I have a Hamiltonian and I want to know the corresponding density matrix. The matrix I'm interested in is the one in this question.


2 Answers 2


There's many different density matrices that can correspond to a given Hamiltonian.

For the 8x8 matrix in your question, there's 8 different "eigenstate" density matrices that can be obtained, one for each of the 8 eigenvectors. The density matrices are constructed by doing the outer product of the eigenvectors. For the $i^{\rm{th}}$ eigenstate of the Hamiltonian, the density matrix $\rho_i$ is:

$ \rho_i = |\psi_i\rangle_ \langle \psi_i| $.

A system can also be in a "pure" superposition of eigenstates, for example:

$|\psi \rangle = \frac{1}{\sqrt{2}}|\psi_1\rangle + \frac{1}{\sqrt{2}}|\psi_2\rangle $.

Then the density matrix is once again made by doing the outer product of the pure wave function $|\psi\rangle$ with itself.

A system can also be in a "mixed" state, which means it's a linear combination of "pure" states.

In this case you would construct the density matrix like this (for example):

$\rho = 0.5 \rho_1 + 0.5\rho_2$,

which descrbes a state which is a 50% mixture of $\rho_1$ and a 50% mixture of $\rho_2$.

  • $\begingroup$ Thanks a mil thats very helpful. So what you're saying is I need to construct all 8 density matrixes, add them together to get the sum density matrix? $\endgroup$ Sep 30, 2018 at 11:48
  • $\begingroup$ It depends what you need the density matrix for :) $\endgroup$ Sep 30, 2018 at 11:51
  • $\begingroup$ I need it to get the purity of my qubit states and the internal energy of the system vs. the magnetisation factor h $\endgroup$ Sep 30, 2018 at 11:53
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    $\begingroup$ The Hamiltonian doesn't tell you the state of the system. It just tells you how the current state of your system changes with respect to time. $\endgroup$ Sep 30, 2018 at 11:54

Your question remains very unclear as to what it actually is that you want to calculate.

There is no direct correspondence between a system Hamiltonian and the quantum state of the system. No matter what the Hamiltonian, any quantum state is a valid state of the system.

Where a Hamiltonian comes in useful is, if you know the state at some time (say, $t=0$), you can find out what the state is at any later time via the Schroedinger equation $$ i\frac{\partial |\psi\rangle}{\partial t}=H(t)|\psi\rangle. $$ If $H$ does not change in time, you get $$ |\psi(t)\rangle=e^{-iHt}|\psi(0)\rangle $$ or, if your initial state is a mixed state, $$ \rho(t)=e^{-iHt}\rho(0)e^{iHt}. $$

Now, there are two reasonable things that might be relevant in terms of a state derived from a Hamiltonian - the thermal state and the ground state (which is the thermal state at 0 temperature). At temperature $T$, the thermal state is $$ \rho_{\text{thermal}}=\frac{e^{-H/(k_BT)}}{\text{Tr}(e^{-H/(k_BT)})}, $$ while the ground state is simply the eigenstate of $H$ with the smallest energy. You can (crudely) think of the thermal state as the best guess about what the state would be if you cooled it to a temperature $T$.

In one of the comments on another answer, you say

I need it to get the purity of my qubit states and the internal energy of the system vs. the magnetisation factor h

Purity has nothing to do with the Hamiltonian. If you know the density matrix $\rho$ of your system, purity is just $\text{Tr}(\rho^2)$. The Hamiltonian will help you with the expected internal energy: $\text{Tr}(\rho H)$ but, again, the state has to be provided from elsewhere, not from the Hamiltonian.

  • $\begingroup$ Thank you for the answer that has cleared things up a little more now. I just need to go back and revise my method I think. I will post a solution when I have my code complete I didn't explain it well but I can provide an answer I feel will suffice $\endgroup$ Oct 1, 2018 at 11:13
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    $\begingroup$ @OisinBrannock It would be better if you could work a bit more on the question to make it clear what it is you're after. An answer is far more relevant once people understand the question. $\endgroup$
    – DaftWullie
    Oct 1, 2018 at 11:40
  • $\begingroup$ Yeah cool I'll work on the question it is a bit all over the place to be fair thanks for the feedback $\endgroup$ Oct 1, 2018 at 11:41
  • $\begingroup$ @DaftWullie: I think the question is completely fine the way it is. "What is the density matrix corresponding to a given Hamiltonian?" the answer is that when the system has Hamiltonian H, there's many different $\rho$ that are valid. I've given some examples and you've given 2 examples. Not any density matrix is okay (for example if the size is different from the size of the Hamiltonian). Further questions about the user's problem can be asked in a new question. $\endgroup$ Oct 1, 2018 at 14:38
  • $\begingroup$ Look, the question the way it is right now, is not too surprising for a beginner. I think a lot of people will have the same question over the coming years. I can't remember but I think I also might have been confused about the relationship between Hamiltonian and state when I was in the very early stages. I think the question can be left how it is, and further questions about the user's goals can go in a new question. $\endgroup$ Oct 1, 2018 at 14:40

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