• Open Access

Andrea De Luca1, Chunxiao Liu2, Adam Nahum3, and Tianci Zhou4

• 1Laboratoire de Physique Théorique et Modélisation, CY Cergy Paris Université, CNRS, F-95302 Cergy-Pontoise, France
• 2Department of Physics, University of California, Berkeley, California 94720, USA
• 3Laboratoire de Physique de l’École Normale Supérieure, CNRS, ENS & Université PSL, Sorbonne Université, Université Paris Cité, 75005 Paris, France
• 4Department of Physics, Virginia Tech, Blacksburg, Virginia 24061, USA

Abstract

We consider the universal aspects of two problems: (i) the singular value structure of a product Mt=mtmt−1…m1 of many large independent random matrices and (ii) the slow purification of a large number of qubits by repeated quantum measurements. The time-evolution operator in the latter case is again a product of matrices mi, representing time steps in the evolution, but the mi are now nontrivially correlated as a result of Born’s rule. Both processes are associated with the decay of natural measures of entropy as a function of time or of the number of matrices in the product. We argue that, for a broad class of models, each process is described by universal scaling forms for purification and that (i) and (ii) represent distinct “universality classes” with distinct scaling functions. Using the replica trick, these universality classes correspond to effective one-dimensional statistical mechanics models for a gas of “kinks,” representing domain walls between elements of the permutation group. This is an instructive low-dimensional limit of the effective statistical mechanics models for random circuits and tensor networks. These results apply to longtime purification in spatially local monitored circuit models on the entangled side of the measurement phase transition.

• Entanglement entropy
• Quantum circuits
• Quantum measurements
• Group theory
• Random matrix theory
• Replica methods

Popular Summary

Measuring a quantum system reveals information about its state, but the act of measurement can also change that state. Consequently, the way in which repeated measurements extract information about a state is surprisingly rich, especially if the system continues to evolve between measurements. However, when measurements are infrequent, this “learning” process slows dramatically. In quantum physics, this process is called purification: The system’s uncertainty, measured by its entropy, decreases as we gather information. In this work, we show that the slow purification process is universal: It follows the same mathematical pattern for a wide range of quantum systems.

We model purification using products of random matrices, a mathematical tool that here finds new use in quantum dynamics. Remarkably, the complex behavior of a general chaotic quantum system can be mapped onto a simpler model: a 1D dilute gas. In this mapping, the spatial dimension of the gas corresponds to time in the quantum system, and the gas particles, which come in many species, represent relationships (or pairings) between different copies of the system. As time goes on, the gas becomes increasingly sparse, somewhat like the ideal gas limit. In this limit, we can calculate its free energy using combinatorics. This reveals a universal scaling law for how entropy decays in the original quantum problem.

Our findings apply broadly to quantum systems under measurement and give physicists new tools for exploring longtime behavior.

Article Text

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In the Ginibre model, these are the configurations with the lowest energy; i.e., all other configurations incur a larger power of 1/q in the Boltzmann weight. Therefore, other configurations are suppressed until t becomes of the order of q (at which point the entropy gain from additional kinks can compete with the energy cost). Though we have not proved it, we guess that configurations outside this set are energetically suppressed for t≪q in a larger class of models, given mild assumptions on the distribution of the blocks. 1.

In this approach, we work with a fixed Hilbert space, H⊗2⊗H*⊗2, which is sufficient to express e−S2(t) in a fixed realization of randomness. We then introduce the permutation states by introducing resolutions of the identity in this Hilbert space at each time step: 1=∑σ∈S2|σ⟫⟪σ*|+P⊥. Here, P⊥ is a projector onto the subspace orthogonal to the permutation states. In the present regime of times t≪q, we expect that terms with P⊥ can be neglected [49]. This leads to the rhs of Eq. (60). 1.

A given transfer matrix element such as TI,(12) depends nontrivially on N (for example, the local properties of the effective 1+1D statistical mechanics problem depend strongly on N), but we assume it can be continued to N→1 (if we are considering measurements) or N→0 (if we are considering forced measurements, or a network of random tensors).

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We thank an anonymous referee for pointing this out.

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