|Produces a Breuer state of even dimension ≥ 2|
|Other toolboxes required||none|
|Function category||Special states, vectors, and operators|
BreuerState is a function that produces a two-qudit "Breuer state" (i.e., the state defined in ). These states are of interest because they are bound entangled. The output of this function is sparse.
- BREUER_STATE = BreuerState(DIM,LAMBDA)
- DIM: The local dimension (must be ≥ 2 and even).
- LAMBDA: The weight of the singlet component in the state (see  for details). A positive real number between 0 and 1. The state will be separable if and only if LAMBDA = 0 and it will have positive partial transpose if and only if LAMBDA <= 1/(DIM + 2)).
A 4 ⊗ 4 bound entangled state
The following code generates a bound entangled Breuer state in $M_4 \otimes M_4$ and then verifies that has positive partial transpose and is entangled (and is thus bound entangled):
Click on "expand" to the right to view the MATLAB source code for this function.
%% BREUERSTATE Produces a Breuer state of even dimension >= 2
% This function has two required arguments:
% DIM: the local dimension (for dim >= 2 and even)
% LAMBDA: describes the weight of the singlet component
% BREUER_STATE = BreuerState(DIM,LAMBDA) gives a Breuer bound entangled
% state for two qudits of local dimension DIM, with the LAMBDA parameter
% describing the weight of the singlet component, as described in .
% The Breuer state that is returned is sparse.
%  H-P. Breuer. Optimal entanglement criterion for mixed quantum
% states. E-print: arXiv:quant-ph/0605036, 2006.
% URL: http://www.qetlab.com/BreuerState
% requires: MaxEntangled.m, SymmetricProjection.m
% authors: Vincent Russo (firstname.lastname@example.org)
% Nathaniel Johnston (email@example.com)
% package: QETLAB
% last updated: December 15, 2014
function breuer_state = BreuerState(dim, lambda)
% Make sure that the dimension is even.
if mod(dim,2) == 1 || dim <= 0
error('BreuerState:InvalidDim','DIM must be an even positive integer.');
% Start by generating a specific maximally-entangled state PSI.
psi = kron(speye(dim),V)*MaxEntangled(dim,1);
% Mix the maximally-entangled state PSI with the normalized projector to
% the symmetric subspace.
breuer_state = lambda * (psi*psi') + (1-lambda) * 2*SymmetricProjection(dim)/(dim*(dim+1));
- H.-P. Breuer. Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett., 97:080501, 2006. E-print: arXiv:quant-ph/0605036