ChessboardState
From QETLAB
ChessboardState | |
Produces a chessboard state | |
Other toolboxes required | none |
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Related functions | BreuerState IsPPT PartialTranspose |
Function category | Special states, vectors, and operators |
ChessboardState is a function that produces a two-qutrit "chessboard state", as defined in ^{[1]}. These states are of interest because they are bound entangled.
Contents
Syntax
- RHO = ChessboardState(A,B,C,D,M,N)
- RHO = ChessboardState(A,B,C,D,M,N,S,T)
Argument descriptions
- A,B,C,D,M,N: Six parameters that define chessboard states, as in ^{[1]}, with S = A*conj(C)/conj(N) and T = A*D/M. If C*M*conj(N) does not equal A*B*conj(C) then RHO is entangled. If each of A,B,C,D,M,N are real then RHO has positive partial transpose, and is hence bound entangled.
- S,T: Additional (optional) parameters of the chessboard states, also as in ^{[1]}. Note that, for certain choices of S and T, this state will not have positive partial transpose, and thus may not be bound entangled – a warning will be produced in these cases.
Examples
Generating bound entangled states
Chessboard states are useful because they form a wide family of bound entangled states. The following code generates a random chessboard state and verifies that it is entangled yet positive-partial-transpose (and hence bound entangled).
>> rho = ChessboardState(randn(1),randn(1),randn(1),randn(1),randn(1),randn(1)); >> IsSeparable(rho) Determined to be entangled via the Filter Covariance Matrix Criterion. Reference: O. Gittsovich, O. Gühne, P. Hyllus, and J. Eisert. Unifying several separability conditions using the covariance matrix criterion. Phys. Rev. A, 78:052319, 2008. ans = 0 >> IsPPT(rho) ans = 1
When specifying S and T
If you specify S and T manually, it is possible that the resulting state will not have positive partial transpose – a warning is produced in these cases.
>> rho = ChessboardState(1,2,3,4,5,6,7,8); Warning: The specified chessboard state does not have positive partial transpose. > In ChessboardState at 45
Source code
Click on "expand" to the right to view the MATLAB source code for this function.
%% CHESSBOARDSTATE Produces a chessboard state
% This function has six required arguments:
% A,B,C,D,M,N: parameters of the chessboard state, as in [1]
%
% RHO = ChessboardState(A,B,C,D,M,N) is the chessboard state defined in
% [1], with S = A*conj(C)/conj(N) and T = A*D/M. If C*M*conj(N) ~=
% A*B*conj(C) then RHO is entangled. If each of A,B,C,D,M,N are real then
% RHO has positive partial transpose, and is hence bound entangled.
%
% This function has two optional arguments:
% S (default A*conj(C)/conj(N))
% T (default A*D/M)
%
% RHO = ChessboardState(A,B,C,D,M,N,S,T) is the chessboard state defined
% in [1]. Note that, for certain choices of S and T, this state will not
% have positive partial transpose, and thus may not be bound entangled --
% a warning will be produced in these cases.
%
% URL: http://www.qetlab.com/ChessboardState
%
% References:
% [1] D. Bruss and A. Peres. Construction of quantum states with bound
% entanglement. Phys. Rev. A, 61:30301(R), 2000.
% requires: IsPPT.m, IsPSD.m, opt_args.m, PartialTranspose.m,
% PermuteSystems.m
%
% author: Nathaniel Johnston (nathaniel@njohnston.ca)
% package: QETLAB
% last updated: March 13, 2013
function rho = ChessboardState(a,b,c,d,m,n,varargin)
% set optional argument defaults: s = ac*/n*, t = ad/m
[s,t] = opt_args({ a*conj(c)/conj(n), a*d/m },varargin{:});
v1 = [m,0,s,0,n,0,0,0,0];
v2 = [0,a,0,b,0,c,0,0,0];
v3 = [conj(n),0,0,0,-conj(m),0,t,0,0];
v4 = [0,conj(b),0,-conj(a),0,0,0,d,0];
rho = v1'*v1 + v2'*v2 + v3'*v3 + v4'*v4;
rho = rho/trace(rho);
if(~IsPPT(rho))
warning('ChessboardState:NotPPT','The specified chessboard state does not have positive partial transpose.');
end
References
- ↑ ^{1.0} ^{1.1} ^{1.2} D. Bruss and A. Peres. Construction of quantum states with bound entanglement. Phys. Rev. A, 61:30301(R), 2000. E-print: arXiv:quant-ph/9911056