Topological order for mixed states

In this post I’d like to take the opportunity to talk about the problem of defining what is meant by topological order at finite temperature. I hope to convince you that there is a natural, operationally well motivated, definition for what this means.

But first let’s recall what is meant by topological order.

There appear to be quite a few related yet not quite obviously equivalent definitions floating around: some define it with reference to edge modes, some define it in terms of the topological entanglement entropy, and yet another definition concerns ground state degeneracy on a surface of genus {g}.

The definition I like the most, and which I’ll exploit for the rest of this post is due variously to Xiao-Gang Wen and coauthors and is in terms of local deformation operations. (I’m using the terminology of “local deformation operations” (LDO) instead of Wen’s “LU” to avoid some confusion later on.) To explain this definition we’ll need to work with infinite quantum lattice systems.

1. Local deformation operations

Suppose we have some quantum lattice system defined with respect to a reasonably regular lattice of dimension {d} (say the integer lattice {\mathbb{Z}^d} for concreteness): we attach a local quantum degree of freedom, say a qudit with local hilbert space {\mathbb{C}^D}, to each site of the lattice. Thus the hilbert space of the lattice can be thought of as the infinite tensor product

\displaystyle \mathcal{H} \approx \bigotimes_{x\in \mathbb{Z}^d} \mathbb{C}^D. \ \ \ \ \ (1)

It is more subtle than this, though, as this space, interpreted naively, isn’t separable. The better way to work with these infinite systems is rather to focus on the algebra of allowed observables, the quasi-local algebra {\mathcal{A}(\mathbb{Z}^d)} which is, very roughly speaking, spanned by all the operators which act on at most a finite number of sites. (This is almost accurate: we also throw in operators which are almost local, i.e., operators which are an infinite sum of strictly finite terms but with coefficients decaying with size.) A really useful quantity is the support of an observable {A\in \mathcal{A}(\mathbb{Z}^d)}, denoted {\mbox{supp}(A)}, which is the set of sites that {A} acts nontrivially on. E.g., {\mbox{supp}(\sigma_j^x) = \{j\}}, i.e., the support of the pauli {x} operator on site {j} is simply {\{j\}}.

We now have enough to define a local deformation operation. This is a finite-depth local unitary quantum circuit {\mathcal{U}}. (Such operations are also known as locally implementable.) More precisely, an LDO is a unitary circuit which is a product {\mathcal{U} = \prod_{j=1}^m U_j} of finitely many quantum circuits where each {U_j} itself is an (infinite) product of pairwise disjoint local gates acting on at most a finite number of neighbouring sites. Here is a schematic of such an operation in one dimension:


A crucial property of an LDO is that the support of a localised operator doesn’t spread too much: there is a bounded set {\Lambda\subset \mathbb{Z}^d} localised at the origin such that for all {A} we have {\mbox{supp}(\mathcal{U}^\dag A \mathcal{U}) = \mbox{supp}(A) + \Lambda}, where the addition symbol is interpreted entrywise:

\displaystyle X+Y = \{x+y\,|\, x\in X, y\in Y\}. \ \ \ \ \ (2)

In the example LDO above, and for an operator {A} located at site {14} the support of {\mathcal{U}^\dag A U} is the subset {{7, 8, \ldots, 20}}, which is a subset of, e.g., {\mbox{supp}(A) + \Lambda} with {\Lambda = \{-10, -9, \ldots, 9, 10\}}:


We thus say that LDOs have a lightcone of width {|\Lambda|}, meaning that information cannot propagate, under the action of an LDO {\mathcal{U}}, more that {|\Lambda|} sites away from a disturbance localised at some site. (The set {\Lambda} is determined by the widths of the largest gates in {U_j}.) Note that we do not admit approximately local unitaries as LDOs. This is a subtle point related to the topology we place on the set of LDOs. The set of all LDOs forms an infinite group {G}, and is meant to capture the set of all operations which locally mess up, or locally deform, the lattice system. (The group {G} may be topologised in a variety of inequivalent ways, each of which have drastic consequences for the following definition. I’ll elide this point here and rely upon our physical intuition to carry us forward.)

Now the set of all LDOs is meant to capture all the ultraviolet scale, irrelevant, stuff that is going on all the time in your typical lattice system. The key physical intuition here is that an operation {\mathcal{U} \in G} cannot “see” the topology of the lattice, because there is a finite light cone bounding the rate at which information can propagate through the lattice under the action of {\mathcal{U}}.

It turns out that many familiar operations are approximately captured by elements of the LDO group {G}, including, constant time evolution according to any local lattice hamiltonian, and also adiabatic evolution through a gapped path of local hamiltonians.

2. Topological order for pure states

Now we finally come to the quantitative definition of topological order. (Actually, to be more precise, we come to the definition of what is not topologically ordered.)

There are two key ideas underlying the definition of topological order: (1) it is a property of states alone; and (2) whatever it is, it shouldn’t depend on whether an LDO has been applied, i.e., the topological order of a state {|\Omega\rangle} should be the same as that of {\mathcal{U}|\Omega\rangle}, {\forall \mathcal{U}\in G}.

The first condition allows us to dispense with hamiltonians and focus just on states and the second condition expresses the topological nature of topological order: it should be insensitive to ultraviolet lattice-scale deformations.

Thus we arrive at the following definition:

Definition 1 A state {|\Omega\rangle} is topologically trivial (TT) if there exists {\mathcal{U}\in G} such that {\mathcal{U}|\Omega\rangle} is a product state.

This definition is meant to capture what topological order isn’t: a state that can’t be mapped to a product state by an LDO must be topologically ordered!

Symmetry protected topological order fits naturally into this framework: all we have to do is replace {G} with the subgroup {G_{\text{SPT}}} of LDOs transforming covariantly with respect to some symmetry operation, e.g., charge conjugation, {\textsl{SU}(2)}, etc.

This is our definition of topological order for pure states. It is worth making a couple of comments at this point about what we can and can’t do with this notion. The first comment is that there are topologically nontrivial states. The simplest example is a ground state of the toric code or any string-net model.

Secondly, because {G} is a group, it gives us an equivalence relation on states: two states {|\Omega\rangle} and {|\Omega'\rangle} are equivalent if there is {\mathcal{U}\in G} such that {|\Omega'\rangle = \mathcal{U}|\Omega\rangle}. (You can check that the group property ensures that we get an equivalence relation {\sim}.) Thus we can take a quotient of state space {\mathcal{S}} with respect to this equivalence: {\mathcal{S}_{\mathrm{top}} = \mathcal{S}/\sim}. This observation tempts us to contemplate a full classification program for topological which would go roughly as follows: (1) find a succinct description of the quotient {\mathcal{S}_{\mathrm{top}}}; and then (2) work out the representative of any state in {|\Omega\rangle} in {\mathcal{S}_{\mathrm{top}}}.

However, you must resist this temptation as it is doomed to failure. The space {\mathcal{S}_{\mathrm{top}}} is truly wild as it contains all kinds of translation non-invariant beasties. However, one shouldn’t give up on a classification program altogether. The next best thing would be to understand as many invariants of the action of {G} on {\mathcal{S}} as possible. Each new invariant at least provides us with a coarse-grained picture of {\mathcal{S}_{\mathrm{top}}}. The situation here is not entirely unlike, e.g., the classification program for manifolds up to diffeomorphism.

The set of such invariants is nonempty because, at least in two dimensions, the topological entanglement entropy {S_{\text{top}}} is, by construction, an invariant of the action of {G} on {\mathcal{S}}. I’m not sure how many other obvious invariants there are: the quantum dimensions ought to be; the {S}-matrix ought to be, and so on.

3. Analogy with entanglement theory

There is a fairly striking and useful analogy between some of the notions involved in the definition of topological order and those appearing in pure-state bipartite entanglement theory.

  1. Everytime I say topologically trivial, think “product state”.
  2. Everytime I say LDO {\mathcal{U}}, think “local unitary” LU {U\otimes V}.
  3. Everytime I say topologically ordered, think “entangled”.

This nice thing about this analogy is that we can replay many of the ideas initially developed for the classification of quantum entanglement in this new setting. Some example ideas:

  1. How about “topological order measures” which measure the distance from a state to the set {\mathcal{S}_{\text{TT}}} of TT states?
  2. What about interconvertibility of topological order with respect to LDOs? Distillation?

4. Topological order for mixed states

The situation for mixed states is perhaps not as clear as it could be. The definition which I like most, and the one that got me thinking about the problem of defining topological order for mixed states, is due to Matt Hastings. It is really the ideal counterpart for what I’ve discussed above in section 2; all I’m doing in this post is interpreting it in a slightly different way and perhaps generalising it slightly on the basis of the entanglement theory analogy I described above.

In the light of the analogy sketched in section 3, let’s now describe what I feel is the simplest notion of topological triviality:

Definition 2 A state {\rho} is topologically trivial if it can be written as a convex combination of TT pure states, i.e.

\displaystyle \rho = \sum_{j} p_j |\phi_j\rangle \langle \phi_j|, \ \ \ \ \ (3)

where {|\phi_j\rangle \in \mathcal{S}_{\text{TT}}}, for all {j}.

This definition has the pleasing operational interpretation that {\rho} is TT if it can be prepared by flipping some coins and applying LDO operations conditional on the result of the coin flips. Surely any {\rho} prepared in such a way is TT!

Let’s compare this definition with an alternative:

Definition 3 A state {\rho} is topologically trivial if the local purification {|\rho\rangle} of {\rho} is TT.

What does “local purification” mean? My suggestion would be to take the canonical “doubled” purification where the local hilbert space is doubled (e.g., local qudits are doubled).

The definition provided by Hastings is much closer in spirit with this second definition. (It’s worth noting that by working in the infinite lattice size limit we can avoid lots of {\epsilon}s and {\delta}s; we can make our soft analysis definition a hard analysis definition if we like in the standard way.)

Inspired by the analogy between the classification of topological order and the classification of quantum entanglement I’ve argued here there are many suggestive open questions to study. Here are a couple of obvious ones:

  1. What is a good measure of mixed-state topological order?
  2. Can we distill pure-state topological order with LDOs from many copies of a mixed topologically ordered state?
  3. Can we construct local witnesses of topological order? (By analogy with the theory of entanglement witnesses?)
  4. What is the “ebit” of topological order? Or is there no “ebit”?
  5. What systems are naturally topologically ordered at finite temperature, or in the presence of dissipation?

I hope the simplicity of the definition of topological order I provided here is clear enough to inspire you to think about these problems and more!


4 Responses to Topological order for mixed states

  1. sflammia says:

    What happens if you insist that the state is invariant under translation of the unit cell? Since there is no Hamiltonian around in this picture, we don’t have to worry about ground states with broken symmetry, we can just insert the translation invariance by hand. Do you think that would make a complete classification any more feasible? (I guess not, but I am curious about your opinion.)

  2. tobiasosborne says:

    Dear Steve,

    Many thanks for your comment! Even with translation invariance enforced there are serious difficulties. Basically the problem is that (a little ball around) every generic state in hilbert space will lead to a different point in S_top. Translation invariance gives us a bunch of constraints, but it doesn’t really cut down the vastness of hilbert space quite enough because a generic quantum state can easily be made translation invariant by summing over translates.

    (My intuition here comes primarily from, e.g., your paper; here translation invariance will not help either.)

    I would even be willing to conjecture that scale invariance and translation invariance are of no help in 3 spatial dimensions and above: the MERA construction strongly suggests that there is a wild zoo in higher dimensions.


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