BreuerState
From QETLAB
BreuerState | |
Produces a Breuer state of even dimension ≥ 2 | |
Other toolboxes required | none |
---|---|
Related functions | ChessboardState |
Function category | Special states, vectors, and operators |
BreuerState is a function that produces a two-qudit "Breuer state" (i.e., the state defined in [1]). These states are of interest because they are bound entangled. The output of this function is sparse.
Contents
Syntax
- BREUER_STATE = BreuerState(DIM,LAMBDA)
Argument descriptions
- DIM: The local dimension (must be ≥ 2 and even).
- LAMBDA: The weight of the singlet component in the state (see [1] 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)).
Examples
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):
>> rho = full(BreuerState(4,0.1)); >> IsPPT(rho) ans = 1 >> IsSeparable(rho) Determined to be entangled by not having a 2-copy PPT symmetric extension. Reference: A. C. Doherty, P. A. Parrilo, and F. M. Spedalieri. A complete family of separability criteria. Phys. Rev. A, 69:022308, 2004. ans = 0
Source code
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 [1].
% The Breuer state that is returned is sparse.
%
% References:
% [1] 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 (vrusso@uwaterloo.ca)
% Nathaniel Johnston (nathaniel@njohnston.ca)
% 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.');
end
% Start by generating a specific maximally-entangled state PSI.
V = fliplr(sparse(diag((-1).^mod(1:dim,2))));
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));
References
- ↑ 1.0 1.1 H.-P. Breuer. Optimal entanglement criterion for mixed quantum states. Phys. Rev. Lett., 97:080501, 2006. E-print: arXiv:quant-ph/0605036