The fourth and final day of the conference. **Update Oct 26:** The slides from the talk will be posted online here.

**Quantum Information in Quantum Many-Body Physics**

*Roger Melko, Entanglement entropy in exotic phases of quantum matter.*

Review of quantum Monte Carlo (QMC). Sign problem. Want to compute Rényi entanglement entropies for frustrated spin systems. He states that the record for Lanczos diagonalization for just the ground state is… 48 spins. (Wow! Presumably this is exploiting some specific symmetries, but still, that’s impressive.) Worried about “weak” critical points where the correlation length is merely larger than the maximum simulable system size, but doesn’t actually diverge in the thermodynamic limit. Works on trying to distinguish these possibilities. In systems with long-range entanglement or topological order, the entanglement scaling can give a picture “Landau-like” order parameter. Can also give a picture of “non-Landau” (deconfined) quantum critical points. Subleading order correction to the area law contains universal information. We will compute Rényi entanglement entropy. Introduce the valence bond basis for QMC sampling (Sandvik PRL 2005). In particular, use projector QMC. The estimator that they use for estimating expectation values is weighted by the action of a projector on a trial (valence bond) state. In this way, many quantities can be estimated such as energies, order parameters, real-space correlation functions, etc. In order to measure entanglement, we can use the swap test. We can use these techniques to study many models: Heisenberg (Néel ordered ground state), resonating valence bond spin liquids, J-J’ models (conventional quantum critical points), J-Q models (unconventional, deconfined quantum critical points). Showed some results for Heisenberg model on a torus. Stochastic series expansion: expand the finite-temperature partition function by expanding the exponential in its Taylor series. You get powers of $latex -beta H$, which you can sample in various ways and using various update rules. (Several buzzwords that I didn’t understand: worm algorithms, directed loop algorithms?) To access Rényi entropies, we can use the replica trick by gluing together several Riemann sheets (depending on which order of the Rényi entropy is desired). Can also compute (Rényi) mutual information. Want to study spin liquids. One possible model, kagome lattice Bose-Hubbard model supports a $latex mathbb{Z}_2$ quantum spin liquid. This measure is a loop gas similar to the toric code. When computing (Rényi) topological entanglement entropy there is a finite-size “chord length” term that doesn’t exactly cancel. Showed some numerics for this model.

*Frank Verstraete, The entanglement spectrum of strongly correlated quantum many-body systems.*

Began with an overview of what questions people care about with regard to entanglement spectrum. Area laws, universal terms, dimensional reduction and holography. Says the following (I’m paraphrasing except for the quotes): Since I was reluctant to prepare my talk, I’ve decided, “with a stroke of genius” to turn this talk around and ask the audience what they would like to hear about so that I don’t disappoint anyone. He discusses the notion of PEPS as dimensional reduction: we argue that we can map 2D quantum systems to 2D classical systems (the PEPS), even though this is potentially difficult (complexity-theoretic obstacles in general) and we know that 2D classical systems can be mapped to 1D quantum systems. But we know how to deal with 1D quantum systems if locality is maintained throughout this series of mappings. These methods should work for gapped systems, but will breakdown for critical systems. A long review of why we care about area laws with some examples and numerics for gapped and critical systems. We can only get area laws iff there is very little weight in the tail of the Schmidt spectrum. But you also need to show that cutting small Schmidt coefficients in one cut that you don’t truncate large Schmidt coefficients across another cut. “Because there is an isometry that maps virtual degrees of freedom to physical ones in an MPS, this proves that the virtual degrees of freedom are real. This is the main message of this talk.” (?) Symmetries present in the state are inherited in the tensor network description, for example in the AKLT spin chain. Review of cMPS. Showed some numerical calculations of the entanglement spectrum in the Lieb-Liniger model using cMPS. Even though this method is known to be critical we see fast decay of the Schmidt coefficients.

*Stephan Inglis, Measuring Rényi entropies in finite-temperature QMC using Wang-Landau sampling.*

Review of definition of Rényi entropy and why we care about it. Review of classical MC with Metropolis update rule. The Wang-Landau (WL) algorithm calculates the density of states instead. Assume an initial DOS, generate a new state and visit with probability $latex p$, then update the DOS by multiplying by some function $latex f$ so that $latex g_{i+1}(E) = g_i(E) times f$. The probability is $latex min(g(E_i)/g(E_f),1 )$ where $latex g(E_i)$ is the DOS at the $latex i$th time step. WL works by calculating the DOS as a function of the energy. In stochastic series expansion (SSE) quantum Monte Carlo (QMC), the energy of the system is related to the number of operators sampled, so a more natural way to parameterize things is in terms of $latex beta$ and a power series expansion. WL gives a much simpler way of computing Rényi entropies because there is no integration needed. At finite temperature, we can use mutual information instead. Showed a few examples. Now a discussion of the noise in WL. Is there a self-consistent way to check the noise? Check the first derivative of the DOS. The noise in the derivative seems to be uncorrelated, so we can just average several simulations to converge to accurate results. The advantages of WL: only one simulation is required rather than many, there is no explicit cumulative integration error and results can be improved building on old data rather easily. The difficulties are that a single simulation takes longer and it is harder to measure observables that are not related to energy.

*Jutho Haegeman, Time-dependent variational principle for quantum many-body systems.*

Time dependent methods for quantum lattice systems include time-evolving block decimation (Vidal). There are several disadvantages, however, such as the method breaks symmetries, the state temporarily leaves the variational manifold, the truncation is suboptimal for infinite-sized systems, and it is manifestly discrete in time. Can we implement this with cMPS? We have to use some results from Dirac. We project onto the tangent plane of the variational manifold at each point in time. If the time-dependent evolution never leaves the manifold, then you get a non-linear differential equation within the manifold. This method inherently respects all symmetries and there is no Trotter error. It is globally optimal (spatially… it is still local in time) and there is no truncation. We will function on uniform MPS. We need to compute overlaps of tangent vectors, and this introduces a metric. We can simplify this through gauge fixing. MPS are gauge invariant in the sense that if we conjugate the bond indicies we get the same physical state. Taking tangent vectors, we get an additive correction which depends only on the choice of gauge, so we can pick the most convenient choice. Describes (pictorially) some gauge fixing conditions which make things particularly nice. Can use this for both imaginary and real time evolution. The method also provides additional checks on the convergence of the method by checking the length of the tangent vector. Showed some examples of numerics. Near a steady state solution, you can linearize the differential equation to simplify things. This leads to the linear (tangent) subspace around the stable point (Rayleigh-Ritz equations). Can study excitations with momentum $latex k$ by combining the Östlund-Rommer ansatz and the Feynman-Bijl ansatz (single-mode approximation). More numerics. Could also study domain walls with momentum $latex k$ using the Mandelstam ansatz. More numerics.

*Ann Kallin, Scaling of entanglement entropy in the 2D Heisenberg ground state.*

QMC in the valence bond basis for measuring the Rényi 2-entropy. Spin-1/2 antiferromagnetic Heisenberg on a square lattice with only nearest-neighbor interactions. Use the power method to get something close to the ground state. Showed some numerics… the method converges well on up to a 20 by 20 lattice. They do QMC sampling with importance weighting. The power of the Hamiltonian acting on the trial state can be written as a sum over products and this is what they sample. Then they use an update rule to switch the valence bonds to do the sampling. Computing the overlap between two valence bond states is easy, and just amounts to counting the number of loops in the resulting diagram when you superimpose both valence bond patterns on the same lattice. This is the “regular” VB QMC. There is also a “loop” version of the algorithm. Now you have two trial states and sandwich your $latex m$ different operators from the Hamiltonian. This method is similar to the SSE method discussed earlier in the conference, but with different boundary conditions. Then some numerics with 100 sites on a 1D Heisenberg chain. The loop algorithm converges quite quickly as long as the size of the region isn’t too large (as a fraction of the total system size). One further improvement is to use a ratio trick, namely, measure ratios of entropies for different region sizes. In 2D, they wanted to look at the effect of corners in the region on the entropy. Chose square-shaped regions and stripes around the torus. Plot scaling of the 2-entropy as a function of the subsystem size. They fit to a function of the form $latex a L+b log L + c$. Showed some nice pictures of “Rényibows” which simultaneously plot the entropy over several system sizes and subsystem sizes. (The name comes from the colors used in the plot.) One of the things they observed was a 1D CFT-like chord-length correction in 2D systems. Also saw some effects due to corners.

*Johannes Wilms, Mutual information in the Lipkin-Meshkov-Glick model.*

We will work with mutual information because it is a quantity which describes mixed states, and we will consider finite temperature. Define the LMG model: every spin interacts with every other spin ($latex XX$ coupling) together with a transverse field. This model was originally studied in the context of atomic nuclei, and more generally for shells of finite Fermi systems and metal clusters, He-3 droplets, “magnetic molecules”, also mappings to BEC and cavity QED. More recently people have loked at ground state and thermal properties. Shows the phase diagram, temperature versus magnetic field strength. We can compute many things in this model by exploiting symmetry. For example, total angular momentum is a good quantum number. The Hamiltonian decomposes into polynomial-sized blocks which have exponential multiplicity. The best way to study the problem is to use Clebsch-Gordan coefficients. Showed a plot of concurrence in the phase diagram… it doesn’t work so well. On the other hand, the mutual information perfectly tracks the phase transition. They looked at the scaling of the mutual information at criticality. They find that it scales linearly with the log of the system size with a prefactor of 1/4 in the finite temperature case (as opposed to 1/3 which is what you expect at zero temperature).

*Winton Brown, Quantum Hammersley-Clifford theorem.*

The classical HCT is a standard representation theorem for positive classical Markov networks. Recently, quantum Markov networks have been used in approximation methods for lower bounds on the free energy of quantum many-body systems. Defined conditional mutual information and the strong subadditivity inequality. The Markov condition in the classical case is given by $latex I(A:C|B) = 0$, which implies that $latex p_{ABC} = p_{A|B} p_B p_{C|B}$ where $latex p_{A|B} = p_{AB}/p_B$. A Markov network is a generalization of a Markov chain to an arbitrary graph $latex G$. The Markov condition just says that whenever I partition the graph into three sets of vertices $latex A, B, C$ with $latex B$ separating $latex A,C$, then $latex I(A:C|B) = 0$. The classical HCT says that any positive probability distribution factorizes over the cliques of $latex G$. The proof is simple. Take the log of the probability distribution. From the conditional independence, we find that $latex p$ is a sum of two terms that act trivially on either $latex A$ or $latex C$. … couldn’t get the rest down in time. For quantum states, we just use the quantum conditional mutual information. What quantum states can be supported in this framework? We immediately find that $latex rho = rho_{AB} rho_{BC}$ where the two factors commute. Then repeating the proof from the classical case runs into trouble because we don’t know if the various terms commute. Consider a graph which has no 3-cliques. On these graphs we have enough structure that each two-body operator has to be zero. This is almost enough, but some 1-body operators might remain. But it turns out this is not an obstacle. Unfortunately, there is no simple general result. A counterexample exists with a five-vertex graph with a single fully connected central node and a square of four nodes surrounding it. Now consider a connection to PEPS. We can construct a PEPS by applying a linear map to each site of virtual spins. If the linear map is unitary, then the PEPS built in this way are quantum Markov networks. Under what condition can we show the reverse implication? For a non-degenerate eigenstate of a quantum Markov network, the Markov properties imply an entanglement area law. Thus, an HC decomposition implies a PEPS representation of some fixed bond dimension. Open problem: show under what conditions quantum Markov networks which are pure states are have an HC decomposition. There exist non-factorizable pure state quantum Markov networks (simple example).

Thank you again, Steve for this outstanding commentary.

Regarding quantum effects in condensed matter theory, I would be interested/grateful for any remarks relating to discussions of transport theory, appearing in the context of any of the presentations at Montreal.

The guiding intuition (as usual in transport theory) is that we can look for nontrivial transport phenomena in any Hamiltonian system that has one or more globally conserved quantities (one of which can be the total energy) that can be represented as spatial integrals over local densities (and here the notion of “spatial integral” can apply equally to physical or abstract notions of “space”). Once these local densities are specified, we can estimate/simulate their local microscopic fluctuations, which are linked to global macroscopic transport coefficients by the formalism of Onsager, Green-Kubo, etc. And from a practical quantum informatic point-of-view, we can seek physical systems that have globally conserved measures that have no local microscopic fluctuations, with view toward robust thermal isolation.

My preliminary impression, though, is that not too many presentations at Montreal discussed quantum informatic aspects of transport dynamics … maybe this impression is mistaken?

Most of the presentations focused on time-independent aspects of many-body systems, or on quantum adiabatic evolution. The only talk that might be more than just a bit relevant for you was the talk of Jutho Haegeman on variational methods for time-dependent Hamiltonians.

Yes, it was eye-opening to see how thoroughly this particular many-body conference was dominated by time-independent formalisms (partition functions, ground states, topological measures, etc.).

In contrast, when one goes to a many-body conference focusing on (for example) synthetic biology, then time-dependent methods are dominant (trajectory ensembles, thermostats, Green-Kubo theorems, etc.).

In principle, it would be perfectly feasible for each conference to exchange methods and yet derive pretty much the same results, and perhaps this synthesis represents one future path for condensed matter physics. Chapter 13 of Frenkel and Smit’s

Understanding Molecular Simulationpresents a student-friendly overview of these issues, which apply similarly to both classical and quantum dynamical systems.