Fast Pre-scramblers Oleg Kaikov12 1Arnold Sommerfeld Center Ludwig-Maximilians-Universität

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Fast Pre-scramblers

Oleg Kaikov∗1,2

1Arnold Sommerfeld Center, Ludwig-Maximilians-Universität,

Theresienstraße 37, 80333 München, Germany

2Max-Planck-Institut für Physik,

Föhringer Ring 6, 80805 München, Germany

January 26, 2023

Abstract

We consider the process of diﬀusion or “pre-scrambling” of information in a quantum

system. We deﬁne a measure for this spreading or “pre-scrambling” of the wave-

function in terms of a minimum probability threshold for the states in the system’s

Hilbert space. We illustrate our ﬁndings on the example of a prototype model with

enhanced memory capacity. We conjecture:

(1) The fastest pre-scramblers require a time logarithmic in the number of degrees

of freedom.

(2) The investigated enhanced memory capacity model is a fast pre-scrambler.

(3) (Fast) pre-scrambling occurs not later than (fast) scrambling.

(4) Fast scramblers are fast pre-scramblers.

(5) Black holes are fast pre-scramblers.

1 Introduction

1.1 Scrambling

Every physical system carries information. Due to the time evolution of the system,

the state of the information within it also evolves. Consider a quantum system of K

degrees of freedom, prepared in a pure state and evolved unitarily in time. Even though

the state of the system remains pure throughout its unitary evolution, nevertheless, the

∗Kaikov.Oleg@physik.uni-muenchen.de

arXiv:2210.02312v2 [hep-th] 25 Jan 2023

system thermalizes after some time. This thermalization takes place in the following sense:

although the time evolution is unitary, over time, the initial state becomes entangled with

the other states in the Hilbert space of the system. The information initially contained

only in the original state is distributed over the other states and is therefore mixed up.

Thus, the system “scrambles” [1,2] information. The characteristic time scale associated

with this process is called the scrambling time ts[1,2].

Before characterizing the scrambling time, we need to ﬁrst address scrambling itself.

Following [1–3], which introduced the concept of scrambling, we call a system scrambled if

the information within it (i.e. the state of the system) is suﬃciently distributed over the

entire Hilbert space of the system with respect to some chosen measure.

Note that we keep the deﬁnition general on purpose: we do not limit the formulation to

a speciﬁc measure of the uniformity of the state distribution. There exist multiple distinct

such measures deﬁned in the literature. Some of the ﬁrst explicit deﬁnitions include the

Haar measure in application to the choice of the mixing unitary transformation [1], or to

states (“Haar-scrambled”) [2]. Following [4], for a model of Kqubits in [2], the authors also

deﬁne a system to be “Page-scrambled” when the entanglement entropy of any subsystem

of n < K/2qubits satisﬁes Sn=n−O(exp[2n−K]). In general, a detailed picture

of scrambling, especially for black holes, requires an understanding of scrambling at a

microscopic level. In particular, the microscopic picture of scrambling, introduced in [5–9]

(see also further references therein), establishes the concept of “memory modes” (see the

discussion below). Moreover, it discusses the concepts and the corresponding time-scales

of “one-particle entanglement” [8] and “maximal entanglement” [9] for microscopic models

of black holes.

However, in the present work we introduce a novel regime, which takes place on time-

scales smaller than the various aforementioned scrambling time-scales. We thus establish

a new pre-scrambling stage of a system’s evolution, which requires quantiﬁcation. In

addition, we note that the results of the present paper are independent of the choice of a

speciﬁc measure for scrambling.

What is the shortest possible scrambling time that a system can have? Following [1–

3], systems that scramble information in a time logarithmic in the number of degrees

of freedom are the quickest scramblers. As in [1–3], we refer to such systems as fast

scramblers. We broaden the above deﬁnition to also include the number of sites, modes,

or generic quantum labels in a system, in addition to the number of degrees of freedom.

Furthermore, to keep the deﬁnition general, we do not include any temperature dependence

or any bound due to the latter, as is, for example, done in [2,10–12] among many other

works. The results of the present paper apply in general, and require only that fast

scramblers have a scrambling time that is logarithmic in the number of degrees of freedom

K. The temperature dependent scrambling time for the model considered in section 2will

be analyzed elsewhere [13]. In this paper we investigate the behavior of the above model

at times preceding its scrambling time.

1.2 Enhanced memory capacity

Before considering the model of section 2we ﬁrst need to introduce the concepts that

are at its foundation. A sub-direction of the larger program of [5–7,14–30] (see also further

references therein) investigates general phenomena of systems that possess states with a

high capacity to store information. As in [6,7,14–25], we refer to such systems as those

with enhanced memory capacity. One of the above phenomena is the eﬀect of “memory

burden” [14,15], which we brieﬂy summarize below along with other terminology. For this,

we largely rely on the recent detailed review included in [25].

The degrees of freedom of a physical system are commonly described as quantum oscil-

lators. The states of the system are then labeled by distinct sequences of the Koscillators’

occupation numbers |n1, . . . , nKi. We refer to each such sequence as a memory pattern

which stores quantum information. The number of distinct patterns that can be stored

within a microscopically narrow energy gap are a measure for the memory capacity of a

system [19,21]. If the N-many microstates |n1, . . . , nKidescribing distinct patterns are

degenerate in energy, they contribute to the microstate entropy S= ln(N). If a mem-

ory pattern contains a large amount of quantum information, it stabilizes the system in

the state of enhanced memory capacity and slows down the system’s evolution. Follow-

ing [14,15], we refer to this as the memory burden eﬀect.

A system can reach a state of enhanced memory capacity by the eﬀect of assisted

gaplessness [21]. In its essence, assisted gaplessness occurs when a highly occupied master

mode interacts attractively with a set of memory modes, lowering their energy gaps. The

memory modes become eﬀectively gapless and degenerate in energy, and can store large

amounts of information. In their own turn, the memory modes backreact on the master

mode via the memory burden eﬀect [14,15] and slow down the change in its occupation

number. As proposed by [14] and investigated in [25], the memory burden eﬀect can

be avoided if the system possesses another set of modes K0to which it can rewrite the

information stored in the modes K.

2 A prototype model

We consider a speciﬁc prototype model of a system with enhanced memory capacity

given in Eq. (34) of [25], which is constructed in correspondence to the black hole’s quantum

N-portrait [5]. The model is given in Eq. (6) with slight changes in notation for the sake of

brevity. It contains two sets of bosonic memory modes Kand K0, with the corresponding

creation and annihilation operators ˆa†

k,ˆakand ˆa†

k0,ˆak0, respectively, for k(0)= 1, . . . , K(0).

The operators ˆa(†)

k(0)obey the standard commutation relations (here and throughout we set

~≡1)hˆaj(0),ˆa†

k(0)i=δj(0)k(0),hˆaj(0),ˆak(0)i= 0,hˆa†

j(0),ˆa†

k(0)i= 0.(1)

The corresponding occupation number operators are given by ˆnk(0)= ˆa†

k(0)ˆak(0)with eigen-

values nk(0)and eigenstates |nk(0)i. Additionally, the model contains two more bosonic

modes ˆna(the master mode) and ˆnb, with creation and annihilation operators ˆa†,ˆaand

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摘要：

FastPre-scramblersOlegKaikov*1;21ArnoldSommerfeldCenter,Ludwig-Maximilians-Universität,Theresienstraÿe37,80333München,Germany2Max-Planck-InstitutfürPhysik,FöhringerRing6,80805München,GermanyJanuary26,2023AbstractWeconsidertheprocessofdiusionorpre-scramblingofinformationinaquantumsystem.Wedeneame...

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Fast Pre-scramblers Oleg Kaikov12 1Arnold Sommerfeld Center Ludwig-Maximilians-Universität

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