# How to do partial trace in a multi-component system

edited

Supposing the system is a spin-1 boson in 1-D optical lattice with three sublevel

$$m_F=0, \pm 1$$

I finally got the ground state as a MPS (or IQMPS). Say I want to trace out the @@m_F=0@@ component and got the reduced density matrix for @@\pm 1@@ mode, how to do it with itensor ?

In fact, I want to calculate the VN-entropy of @@mF=\pm 1@@ sub-system after I partially trace out @@mF=0@@. Thus any other indirect methods to obtain VN-entropy are also great.

commented by (70.1k points)

For your information, the only kind of vN entanglement that is straightforward to compute from an MPS is that of a left-right bipartition of the site indices of the MPS. In principle entanglement for other partitions and in other bases can be computed, but it is somewhat of an open research topic how best to do so.
commented by (650 points)
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Hi, miles. You can think of three sublevels as three components or three different species of atoms. That is, the model is a 1-D optical lattice filled with three different atoms and there are inter and intra-interaction among them. I want to trace one or two species of atom component out to get the reduced density matrix for the remain part.

In fact, the vN-entropy of bond can be interpreted as the entanglement between atoms on different sites, while I want to know the entanglement between different modes.

Hi Junjie,
So the most straightforward answer to your question is that you cannot efficiently measure this particular quantity from an MPS representation of the ground state. Sorry.

One way to see this is to think of your sites not as 3*N dimensional sites, but as separate sites for each of your sublevels. So this would be a representation of your problem having a unit cell, with the first site being sublevel -1, the second being sublevel 0, the third being sublevel +1, the fourth being sublevel -1, etc.

From this point of view, the entanglement you are asking for is like the entanglement of every third site, starting with site 2, relative to all the other sites. But as you probably know, the only entanglement "cut" which is efficient to obtain from an MPS is a left-right bipartition.

Ok that's the straightforward answer. The more complicated answer is that you may be able to obtain this quantity if you "swap" the sites or reorder them. So if you apply swap gates to move all of the sublevel 0 sites to the right of the sublevel +1 and -1 sites then it would be a left-right bipartition. But this is unlikely to scale well, so it's probably inefficient.

Another idea is that you could perform sampling to obtain the second Renyi entropy of the sublevel 0 site entanglement.

Best regards,
Miles

commented by (650 points)
thanks miles, I got this. Then I have the following question. Supposing I rearrange my model as follows : originally I got N sites filled with spin-1 boson(3 components), and MPS generating from this kind of sites is hard to do partial trace on component. What if I change the model to be a 3*N sites with number 1 to N sites filled with mF=1 boson and N+1 to 2*N filled with mF=0 boson and 2*N+1 to 3*N filled with mF=-1 boson. I think these 2 models are identical, though the 2-nd one might be harder to caculate, especially it is hard to conserve quantum number using IQMPS.
commented by (70.1k points)
Hi, yes you could in principle set up a model like this. But yes you may find it's harder (though not much) to arrange the quantum number sectors of the sites to conserve the property you want. More importantly, the physics of your system is probably such that splitting up the different levels spatially is likely to make the MPS much more entangled, but you'll have to guess that for yourself as I don't know what the Hamiltonian is.