I am concerned about the theorem saying that there is no topological order in 1d. According to the seminal paper https://arxiv.org/pdf/1008.3745.pdf, there are no non-trivial topological phases in 1d (we talk about systems with local gapped Hamiltonians).

According to the definition, the topological phase is an equivalence class of ground states under local-unitary transformation. To my intuition, if two ground states belong to the same phase, their parent Hamiltonians lie in the same connected component of parameter space. So, I can continuously deform one Hamiltonian to get another one without closing the gap. Then, we can solve an ODE and find the unitary transformation relating two ground states.

So, what is the algorithm of defining whether two given states in the same phase or not? Let us take 1d spin chain with N sites and two arbitrary gapped ground states. If I consider transformations acting only at one site at once, I am coming to the QI classification of states. There are many different classes under LOCC, and LU-classes are even richer. Should we consider transformations acting on two consecutive sites after that? k-sites transformations? (k << N) Did anybody apply that definition in practice? My question is: for two given states, how we understand, whether an LU-transformation/constant depth quantum circuit connecting them exists or not?

I know that it is more physical and more computationally efficient to look at the fixed points of RG flow. I would appreciate your help.

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    $\begingroup$ I am unclear on what the question is. Is it "Given two (gapped) ground states in the same topological phase, how does one construct a LU transformation/constant-depth quantum circuit connecting the states?" $\endgroup$ – d_b Jun 27 '19 at 0:23
  • $\begingroup$ This is an interesting question. However, what I am trying to ask is simpler "How we can understand, that an LU transformation/quantum circuit exists/ does not exist?" $\endgroup$ – RomaNVKZ Jun 27 '19 at 1:32
  • $\begingroup$ @RomaNVKZ In that case, it might help if you edit your actual question to make it more precise. However, it still seems unclear what you ask: Is it about exact LU-equivalence of finite many-qubit systems, as in quantum information? Or is it about the question whether a two families of states - the ground states of two Hamiltonians for system sizes $N\in \mathbb N$ -- can be transformed into each other (approximately) by constant-depth (or low-depth) circuits, as it is the case in the classification of phases? These seem two quite different settings! $\endgroup$ – Norbert Schuch Jun 27 '19 at 16:15
  • $\begingroup$ Could you tell me, what is the difference in these settings? All the notions like the gap and the quantum phase make sense in the thermodynamic limit, of course. Is it not true that the difference between those two settings in the number of sites? $\endgroup$ – RomaNVKZ Jun 27 '19 at 23:48
  • $\begingroup$ @RomaNVKZ Please use @[username] to ping me. The difference is that in the first setting things could depend on the number of sites, while in the second setting, they don't depend on the number of sites - the low-depth circuit disentangles the system independent of the system size. It's like $\forall N\exists U$ vs. $\exists U\forall N$: a big difference. $\endgroup$ – Norbert Schuch Jun 28 '19 at 2:13

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