Universality in quantum snapshots

This is a Perspective on "Solvable model of deep thermalization with distinct design times" by Matteo Ippoliti and Wen Wei Ho, published in Quantum 6, 886 (2022).

By Pieter W. Claeys (Max Planck Institute for the Physics of Complex Systems, 01187 Dresden, Germany).

Published:	2023-01-27, volume 7, page 71
Doi:	https://doi.org/10.22331/qv-2023-01-27-71
Citation:	Quantum Views 7, 71 (2023)

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Thermalization and random matrices

One of the main challenges in quantum mechanics lies in understanding the dynamical behaviour of many-body systems. How do isolated quantum systems relax to equilibrium? What universal features emerge at late times? Historically, the study of quantum dynamics has been strongly intertwined with the study of random matrices. Much of our understanding of chaotic quantum dynamics originates from identifying the ways in which highly structured Hamiltonians exhibit universal features from random matrix theory. One of the greatest successes of this connection was in explaining the phenomenon of thermalization: for generic isolated quantum systems left to evolve under Hamiltonian dynamics, late-time expectation values of observables become stationary and take universal values consistent with statistical mechanics (e.g. thermal values consistent with a Gibbs state) [1]. Here the crucial observation was that the matrix representation of local observables within the eigenbasis of a Hamiltonian can be roughly treated as a random matrix. Locality plays an important role, both in the Hamiltonian and the observable: global observables are not expected to relax to equilibrium, but for local observables the remainder of the system can act as a thermal bath and arguments from statistical physics apply [2].

In a complementary approach, purely random matrices can be used as minimal toy models in which the quantum dynamics remains tractable and such universal features can be exactly quantified. Aspects of quantum dynamics can be understood by directly starting from a random matrix model: the unitary dynamics generated by a quantum Hamiltonian can be mimicked using random unitary matrices, where the locality of physical interactions can be taken into account by considering local unitary matrices. This research program has been widely successful in shedding light on generic features of many-body dynamics (including thermalization as well as entanglement growth, operator spreading, etc.) [3].

In a work by Ippoliti and Ho recently published in Quantum, these two aspects are combined and a random matrix model for deep thermalization is introduced and analyzed [4]. The authors consider higher notions of thermalization, each with their own distinct time scale, and present a random matrix model for which the corresponding time scales can be exactly obtained. Using this model, the authors relate such higher notions of thermalization to regular thermalization and highlight the importance of locality. Despite this being a minimally structured model, the observed phenomenology can be expected to appear in a wide range of settings and could result in bounds for more realistic models of quantum dynamics.

Deep thermalization

Regular thermalization is typically observed on the level of local observables. Suppose that we are interested in observables that act only within some subsystem $A$, we can then treat the rest of the system as a bath $B$. Given a wave function $|\psi\rangle_{AB}$ for the full system, the expectation value of all such local observables follows directly from the reduced density matrix $\rho_A$, defined as

\begin{equation}
\rho_A = \mathrm{Tr}_B \left[|\psi_{AB} \rangle \langle \psi_{AB}| \right] = \sum_{z_B} p(z_B) |\psi_{A}(z_B) \rangle \langle \psi_{A}(z_B)|\,.
\nonumber
\end{equation}

The first equality is simply the definition of the reduced density matrix, whereas in the second equality the implicit averaging over bath states has been made explicit. Here $z_B$ labels a set of complete basis states for the bath, $p(z_B)$ is the probability of measuring the state $z_B$ if we only measure the bath state, and $|\psi_A(z_B)\rangle$ is the resulting (pure) state for the subsystem $A$:

\begin{equation}
p(z_B) = \langle \psi_{AB}| \left(𝟙_A \otimes |{z_B}\rangle\langle z_B | \right) |\psi_{AB}\rangle, \qquad |\psi_A(z_B)\rangle = \left(𝟙_A \otimes \langle z_B|\right)|\psi_{AB}\rangle/\sqrt{p(z_B)}.
\nonumber
\end{equation}

Within thermalization, if a quantum state is left to evolve for a sufficiently long time then the reduced density matrix must reproduce a `thermal density matrix’ — in the absence of any conservation laws simply proportional to the identity matrix $𝟙_A$. During the dynamics the subsystem becomes increasingly entangled with its environment, such that tracing out the environment effectively results in a maximal loss of information about the microscopic degrees of freedom.

As apparent from the second equality, the density matrix can also be interpreted in a different way. Consider the scenario where we measure the bath state with measurement outcome $z_B$, construct the density matrix for the subsystem $A$ as $|\psi_{A}(z_B) \rangle \langle \psi_{A}(z_B)|$, and average this density matrix over all possible measurement outcomes with corresponding probability $p(z_B)$. The reduced density matrix follows as the `mean’ density matrix for the subsystem $A$. Far from just being a theoretical tool, this interpretation is particularly appealing given recent advances in quantum simulators. In many physical realizations (e.g., ultracold atoms with quantum gas microscopes, individually trapped ions or Rydberg atoms, superconducting qubits etc.), measurements involve the simultaneous read-out of the state of both the subsystem and the bath, leading to snapshots containing microscopically resolved information about the state.

A new type of universality known as deep thermalization has recently been identified in the statistical properties of these snapshots [5,6,7,8,9,10,11]. Specifically, if we take an initial state and let it evolve under quantum dynamics, after sufficiently long times any average of subsystem properties over measurement outcomes is expected to become equivalent to an average over purely random states. In other words, the quantum state on the subsystem $A$ can be effectively treated as a purely random state after a measurement on the bath $B$. This can be seen as a variant of the ergodic hypothesis — every quantum state for the subsystem is equally likely. This behaviour is quantified by considering higher moments of the statistical ensemble, i.e. defining

\begin{equation}
\rho^{(k)} = \sum_{z_B} p(z_B) \left(|\psi_{A}(z_B) \rangle \langle \psi_{A}(z_B)|\right)^{\otimes k}\,.
\nonumber
\end{equation}

Deep thermalization posits that, for generic quantum dynamics, at late times these moments will reproduce those of the Haar distribution:

\begin{equation}
\rho^{(k)}_{\textrm{Haar}} = \int d\phi_A \left(|\phi_{A} \rangle \langle \phi_{A}|\right)^{\otimes k}\,.
\nonumber
\end{equation}

In this distribution states are drawn completely randomly instead of being obtained by measurements on a bath.
For $k=1$ the thermal behaviour of the reduced density matrix is reproduced since $\rho^{(1)}_{\textrm{Haar}} \propto 𝟙_A$. In this way, thermalization can be extended to higher moments $k>1$.
This phenomenon was first predicted in [5] and has already been observed experimentally in a Rydberg quantum simulator [6]. Exact results have been obtained for circuit models exhibiting a specific space-time symmetry [7,8,9], in thermalizing systems with random measurement bases [10], and in free-fermion models [11]. On the practical level, the emergence of state designs following projective measurements has found an application in the design of efficient protocols for state tomography [12].
This emergent randomness at long times can be contrasted with the behaviour at short times, where the system state $|\psi_{A}(z_B)\rangle$ is strongly correlated with the bath state $z_B$. At longer times the distribution then becomes more and more random, and the moments for increasingly higher $k$ will agree with those of the Haar random distribution. The specific time it takes to have agreement between these moments is known as the design time $t_k$, since the distribution over states then forms an approximate quantum state $k$-design. It is now a natural question to ask if the thermalization of these higher moments and the corresponding design times are independent of the thermalization of the reduced density matrix.

Schematic of the setup. Subsystem $A$ interacts with a bath $B$ through a small number of bath degrees of freedom $B_1$, modelling a “local bottleneck”. Subsystem $A$ and bath $B$ are prepared in an initial state $|0\rangle$ and evolve using random unitary matrices $U_t$ and $V_t$ respectively. Measuring the state of the bath $B$ returns a state for $A$, and the statistical properties of the resulting distribution can be captured by the Haar-random distribution after sufficiently many time steps (i.e., applications of the unitaries $U_t$ and $V_t$).

Random matrix model for deep thermalization

In [4] the authors present an exactly solvable model for deep thermalization by considering a small local subsystem coupled to an infinite bath through local interactions. Random unitary matrices model the dynamics of the subsystem, the bath, and the bath-subsystem interactions.

By performing appropriate averaging, the authors are able to make exact predictions for the design times, which depend nontrivially on the order $k$. While previous analytical results for deep thermalization were known, these were pathological in the sense that all design times were identical [7,9,8]. One of the main results of the paper is in relating approximate thermalization, i.e. how well the reduced density matrix reproduces the thermal one, to deep thermalization. Specifically,

\begin{equation}
\Delta^{(k)} = \left(\frac{1+d_A}{1+d_A/k}\right)^{(1/2)}\Delta^{(1)},
\nonumber
\end{equation}

where $d_A$ is the dimension of the Hilbert space for the subsystem $A$, and $\Delta^{(k)} = ||\rho^{(k)}-\rho^{(k)}_{\textrm{Haar}}||_2/||\rho^{(k)}_{\textrm{Haar}}||_2$ quantifies how much the moments differ from those of the Haar-random distribution. Mathematically, this result is supported by showing that in this setup the higher moments are given by those of the so-called Scrooge or Gaussian adjusted projected ensemble. In such an ensemble, the higher moments reproduce those of the maximally entropic distribution consistent with a fixed first moment (reduced density matrix). For higher moments the distribution becomes increasingly more distinguishable from the Haar-random distribution. While exact thermalization and $\Delta^{(1)}=0$ leads to exact deep thermalization — explaining the pathological behaviour of previous results — approximate thermalization is generically expected and will result in a hierarchy of different moments and resulting different design times. Interestingly, these can all be related to the time scale for regular thermalization. As one specific prediction, in the limit of an infinite bath the design time $t_k$ required to achieve an approximate quantum state $k$-design is found as

\begin{equation}
t_k = t_1 + \frac{1}{v_E}\log_2\left(\frac{1+d_A}{1+d_A/k}\right),
\nonumber
\end{equation}

in which $v_E$ is the so-called entanglement velocity quantifying entanglement growth. The design times scale logarithmically with the order $k$ for sufficiently small $k$, such that higher-order designs are quickly achieved.

This result is remarkable in that different notions of thermalization are unified and a quantitative prediction can be made for the design times, which are shown to be consistent with numerical simulations. The next challenge is now to quantify which aspects of the random matrix model reproduce those of more structured models, whether the presented results effectively present optimal bounds for design times, and how these results extend to systems with conservations laws, where additional bottlenecks can appear.

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