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Mar
8
comment Phase Transition at Zero Temperature (Not QPT)
I think the OP asks exactly for systems which have some kind of order at $T=0$ which immediately disappears at $T>0$, and I would say the Heisenberg model is exactly such an example. Whether this should be called a phase transition, I don't know.
Mar
7
answered Phase Transition at Zero Temperature (Not QPT)
Feb
26
comment Time evolution of a discrete 1-d lattice of spin-(1/2) particles under a given Hamiltonian, or special cases thereof
This makes this a rather broad question. If you only have nearest neighbor interactions, this is can be transformed to the XX model (or directly solved by mapping it to free fermions). If the $r_{ij}$ only act between even and odd sites, you can transform this to a model of hopping particles (but this can be hard to solve). --- In any case: Why do you call this a 1-d lattice? The way you write it there is no 1-d structure.
Feb
26
comment Time evolution of a discrete 1-d lattice of spin-(1/2) particles under a given Hamiltonian, or special cases thereof
Do you know anything about $r_{ij}$? (If not, why is it a 1D lattice?)
Feb
26
comment How to connect these two formulations regarding the need for a density matrix in quantum mechanics?
en.wikipedia.org/wiki/Purification_of_quantum_state
Feb
24
comment Expression of density operator
@Urgje I see. How can you avoid that your basis $\lvert u_k\rangle$ of $\mathcal H$ contains un-normalizable states? What would happen if you would choose plain waves as your basis?
Feb
23
comment Tensor product notation in quantum mechanics
Well, if $1$ and $2$ stood for the electrons in state $a$ and $b$, respectively, this would make perfect sense.
Feb
23
comment Tensor product notation in quantum mechanics
Where did you see that expression? What is the meaning of the subscripts $1$ and $2$ for the kets?
Feb
23
comment Expression of density operator
Isn't the whole discussion about forms in which operators can be specified which allow for the efficient computation of properties? So the best form for $\rho$ will likely be related to the form in which $A$ is given. E.g. for qubits, would you assume that $A$ is expressed in terms of Paulis as well?
Feb
22
comment Expression of density operator
@Urgje "not all states can be represented as density operators" -- could you elaborate?
Feb
22
comment Expression of density operator
@Apogee: Admitted. But I suspect the OP had a fixed basis in mind, not one which depends on $\rho$. Otherwise, choosing a basis which contains $\rho$ would make things even simpler!
Feb
22
comment Expression of density operator
Regarding your edit: How is $A$ specified? And what is wrong with a computational basis $\lvert i\rangle\langle j\rvert$? From a computational basis, you can "easily" compute $\mathrm{tr}(A\rho)$ if you can evaluate the matrix elements of $A$ ("easily" because it is an infite sum, but this is anyways unavoidable).
Feb
22
comment Expression of density operator
I tend to be careful with infinite dimensions, there are just too many pitfalls. I doubt, however, that such tensor products would be well-defined in infinite dimensions.
Feb
22
comment Expression of density operator
@Phoenix87 For powers of two, one can use tensor products of Paulis (and this done).
Feb
22
comment Expression of density operator
??? Why not? Do you want a form where you can easily check whether $\rho\ge0$ and/or $\mathrm{tr}(\rho)=1$? It would be useful if you could specify which properties you like about this form.
Feb
22
comment Expression of density operator
What would be wrong e.g. with an expansion in a computational basis $\lvert i \rangle\langle j\rvert$? This is also quite useful for explicit calculations ...
Feb
22
comment Expression of density operator
What would make an expression "similar"? In principle, you can pick any basis of $\mathcal B(\mathcal H)$ and expand $\rho$ in it. Which property do you want?
Feb
22
comment Expression of density operator
This is incorrect. E.g., for $\mathbb C^2$ your basis would only have two elements.
Feb
22
comment Expression of density operator
Why should $\rho$ be idempotent?
Feb
21
revised What exactly happens at the second-order phase transition of the 2D Toric code?
added tag (topological-order)