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arXiv:1805.03813 · Journal of the Physical Society of Japan

Theorems on Entanglement Typicality in Non-equilibrium Dynamics

Koji Yamaguchi

Published May 9, 2019• 1 citation

Abstract

The notion of typicality in statistical mechanics is essential to characterize a macroscopic system. An overwhelming majority of the pure state looks almost identical if we neglect macroscopic non-local correlations, suggesting that thermal equilibrium is the collection of the typical properties. Quantum entanglement, which characterizes a non-local correlation, also has a typical behavior in equilibrium systems. However, it remains elusive whether there is a typical behavior of entanglement in dynamical non-equilibrium systems. To investigate the typicality, we consider a situation where a system in a pure state starts to share entanglement with its environment system due to the interaction between them. Assuming the initial state is randomly chosen from an ensemble of pure states, a criteria for the typicality of the Rényi entropies is presented. In addition, it is analytically proven that the second Rényi entropy has a typical behavior in two cases. The first one is an energy dissipation process in a multiple-qubit system which is initially in a random pure state in an energy shell. Since the typical behavior is qualitatively the same as the prediction of the Page curve conjecture, it gives the first proof of the Page curve conjecture in a dynamical process. In the second case, the typicality is proven for any dynamics described by a multiple-product of a single-qudit channel when the system is initially in a pure state randomly chosen from the whole Hilbert space. This result shows that entanglement typicality is not a specific feature of energy dissipating processes.

DOI: 10.7566/jpsj.88.064002

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The notion of typicality in statistical mechanics is essential to characterize a macroscopic system. An overwhelming majority of the pure state looks almost identical if we neglect macroscopic non-local correlations, suggesting that thermal equilibrium is the collection of the typical properties. Quantum entanglement, which characterizes a non-local correlation, also has a typical behavior in equilibrium systems. However, it remains elusive whether there is a typical behavior of entanglement in dynamical non-equilibrium systems. To investigate the typicality, we consider a situation where a system in a pure state starts to share entanglement with its environment system due to the interaction between them. Assuming the initial state is randomly chosen from an ensemble of pure states, a criteria for the typicality of the Rényi entropies is presented. In addition, it is analytically proven that the second Rényi entropy has a typical behavior in two cases. The first one is an energy dissipation process in a multiple-qubit system which is initially in a random pure state in an energy shell. Since the typical behavior is qualitatively the same as the prediction of the Page curve conjecture, it gives the first proof of the Page curve conjecture in a dynamical process. In the second case, the typicality is proven for any dynamics described by a multiple-product of a single-qudit channel when the system is initially in a pure state randomly chosen from the whole Hilbert space. This result shows that entanglement typicality is not a specific feature of energy dissipating processes.

In a quantum system in a pure state, a subsystem generally has a non-zero entropy because of entanglement with the rest of the system. Is the average entanglement entropy of pure states also the typical entropy of the subsystem? The exact formula for the average

\langle{S_{A}}\rangle

was conjectured by Page in 1995 and later proved. Here we compute the exact formula for the average entropy and variance

(\Delta S_A)^2 = \langle{S_A}^2\rangle - \langle{S_A}\rangle{}^2

both for random pure states and for energy eigenstates of a non-interacting system. We show that for large systems the variance is small,

\Delta S_A/\langle{S_{A}}\rangle\ll 1

, and therefore the average entanglement entropy is typical. We compare exact and numerical results for the probability distribution

P(S_A) \mathrm{d}S_A

, and comment on the relation to previous results on concentration of measure bounds. We discuss the application to physical systems in an energy eigenstate. In particular, for a system of non-interacting spins in a magnetic field, we show how the thermal entropy arises as the typical entanglement entropy of a subsystem.

Abstract In a 1+1 dimensional QFT on a circle, we consider the von Neumann entanglement entropy of an interval for typical pure states. As a function of the interval size, we expect a Page curve in the entropy. We employ a specific ensemble average of pure states, and show how to write the ensemble-averaged Rényi entropy as a path integral on a singular replicated geometry. Assuming that the QFT is a conformal field theory with a gravitational dual, we then use the holographic dictionary to obtain the Page curve. For short intervals the thermal saddle is dominant. For large intervals (larger than half of the circle size), the dominant saddle connects the replicas in a non-trivial way using the singular boundary geometry. The result extends the ‘island conjecture’ to a non-evaporating setting.

Typicality arguments replace the postulated mixed state ensembles of statistical mechanics with pure states sampled uniformly at random, explaining why most microstates of large systems exhibit thermal behavior. This paradigm has been revived in quantum contexts, where entanglement is deemed essential, but no clear quantitative link between entanglement structure and typicality has been established. Here, we study pure quantum states with controlled multipartite entanglement and show that when entanglement grows with system size

N

, fluctuations in macroscopic observables decay exponentially with

N

, whereas if entanglement remains finite, one recovers only the classical

1/\sqrt{N}

suppression. Our work thus provides a quantitative connection between entanglement structure and the emergence of typicality, demonstrating that entanglement is crucial for thermalization in small quantum systems but unnecessary to justify equilibrium in macroscopic ensembles. This unifies classical and quantum foundations of statistical mechanics by pinpointing exactly when and why entanglement matters.

The characterization of ensembles of many-qubit random states and their realization via quantum circuits are crucial tasks in quantum-information theory. In this work, we study the ensembles generated by quantum circuits that randomly permute the computational basis, thus acting classically on the corresponding states. We focus on the averaged entanglement and present two main results. First, we derive generically tight upper bounds on the entanglement that can be generated by applying permutation circuits to arbitrary initial states. We show that the late-time “entanglement Page curves” are bounded in terms of the initial state participation entropies and their overlap with the “maximally antilocalized” state. Generally speaking, this result states that the quantum correlations generated by classical circuits are bounded in terms of some quantum property of the initial state (namely, the degree to which it can be written as a superposition of classical states). Second, comparing the averaged Rényi-2 entropies generated by ( 1 ) an infinitely deep random circuit of two-qubit gates and ( 2 ) global random permutations, we show that the two quantities are different for finite N but the corresponding Page curves coincide in the thermodynamic limit. We also discuss how these conclusions are modified by additional random phases or considering circuits of k -local gates with k ≥ 3 . Our results are exact and highlight the implications of classical features on entanglement generation in many-body systems.

Characterizing the time evolution of generic quantum many-body systems is a fundamental challenge, as representing the exact state requires exponentially scaling computational resources. While hydrodynamics and statistical mechanics successfully simplify this task by predicting the expectation values of local observables, these macroscopic frameworks provide no information about nonlinear characteristics of the quantum state. In this work, we demonstrate that for systems exhibiting a timescale separation, with dynamics governed by the transport of conserved charges, this lost information can be systematically recovered. By constraining the maximum-entropy Scrooge ensemble solely by the system's slow modes, we accurately reconstruct complex, nonlinear quantum properties of the global time-evolved state, including the half-chain entanglement entropy and the participation entropy in the computational basis. Our results generalize the paradigm of canonical typicality, revealing that a hydrodynamically bottlenecked system behaves as a typical pure state within a dynamically restricted submanifold of the Hilbert space - a phenomenon we term global restricted typicality.

We investigate the decay of the global entanglement, due to decoherence, of multiqubit systems interacting with a reservoir in a non-Markovian regime. We assume that during the decoherence process each qubit of the system interacts with its own, independent environment. Most previous works on this problem focused on particular initial states or families of initial states amenable of analytical treatment. Here we determine numerically the typical, average behavior of the system corresponding to random initial pure states uniformly distributed (in the whole Hilbert space of n-qubit pure states) according to the Haar measure. We study systems consisting of 3, 4, 5, and 6 qubits. In each case we consider also the entanglement dynamics corresponding to important particular initial states, such as the GHZ states or multiqubit states maximizing the global entanglement, and determine in which cases any of these states is representative of the average behavior associated with general initial states.