Remarks on the Time Asymptotics of Schmidt Entropies

Abstract

Schmidt entropy is used as a common denotation for all Hilbert space entropies that can be defined via the Schmidt decomposition theorem; they include quantum entanglement entropies and classical separability entropies. Exact results about the asymptotic growth in time of such entropies (in the form of Renyi entropies of any order $\ge 1$) are directly derived from the Schmidt decompositions. Such results include a proof that pure point spectra entail boundedness in time of all entropies of order larger than 1; and that slower than exponential transport forbids faster than logarithmic asymptotic growth. Applications to coupled Quantum Kicked Rotors and to Floquet systems are presented.

1. Introduction

Understanding the time evolution of quantum entanglement is central to the foundations of statistical mechanics. A crucial issue is the growth in time of the entanglement entropy (EE) of two interacting subsystems of a closed system. Linear growth (apart from fluctuations, which also convey important physics [1]) is connected to “quantum chaos”. In several model systems, it has been numerically observed, theoretically supported, and related to classical exponential instability [2,3]. In special cases [2], which are dynamically akin to multi-dimensional inverted harmonic oscillators, such linear growth persists forever; otherwise, it is observed over possibly quite long but nevertheless finite time scales that increase with the size of the considered many body systems, or in the thermodynamic limit [4,5,6]. Saturation, or not faster than logarithmic growth, is thereafter observed. Though the $\hslash \to 0$ limit of the time evolution of quantum entanglement has some nontrivial aspects [7], entanglement has no classical counterpart. However, from a mathematical viewpoint, EE, when the full system is in a pure state, is a special case of an abstract Hilbert space construction that works unaltered also in the Koopman formulation of classical mechanics [8]. This well known construction is based on the Schmidt decomposition theorem, and the entropies it produces are thereby denoted “Schmidt entropies” in the present paper. The classical Schmidt entropy was introduced in [9] and it was dubbed “separability entropy” [10]. In this paper, it is shown that the Schmidt decomposition affords straightforward access to asymptotic upper bounds on the growth of Schmidt entropies, bypassing the customary approach based on first tracing out either subsystem. Thanks to a minimum property of the Schmidt decomposition (Proposition 1), boundedness in time of all Schmidt entropies of the Renyi type of order larger than 1 is quite easily proven, whenever the unitary evolution has a pure point spectrum—in particular, in the presence of dynamical localization (Proposition 2). In the Section Setup upper logarithmic bounds for the growth of Schmidt entropies are proven in special cases, when the dynamics satisfy a technical condition (Proposition 3) that, roughly speaking, rules out faster than power law transport. Applications are presented to a system of two coupled Quantum Kicked Rotors (Section 2.2) and to the extended Hilbert space Floquet dynamics of periodically driven systems (Section 2.3).

Setup

The basic Hilbert space framework for the study of two interacting, finite dynamical systems, whether quantum or classical, consists of separable Hilbert spaces ${\mathcal{H}}_1$,${\mathcal{H}}_2$, $\mathcal{H}={\mathcal{H}}_1\otimes {\mathcal{H}}_2$, and a unitary group ${\left\{{\widehat{U}}^t\right\}}_{t\in \mathbb{Z}}$ in $\mathcal{H}$. Only the case when all such spaces are infinite dimensional is considered here. Let $B_i={\left\{u_n^{(i)}\right\}}_{n\in \mathbb{N}}$, $(i=1,2)$ be Hilbert bases in ${\mathcal{H}}_i$. Using a single integer to label the base vectors in $B_i$ is not a limitation of generality, because any Hilbert base in a separable Hilbert space is countable. In concrete examples, it is often expedient to resort to multiple labels reflecting the physical structure of the subsystems. Then, $B\equiv {\{u_{n_1}^{(1)}\otimes u_{n_2}^{(2)}\}}_{n_1,n_2\in \mathbb{N}}$ is a Hilbert base in $\mathcal{H}$ that will be called a bipartite base. The components of a vector w on such a base will be denoted $w(B,n_1,n_2)$. The Schmidt decomposition theorem states that bipartite bases $B=B_w$ exist, such that $\forall n_1,n_2$:

$$|w(B_w,n_1,n_2)|={\lambda }_{n_1}{\delta }_{n_1{n}_2},$$

(1)

where ${\lambda }_n$ are nonnegative numbers; the positive ones have finite multiplicity at most, and ${\sum }_n{\lambda }_n^2={\parallel w\parallel }^2$. Given w, the values ${\lambda }_n$ are unique, but the base $B_w$ is not, due to arbitrary phase factors, and possibly to multiplicities. In the following, all bases $B_w$ with the property (1) are dubbed Schmidt bases for the vector w. If $\parallel w\parallel =1$ and $A=\{v_{n_1}^{(1)}\otimes v_{n_2}^{(2)}\}$ is any bipartite base (not necessarily a Schmidt base for w), then the numbers $|w(A,n_1,n_2){|}^2$ define a probability distribution on ${\mathbb{N}}^2$. The “spread” of this distribution can be measured by the Reny entropies: for $0<\theta \ne 1$, these are defined as

$$H_{\theta }(w,A)={\displaystyle \frac{1}{1-\theta }}ln(\sum _{n_1,n_2}|w(A,n_1,n_2{|}^{2\theta }).$$

(2)

The entropy $H_1(w,A)$ is defined via the limit $\theta \to 1$ (whenever the limit exists) and is equal to the Shannon entropy. It is immediate that

$$\begin{array}{c}\hfill H_{\theta }(w,B_w)={\displaystyle \frac{1}{1-\theta }}ln(\sum _n{\lambda }_n^{2\theta })(\theta \ne 1)\\ \hfill H_1(w,B_w)=-\sum _n{\lambda }_n^{2}ln({\lambda }_n^2).\end{array}$$

(3)

In the quantum case, the entropies $H_{\theta }(w,B_w)$ are called “entanglement entropies”, and in the classical case, “separability entropies” [9]. Here, they will share the denotation “Schmidt entropies” and the symbol ${\mathcal{S}}_{\theta }$.

2. Results

The Schmidt bases of a vector w can be characterized as those bipartite bases that make w maximally close to being separable. More precisely,

Proposition 1.

Whenever $\theta \ge 1$, the entropy $H_{\theta }(w,B_w)$ is a lower bound for the entropies $H_{\theta }(w,A)$ relative to arbitrary bipartite bases A; and this bound is attained if and only if A is a Schmidt base.

Proof. 

Let $\widehat{\rho }$ be a unit trace positive operator in a separable Hilbert spaces $\mathcal{H}$. If $A:=\left\{w_n\right\}$ is a Hilbert base, and $f:(0,1]\to \mathbb{R}$ is a strictly convex function, then the following inequality follows from a simple convexity argument, or else from general results in [11]:

$$\sum _{n}f\left(\langle w_n\left|\widehat{\rho }\right|w_n\rangle \right)\le \mathrm{Tr}(f(\widehat{\rho }))$$

(4)

with the equality holding if and only if A is an eigenbase for $\widehat{\rho }$. Then, let $f(x)=x^{\theta }$, A any bipartite base, and $\widehat{\rho }:={\widehat{\rho }}^{(1)}\otimes {\widehat{\rho }}^{(2)}$, where ${\widehat{\rho }}^{(i)}$ are operators in ${\mathcal{H}}_i$ whose matrix elements on the bases $A_i$ are given by the “partial traces” of the operator $|w\rangle \langle w|$:

$${\widehat{\rho }}^{(1)}(n_1,n_1^{\prime })=\sum _{n_2}w(n_1,n_2)w^{*}(n_1^{\prime },n_2),{\widehat{\rho }}^{(2)}(n_2,n_2^{\prime })=\sum _{n_1}w(n_1,n_2)w^{*}(n_1,n_2^{\prime }).$$

(5)

Using the Schmidt decomposition, it is easily seen that both operators share the eigenvalues ${\lambda }_n^2$. Inequality (4) translates to

$$\sum _{n_1,n_2}{\left[\left({\widehat{\rho }}^{(1)}\otimes {\widehat{\rho }}^{(2)}\right)(n_1,n_2,n_1,n_2)\right]}^{\theta }\le {(\sum _n{\lambda }_n^{2\theta })}^2.$$

(6)

On the other hand,

$$\begin{array}{c}\hfill \sum _{n_1,n_2}{\left[\left({\widehat{\rho }}^{(1)}\otimes {\widehat{\rho }}^{(2)}\right)(n_1,n_2,n_1,n_2)\right]}^{\theta }=\sum _{n_1,n_2}{(\sum _r{\left|w(n_1,r)\right|}^2)}^{\theta }{(\sum _s{\left|w(s,n_2)\right|}^2)}^{\theta }\\ \hfill \ge {(\sum _{n_1,n_2}{\left|w(n_1,n_2)\right|}^{2\theta })}^2\end{array}$$

(7)

(the last estimate, using that ${\ell }^{\theta }$ norms are non-increasing with $\theta$). The claim $H_{\theta }(w,A)\ge H_{\theta }(w,B_w)$ follows, from putting together (3), (6) and (7). The case $\theta =1$ can be dealt with along similar lines using $f(x)=xln(x)$, or else by taking the limit $\theta \searrow 1$.

□

This result provides a clue to estimating the growth of the Schmidt entropy of a state ${\widehat{U}}^{t}w$ over its Schmidt bases, where $t\in \mathbb{Z}$ and $\widehat{U}$ is a unitary operator in $\mathcal{H}$, just because it cannot grow faster, than entropies related to fixed (i.e., time independent) bipartite bases.

For $\theta >1$, the entropy $H_{\theta }(w,A)$ of a state w can be estimated by means of the following inequality:

$$H_{\theta }(w,A)\le \theta {(1-\theta )}^{-1}ln(1-{ϵ}^2)+ln(N_{ϵ}(w,A)),$$

(8)

where $ϵ$ is arbitrary in $(0,1)$ and $N_{ϵ}(w,A)$ is an integer that is defined as follows. A minimal ϵ-support of the probability distribution ${\left|w(\mathbf{n},A)\right|}^2$ is a finite set ${\mathcal{F}}_{ϵ}\subset {\mathbb{N}}^2$ of sites such that (i) it is an $ϵ$-support for the distribution; that is, ${\sum }_{\mathbf{n}\in {\mathcal{F}}_{ϵ}}{\left|w(\mathbf{n},A)\right|}^2\ge 1-{ϵ}^2$, and (ii) ${\sum }_{\mathbf{n}\in {\mathcal{F}}^{\prime }}{\left|w(\mathbf{n},A)\right|}^2<1-{ϵ}^2$ for any finite set of sites ${\mathcal{F}}^{\prime }$ with $♯({\mathcal{F}}^{\prime })<♯(\mathcal{F})$ (where ♯ denotes cardinality). The distribution may have different minimal $ϵ$-supports (for the same $ϵ$), but all of them must have the same size $♯({\mathcal{F}}_{ϵ})$, and this is how the integer $N_{ϵ}(w,A)$ is defined. Inequality (8) is proven in a slightly more general form in Appendix A. It cannot be carried to the limit $\theta \searrow 1$, because it is established for arbitrary $w\in \mathcal{H}$, for which $H_1(w)$ is generically undefined. A first application is the following Proposition. The space $\mathcal{H}$ being bipartite is irrelevant to this Proposition, which is of general validity; so, to simplify notations, the base vectors are now labeled by a single integer.

Proposition 2.

Let A be any (not necessarily bipartite) Hilbert base ${\left\{u_n\right\}}_{n\in \mathbb{N}}$ in $\mathcal{H}$. The Renyi entropies $H_{\theta }({\widehat{U}}^{t}w,A)$ with $\theta >1$ are bounded in time t whenever w, ($\parallel w\parallel =1$), belongs in the pure point spectral subspace ${\mathcal{H}}_{pp}(\widehat{U})$. Hence, the same is true of the Schmidt entropies ${\mathcal{S}}_{\theta }({\widehat{U}}^{t}w)$.

Proof. 

Let $E:={\left\{e_n\right\}}_{n\in \mathbb{N}}$ be an eigenbase for $\widehat{U}$. For integers $M>0$, $N>0$, let ${\mathcal{H}}_N^{(A)}$ and ${\mathcal{H}}_M^{(E)}$ be the subspaces of $\mathcal{H}$ which are respectively spanned by $\{u_1,...,u_N\}$ and by $\{e_1,...,e_M\}$, and let ${\widehat{P}}_M^{(E)}$, ${\widehat{P}}_N^{(A)}$ denote the corresponding projectors. The subspaces ${\mathcal{H}}_M^{(E)}$ are invariant under $\widehat{U}$. Assuming $\parallel w\parallel =1$ and $ϵ\in (0,1)$, let $M=M_{ϵ}$ be chosen such that $\parallel (\widehat{1}-{\widehat{P}}_{M_{ϵ}}^{(E)})w\parallel <ϵ/3$, and denote $w_t={\widehat{U}}^{t}w$, $w_{ϵ,t}:={\widehat{P}}_{M_{ϵ}}^{(E)}{\widehat{U}}^{t}w$. The set $\mathcal{K}:={\left\{w_{ϵ,t}\right\}}_{t\in \mathbb{Z}}$ is a bounded subset of the unit sphere in the finite dimensional space ${\mathcal{H}}_{M_{ϵ}}^{(E)}$, so its closure $\overline{\mathcal{K}}$ is compact. The functions $f_N:\mathcal{H}\to {\mathbb{R}}_{+}$ which are defined by $f_N(.):=\parallel (\widehat{1}-{\widehat{P}}_N^{(A)})(.)\parallel$ monotonically decrease to 0 as $N\to \infty$. Therefore, thanks to Dini’s theorem, such convergence is uniform in the compact set $\overline{\mathcal{K}}$; so, a t-independent integer $N_{ϵ}^{\prime }$ exists such that $\parallel (\widehat{1}-{\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)})w_{ϵ,t}\parallel <ϵ/3$, $\forall t\in \mathbb{Z}$. Therefore,

$$\begin{array}{c}\hfill \parallel w_t-{\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_t\parallel <\parallel w_t-{\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_{ϵ,t}\parallel +\parallel {\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_{ϵ,t}-{\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_t\parallel \end{array}$$

$$\begin{array}{c}\hfill \le \parallel w_t-w_{ϵ,t}\parallel +\parallel w_{ϵ,t}-{\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_{ϵ,t}\parallel +\parallel {\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_{ϵ,t}-{\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_t\parallel \\ \hfill \le 2\parallel w_t-w_{ϵ,t}\parallel +\parallel w_{ϵ,t}-{\widehat{P}}_{N_{ϵ}^{\prime }}^{(A)}{w}_{ϵ,t}\parallel \end{array}$$

(9)

$$\begin{array}{l}\le ϵ\forall t\in \mathbb{Z}.\end{array}$$

(10)

Therefore, the states $w_t$ are supported within a maximum error $ϵ$ (in norm) by the first $N_{ϵ}^{\prime }$ vectors of the base A; hence, the integer $N_{ϵ}^{\prime }$ is a t-independent upper bound for the integer $N_{ϵ}(w_t,A)$ in the inequality (8). The claim follows from (8) and Proposition 1. □

The case $\theta =1$ is not covered by the above result. Pure point spectrum (“spectral localization”) does not guarantee “dynamical localization”, that is, boundedness in time of the moments of the distribution ${\left|w(\mathbf{n},A)\right|}^2$, and not even of its Shannon entropy: some information about eigenstates is needed for that. As a matter of fact, dynamical localization is a stronger property than spectral localization. In ref. [4], unbounded logarithmic growth has been exposed in the presence of many-body localization. This is surmised to occur in the limit of an infinite system, so it is in no contradiction to the present analysis.

2.1. Ballistic Bounds

Unlike the previous sections, the bipartite base B is here parametrized by ${\mathbb{Z}}^2$ rather than ${\mathbb{N}}^2$, with no significant differences; moreover, it is held fixed, and is no longer specified in notations. Asymptotic upper estimates for the growth of entropies with $\theta >1$ based on (8) require estimates for the growth of $N_{ϵ}(U^{t}w)$. Such estimates can be obtained via Chebyshev’s inequality from bounds on the growth of moments of the distribution ${\left|w(\mathbf{n})\right|}^2$, whenever available. In cases when such growth is power-like—that is, generalized “ballistic bounds” hold—inequality (8) implies that the asymptotic growth of entropies with $\theta >1$ is not faster than logarithmic. Entropies $H_1({\widehat{U}}^{t}w)$ require a separate analysis, as they are only defined for w in a special subset (actually a subset of the 1st Baire category [11]) of $\mathcal{H}$; however, such entropies, too, cannot grow faster than logarithmically in the presence of ballistic bounds for the growth of moments. In several cases, such bounds can established by the following strategy, which extends a result in [12]. For integer $\nu \in \mathbb{N}$, define $X_{\nu }:={\{w\in \mathcal{H}:\parallel w\parallel }_{\nu }<+\infty \}$, where

$${\parallel w\parallel }_{\nu }:=sup\left\{\right|w(\mathbf{n})|e^{\nu (|n_1|+|n_2|)},\mathbf{n}\in {\mathbb{Z}}^2\},$$

(11)

is a norm in $X_{\nu }$, stronger than the $\mathcal{H}$-norm $\parallel .\parallel$.

Proposition 3.

Let $\widehat{U}(X_{\nu })\subseteq X_{\nu }$ for some ν. Then, $\forall w\in \mathcal{H}$, $\forall \theta >1$,

$$H_{\theta }({\widehat{U}}^{t}w)=O(ln(t))\mathrm{as}t\to \infty .$$

(12)

Proof. 

The proof will be attained in steps (0)–(3).

(0) there is $C:\mathbb{N}\to {\mathbb{R}}_{+}$ so that $\forall w\in X_{\nu }$,

$$\parallel \widehat{U}{w\parallel }_{\nu }\le C(\nu ){\parallel w\parallel }_{\nu }$$

(13)

$\widehat{U}$ is a linear operator in the Banach space $X_{\nu }$, and is continuous in the weaker $\mathcal{H}$ topology. Thanks to the Closed Graph theorem, it is also continuous in $X_{\nu }$.

(1) One can find $w_{ϵ}\in X_{\nu }$ with $\parallel w_{ϵ}\parallel =1$, such that $\parallel w_{ϵ}-w\parallel <ϵ$. If $\widehat{U}(X_{\nu })\subseteq X_{\nu }$, then $\forall t$, $N_{ϵ}({\widehat{U}}^{t}w)\le N_{ϵ/2}({\widehat{U}}^t{w}_{ϵ})$. $X_{\nu }$ is dense in $\mathcal{H}$ and existence of $w_{ϵ}$ easily follows. Since $\parallel U^t{w}_{ϵ}-{\widehat{U}}^{t}w\parallel <ϵ$, any $ϵ-$support for $U^t{w}_{ϵ}$ is a $2ϵ-$support for ${\widehat{U}}^{t}w$. Claim (1) follows.

(2) $\forall w\in X$, $\forall ϵ\in (0,1)$

$$N_{ϵ}(w)<{\nu }^{-2}{ln}^2\left({ϵ}^{-2}{A}_{\nu }{\parallel w_{ϵ}\parallel }_{\nu }^2\right),$$

(14)

where $A_{\nu }=16e(e^{2\nu }+1){(e^{2\nu }-1)}^{-2}$. For $L\in \mathbb{N}$, let $Q_L$ be the square box defined in ${\mathbb{Z}}^2$ by $-L\le n_1,n_2\le L$. The probability in which the distribution $|w_{ϵ}{(\mathbf{n})|}^2$ locates outside of $Q_L$ is estimated using the norm $\parallel w_{ϵ}{\parallel }_{\nu }$, and then $L=L_{ϵ}$ can be found via a trivial albeit boring computation, such that $Q_{L_{ϵ}}$ is a $ϵ-$support for $w_{ϵ}$. As $N_{ϵ/2}$ is the minimal size of $ϵ/2-$ supports, $N_{ϵ}(w)\le ♯(Q_{L_{ϵ/2}})={(2L_{ϵ/2}+1)}^2$. Replacing the just obtained explicit expression of $L_{ϵ/2}$ yields (14).

(3) The proof is concluded by replacing w by ${\widehat{U}}^{t}w$ in (14), using (13) to estimate $\parallel {\widehat{U}}^t{w}_{ϵ}{\parallel }_{\nu }$, and finally resorting to (8). □

The case $\theta =1$ cannot be handled by (8) for arbitrary states w. The following weaker statement holds:

Proposition 4.

If $\widehat{U}(X_{\nu })\subseteq X_{\nu }$, then (12) is also true $\forall w\in X_{\nu }$, when $\theta =1$.

Proof. 

Thanks to inequalities of “thermodynamic” type, the Shannon entropy of the distribution $|{\widehat{U}}^t{w(\mathbf{n})|}^2$ cannot grow faster than the logarithm of the square of the moment $m_1({\widehat{U}}^{t}w):={\sum }_{\mathbf{n}}(|n_1|+|n_2|)|U^{t}w(\mathbf{n}){|}^2$. For $w\in X_{\nu }$ the asymptotic growth of such a moment cannot be faster than linear. □

2.2. Coupled Quantum Kicked Rotors (QKRs)

The Kicked Rotor model is an iconic model in classical and quantum chaos. The fundamental effect of quantum suppression of classical chaotic diffusion, nowadays known as dynamical localization, was first detected in the QKR. A two-dimensional version of the QKR was introduced long ago in [13] in order to test dynamical localization in dimension 2. That two-dimensional QKR is equivalent to two coupled one-dimensional QKR. In the formalism that was established in Section Setup, this model is described as follows: ${\mathcal{H}}_i=L^2(S_i^1)$, where $S_i^1$ are circles parametrized by angles ${\theta }_i$, with the Fourier bases $u_{n_i}({\theta }_i)={(2\pi )}^{-1/2}exp(\mathit{ı}{n}_i{\theta }_i)$. The unitary $\widehat{U}$ is defined in $\mathcal{H}={\mathcal{H}}_1\otimes {\mathcal{H}}_2$ as $\widehat{U}=\widehat{K}({\widehat{R}}^{(1)}(T_1)\otimes {\widehat{R}}^{(2)}(T_2))$, where each ${\widehat{R}}^{(i)}(T_i)$ is diagonal on the Fourier base, with diagonal elements $R^{(i)}{(T_i)}_{n_i{n}_i}=exp(-\frac{\mathit{ı}}{2}(T_i{n}_i^2))$, with $T_1,T_2$ positive reals. In coordinate representation, the operator $\widehat{K}$ is the multiplication operator by $exp(\mathit{ı}V({\theta }_1,{\theta }_2))$, with $V({\theta }_1,{\theta }_2)$ a real valued function. Units are chosen such that all parameters are dimensionless and $\hslash =1$.

Proposition 5.

If the function of two complex variables $z_{1,2}$, which is defined by $V({\theta }_1,{\theta }_2)$ on the unit di-circle in ${\mathbb{C}}^2$, can be analytically continued in $(\mathbb{C}\setminus \{0\})\times (\mathbb{C}\setminus \{0\})$, then the estimate (12) holds true for the QKR model.

Proof. 

It will be shown that Proposition 3 applies. The spaces $X_{\nu }$ can be interpreted as follows: $w\in \mathcal{H}$ is represented by the function $\mathfrak{w}(z_1,z_2):={(2\pi )}^{-1}{\sum }_{\mathbf{n}\in {\mathbb{Z}}^2}w(\mathbf{n})z_1^{n_1}{z}_2^{n_2}$ of the complex variables $z_i$, read on the unit di-circle. If $w\in X_{\mathbf{\nu }}$, then $\mathfrak{w}$ is analytically continued in the region $e^{-\nu }<\left|z_i\right|<e^{\nu }$. Thanks to the assumption about V, $\widehat{U}$ does not change the domain of analiticity of the function $\mathfrak{w}$, so it maps each $X_{\nu }$ in itself for all $\nu$; hence, $H_{\theta }({\widehat{U}}^{t}w)=O(ln(t))\mathrm{as}t\to \infty$ follows from Proposition 3. □

Dynamical localization (hence pure point spectrum) is supported by compelling numerical evidence whenever the triple $T_1,T_2,2\pi$ is sufficiently incommensurate. In such cases, the bound (12) is irrelevant, as $H_{\theta }(U^{t}w)$ is bounded in time due to Proposition 2. However, Propositions 3, 4 hold true in all cases, even when the system is delocalized, due to insufficient incommensurability; in particular, in the case of resonances. QKR resonances occur whenever $T_1/\pi$ or $T_2/\pi$ or both are rational numbers; they generalize to the 2-QKR model the resonances of the single QKR [14] that produce a continuous spectrum [15] leading to quadratic growth of the rotors’ energy. Then, unbounded logarithmic growth can be suspected. Indeed, for the “main resonance” $T_1=T_2=4\pi$, this can be directly proven, choosing e.g.,$V({\theta }_1,{\theta }_2)=kcos({\theta }_1)cos({\theta }_2)$ like in [13]. The upper bound (12) holds (in this case, this bound can also be checked by a simple direct calculation), and a lower bound is provided by the following result:

Proposition 6.

For the QKR model with $T_1=T_2=4\pi$ and $V({\theta }_1,{\theta }_2)=kcos({\theta }_1)cos({\theta }_2)$ and initial state given in coordinate representation by $\check{w}({\theta }_1,{\theta }_2)=$ const.$={(2\pi )}^{-1}$:

$$\forall \theta \in [1,2],{\mathcal{S}}_{\theta }({\widehat{U}}^{t}w)>{\displaystyle \frac{1}{2}}ln(t)+O(1)\mathrm{as}t\to \infty .$$

Proof. 

Thanks to monotonicity of Renyi entropies, it is sufficient to prove the thesis for $\theta =2$. A well known characterization of the Schmidt entropy ${\mathcal{S}}_2$ is instrumental:

$${\mathcal{S}}_2(w_t)=-ln(\mathrm{Tr}({\widehat{\rho }}_t^2))$$

(15)

where ${\rho }_t$ is the reduced state, and $\mathrm{Tr}({\widehat{\rho }}_t^2)$ is its “purity”:

$$\mathrm{Tr}({\widehat{\rho }}_t^2):=\sum _{r,s,n,m}{w}_t(n,r,A)w_t^{*}(m,r,A)w_t^{*}(n,s,A)w_t(m,s,A).$$

(16)

where A is an arbitrary bipartite base. With the chosen initial state, the state at time t is, in the coordinate representation, $(w_t)({\theta }_1,{\theta }_2)={(2\pi )}^{-1}exp(\mathit{ı}tkcos({\theta }_1)cos({\theta }_2))$. Choosing as A the double Fourier base and implementing the integral representation of the Bessel functions $J_n(.)$ of first kind and integer order (16) yields

$$\mathrm{Tr}({\widehat{\rho }}_t^2)={\int }_0^{2\pi }{\int }_0^{2\pi }d\theta d{\theta }^{\prime }{J}_0^2\left(kt(cos(\theta )-cos({\theta }^{\prime }))\right)$$

(17)

For $ϵ>0$, let $B_{ϵ}:=\{(\theta ,{\theta }^{\prime })\in (0,2\pi )\times (0,2\pi ):|cos(\theta )-cos({\theta }^{\prime })|<ϵ\}$; then, let the integral in (17) be split in the sum of the integrals over $B_{ϵ}$ and its complement. The former integral can be estimated using $J_0^2\le 1$, and the latter by means of the large argument asymptotics of Bessel functions for $x\to +\infty$:

$$J_0^2(x)\sim {\displaystyle \frac{1}{\pi x}}(1+sin(2x))+o(x^{-1})$$

Doing so, one obtains

$$\mathrm{Tr}({\widehat{\rho }}_t^2)<c_1ϵ+c_2{(tϵ)}^{-1}+o(t^{-1})$$

with $c_1,c_2$ positive constants. This estimate is optimized in the leading order by choosing $ϵ={(c_2/c_{1}t)}^{1/2}$:

$$\mathrm{Tr}({\widehat{\rho }}_t^2)<2\sqrt{c_1{c}_2}{t}^{-1/2}+o(t^{-1}).$$

The proof is concluded on replacing this estimate in (15). □

2.3. Entangling with Clocks

Let a quantum system with Hilbert space ${\mathcal{H}}_1$ evolve in continuous time under a time-periodic Hamiltonian $\widehat{H}(t)$, with period T. Its dynamics can be imbedded in the autonomous dynamics of a larger system. This “extended Hilbert space method” [16,17] can be depicted as the coupling to a fictitious system, which acts as a timer. The time t in the original Hamiltonian is replaced by an observable $\tau$ of the other system, whose internal dynamics are chosen such that this observable increases exactly like time. A system behaving like that is just a clock that measures time modulo T. It can be thought of as a rotor, i.e., a particle moving on a circle at a constant speed. The Hilbert space of the clock is ${\mathcal{H}}_2=L^2({\mathbb{T}}_T)$, where ${\mathbb{T}}_T:=\mathbb{R}/(T\mathbb{Z})$ can be identified with the circle. As a Hamiltonian of the clock, one chooses ${\widehat{H}}_c:=-i\partial /\partial \tau$, with periodic boundary conditions at $\tau =0,T$. It is linear and not quadratic in momentum because the beat of the clock is not supposed to depend on its state. The (system+clock) Hilbert space is then $\mathcal{H}={\mathcal{H}}_1\otimes {\mathcal{H}}_2$ with ${\mathcal{H}}_2=L^2({\mathbb{T}}_T)$, and the complete Hamiltonian is the Floquet Hamiltonian ${\widehat{\mathbb{H}}}_F={\widehat{H}}^c+\widehat{H}(\tau )$, which generates a unitary group ${\widehat{\mathbb{U}}}_F(t)=exp(\mathit{ı}t{\widehat{\mathbb{H}}}_F)$. A state $w\in \mathcal{H}$ is conveniently figured as a function $\mathbf{w}:{\mathbb{T}}_T\to {\mathcal{H}}_1$, and then,

$$({\widehat{\mathbb{U}}}_F(T)\mathbf{w})(\tau )=\widehat{U}(\tau ,\tau +T)\mathbf{w}(\tau ).$$

(18)

This equation connects the Floquet propagator (left) to the family of one-period propagators of the original time-dependent Hamiltonians (right). Since $\widehat{U}(0,T)={\widehat{U}}^{†}(0,\tau )\widehat{U}(\tau ,\tau +T)\widehat{U}(0,\tau )$, all one-period propagators are unitarily equivalent to one another, so they share the same spectrum; hence, so does the Floquet propagator, with infinite multiplicity.

A convenient testing ground is provided by the QKR model: a quantum rotor, subjected to a time-periodic sequence of $\delta$-kicks. The singular nature of the driving commands care in defining the domain of the Floquet Hamiltonian, but this is not necessary for the present purposes. Here, ${\mathcal{H}}_1=L^2(S^1)$, and the bipartite base is the double Fourier base: $u_{n_1}^{(1)}(\theta )={(2\pi )}^{-1/2}exp(\mathit{ı}{n}_1\theta )$, $u_{n_2}^{(2)}(\tau )=T^{-1/2}exp(2\pi \mathit{ı}{n}_2\tau /T)$. The family of one-period propagators is, in this case,

$$\widehat{U}(\tau ,\tau +T)=\widehat{R}(\tau )\widehat{K}\widehat{R}(T-\tau )$$

(19)

where $\widehat{K}=exp(-\mathit{ı}kcos(\widehat{\theta }))$, and $\langle u_n^{(1)}|\widehat{R}(.)|u_m^{(1)}\rangle ={\delta }_{nm}exp(-\mathit{ı}{n}^2(.)/2)$.

Proposition 7.

The asymptotic estimate (12) holds true for the system QKR+clock, with $\widehat{U}$ the one-period Floquet propagator.

Proof. 

This result does not follow from Proposition 12, because the hypothesis is violated, as will be apparent in a few lines. With t a nonnegative integer,

$${\mathbf{w}}_t(\tau )=\widehat{U}(\tau ,\tau +tT){\mathbf{w}}_0(\tau )=\widehat{R}(\tau ){(\widehat{U}(0,T))}^t\widehat{R}(-\tau ){\mathbf{w}}_0(\tau ).$$

With the choice $w_0(\theta ,\tau )=$ const $={(2\pi T)}^{-1}$,

$$|w_t(n_1,n_2){|}^2=\frac{1}{2{\pi }^3}{\left|Q(n_1,t)\right|}^2{sin}^2(n_1^{2}T/4){(n_2+n_1^{2}T/(4\pi ))}^{-2},$$

where $Q(n,t):=\langle u_n^{(1)}|{(\widehat{K}\widehat{R}(T))}^t|u_0^{(1)}\rangle$. Next, for M a positive integer, let $B_M:=\{(n_1,n_2)\in {\mathbb{Z}}^2:|n_1|\le M,D(n_1,n_2)\le M\}$, where $D(n_1,n_2)=|n_2+n_1^{2}T/(4\pi )|$. This is a finite set of ${(2M+1)}^2$ points, and $M=M(ϵ,t)$ will now be chosen such that it supports at least $1-{ϵ}^2$ of the probability distribution $|w_t(n_1,n_2){|}^2$. To this end,

$$P(B_M^c)\le \sum _{n_1>M}|Q(n_1,t){|}^2+\frac{1}{2{\pi }^3}\sum _{|n_1|\le M}{\left|Q(n_1,t)\right|}^2(\sum _{D(n_1,n_2)>M}D{(n_1,n_2)}^{-2}).$$

The ballistic bound on the QKR dynamics (see, e.g., [12]) entails that propagation on the Fourier base is not faster than quadratic in time. So, the Chebyshev inequality can be implemented on the first term on the rhs, leading to

$$P(B_M^c)\le c_1{t}^2{M}^{-2}+c_2{M}^{-1}$$

where $c_1,c_2$ are numerical constants. Therefore, if $M^2\ge M(ϵ,t)\approx 2c_2{t}^2/{ϵ}^2$ and t is sufficiently large, then $B_M$ is a $ϵ$-support for the distribution $|w_t(n_1,n_2){|}^2$, so $N_{ϵ}(w_t)\le ♯(B_M)\le {(2M(ϵ,t)+1)}^2$, and the thesis follows from Proposition 8. □

If T is sufficiently incommensurate to $2\pi$, then dynamical localization occurs in the QKR, so $\widehat{U}(0,T)$ has a pure point spectrum. The same is true of ${\widehat{\mathbb{U}}}_F(T)$, and Proposition 2 enforces boundedness of Renyi entropies in the discrete time evolution. At resonance, the period T is commensurate to $2\pi$, and then the spectrum is absolutely continuous. For the main resonance $T=4\pi$, the evolution of ${\mathcal{S}}_2(w_t)$ can be explicitly computed, leading to a lower bound similar to Proposition 6. The proof is similar though not identical to the proof of that Proposition and is omitted here.

Entanglement of the rotor with the clock, which is basically a classical dynamical system, may appear sort of eccentric.

2.4. Dynamical Chaos

For quantum bipartite systems which are chaotic in the classical limit, quasi-classical approximations have proven remarkably successful in exposing linear growth of entanglement entropies [3] over intermediate time scales. The smaller ℏ is, the longer such quasi-classically linear eras are; however, they eventually come to an end due to quantum interference, so the asymptotics of quantum entanglement as $t\to \infty$ are typically out of the reach of quasi-classical approximations. For the double QKR model studied in this paper, the asymptotic regime is dominated by purely quantum effects: dynamical localization and quantum resonances, which have no classical counterpart and are in fact decided by the arithmetic type of the effective Planck constant. Dynamical localization and ballistic bounds are fairly general occurrences in the quantum mechanics of finite systems, so not faster than logarithmic asymptotic growth of entanglement entropy, as established in the present paper, has some generality, too. This is not the case with separability entropies of classical dynamical systems. Though such entropies are not in the main focus of the present paper, it is worth recalling that in a 1997 paper by Pattanayak et al. [18], it was shown that, for a chaotic dynamical system defined by a group of diffeomorphisms over a differentiable manifold, the evolving phase space distributions acquire structure on smaller and smaller scales, at an exponential rate typically larger than the maximal Lyapunov exponent. In the case of a bipartite system, this implies that the corresponding distribution over wave numbers (e.g., over bipartite Fourier bases) includes exponentially larger wave numbers. According to general arguments in the present paper, this may result in asymptotic linear growth of the separability entropy.

3. Discussion

Most investigations of the time evolution of EE in bipartite systems, when the total system is in a pure state, use the definition of EE as von Neumann entropy of the reduced density matrix that is obtained by tracing out one subsystem. In this paper, its alternative definition is used, based on the Schmidt decomposition theorem. This is also the definition of the classical separability entropy. Supplemented with a minimum property of the Schmidt base, this definition circumvents the need for partial traces and makes the search for upper estimates simpler, on the border of triviality. For instance, it allows for a simple yet exact proof that Renyi entropies are bounded in time whenever the full system dynamics are localized; and it makes it likewise obvious that linear growth is only possible in the presence of unbounded, exponentially fast expansion over bipartite bases. Whether the Schmidt definition will grant easier access to lower asymptotic bounds is an open question: in the present paper, lower bounds have been worked out only for special models, resorting to the partial trace definition. No application has been given to the classical separability entropy. The examples of unbounded EE growth which are presented in this paper bear no relation to classical chaotic diffusion and are due to QKR resonances, which are purely quantum effects with no classical counterpart.

In closing, it should be stressed that (i) the presented analysis only applies to systems (quantum or classical) with a finite number of degrees of freedom; and not to infinite systems, such as spin chains or the like, which are frequently used in studies of quantum entanglement; and that (ii) the interacting subsystems can have any finite number of degrees of freedom, even though in the presented examples, they are “single-body” systems.