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| Fig. 1: This shows pictorially the argument for why Rényi entropies with α >1 will grow slowly in certain systems with conserved quantities, as discussed in the text. In this case, the system is a spin-1/2 chain with total spin conserved. Above is a typical product state of a spin chain in black, and below is a charge-protected mode of this state. The charge density shown in red is the charge associated with the conserved spin. Because charge is conserved, a region with no spin can only have trivial dynamics unless there is an inward flux of charge from some other region. Thus entanglement cannot be generated across the cut for the charge-protected mode due to its void of charge around the cut, until such time has passed that the charge has had time to diffuse into the middle of the chain. (Source: S. Cree) |
In every branch of physics, major developments often arise after the introduction of new theoretical or experimental tools that allow us to ask new questions about physical systems. A number of recent developments has allowed us to ask many more theoretical questions about quantum many-body dynamics, which can be said to fall broadly into two categories - tools that allow the study of a greater variety of systems (such as tensor network methods, and analytical tools for studying random circuits), and tools that allow us to ask new questions of familiar systems (such as probing quantum chaos via out-of-time-ordered correlators). Here we investigate a recent development of the latter category, namely the study of entanglement dynamics of many-body quantum systems.
Entanglement is a distinct type of correlation present in quantum systems, and is known to play a key role in many parts of physics - including quantum foundations, black hole information theory, thermodynamics, quantum computing and many more. There are many more ways for a system to be entangled than to be unentangled, which gives measures of entanglement an entropy-like tendency to grow in time for generic systems. The details of how this entanglement growth occurs can reveal distinct characteristics of the system in question, such as indicating whether a system exhibits localization, distinguishing integrable systems from thermalizing systems, and even signalling the existence of a local conserved charge.
One thing that sets entanglement apart from most other tools for studying quantum many-body dynamics is that it is not quantified via correlation functions of observables - rather, it is an inherently nonlinear quantity. To see why entanglement must be nonlinear, just consider two distinct unentangled states |φ⟩ and |ψ⟩. Their superposition can be strongly entangled despite both individual states being completely unentangled, so for any measure of entanglement S, we have S(|φ⟩) + S(|ψ⟩) = 0 ≠ S(|φ⟩+|ψ⟩).
How do we construct a measure of entanglement for a many-body system? First, we make a "cut" through the system to partition it into two regions. If one has access to only one of these regions, then the state of the sites in that region are not described by a pure quantum state but by a density matrix ρ - which is essentially a probability distribution over a set of pure states, with the randomness arising from the ignorance about the state of the other region. This probability distribution is known as the entanglement spectrum, and completely characterizes the strength of the correlations between the two regions in the state |ψ⟩ . Loosely, the more uncertain this probability distribution, the more entanglement between these regions. There are many different classical ways of characterizing the "uncertainty" (or "entropy") of a probability distribution, each of which corresponds to a measure of entanglement when applied to the entanglement spectrum of a state.
A particularly well-studied family of measures is known as the Rényi entropies, which are a family parametrized by a nonnegative real number α, given by the following formula, [1]
| Sα | = | 1 1-α |
log[ Tr(ρα) ]. |
For α >1, these entropies are primarily indicators of how small the largest probability in the entanglement spectrum is. At α =1 , it reduces to the well-known von Neumann entropy (or "entanglement entropy"),
which measures the "typical order of magnitude" for a probability taken from this distribution. Finally, for smaller α<1 , the Rényi entropies are more dominated by the smaller probabilities in the spectrum, reflecting more the number of probabilities in the distribution rather than their magnitudes.
A typical state (i.e. one chosen at random according to a uniform distribution) will almost always have the maximal amount of entanglement according to all of these Rényi entropies. [2] In a system with a time-independent Hamiltonian, energy will be conserved, which complicates this statement. The entanglement will not grow to the maximum amount, but rather the maximum amount subject to the constraint that energy is conserved from the initial state, which may be at some finite temperature. In fact at low temperatures, the late-time state of the system will look locally similar to the ground state, which has weak entanglement for a non-critical local Hamiltonian - so to see a significant range of entanglement growth we will usually be at high energies/temperatures - much like the entropy of a thermodynamic system. That means that for generic time evolution of a quantum system, we should expect all of the Rényi entropies to increase. The most straightforward way to study the dynamics of entanglement is to begin with a state that is fine-tuned to have minimal initial entanglement - such as product states - and see how it dynamically approaches the equilibrium value.
In this report, we summarize some of the scenarios where entropies with different α play different roles in characterizing the dynamical behaviour of many-body systems. In particular, we will discuss a recent finding that in certain systems with a conserved quantity that spreads out diffusively in time, the Rényi entropies at α >1 grow much more slowly than the the von Neumann entropy (i.e. the α =1 case). [3-5]
The toy model being considered is a spin chain with overall spin conservation, but the conclusions are expected to apply more generally. The argument goes as follows, and is depicted in Figure 1. Since Rényi entropies at α>1 are dominated by a single probability in the entanglement spectrum, their growth is bounded by the effects of a single mode. That is, the Rényi entropies of a state |ψ⟩ can only be so large if there is some product state |φ⟩ with significant inner product |⟨φ|ψ⟩| (a result known as the Eckart-Young theorem). In particular, for a typical initial product state |ψ(0)⟩, there will be some "charge-protected mode" |φ (0)⟩ with significant overlap |⟨φ|ψ⟩|, such that |φ (0)⟩ has a large void of charge on either side of the entanglement cut. Because of the void of charge, there is no possibility of entanglement being generated across the entanglement cut unless charge manages to "leak" in from outside the void. Thus, until t is large enough that charge has leaked into the centre where the cut is, this charge-protected mode |φ (t)⟩ will remain unentangled across the cut, and continue to provide an upper bound on the Rényi entropies of |ψ(t)⟩ due to the Eckart-Young theorem.
One can consider modes with larger and larger voids, which will delay the leakage of charge into the central cut at the expense of having smaller overlap with |ψ(0)⟩. For fixed t, the appropriately sized void that balances these effects will have a length of order √t due to the diffusive nature of the charge transport, which will give an overlap on the order of |⟨φ|ψ⟩| ~ O(e-√t). Thus, it follows from the Eckart-Young theorem that the Rényi entropies will increase at a rate of at most O(√t). This is in contrast to the von Neumann entropy, which is known to increase linearly for typical chaotic systems! This is a surprising example in which the different Rényi entropies convey different essential features of the system's dynamics.
© Sam Cree. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
[1] A. Rényi, "On Measures of Entropy and Information," in Proc. Fourth Berkeley Symp. on Math. Statist. and Prob., Vol. 1 (University of California Press, 1961), pp. 547-561.
[2] D. N. Page, "Average Entropy of a Subsystem," Phys. Rev. Lett. 71, 1291 (1993).
[3] T. Rakovszky, F. Pollmann, and C. W. Von Keyserlingk, "Sub-Ballistic Growth of Rényi Entropies due to Diffusion," Phys. Rev. Lett. 122, 250602 (2019).
[4] Y. Huang, "Dynamics of Rényi Entanglement Entropy in Qudit Systems," IOP SciNotes 1, 035205 (2020).
[5] T. Zhou and A. W. W. Ludwig, "On the Diffusive Scaling of Rényi Entanglement Entropy," Phys. Rev. Research 2, 033020 (2020).