# How to calculate flux of a reduced density matrix?

**URL:** <https://itensor.discourse.group/t/how-to-calculate-flux-of-a-reduced-density-matrix/243>\
**Category:** ITensor Julia Questions\
**Tags:** julia, dmrg, mps\
**Created:** [July 10, 2022, 4:21pm UTC](https://itensor.discourse.group/t/how-to-calculate-flux-of-a-reduced-density-matrix/243 "2022-07-10T16:21:46Z")\
**Posts on this page:** 1\
**Showing post:** 7

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**Author:** ![miles](https://yyz2.discourse-cdn.com/free1/user_avatar/itensor.discourse.group/miles/32/6_2.png) [@miles](https://itensor.discourse.group/u/miles)\
**Post date:** [July 14, 2022, 1:31pm UTC](https://itensor.discourse.group/t/how-to-calculate-flux-of-a-reduced-density-matrix/243/7 "2022-07-14T13:31:42Z")

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Hi Zach,  
Thanks for the follow-up question here. So for the case of a pure state which is an MPS, there is a shortcut to getting the kind of rdm you want (where sites `1:k` are traced over). The shortcut is to “gauge” the MPS so that the tensors on sites 1,2,…,k are all “left-orthogonal”. This can be done in ITensor by calling `orthogonalize!(psi,k+1)` on an MPS `psi`. Afterward, the “orthogonality center” will be at site `k+1`, meaning that the tensors 1,2,…,k will all be left-orthogonal (and tensors `k+2,k+3,...,N` will also be right-orthogonal).

After that step, you can show that tracing sites 1:k just amounts to leaving out those tensors (they cancel) and just contracting MPS tensor number k+1 with its primed conjugate over the virtual or link index on bond k. If we call the virtual index for bond k+1 b\_{k+1} and the site index for site k+1 s\_{k+1} then the resulting rdm will have indices s\_{k+1},b\_{k+1}, s'\_{k+1},b'\_{k+1}. Then you can input this rdm to our `eigen` function in ITensor and factorize it to get its eigenvalues.

I’d have to play around with the code to remember how to extract the QNs. Once you have the QN objects themselves, you can call the [`val` function](https://itensor.github.io/ITensors.jl/stable/QN.html#ITensors.val-Tuple%7BQN,%20Any%7D) on them to get their numerical values. Say the name of the QN you want is `"Sz"` so like inside of a QN object like `q-QN("Sz",-1,"Nf",1)`. Then you do `val(q,"Sz")` and it will return the number `-1`.

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