486 research outputs found
On the fidelity of mixed states of two qubits
We consider a single copy of a mixed state of two qubits and show how its
fidelity or maximal singlet fraction is related to the entanglement measures
concurrence and negativity. We characterize the extreme points of the convex
set of states with constant fidelity, and use this to prove tight lower and
upper bounds on the fidelity for a given amount of entanglement.Comment: 4 pages; part I of quant-ph/0203073v2; see quant-ph/0303007 for part
I
Diagonalizing transfer matrices and matrix product operators: a medley of exact and computational methods
Transfer matrices and matrix product operators play an ubiquitous role in the
field of many body physics. This paper gives an ideosyncratic overview of
applications, exact results and computational aspects of diagonalizing transfer
matrices and matrix product operators. The results in this paper are a mixture
of classic results, presented from the point of view of tensor networks, and of
new results. Topics discussed are exact solutions of transfer matrices in
equilibrium and non-equilibrium statistical physics, tensor network states,
matrix product operator algebras, and numerical matrix product state methods
for finding extremal eigenvectors of matrix product operators.Comment: Lecture notes from a course at Vienna Universit
Fermionic Implementation of Projected Entangled Pair States Algorithm
We present and implement an efficient variational method to simulate
two-dimensional finite size fermionic quantum systems by fermionic projected
entangled pair states. The approach differs from the original one due to the
fact that there is no need for an extra string-bond for contracting the tensor
network. The method is tested on a bi-linear fermionic model on a square
lattice for sizes up to ten by ten where good relative accuracy is achieved.
Qualitatively good results are also obtained for an interacting fermionic
system.Comment: As published in Phys. Rev.
Variational Numerical Renormalization Group: Bridging the gap between NRG and Density Matrix Renormalization Group
The numerical renormalization group (NRG) is rephrased as a variational
method with the cost function given by the sum of all the energies of the
effective low-energy Hamiltonian. This allows to systematically improve the
spectrum obtained by NRG through sweeping. The ensuing algorithm has a lot of
similarities to the density matrix renormalization group (DMRG) when targeting
many states, and this synergy of NRG and DMRG combines the best of both worlds
and extends their applicability. We illustrate this approach with simulations
of a quantum spin chain and a single impurity Anderson model (SIAM) where the
accuracy of the effective eigenstates is greatly enhanced as compared to the
NRG, especially in the transition to the continuum limit.Comment: As accepted to PRL. Main text: 4 pages, 4 (PDF) figures;
Supplementary material: 4 pages, 6 PDF figures; revtex4-
On the geometry of entangled states
The basic question that is addressed in this paper is finding the closest
separable state for a given entangled state, measured with the Hilbert Schmidt
distance. While this problem is in general very hard, we show that the
following strongly related problem can be solved: find the Hilbert Schmidt
distance of an entangled state to the set of all partially transposed states.
We prove that this latter distance can be expressed as a function of the
negative eigenvalues of the partial transpose of the entangled state, and show
how it is related to the distance of a state to the set of positive partially
transposed states (PPT-states). We illustrate this by calculating the closest
biseparable state to the W-state, and give a simple and very general proof for
the fact that the set of W-type states is not of measure zero. Next we show
that all surfaces with states whose partial transposes have constant minimal
negative eigenvalue are similar to the boundary of PPT states. We illustrate
this with some examples on bipartite qubit states, where contours of constant
negativity are plotted on two-dimensional intersections of the complete state
space.Comment: submitted to Journal of Modern Optic
Boundary-field-driven control of discontinuous phase transitions on hyperbolic lattices
The multistate Potts models on two-dimensional hyperbolic lattices are
studied with respect to various boundary effects. The free energy is
numerically calculated by Corner Transfer Matrix Renormalization Group method.
We analyze phase transitions of the Potts models in the thermodynamic limit
with respect to contracted boundary layers. A false phase transition is present
even if a couple of the boundary layers are contracted. Its significance
weakens, as the number of the contracted boundary layers increases, until the
correct phase transition (deep inside the bulk) prevails over the false one.
For this purpose we derive a thermodynamic quantity, the so-called bulk excess
free energy, which depends on the contracted boundary layers and memorizes
additional boundary effects. In particular, the magnetic field is imposed on
the outermost boundary layer. While the boundary magnetic field does not affect
the second-order phase transition in the bulk if suppressing all the boundary
effects on the hyperbolic lattices, the first-order (discontinuous) phase
transition is significantly sensitive to the boundary magnetic field. Contrary
to the phase transition on the Euclidean lattices, the discontinuous phase
transition on the hyperbolic lattices can be continuously controlled (within a
certain temperature coexistence region) by varying the boundary magnetic field.Comment: 12 pages, 13 figure
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