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Liu, Zi-Wen, Lloyd, Seth, Zhu, Elton and Zhu, Huangjun (2018). Entanglement, quantum randomness, and complexity beyond scrambling. J. High Energy Phys. (7). NEW YORK: SPRINGER. ISSN 1029-8479

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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 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. This paper is an extended version of [1].

Item Type:	Journal Article
Creators:	Creators Email ORCID ORCID Put Code Liu, Zi-Wen UNSPECIFIED UNSPECIFIED UNSPECIFIED Lloyd, Seth UNSPECIFIED UNSPECIFIED UNSPECIFIED Zhu, Elton UNSPECIFIED UNSPECIFIED UNSPECIFIED Zhu, Huangjun UNSPECIFIED UNSPECIFIED UNSPECIFIED
URN:	urn:nbn:de:hbz:38-179873
DOI:	10.1007/JHEP07(2018)041
Journal or Publication Title:	J. High Energy Phys.
Number:	7
Date:	2018
Publisher:	SPRINGER
Place of Publication:	NEW YORK
ISSN:	1029-8479
Language:	English
Faculty:	Unspecified
Divisions:	Unspecified
Subjects:	no entry
Uncontrolled Keywords:	Keywords Language AVERAGE ENTROPY; PAGES CONJECTURE; UNITARY; PROOF Multiple languages Physics, Particles & Fields Multiple languages
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/17987