HermAlgLab

Hermitian Algebras MatLab library (HermAlgLab) is a collection of functions to test the behaviour of matrix algebras built up from Hermitian matrices. Currently, the library has support for

• Finding the anti-commutator (Jordan) closure of a set of Hermitian matrices
• Finding the commutator (Lie) closure of a set of Hermitian matrices
• Finding the tetrad closure of a set of Hermitian matrices
• Finding the free algebra closure of a set of Hermitian matrices
• Finding the Hermitian part of any of these closures
• Simulating quadratic systems of ODEs over the underlying algebra

Documentation is under development

Quickstart

In this example we can test the Jordan and Lie closures of the set of Hermitian matrices composed of tensors of Pauli matrices F={II, XI, XX, ZZ}

I = [1,0;0,1]; X = [0,1;1,0]; Y = [0,-1i;1i,0]; Z = [1,0;0,-1];
F(:,:,1)     = kron(I,I);
F(:,:,end+1) = kron(X,I);
F(:,:,end+1) = kron(X,X);
F(:,:,end+1) = kron(Z,Z);

[J, J_level] = JORDAN(F, tol);
[L, L_level] =    LIE(F, tol);

Cohn Reversible Theorem

Testing a theorem like the Cohn Reversible Theorem (citation needed) is easy using the b_* functions and the various CLOSURE functions.

% Set the tolerance to 1e-4
tol = 1e-4;

% Let's use the test set of two non-orthogonal states
F = F2nonorth(1i/3);

% Cohn's reversibility theorem
A = HERM(FREE(F,tol));
B = b_join(JORDAN(F,tol), TETRADS(F,tol));
if b_eq(A,B)
  printf('Cohn reversibility theorem holds true for F.\n')
else
  printf('Cohn reversibility theorem does not hold true for F.\n')
end