This is very interesting and beautiful. One comment which I mentioned to the authors, it seems like it might be possible to "mark" this money. If the money passes through your hands (someone gives it to you in exchange for goods and you then use to to purchase goods from someone else), you can measure some other knot invariant, other than the Alexander polynomial, and then use the knowledge of that invariant later to demonstrate that you had indeed had access to the money.

Thanks, Jonas. Good to know. Thanks very much for answering this question.

However, do you know the answer to a related question: does there exist a complete, orthonormal basis such that both the operators x and \partial_x are close (in operator norm) to diagonal in this basis? I think that this is really the basis that one wants (and the negative result for the finite dimensional case I mention is with respect to this question). This is a stronger requirement than just having bounded variance.

Thanks,
Matt

I may be misunderstanding this paper, or may be confused about one theorem, but I think one of the claims is wrong. On the top of page 4, he states that there exists a complete orthonormal system (presumably he means a complete orthonormal basis for square-integrable functions) consisting of functions which have bounded variance and whose Fourier transforms also have bounded variance. However, I think that no such basis exists. A few cases I know of:



1)If the orthonormal basis is a Gabor basis (all functions in the basis are obtained by translates in time or frequency of a certain elementary function), then the Balian-Low theorem says that this is impossible. I think that there is a generalization of Balian-Low to arbitrary basis.



2)[This part of remark removed because I realized I misstated the result]


3)Finally, the finite dimensional case, I do know. If we have a finite dimensional Hilbert space and define the "shift" and "clock" operators as unitary matrices (shift is 1 above the main diagonal, clock is \exp(i 2pi k/N) along the main diagonal), then these matrices are discrete analogues of position and momentum and there is no way, as N gets large, to find a basis in which both operators are close to diagonal (close meaning distance from a diagonal matrix goes to zero as N gets large). This is a known result in the theory of almost commuting matrices.



Maybe I misunderstand the claim in this paper? Or maybe I am mistaken and there indeed is no version of Balian-Low for arbitrary bases?

Thanks for the detailed reply Ari. As I say, I really think this is great if it works. Regarding the correlation decay theorems, for any Hamiltonian H, and any operators A and B, we need to know 4 things to bound the correlation function of A and B. For definiteness, let me assume that the Hamiltonian acts on qubits on a two-dimensional lattice; I understand that you say that the lattice changes, but in fact this should just mean that the interactions change with L: as L increases, we increase the range of interactions beyond nearest neighbor.

We write the Hamiltonian H as a sum of term of sets of diameter at most R, where R is the interaction range. We need to know:
1)How does R depends on L. I believe that you are saying that R can scale as polylog(L). That is, the interactions can be embedded in this two-dimensional lattice with a range that is a polylog of L.

2)A bound, J, on the norm of these terms in the Hamiltonian. I believe that this is O(1) for your system.

3)The support of A and B. I believe that they both include at most polylog(L) sites, and that they are a distance L away from each other.

4)The spectral gap. The claim is that this is bounded below by an L-independent constant.

Given (1) and (2), we can bound the Lieb-Robinson velocity by polylog(L). Then, we can bound the correlation of A and B by ||A|| ||B|| polylog(L) exp(-L/polylog(L)), which goes to zero for large L.

So, one of those 4 claims I made about the Hamiltonian above is inconsistent with having correlation between the spins. I'd be very interested to see where I misunderstood your construction. Perhaps you require longer range interactions? If you prefer to talk over email, let me know.

If you prefer to discuss over email

Indeed, as I think about it more, I would really like to see the error correction explained better. The example I gave above shows that, if in fact it is possible to implement an arbitrary circuit with a size-independent gap, this can only be done using non-local interactions in the Hamiltonian. Again, I think this is really great if it works.

This is really an amazing advance if it works. I still do not quite understand it yet, though, so I am not sure of all the details.



In particular, I don't quite understand the following: suppose I imagine my qubits arranged in, for example, 2-dimensional space on a square lattice, for example, with coordinates (x,y). Let q_i denote a qubit with coordinates (i,1). Let the first round of the quantum circuit put q_1 and q_2 in a singlet state. Then, the next round of the quantum circuit swaps qubits q_2 and q_3, so that after that round, q_1 and q_3 are in a singlet, and all others are in the |0> state. The round after that swaps q_3 and q_4, and so on. So, after L-1 rounds, q_0 and q_{L-1} are in a singlet state. So, now we translate this quantum circuit into a GSQC. Quantum error correction can be done in the gate model using local measurements and gates in two dimensions...and I think it can be done using local classical control. If this is true, that we can do all the quantum error correction using local measurements, gates, and control, then we should maybe expect that the GSQC would have local interactions. However, given local interactions and a spectral gap, we cannot have long-range correlations between two far separated qubits (q_1 and q_{L-1} in the case above). So, perhaps it is not possible to implement the quantum error correction using local gates in this GSQC model. Indeed, even if we have interactions up to a range which is L^{\alpha} for some \alpha less than unity, the same bound on correlations holds (the exact power alpha depends upon the cardinality of the support of interactions)

I don't see it either, Dave. As far as I understand it, it seems to be obvious geometrically: there is a space of density matrices, divided into entangled and unentangled regions. If the density matrix starts somewhere in the entangled region at time t=0, and follows a path which asymptotically approaches a matrix, such as the maximally mixed state, which lies _within_ the unentangled region rather than on the border between regions, then at some finite time it crosses the border.

You know, one could define "LOCC sudden death", "separable sudden death", "NPT sudden death"! The scary thing is that I can just see someone, someday writing a paper about those ideas...

The paper is now considerably improved and presents an interesting model system with confined defects and a finite excitation spectrum despite the presence of divergent couplings.

In fact, even if we consider this Hamiltonian as reasonable, there's still a problem. The model doesn't do what is claimed. Even with the logarithmic renormalization of the chemical potential (basically, adding a logarithmically large penalty for defects), it is favorable to put a large number of defects near each other (the interaction energy goes as the square of the number of defects, while the penalty goes linearly). On the other hand, since the authors remove the k=0 mode from the interaction, this means that at long range, the interaction actually becomes repulsive. So, the result is that the system will phase separate. You will get one half the system full of defects and the other half with no defects, which is a very different state.

It's nice to have some "scites" on my paper with Daniel, but everyone interested in my paper with Daniel should also check out this paper by Irani since very similar results are obtained, with a few difference as explained in our papers (we have a slightly simpler construction and a better scaling of entropy with gap, but Irani shows the important result that this can be done even in a translationally invariant system).

This is a problem that many people have worked on, namely how to construct a model with topological order at non-zero temperature in 2 or 3 dimensions. This problem has deep theoretical and physical implications. However, I don't think this paper help solve this problem. Because of the importance of the problem, I'd like to explain my concerns to hopefully clarify what I think needs to be accomplished. Any comments on these concerns are very welcome.

I feel that the right problem is to construct such a model using finite range and finite strength interactions. My first concern is that while the model considered here is finite range, the interaction is not finite strength (phi has unbounded operator norm). Once you allow interaction terms with unbounded norm, it seems one might as well just allow Hamiltonians which include very large penalties for defects, which would accomplish the desired result much more simply (in fact, as explained below, it seems that that is basically what the authors do at one point). So, really we should consider only Hamiltonians with terms bounded in operator norm.

Going along with the above concern is that the given Hamiltonian doesn't seem to be one that can easily be realized. While acoustic phonons certainly exist in nature, the physical coupling to phonons is to the gradient of the field phi, not to the field phi itself. Another way to realize a gapless bosonic mode is by an XY model below the Kosterlitz-Thouless transition temperature. However, in that case, the natural coupling is to cos(phi) or sin(phi).

The final concern is that the term that the authors add, coupling phi to the defect, must lower the energy of a defect (usual second order perturbation theory result). So, while adding such a term on its own does make the energy of a pair of defects separated by distance 1 much less than the energy of a pair of defects separated by distance L (in fact, reducing the energy by log L), it does this by making the energy of a pair of defects at distance 1 very negative (roughly, -log L). So, such a Hamiltonian will favor forming defects which should destroy topological order. The authors get around this by commenting (bottom left-hand column, page 3) that they absorb certain terms into a redefinition of the defect chemical potential. However, what this means, as far as I can see, is that they add a penalty term of order log L for each defect. However, doing this on its own would very easily realize the desired topological order without any need for any complicated constructions. So, this gets back to my first point-it seems that the real reason for getting the topological order at non-zero temperature in this case is actually the logarithmically large penalty term added for each defect.

This is an excellent review of area laws that covers all the major topics: gapped and gapless systems; bosonic, fermionic, and spin systems; one-dimensional and higher-dimensional problems; and so on. While many rigourous results are given, the presentation is intuitive and clear.

This paper is worth reading just to see the bound Eq. (2) on the number of solutions to the quantum marginal problem. The calculation is just a few lines and very clean.

DMRG and other MPS methods work very well in practice to find ground states of 1d problems, but in some practical cases can get stuck in false minima. This is a separate question from the question of whether the desired ground state does indeed have a matrix product state representation. The question is whether one can find the MPS. Recently Eisert presented evidence that the problem of finding the best MPS can be NP-hard. Eisert varied over certain matrices while leaving other matrices fixed and showed that this problem was NP-hard. Thus, Eisert's result left open the question of whether finding the ground states of 1d Hamiltonians with MPS ground states of polynomial bond dimension really is a hard problem or not. The present paper answers this question.

Matrix product states/PEPS are much better understood in 1d than in 2d. In 1d, there is a well-developed theory to prove gaps in local Hamiltonians with matrix product ground states such as the AKLT Hamiltonian. This paper presents some valuable steps towards a similar theory in 2d. Personally, I believe that any gapped local Hamiltonian in 2d will have a ground state that is well described by a matrix product state, and so I feel that this paper represents a valuable study of a very important class of states.

Chiral spin liquids have been conjectured in condensed matter physics for some time. Kalmeyer and Laughlin proposed such a state for the antiferromagnet on the triangular lattice, and Marston and Zeng for the kagome lattice antiferromagnet. However, neither of those states seems to actually be realized in the given Hamiltonians. Thus, this paper is interesting as it appears to be an example of a actual chiral spin liquid state in an SU(2) invariant system.



From one point of view, the presence of a chiral spin liquid is not surprising for this Hamiltonian. Consider a triangle with one spin up and the other two down. Then, the up spin can hop around the triangle as if it feels a magnetic field due to the chiral interaction term. So, the system looks like hardcore particles in a magnetic field. However, there still is an exact SU(2) symmetry in this system, so it is a little different from the usual quantum Hall state.



One can modify the Hamiltonian to make the PT symmetry spontaneously broken instead of explicitly broken. Long ago, Haldane suggested a model with an interaction that coupled the chirality of neighboring plaquettes, in a manner that favored all the chiralities being the same. This model spontaneously broke chirality. By coupling this model of Haldane to a ferromagnetic Heisenberg model on the triangular lattice, with the coupling term coupling the chirality in a plaquette on the triangular lattice to the chirality in a plaquette in the Haldane model, one arrives at the model with spontaneously broken PT symmetry that still looks exactly like the model the authors consider.



One very interesting thing to do would be to study the Chern number in this system, and to compare to studies of the Chern number in the kagome antiferromagnet as done by Waldtmann, Lhuillier and collaborators.