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

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Vikash
Vikash 2025-8-18，5:46
评论： Vikash 2025-8-19，5:31
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?

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回答（1 个）

Matt J
Matt J 2025-8-18，10:58
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';
  1 个评论
Vikash
Vikash 2025-8-19，5:31
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.

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