# Whether TDVP can be used to calculate lindblad master equation

**URL:** <https://itensor.discourse.group/t/whether-tdvp-can-be-used-to-calculate-lindblad-master-equation/813>\
**Category:** DMRG and Numerical Methods\
**Tags:** julia, mps\
**Created:** [April 27, 2023, 1:16pm UTC](https://itensor.discourse.group/t/whether-tdvp-can-be-used-to-calculate-lindblad-master-equation/813 "2023-04-27T13:16:53Z")\
**Posts on this page:** 5\
**Page:** 1

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**Author:** ![jzHan](https://avatars.discourse-cdn.com/v4/letter/j/45deac/32.png) [@jzHan](https://itensor.discourse.group/u/jzHan)\
**Post date:** [April 27, 2023, 1:16pm UTC](https://itensor.discourse.group/t/whether-tdvp-can-be-used-to-calculate-lindblad-master-equation/813/1 "2023-04-27T13:16:53Z")

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Hello, I am currently learning how to use itensor to compute open systems. I found that TEBD and TDVP are the methods for time evolution in the itensor, but the lindblad master equation used in my model is dissipative. I want to ask whether TDVP has the ability to evolve master equation.

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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:** [April 28, 2023, 1:49pm UTC](https://itensor.discourse.group/t/whether-tdvp-can-be-used-to-calculate-lindblad-master-equation/813/2 "2023-04-28T13:49:01Z")

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Yes, the TDVP method can be used to simulate open quantum systems. When I searched the literature for applications of TDVP to these systems I found several papers, for example this one:  
[https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.062115](https://journals.aps.org/pra/abstract/10.1103/PhysRevA.86.062115)

Our TDVP code assumes one is evolving a state by some MPO H as e^{-i H t} \ket{\Psi} however \ket{\Psi} here could be a “purified” density matrix so it could represent an open system.

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**Author:** ![jzHan](https://avatars.discourse-cdn.com/v4/letter/j/45deac/32.png) [@jzHan](https://itensor.discourse.group/u/jzHan)\
**Post date:** [April 28, 2023, 2:15pm UTC](https://itensor.discourse.group/t/whether-tdvp-can-be-used-to-calculate-lindblad-master-equation/813/3 "2023-04-28T14:15:42Z")

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I really appreciate the detailed replies.  
I will read this paper carefully.

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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:** [April 28, 2023, 9:36pm UTC](https://itensor.discourse.group/t/whether-tdvp-can-be-used-to-calculate-lindblad-master-equation/813/4 "2023-04-28T21:36:37Z")

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Great. There might be papers on this which are more accessible and easier to read, so I’d suggest you look for a few different ones and compare their approaches. Good luck!

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**Author:** ![Meng\_Xu](https://avatars.discourse-cdn.com/v4/letter/m/8dc957/32.png) [@Meng\_Xu](https://itensor.discourse.group/u/Meng_Xu)\
**Post date:** [June 22, 2023, 2:55pm UTC](https://itensor.discourse.group/t/whether-tdvp-can-be-used-to-calculate-lindblad-master-equation/813/5 "2023-06-22T14:55:58Z")

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That’s works with TDVP or TEBD to propagate your Lindblad master equation but not directly. As the first step, you need to reshape the density matrix equation into wave-like (1d vector) equation, e.g., \hat{a}\hat{\rho}\hat{a}^\dagger to \hat{a}\hat{b} W(t) (Purification or kronecker product) with operator \hat{b} considered as fictitious modes, functioning as counterpart to the operator \hat{a} modes.

Then you can perform the time evolution with TDVP, the reduced dynamics can be obtained by another trace or not.