How can I perform partial trace for a pure state with four subsystems?

Hello everyone,

I am working a quantum system of Hilbert space 2X2XNXN and investigating the dynamical evolution of the entanglement between subsystems (1,3) and (2,4). At every time step, I have a pure state \ket{psi} of dimension 2X2XNXN, and I want to find a reduced density matrix after tracing out subsystems 2nd and 4th, i.e., the reduced density matrix should be 2NX2N dimensional.

I looked at some functions which are available online for doing these (one of the very famous one is from QETLAB) but they are very slow and I need to call them after each time step.

Given that I have pure state at all time steps, is there an efficient way to do this partial trace without using those complicated functions?

Answer 1

[Matt J]

Does this do what you want?

% reshape |psi> into tensor (s1,s2,s3,s4)
psi_tensor = reshape(psi,[2,2,N,N]);

% reorder to group (s1,s3) vs (s2,s4), then flatten
psi_mat = reshape(permute(psi_tensor,[1,3,2,4]), [2*N, 2*N]);

% reduced density matrix rho_13 = psi_mat * psi_mat'
rho_13 = psi_mat * psi_mat';

[Vikash]

Hi Matt J,

Thanks for your reply. I did try that and unfortunately it is not working. I also thought it should be straighforward.

For example, I am doing something like this.

% Standard method using a function from QETLAB
rho = sparse(psif*psif');
rhoC = sparse(PartialTrace(rho, [2 4], [2 2 dimL dimL]));

% The method you suggested
% reshape |psi> into tensor (s1,s2,s3,s4)
psi_tensor = reshape(full(psif),[2,2,dimL,dimL]);
% reorder to group (s1,s3) vs (s2,s4), then flatten
psi_mat = reshape(permute(psi_tensor,[1,3,2,4]), [2*dimL, 2*dimL]);
% reduced density matrix rho_13 = psi_mat * psi_mat'
rho_13 = psi_mat * psi_mat';

where I took dimL = 5 as an example and psif is column vector of dimension 4*dimL*dimL. These two methods produce different output as attached in the image.