# Construction of Matrix product density operator

Hi Miles

           I was wondering how can we construct the matrix product density operator in ITensor since there is not any explicit way of doing that. I need to construct MPDO for the pure state as thermal state. Thanks in advance.


Hi, good question but the important thing to know is that basically the MPS and MPO classes in ITensor are essentially just arrays of ITensors. The only details beyond them just being arrays are that they "remember" their orthogonality center (if they have one) and they have some associated functions that expect those types (MPS, MPO) to be passed.

But you can always put any collection of ITensors into an array and pass this array around, or store this array in a custom type (MPDO, say) then define analogous functions or operations on it as we have for MPS and MPO.

Finally, an MPDO is structurally equivalent to an MPO, correct? If so you could just repurpose the ITensor MPO type for your work. Or are you referring to the network I know as a "locally purified state" which looks more like two MPO's contracted together?

Best regards,
Miles

commented by (360 points)
Hello all,

I'd like to ask a follow up question on this topic since I do not fully understand the answer: suppose I have a set of states obtained with a procedure like the one exemplified here:

https://itensor.org/docs.cgi?vers=cppv3&page=formulas/excited_dmrg

so I would have a number of excited states. Now, how would I go about writing a thermal density matrix where these states are added together with a thermal weight?

It seems to me we have a set of wave functions and that the density matrices for each of them should be written and then summed over, but it's not clear to me how that would be done with the MPS resulting from the excited state search.

Bonus question: assuming I have such a mixed state, how to get the expected value of an operator?

Best wishes,

Rafael