 3 bound entangled states having a PPT. We also present a necessary and sufficient condition for the separability of generic density matrices for which the sum of the ranks of $\varrho$ and $\varrho^{T_A}$ satisfies $r(\varrho)+r(\varrho^{T_A}) \le 2MN-M-N+2$. This separability condition has the form of a constructive check, providing thus also a pure product state decomposition for separable states, and it works in those cases where a system of couple polynomial equations has a finite number of solutions, as expected in most cases.Comment: RevTex, 10 pages, no figure

On the structure of a reversible entanglement generating set for tripartite states

We show that Einstein–Podolsky–Rosen–Bohm (EPR) and Greenberger–Horne–Zeilinger–Mermin (GHZ) states can not generate, through local manipulation and in the asymptotic limit, all forms of three–partite pure–state entanglement in a reversible way. The techniques that we use suggest that there may be a connection between this result and the irreversibility that occurs in the asymptotic preparation and distillation of bipartite mixed states

Fast convergence of imaginary time evolution tensor network algorithms by recycling the environment

We propose an environment recycling scheme to speed up a class of tensor network algorithms that produce an approximation to the ground state of a local Hamiltonian by simulating an evolution in imaginary time. Specifically, we consider the time-evolving block decimation (TEBD) algorithm applied to infinite systems in 1D and 2D, where the ground state is encoded, respectively, in a matrix product state (MPS) and in a projected entangled-pair state (PEPS). An important ingredient of the TEBD algorithm (and a main computational bottleneck, especially with PEPS in 2D) is the computation of the so-called environment, which is used to determine how to optimally truncate the bond indices of the tensor network so that their dimension is kept constant. In current algorithms, the environment is computed at each step of the imaginary time evolution, to account for the changes that the time evolution introduces in the many-body state represented by the tensor network. Our key insight is that close to convergence, most of the changes in the environment are due to a change in the choice of gauge in the bond indices of the tensor network, and not in the many-body state. Indeed, a consistent choice of gauge in the bond indices confirms that the environment is essentially the same over many time steps and can thus be re-used, leading to very substantial computational savings. We demonstrate the resulting approach in 1D and 2D by computing the ground state of the quantum Ising model in a transverse magnetic field.Comment: 17 pages, 28 figure

Simulation of fermionic lattice models in two dimensions with Projected Entangled-Pair States: Next-nearest neighbor Hamiltonians

In a recent contribution [Phys. Rev. B 81, 165104 (2010)] fermionic Projected Entangled-Pair States (PEPS) were used to approximate the ground state of free and interacting spinless fermion models, as well as the $t$-$J$ model. This paper revisits these three models in the presence of an additional next-nearest hopping amplitude in the Hamiltonian. First we explain how to account for next-nearest neighbor Hamiltonian terms in the context of fermionic PEPS algorithms based on simulating time evolution. Then we present benchmark calculations for the three models of fermions, and compare our results against analytical, mean-field, and variational Monte Carlo results, respectively. Consistent with previous computations restricted to nearest-neighbor Hamiltonians, we systematically obtain more accurate (or better converged) results for gapped phases than for gapless ones.Comment: 10 pages, 11 figures, minor change

Explicit tensor network representation for the ground states of string-net models

The structure of string-net lattice models, relevant as examples of topological phases, leads to a remarkably simple way of expressing their ground states as a tensor network constructed from the basic data of the underlying tensor categories. The construction highlights the importance of the fat lattice to understand these models.Comment: 5 pages, pdf figure

Numerical study of the hard-core Bose-Hubbard Model on an Infinite Square Lattice

We present a study of the hard-core Bose-Hubbard model at zero temperature on an infinite square lattice using the infinite Projected Entangled Pair State algorithm [Jordan et al., Phys. Rev. Lett. 101, 250602 (2008)]. Throughout the whole phase diagram our values for the ground state energy, particle density and condensate fraction accurately reproduce those previously obtained by other methods. We also explore ground state entanglement, compute two-point correlators and conduct a fidelity-based analysis of the phase diagram. Furthermore, for illustrative purposes we simulate the response of the system when a perturbation is suddenly added to the Hamiltonian.Comment: 8 pages, 6 figure

Simulation of strongly correlated fermions in two spatial dimensions with fermionic Projected Entangled-Pair States

We explain how to implement, in the context of projected entangled-pair states (PEPS), the general procedure of fermionization of a tensor network introduced in [P. Corboz, G. Vidal, Phys. Rev. B 80, 165129 (2009)]. The resulting fermionic PEPS, similar to previous proposals, can be used to study the ground state of interacting fermions on a two-dimensional lattice. As in the bosonic case, the cost of simulations depends on the amount of entanglement in the ground state and not directly on the strength of interactions. The present formulation of fermionic PEPS leads to a straightforward numerical implementation that allowed us to recycle much of the code for bosonic PEPS. We demonstrate that fermionic PEPS are a useful variational ansatz for interacting fermion systems by computing approximations to the ground state of several models on an infinite lattice. For a model of interacting spinless fermions, ground state energies lower than Hartree-Fock results are obtained, shifting the boundary between the metal and charge-density wave phases. For the t-J model, energies comparable with those of a specialized Gutzwiller-projected ansatz are also obtained.Comment: 25 pages, 35 figures (revised version