[Paper Review] High-temperature thermalization implies the emergence of quantum state designs

Henrik Wilming, Ingo Roth|arXiv (Cornell University)|Feb 3, 2022

Advanced Thermodynamics and Statistical Mechanics4 citations

TL;DR

This paper proves that high-temperature thermalization—where a small subsystem becomes nearly maximally mixed—necessarily leads to the emergence of approximate quantum state $k$-designs in post-measurement states when the complementary subsystem is measured in a random orthonormal basis. The key result is that for any large system where a subsystem is close to maximally mixed, almost all measurement bases (with high probability under the Haar measure) yield $\epsilon$-approximate $k$-designs, establishing a general link between thermalization and quantum design emergence beyond specific models.

ABSTRACT

It was recently observed that in certain thermalizing many-body systems, measuring the complement of a subsystem that thermalized to infinite temperature in a suitable orthonormal basis gives rise to approximate quantum state k-designs as post-measurement states on the subsystem. We prove that this emergence of approximate k-designs holds true in every large system where some small subsystem is close to being maximally mixed. On a technical level we show that any high-dimensional purification of an approximately maximally mixed state induces an approximate quantum state k-design when measured in a suitable orthonormal basis. Moreover, we show that this is true with overwhelming probability for measurement bases chosen uniformly at random from the Haar measure.

Motivation & Objective

• To establish a general connection between high-temperature thermalization and the emergence of quantum state designs in post-measurement ensembles.
• To show that approximate $k$-designs arise not only in specific models but universally when a subsystem is nearly maximally mixed.
• To demonstrate that this phenomenon occurs with overwhelming probability for measurement bases sampled from the Haar measure.
• To provide a rigorous proof that thermalization of the reduced density matrix implies the emergence of higher-order quantum designs in the post-measurement ensemble.

Proposed method

• Prove that any high-dimensional purification of an approximately maximally mixed state induces an approximate $k$-design upon measurement in a suitable orthonormal basis.
• Use random matrix theory and concentration of measure to show that Haar-random measurement bases yield $\epsilon$-approximate $k$-designs with high probability.
• Define an $\epsilon$-approximate $k$-design via trace-distance between the $k$-fold tensor product of the ensemble and the Haar average over pure states.
• Establish a Lipschitz bound on the deviation of the $k$-th moment of the measurement ensemble from the Haar average, using operator norm estimates and trace inequalities.
• Apply the Gaussian concentration inequality to bound the probability that the ensemble deviates from a $k$-design, leading to the required sample size condition on $M = \dim(\mathcal{H}_{\bar{A}})$.
• Derive a sufficient condition on the dimension $M$ of the complement subsystem such that the post-measurement ensemble is an $\epsilon$-approximate $k$-design with probability at least $1 - \Delta$.

Experimental results

Research questions

• RQ1 Does high-temperature thermalization—defined by a subsystem's reduced density matrix being close to maximally mixed—necessarily lead to the emergence of quantum state $k$-designs in post-measurement ensembles?
• RQ2 For which measurement bases does the post-measurement ensemble approximate a $k$-design, and with what probability?
• RQ3 Can the emergence of $k$-designs be guaranteed generically, without relying on specific model assumptions or symmetries?
• RQ4 What is the minimal dimension of the complementary subsystem required to ensure $\epsilon$-approximate $k$-designs with high probability?
• RQ5 How does the probability of failure in achieving a $k$-design scale with system size and $k$?

Key findings

• If the reduced density matrix of a small subsystem $A$ is within trace distance $\delta < \frac{1}{2d_A}$ of the maximally mixed state, then the post-measurement ensemble is an $\epsilon$-approximate $k$-design.
• For any $\epsilon' > 0$, $\Delta > 0$, and $k \in \mathbb{N}$, if the dimension $M$ of the complement subsystem satisfies $M > \frac{4(2k-1)^2 d_A^{2k-1}}{\epsilon'^2} \log\left(\frac{2d_A^{2k}}{\Delta}\right)$, then the post-measurement ensemble is an $\epsilon$-approximate $k$-design with probability at least $1 - \Delta$.
• The result holds with overwhelming probability for measurement bases chosen uniformly at random from the Haar measure on the unitary group.
• The emergence of $k$-designs is not restricted to specific models or bases like the computational basis, but is generic under random measurement.
• The Lipschitz constant of the deviation functional is bounded by $\frac{2(2k-1)}{\sqrt{d_A}}$, enabling concentration of measure arguments.
• The paper establishes that thermalization to infinite temperature is sufficient for $k$-design emergence, even without additional symmetries or integrability conditions.