Entanglement Entropy and Complexity in Random Systems

Zi-Wen Liu, Huangjun Zhu, Elton Yechao Zhu +1 more

Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that hides initially localized quantum information. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Renyi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Renyi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Renyi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Renyi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom.

Durrani, Ijaz

Dominik Janzing

Zi-Wen Liu, Seth Lloyd, Elton Yechao Zhu +1 more

Abstract Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that “hides” initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quantum gravity, many-body physics, quantum statistical mechanics, quantum information etc. Scrambling can exhibit different complexities depending on the degree of randomness it produces. For example, notice that the complete randomization implies scrambling, but the converse does not hold; in fact, there is a significant complexity gap between them. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Rényi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Rényi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom. This paper is an extended version of Phys. Rev. Lett. 120 (2018) 130502 [1].

John Cardy

Jim Dorsey

Abstract Entanglement entropy has long served as a central diagnostic of complexity and information scrambling in quantum many-body systems, and its behavior in random quantum circuits is by now well characterized. However, as a single scalar, entanglement entropy captures only the total uncertainty of the reduced density matrix spectrum while discarding the detailed structure of that spectrum — including the shape of the eigenvalue distribution, level repulsion signatures, and the transition from integrable to chaotic behavior. In this work, we propose spectral entropy — the Shannon entropy of the eigenvalue distribution of the reduced density matrix — as a structured diagnostic that goes beyond standard entanglement entropy. Using numerical simulations, we compute the spectral entropy of subsystem reduced states in Haar-random and Clifford circuits as a function of circuit depth, and compare its behavior against von Neumann entanglement entropy, Rényi-2 entropy, and entanglement spectrum level-spacing statistics. We expect to find three key results: (i) spectral entropy exhibits a growth timescale distinct from that of entanglement entropy, potentially reflecting spectral restructuring more sensitively; (ii) in the depth regime where entanglement entropy has already saturated, spectral entropy may continue to evolve, revealing late-time complexification that standard entanglement measures cannot resolve; (iii) spectral entropy may display a distinctive response to measurement-induced or noise-induced disentanglement that differs qualitatively from entanglement entropy. This work positions spectral entropy as a complementary — not replacement — tool to entanglement entropy, offering a finer spectral lens on information scrambling, thermalization, and complexity growth in random quantum circuits.