# Remarks on the rate of linear vortex symmetrization

Hao Jia  University of Minnesota<EMAIL_ADDRESS>
###### Abstract.

We reformulate results from [9] in simplified forms and derive rigorously the
bounds given in Bassom and Gilbert [1], which provided interesting insights on
the vortex symmetrization phenomenon.

HJ is supported in part by NSF grant DMS-1945179.

##  1\. Introduction and main results

The dynamics of two dimensional incompressible fluid flows is often dominated
by the interaction of vortices, through symmetrization of small perturbations
around a vortex and merging of same sign vortices. These processes are
believed to play a significant role in reverse cascade of energy from small to
large scales in two dimensional flows, which is often observed in numerical
simulations. There is an extensive literature studying the dynamics of vortex
solutions to two dimensional Euler and Navier Stokes equations with small
viscosity, from both applied math and physics perspectives. See for example
[1, 4, 6, 12].

To state our motivations and results precisely, consider a steady state
(vortex) given by a radial vorticity field $\Omega(r)$, with
$\Omega^{\prime}(r)>0$ for $r>0$ using the polar coordinate $(r,\theta)$. Let
$U(r)e_{\theta}$ be the associated velocity field. Then from Biot-Savart law,
$U(r)$ satisfies

$\partial_{r}U(r)+U(r)/r=\Omega(r),\,\,r\in(0,\infty),\quad{\rm
with}\,\,\lim_{r\to 0}U(r)=\lim_{r\to\infty}U(r)=0.$ (1.1)

The linearized two dimensional Euler equation around $U(r)e_{\theta}$ is

$\partial_{t}\omega+\frac{U(r)}{r}\partial_{\theta}\omega-\frac{\Omega^{\prime}(r)}{r}\partial_{\theta}\psi=0,\qquad(\theta,r)\in\mathbb{T}\times(0,\infty),$
(1.2)

with initial data $\omega_{0}(\theta,r)$. The stream function $\psi$ is
determined through

$\left(\partial_{rr}+r^{-1}\partial_{r}+r^{-2}\partial_{\theta}^{2}\right)\psi=\omega,\qquad(\theta,r)\in\mathbb{T}\times(0,\infty).$
(1.3)

We assume the orthogonality conditions

$\int_{\mathbb{R}^{2}}\omega_{0}(\theta,r)(\cos{\theta},\sin{\theta})\,r^{2}drd\theta=0\qquad{\rm
and}\qquad\int_{\mathbb{T}}\omega_{0}(\theta,r)\,d\theta=0.$ (1.4)

The general case can be reduced to the case with the orthogonality conditions
(1.4) by re-centering a slightly modified $\Omega(r)$.

Taking Fourier transform in $\theta$ in the equation (1.2)-(1.3) for $\omega$,
we obtain that

$\partial_{t}\omega_{k}+ik\frac{U(r)}{r}\omega_{k}-ik\frac{\Omega^{\prime}(r)}{r}\psi_{k}=0,\quad(\partial_{r}^{2}+r^{-1}\partial_{r}-k^{2}r^{-2})\psi_{k}=\omega_{k},$
(1.5)

for $k\in\mathbb{Z},t\geq 0,r\in(0,\infty)$, with initial data
$\omega_{0}^{k}$. Here $\omega_{k}$ and $\psi_{k}$ are the $k-$th Fourier
coefficients of $\omega$ and $\psi$. Below we assume that $k\in[1,\infty)$
unless otherwise specified, since the zero mode does not evolve and the case
$k\in(-\infty,-1]$ can be studied similarly using a complex conjugation.

Inspired by recent breakthroughs on asymptotic stability near shear flows for
$2d$ Euler equations, see [2] on Couette flow and more recently [10, 11] on
general stable monotonic shear flow, precise results on the linearized
dynamics around vortex solutions have been obtained in [3, 9, 13]. See also
[7] for a result on nonlinear vortex symmetrization near a point vortex. These
results show that the velocity and stream fields decay over time with explicit
rates, and the vorticity field weakly converges to a radial function, a
phenomenon often termed “vortex symmetrization”. One can also view the
linearized asymptotic results as an essential step towards a proof of full
nonlinear vortex symmetrization, a major open problem in 2d Euler equations.

Linearized stability analysis has been studied, however, much earlier by
Bassom and Gilbert [1]. It appears that, despite recent progresses, the
results obtained in [1], using matched asymptotics method and further verified
by numerical computations, have not been fully proved.

There are two key predictions in [1], which we reformulate slightly as
follows.

  1. (1)

We can decompose the vorticity field for $r>0,t>1$ as

$\omega_{k}(t,r)=f_{0k}(t,r)e^{-ikU(r)t/r}+f_{1k}(t,r),$ (1.6)

where $f_{0k}(t,r)$ vanishes as $r\to 0+$ to the order $r^{\sqrt{k^{2}+8}}$
and $f_{1k}(t,r)$ decays over time with the rate
$r^{-2-\sqrt{k^{2}+8}}t^{-1-\sqrt{k^{2}+8}}$. See (5.14a) in [1].

  2. (2)

For any $\eta\in C_{0}^{\infty}(\mathbb{R}^{2})$, denoting $\eta_{k}(r)$ as
the $k-$th mode of $\eta$ in $\theta$, we have for $t>1$,

$\int_{0}^{\infty}\omega_{k}(t,r)\eta_{k}(r)\,rdr\lesssim
t^{-1-\sqrt{k^{2}+8}}.$ (1.7)

See figure 5 in [1] and the discussion below it.

The predicted decay rates over time in (1.6)-(1.7) are rather strong, while
the existing results in [3, 9] only provide rigorous decay rates given by
$1/t$.

It turns out that one can derive (1.6)-(1.7) from the bounds on spectral
density functions proved in [9] by Ionescu and the author. The paper [9]
however focused its attention more on the optimal Gevrey regularity bounds,
aiming at possible applications in nonlinear vortex symmetrization problem,
and due to various technical reasons including the need to work with very
precise Gevrey and weighted spaces, some of the estimates are quite involved.
In this note we reformulate the main results in [9] in a more concise way
(without insisting on the optimality of bounds at the Gevrey level), and
rigorously justify the formulae (1.6) and (1.7) as a consequence of [9].

To simplify our presentations as much as possible, we allow our implied
constants to depend on the Fourier mode $k$. Our main assumptions on the
background vorticity field $\Omega$ are the following.

######  Assumption 1.1.

We assume that the background radial vorticity profile $\Omega(r)$ satisfies
the following natural conditions. There exist constants
$C_{\ast}\in(0,\infty)$ and $c_{\ast}\in(-\infty,0)$ such that for all
$r\in(0,\infty)$ and $j\in\mathbb{Z}\cap[0,\infty)$ we have

$\begin{split}&0<\Omega(r)\leq\frac{C_{\ast}}{\langle
r\rangle^{6}},\quad\partial_{r}\Omega(r)<0,\quad\Big{|}(r\partial_{r})^{j}\Big{(}\frac{\Omega^{\prime}(r)}{r}-\frac{c_{\ast}}{\langle
r\rangle^{8}}\Big{)}\Big{|}\leq\frac{C_{\ast}^{j+1}(j!)^{2}}{\langle
r\rangle^{10}}r^{2}.\end{split}$ (1.8)

In the above, we used the notation $\langle x\rangle:=\sqrt{x^{2}+2}$ for
$x\in\mathbb{R}$, and the convention that $0!=1$.

We can now state our main result as follows.

######  Theorem 1.2.

Assume that $k\in\mathbb{Z}\cap[1,\infty)$, and that $\omega_{0k}(r)$ is
smooth for $r\in(0,\infty)$ and satisfies the Gevrey regularity bounds in
$v=\log r\in\mathbb{R}$, in (2.20) and (2.26)-(2.27) below. Let
$\omega_{k}(t,r)\in C^{\infty}([0,\infty)\times(0,\infty))$ be the unique
solution to (1.5) with initial data $\omega_{0k}$. Then we have the following
conclusions.

  1. (1)

We can decompose the vorticity field for $r>0,t>1$ as

$\omega_{k}(t,r)=f_{k\rm loc}(t,r)e^{-ikU(r)t/r}+f_{k\rm nloc}(t,r),$ (1.9)

where $f_{k\rm loc}(t,r)$ satisfies the bounds for $r>0,t>1$ and
$j\in[0,\infty)$,

$\big{|}(r\partial_{r})^{j}f_{k\rm
loc}(t,r)\big{|}\lesssim_{j,k}M_{k}\frac{r^{\sqrt{k^{2}+8}}}{1+r^{|k|+8+\sqrt{k^{2}+8}}},$
(1.10)

and $f_{k\rm nloc}(t,r)$ satisfies the bounds for $r>0,t>1$ and
$j\in[0,\infty)$,

$\begin{split}&\big{|}(r\partial_{r})^{j}f_{k\rm nloc}(t,r)\big{|}\\\
&\lesssim_{j,k}M_{k}\frac{\big{(}r+r^{-1}\big{)}^{2}}{t}\min\bigg{\\{}\big{(}r+r^{-1}\big{)}^{2\sqrt{k^{2}+8}}t^{-\sqrt{k^{2}+8}},\,\,1\bigg{\\}}\frac{r^{\sqrt{k^{2}+8}}}{1+r^{|k|+\sqrt{k^{2}+8}+8}};\end{split}$
(1.11)

  2. (2)

For any $\eta\in C_{0}^{\infty}(\mathbb{R}^{2})$, with
$\eta(r,\theta)=O(r^{k})$ as $r\to 0$, we have for $t>1$,

$\Big{|}\int_{0}^{\infty}\omega_{k}(t,r)\eta_{k}(r)\,rdr\Big{|}\lesssim_{j,k}M_{k}\,t^{-1-\sqrt{k^{2}+8}}.$
(1.12)

In the above, $M_{k}$ is the weighted Gevrey-2 norm of $\omega_{0k}$ in the
variable $v=\log r$, given by (2.26)-(2.27). The implied constants depend on
the constants $C_{\ast},c_{\ast}$ in (1.8) and the constants from
(2.26)-(2.27) in the definition of the Gevrey norms, as well as a structure
constant associated with the linearized operator around $\Omega$, but are
independent of $t>1,r\in(0,\infty)$.

The assumptions (2.26)-(2.27) on the initial data are basically that
$\omega_{0k}(r)$ is Gevrey-2 regular in $v=\log r$, that
$\omega_{0k}(r)=\sigma_{k}r^{k}+O(r^{k+2})$ as $r\to 0+$, and that
$\omega_{0k}(r)=O(r^{-k-8})$ as $r\to+\infty$.

Theorem 1.2 provides a rigorous proof of the predictions (1.6)-(1.7) in [1].
Moreover, we have more precise information on the regularity of the “profiles”
$f_{k\rm loc}$ and $f_{k\rm nloc}$ of the vorticity field $\omega_{k}(t,r)$.

We note that one can also prove similar bounds to (1.10)-(1.11) with smooth
but non-Gevrey initial data. Less smooth, such as Sobolev regular, initial
data can also be considered with correspondingly weaker regularity control on
the profiles. We use Gevrey regularity in our presentation since these results
are already proved in [9] and can be used without additional proof.

There is a similar vorticity depletion phenomenon for non-monotonic shear
flows near critical points, investigated first in [5], and established
rigorously in [14] with sharp decay over time at the critical point for
symmetric flows and qualitative bounds for general non-monotonic shear flows.
A more quantitative version of the vorticity depletion for general non-
monotonic shear flows was proved in [8]

The rest of the paper is organized as follows. In section 2 we recall some
general results on (1.5) and essential estimates we need from [9], and then we
give the proof of Theorem 1.2 in section 3.

##  2\. Simplified bounds on the spectral density function

In this section, we introduce the spectral representation formula and recall
the main bounds on the spectral density functions proved in [9] in simplified
forms.

Define for $r\in\mathbb{R}^{+}:=(0,\infty)$,

$b(r):=U(r)/r,\qquad d(r)=\Omega^{\prime}(r)/r.$ (2.1)

It follows from (1.1), (1.8) and (2.1) that for $r\in\mathbb{R}^{+}$,

$b(r)=U(r)/r=\int_{0}^{1}s\,\Omega(rs)\,ds\approx\frac{1}{\langle
r\rangle^{2}},\qquad b^{\prime}(r)\approx-\frac{r}{\langle
r\rangle^{4}},\qquad|b^{\prime\prime}(r)|\lesssim\frac{1}{\langle
r\rangle^{4}}.$ (2.2)

In the above the implied constants depend only on the constants $c_{\ast}$ and
$C_{\ast}$ in (1.8).

Define for $k\in\mathbb{Z}\backslash\\{0\\},r,\rho\in(0,\infty)$, the function

$G_{k}(r,\rho):=\left\\{\begin{array}[]{ll}\frac{\rho}{2|k|}\Big{(}\frac{r}{\rho}\Big{)}^{|k|}&\qquad{\rm
for}\,\,r<\rho,\\\
\frac{\rho}{2|k|}\Big{(}\frac{\rho}{r}\Big{)}^{|k|}&\qquad{\rm
for}\,\,r\geq\rho.\end{array}\right.$ (2.3)

$G_{k}(r,\rho)$ is the Green’s function for the differential operator
$-\partial_{r}^{2}-r^{-1}\partial_{r}+k^{2}r^{-2}$ on $(0,\infty)$ with
vanishing boundary conditions.

For each $k\in\mathbb{Z}\backslash\\{0\\}$, we set for any $f\in
X:=L^{2}\big{(}\mathbb{R}^{+},rdr\big{)}$,

$L_{k}f(r):=\frac{U(r)}{r}f(r)+\frac{\Omega^{\prime}(r)}{r}\int_{0}^{\infty}G_{k}(r,\rho)f(\rho)\,d\rho.$
(2.4)

The equation (1.5) can be reformulated as

$\partial_{t}\omega_{k}+ikL_{k}\omega_{k}=0.$ (2.5)

Since $L_{k}$ is a compact perturbation of the simple multiplication operator
$f\to b(r)f$, by general spectral theory, we can conclude that the spectrum of
$L_{k}$ consists of the continuous spectrum $[0,b(0)]$ and possibly some
discrete eigenvalues in $\mathbb{R}$. Our main assumptions (1.8)-(2.17) imply
that there are no discrete eigenvalues.

By standard theory of spectral projections, we then have

$\begin{split}&\omega_{k}(t,r)=\frac{1}{2\pi i}\lim_{\epsilon\to
0+}\int_{\mathbb{R}}e^{-ik\lambda t}\Big{\\{}\left[(\lambda-i\epsilon-
L_{k})^{-1}-(\lambda+i\epsilon-
L_{k})^{-1}\right]\omega^{k}_{0}\Big{\\}}(r)\,d\lambda\\\ &=\frac{-1}{2\pi
i}\lim_{\epsilon\to
0+}\int_{0}^{\infty}e^{-ikb(r_{0})t}b^{\prime}(r_{0})\Big{\\{}\big{[}(-b(r_{0})-i\epsilon+L_{k})^{-1}-(-b(r_{0})+i\epsilon+L_{k})^{-1}\big{]}\omega^{k}_{0}\Big{\\}}(r)\,dr_{0}.\end{split}$
(2.6)

We then obtain from (1.5) that

$\begin{split}\psi_{k}(t,r)&=\frac{1}{2\pi i}\lim_{\epsilon\to
0+}\int_{0}^{\infty}e^{-ikb(r_{0})t}b^{\prime}(r_{0})\int_{0}^{\infty}G_{k}(r,\rho)\\\
&\qquad\qquad\times\bigg{\\{}\Big{[}(-b(r_{0})-i\epsilon+L_{k})^{-1}-(-b(r_{0})+i\epsilon+L_{k})^{-1}\Big{]}\omega^{k}_{0}\bigg{\\}}(\rho)\,d\rho\,dr_{0}\\\
&=\frac{1}{2\pi i}\lim_{\epsilon\to
0+}\int_{0}^{\infty}e^{-ikb(r_{0})t}b^{\prime}(r_{0})\left[\psi_{k,\epsilon}^{-}(r,r_{0})-\psi_{k,\epsilon}^{+}(r,r_{0})\right]dr_{0}.\end{split}$
(2.7)

In the above,

$\begin{split}&\psi_{k,\epsilon}^{+}(r,r_{0}):=\int_{0}^{\infty}G_{k}(r,\rho)\Big{[}(-b(r_{0})+i\epsilon+L_{k})^{-1}\omega^{k}_{0}\Big{]}(\rho)\,d\rho,\\\
&\psi_{k,\epsilon}^{-}(r,r_{0}):=\int_{0}^{\infty}G_{k}(r,\rho)\Big{[}(-b(r_{0})-i\epsilon+L_{k})^{-1}\omega^{k}_{0}\Big{]}(\rho)\,d\rho.\end{split}$
(2.8)

We note that $\psi_{k,\epsilon}^{+}(r,r_{0}),\psi_{k,\epsilon}^{-}(r,r_{0})$
satisfy for $\iota\in\\{+,-\\}$ and $r,r_{0}\in\mathbb{R}^{+}$,

$\Big{[}k^{2}/r^{2}-r^{-1}\partial_{r}-\partial_{r}^{2}\Big{]}\psi_{k,\epsilon}^{\iota}(r,r_{0})+\frac{d(r)}{b(r)-b(r_{0})+i\iota\epsilon}\psi_{k,\epsilon}^{\iota}(r,r_{0})=\frac{\omega^{k}_{0}(r)}{b(r)-b(r_{0})+i\iota\epsilon}.$
(2.9)

It is more convenient to work in the variable $v$ with $r=e^{v}$ for
$r\in(0,\infty)$. We therefore introduce for
$k\in\mathbb{Z}\backslash\\{0\\},\epsilon\in[-1/4,1/4]\backslash\\{0\\}$ and
$\iota\in\\{+,-\\}$,

$\Pi^{\iota}_{k,\epsilon}(v,w):=\psi_{k,\epsilon}^{\iota}(r,r_{0}),\qquad{\rm
where}\,\,r=e^{v},\,\,r_{0}=e^{w}\,\,{\rm and}\,\,r,r_{0}\in(0,\infty),$
(2.10) $B(v):=b(r),\qquad D(v):=d(r),\qquad{\rm
where}\,\,r=e^{v},\,\,r\in(0,\infty).$ (2.11)

We also define

$f_{0}^{k}(v):=\omega_{0}^{k}(r),\quad\phi_{k}(t,v):=\psi_{k}(t,r),\qquad{\rm
where}\,\,r=e^{v},\,\,r\in(0,\infty).$ (2.12)

It follows from (2.7)-(2.12) that for $\iota\in\\{+,-\\},v,w\in\mathbb{R}$,
$k\in\mathbb{Z}\backslash\\{0\\}$, $\epsilon\in[-1/4,1/4]$,

$k^{2}\Pi^{\iota}_{k,\epsilon}(v,w)-\partial_{v}^{2}\Pi^{\iota}_{k,\epsilon}(v,w)+e^{2v}D(v)\frac{\Pi^{\iota}_{k,\epsilon}(v,w)}{B(v)-B(w)+i\iota\epsilon}=e^{2v}\frac{f_{0}^{k}(v)}{B(v)-B(w)+i\iota\epsilon},$
(2.13)

and the normalized stream function has the representation formula

$\phi_{k}(t,v)=\frac{1}{2\pi i}\lim_{\epsilon\to
0+}\int_{\mathbb{R}}e^{-ikB(w)t}\partial_{w}B(w)\left[\Pi_{k,\epsilon}^{-}(v,w)-\Pi_{k,\epsilon}^{+}(v,w)\right]dw.$
(2.14)

The key is to study the regularity properties of the associated spectral
density functions $\Pi_{k,\epsilon}^{-}(v,w)$ and $\Pi_{k,\epsilon}^{+}(v,w)$
for $v,w\in\mathbb{R}$, and $\epsilon\in[-1/4,1/4]\backslash\\{0\\}$ small.

We summarize our calculations in the following proposition.

######  Proposition 2.1.

Suppose $\Omega(r),r\in\mathbb{R}^{+}$, is a radial function satisfying the
assumption (1.8). Let $U(r),r\in\mathbb{R}^{+}$, be given through

$\partial_{r}U(r)+U(r)/r=\Omega(r),\,\,r\in(0,\infty),\quad{\rm
with}\,\,\lim_{r\to 0}U(r)=\lim_{r\to\infty}U(r)=0.$ (2.15)

Consider $\Omega(r)$ as the steady vorticity profile for the two dimensional
incompressible Euler equation with the associated velocity field
$U(r)e_{\theta}$. The linearized equations around $\Omega$ for the
perturbation vorticity $\omega(t,\theta,r)\in
C^{1}([0,\infty),H^{1}(\mathbb{T}\times\mathbb{R}^{+},r\langle
r\rangle^{8}d\theta dr))$ and the associated stream function
$\psi(t,\theta,r)$, $t\geq 0,\theta\in\mathbb{T},r\in\mathbb{R}^{+}$ are given
by

$\begin{split}&\partial_{t}\omega+\frac{U(r)}{r}\partial_{\theta}\omega-\frac{\Omega^{\prime}(r)}{r}\partial_{\theta}\psi=0,\qquad\left(\partial_{rr}+r^{-1}\partial_{r}+r^{-2}\partial_{\theta}^{2}\right)\psi=\omega.\end{split}$
(2.16)

We assume that the initial vorticity deviation $\omega_{0}(\theta,r)$
satisfies the orthogonality conditions

$\int_{\mathbb{R}^{2}}\omega_{0}(\theta,r)(\cos{\theta},\sin{\theta})\,r^{2}drd\theta=0,\qquad{\rm
and\,\,for\,\,}r\in\mathbb{R}^{+},\,\,\int_{\mathbb{T}}\omega_{0}(\theta,r)\,d\theta=0.$
(2.17)

For $k\in\mathbb{Z}\backslash\\{0\\}$ and $t\geq 0,r\in\mathbb{R}^{+}$,
letting

$\omega_{k}(t,r):=\frac{1}{2\pi}\int_{\mathbb{T}}\omega(t,\theta,r)e^{-ik\theta}\,d\theta,\quad\psi_{k}(t,r):=\frac{1}{2\pi}\int_{\mathbb{T}}\psi(t,\theta,r)e^{-ik\theta}\,d\theta,$
(2.18)

then $\omega_{k}(t,r),\psi_{k}(t,r)$ satisfy for $t\geq 0,r\in\mathbb{R}^{+}$

$\begin{split}&\partial_{t}\omega_{k}+ik\frac{U(r)}{r}\omega_{k}-ik\frac{\Omega^{\prime}(r)}{r}\psi_{k}=0,\quad\left(\partial_{rr}+r^{-1}\partial_{r}-k^{2}r^{-2}\right)\psi_{k}=\omega_{k}.\end{split}$
(2.19)

Define for $t\geq 0,r\in\mathbb{R}^{+},v\in\mathbb{R}$ with $r=e^{v}$, the
functions

$\begin{split}&b(r):=U(r)/r,\quad d(r):=\Omega^{\prime}(r)/r,\quad
B(v):=b(r),\quad D(v):=d(r),\\\ &f_{k}(t,v):=\omega_{k}(t,r),\quad
f_{0}^{k}(v):=\omega_{0}^{k}(r),\quad\phi_{k}(t,v):=\psi_{k}(t,r).\end{split}$
(2.20)

We have the representation formula for $t\geq 0,v\in\mathbb{R}$,

$\phi_{k}(t,v)=\frac{1}{2\pi i}\lim_{\epsilon\to
0+}\int_{\mathbb{R}}e^{-ikB(w)t}\partial_{w}B(w)\left[\Pi_{k,\epsilon}^{-}(v,w)-\Pi_{k,\epsilon}^{+}(v,w)\right]dw,$
(2.21)

where the spectral density functions $\Pi_{k,\epsilon}^{\iota}(v,w)$ satisfy
for $\iota\in\\{+,-\\},v,w\in\mathbb{R}$, and
$\epsilon\in[-1/8,1/8]\backslash\\{0\\}$

$k^{2}\Pi^{\iota}_{k,\epsilon}(v,w)-\partial_{v}^{2}\Pi^{\iota}_{k,\epsilon}(v,w)+e^{2v}D(v)\frac{\Pi^{\iota}_{k,\epsilon}(v,w)}{B(v)-B(w)+i\iota\epsilon}=\frac{e^{2v}f_{0}^{k}(v)}{B(v)-B(w)+i\iota\epsilon}.$
(2.22)

We also record the following bounds for later applications. There exists
$C^{\ast}\in(0,\infty)$, depending on $C_{\ast},c_{\ast}$ in (1.8) such that
for $v\in\mathbb{R}$ and $j\in\mathbb{Z}\cap[1,\infty)$,

$\begin{split}&\Big{|}D(v)-\frac{c_{\ast}}{(1+e^{2v})^{4}}\Big{|}\leq
C^{\ast}\frac{e^{2v}}{(1+e^{2v})^{5}},\quad\big{|}\partial^{j}_{v}D(v)\big{|}\leq(C^{\ast})^{j}(j!)^{2}\frac{e^{2v}}{(1+e^{2v})^{5}},\\\
&B(v)\approx\frac{1}{1+e^{2v}},\quad\partial_{v}B(v)\approx-\frac{e^{2v}}{(1+e^{2v})^{2}},\quad\big{|}\partial^{j}_{v}B(v)\big{|}\lesssim(C^{\ast})^{j}(j!)^{2}\frac{e^{2v}}{(1+e^{2v})^{2}},\end{split}$
(2.23) $\partial_{v}B(v)=(c_{\ast}/4)e^{2v}+O\big{(}e^{4v}\big{)}\qquad{\rm
for}\,\,v<0.$ (2.24)

Denote $\alpha\wedge\beta:=\min\\{\alpha,\beta\\}$ for
$\alpha,\beta\in\mathbb{R}$. Define for $v,w\in\mathbb{R}$ the function
$d(v,w)$ as

$d(v,w):=\big{|}\big{[}\min\\{v,w\\},\max\\{v,w\\}\big{]}\cap\big{[}\min\\{w,0\\},0\big{]}\big{|}.$
(2.25)

Fix $\Phi_{0}(v)\in C^{\infty}(-\infty,-1)$ such that $\Phi_{0}\equiv 1$ on
$(-\infty,-2]$. We choose also a smooth cutoff function
$\Phi^{\ast}:\mathbb{R}\to[0,1]$ satisfying $\Phi^{\ast}\in
C_{0}^{\infty}(-4,4)$ and $\Phi^{\ast}\equiv 1$ on $[-2,2]$.

We make the following assumptions on the initial data. Recall that
$c_{\ast}\in(-\infty,0)$ is from Assumption 1.1.

######  Assumption 2.2.

There exist coefficients $\sigma_{k}\in\mathbb{R}$ for each
$k\in\mathbb{Z}\backslash\\{0\\}$, and constants
$M^{\dagger}_{k}\in(2\sigma_{k},\infty),C_{0}>1$, such that the following
statement holds. Define for $k\in\mathbb{Z}\backslash\\{0\\}$ and
$v\in\mathbb{R}$,

$F_{0k}(v):=f_{0}^{k}(v)-(\sigma_{k}/c_{\ast})D(v)e^{|k|v}\Phi_{0}(v),$ (2.26)

then $F_{0k}$ satisfies the bounds for all $k\in\mathbb{Z}\backslash\\{0\\}$
and $v\in\mathbb{R}$,

$\big{|}\partial_{v}^{j}F_{0k}(v)\big{|}\leq
M_{k}C_{0}^{j}(j!)^{2}\frac{e^{(|k|+2)v}}{1+e^{(2|k|+10)v}}.\\\ $ (2.27)

Decompose for
$k\in\mathbb{Z}\backslash\\{0\\},\epsilon\in[-1/8,1/8]\backslash\\{0\\}$ and
$v,w\in\mathbb{R}$,

$\Pi_{k,\epsilon}^{\iota}(v,w):=(\sigma_{k}/c_{\ast})e^{|k|v}\Phi_{0}(v)+\Gamma_{k,\epsilon}^{\iota}(v,w).$
(2.28)

It follows from (2.22) that $\Gamma_{k,\epsilon}^{\iota}(v,w)$ satisfies the
equation for $v,w\in\mathbb{R},\epsilon\in[-1/8,1/8]\backslash\\{0\\},$
$\iota\in\\{+,-\\}$,

$\begin{split}&(k^{2}-\partial_{v}^{2})\Gamma_{k,\epsilon}^{\iota}(v,w)+\frac{e^{2v}D(v)}{B(v)-B(w)+i\iota\epsilon}\Gamma_{k,\epsilon}^{\iota}(v,w)\\\
&=\frac{e^{2v}F_{0k}(v)}{B(v)-B(w)+i\iota\epsilon}+(\sigma_{k}/c_{\ast})\big{(}2|k|e^{|k|v}\partial_{v}\Phi_{0k}(v)+e^{|k|v}\partial_{v}^{2}\Phi_{0k}(v)\big{)}.\end{split}$
(2.29)

The main results we need below are the following precise estimates on the
spectral density functions.

######  Theorem 2.3.

Assume that $k\in\mathbb{Z}\backslash\\{0\\}$, $f_{k}(t,v),\phi_{k}(t,v)$ for
$t\geq 0,v\in\mathbb{R}$ are as in Proposition 2.1, and that the assumption
2.2 holds. Recall the definition (2.25) for $d(v,w)$ for $v,w\in\mathbb{R}$.
Then we have the following conclusions. The limiting spectral density function

$\Gamma_{k}(v,w):=(-i)\lim_{\epsilon\to
0+}\big{[}\Gamma_{k,\epsilon}^{+}(v,w)-\Gamma_{k,\epsilon}^{-}(v,w)\big{]}=2\lim_{\epsilon\to
0+}\imaginary\,\Gamma_{k,\epsilon}^{+}(v,w)$ (2.30)

exists, as limit of functions in $L^{2}_{\rm loc}(\mathbb{R}^{2})$. Define for
$v,w\in\mathbb{R}$, the “profile” $\Theta_{k}(v,w)$ for $\Gamma_{k}(v,w)$,

$\Theta_{k}(v,w):=\Gamma_{k}(v+w,w).$ (2.31)

Then $\Gamma_{k}(v,w)$ and $\Theta_{k}(v,w)$ satisfy the following properties.

(i) Bounds when $v$ is away from $w$. For $v\in\mathbb{R}$ with $|v-w|\geq
1/2$ and $w\in\mathbb{R}$, $\Gamma(v,w)$ is smooth in $v,w$, and for any
$j_{1},j_{2}\in\mathbb{Z}\cap[0,\infty)$,

$\begin{split}&\big{|}\partial_{v}^{j_{1}}\partial_{w}^{j_{2}}\Gamma_{k}(v,w)\big{|}\lesssim_{j_{1},j_{2},k}M_{k}e^{-|k||w|-(\sqrt{k^{2}+8}-|k|)d(0,w)-4\max\\{w,0\\}}e^{-|k||w-v|-(\sqrt{k^{2}+8}-|k|)d(v,w)}.\end{split}$
(2.32)

(ii) Bounds when $v$ is close to $w$. For $w\in\mathbb{R}$ and
$j\in\mathbb{Z}\cap[0,\infty)$, we have

$\begin{split}&\Big{\|}\partial^{j}_{w}\Theta_{k}(v,w)\Big{\|}_{H^{1}(v\in[-2,2])}\lesssim_{j,k}M_{k}e^{-|k||w|-(\sqrt{k^{2}+8}-|k|)d(0,w)-4\max\\{w,0\\}}.\end{split}$
(2.33)

(iii) Equation for $\Gamma_{k}(v,w)$. In addition, $\Gamma_{k}(v,w)$ satisfies
for $v,w\in\mathbb{R}$ the equation

$(k^{2}-\partial_{v}^{2})\Gamma_{k}(v,w)+{\rm
P.V.}\frac{e^{2v}D(v)\Gamma_{k}(v,w)}{B(v)-B(w)}=-2\pi\frac{e^{2w}\big{(}D(w)\digamma_{k}(w)-F_{0k}(w)\big{)}}{\partial_{w}B(w)}\delta(v-w),$
(2.34)

where ${\rm P.V.}$ represents principal value and $\digamma_{k}\in
C^{\infty}(\mathbb{R})$ satisfies the bound for $w\in\mathbb{R}$ and
$j\in\mathbb{Z}\cap[0,\infty)$,

$\big{|}\partial_{w}^{j}\digamma_{k}(w)\big{|}\lesssim_{j,k}M_{k}e^{-|k||w|-(\sqrt{k^{2}+8}-|k|)d(w,0)}.$
(2.35)

(iv) Refined regularity property of $\Theta_{k}(v,w)$ in $v$. For any smooth
$\zeta:[-10,10]\to\mathbb{R}$ and $|\zeta^{\prime}|\gtrsim 1$ on $[-10,10]$,
define

$\mathcal{H}(v,\rho,w):=\Theta(\zeta(v+\rho)-\zeta(\rho),w),$ (2.36)

we have for $j_{1},j_{2}\in[0,\infty),\rho\in[-10,10]$, $w\in\mathbb{R}$,

$\begin{split}&\big{\|}\partial_{\rho}^{j_{1}}\partial_{w}^{j_{2}}\mathcal{H}(v,\rho,w)\big{\|}_{H^{1}(v\in[-1,1])}\lesssim_{j_{1},j_{2},k,\zeta}M_{k}e^{-|k||w|-(\sqrt{k^{2}+8}-|k|)d(w,0)-4\max\\{w,0\\}}.\end{split}$
(2.37)

(v) The explicitly solvable case $k=1$. Assume that $k=1$. For
$v,w\in\mathbb{R}$, we have

$\begin{split}\Gamma_{k}(v,w)&=2\pi\frac{B(v)-B(w)}{(\partial_{w}B(w))^{2}}e^{v+w}\mathbf{1}_{v<w}\bigg{\\{}f_{0}^{k}(w)-e^{-w}\frac{D(w)}{\partial_{w}B(w)}\int_{-\infty}^{w}f_{0}^{k}(\rho)e^{3\rho}\,d\rho\bigg{\\}}.\end{split}$
(2.38)

(vi) Representation formula for the stream function and vorticity function.
Finally, we have the representation formula

$\begin{split}&\phi_{k}(t,v)=-\frac{1}{2\pi}\int_{\mathbb{R}}e^{-ikB(w)t}\Gamma_{k}(v,w)\partial_{w}B(w)\,dw,\qquad
f_{k}(t,v)=-e^{-2v}(k^{2}-\partial_{v}^{2})\phi_{k}(t,v).\\\ \end{split}$
(2.39)

Theorem 2.3 is a simplified form of Theorem 1.4 in [9] (see also Proposition
6.1 in [9] for the formula (2.38)), with a stronger decay as $w\to+\infty$.
The stronger decay can be seen in the factor $e^{-4\max\\{w,0\\}}$ from
(2.32), (2.33) and (2.37). Although the stronger bounds were not stated
explicitly in [9] where it is not needed to get the slower $1/t$ decay, the
proof follows from the same arguments for the proof of Proposition 8.1. The
decay rate of the spectral density function $\Gamma_{k}(v,w)$, as
$w\to\infty$, depends naturally on the decay rate of the background vortex and
the initial data, and may decay very fast so that it is not a serious concern
for us. The order of vanishing as $w\to-\infty$, corresponding to $r\to 0+$
which ultimately leads to “vorticity depletion”, is a much more delicate
issue, due to the nonlocal term in the linearized equation (1.5), and is one
of the main questions that is addressed here and in [9], as well as in other
works [3, 13].

The bounds (2.32) and the representation formula (2.39) show that the main
contribution to the vorticity and stream functions $\omega_{k},\psi_{k}$ are
given by the spectral density function when the spectral parameter $w$ is
close to the physical space parameter $v$, as expected.

We refer the detailed proof of Theorem 2.3 to [9], and only highlight two
important observations used in the proof. The first key insight in the proof
is to notice that for $w<<-1$ and $v\in[w+2,-10]$, the potential term in
equation (2.29) satisfies

$\frac{e^{2v}D(v)}{B(v)-B(w)+i\iota\epsilon}\approx 8.$ (2.40)

The long range effect of the potential results in the faster decay described
(2.33) than $e^{-|k||w|}$ as $w\to-\infty$. The second key insight is to
notice that $\Gamma_{k,\epsilon}^{+}(v,w)$ and $\Gamma_{k,\epsilon}^{-}(v,w)$
have significant cancellations, resulting in an extra decay when $v$ is away
from $w$, as can be seen from the equation (2.34), which roughly speaking
shows that $\Gamma_{k}(v,w)$ is “generated” by the source term
$-2\pi\frac{e^{2w}\big{(}D(w)\digamma_{k}(w)-F_{0k}(w)\big{)}}{\partial_{w}B(w)}\delta(v-w)$.
The decay rates for $F_{0k}(w),D(w),\partial_{w}B(w)$ and $\digamma_{k}(w)$
imply that for $w>1$,

$\Big{|}\frac{e^{2w}\big{(}D(w)\digamma_{k}(w)-F_{0k}(w)\big{)}}{\partial_{w}B(w)}\Big{|}\lesssim
e^{-(|k|+4)|w|},$

which heuristically explains the extra decay factor $e^{-4\max\\{w,0\\}}$ that
we included in Theorem 2.3 in various bounds, in comparison with Theorem 1.4
of [9].

##  3\. Proof of the main results

We are now ready to give the proof of Theorem 1.2, using Theorem 2.3. Consider
first the bounds (1.10)-(1.11). It follows from the representation formula
(2.39) and equation (2.34) that for $t>1,v\in\mathbb{R}$,

$\begin{split}f_{k}(t,v)=&\frac{1}{2\pi}\int_{\mathbb{R}}e^{-ikB(w)t}{\rm
P.V.}\frac{D(v)\Gamma_{k}(v,w)}{B(v)-B(w)}\partial_{w}B(w)\,dw\\\
&+\Big{[}D(v)\digamma_{k}(v)-F_{0k}(v)\Big{]}e^{-ikB(v)t}.\end{split}$ (3.1)

We define for $t>1,v\in\mathbb{R}$,

$\begin{split}F_{k\rm
loc}(t,v):=&\frac{1}{2\pi}D(v)\int_{\mathbb{R}}e^{-ik(B(w)-B(v))t}{\rm
P.V.}\frac{\Gamma_{k}(v,w)}{B(v)-B(w)}\Phi^{\ast}(v-w)\partial_{w}B(w)\,dw\\\
&+\Big{[}D(v)\digamma_{k}(v)-F_{0k}(v)\Big{]},\end{split}$ (3.2)

and

$\begin{split}F_{k\rm
nloc}(t,v):=\frac{1}{2\pi}D(v)\int_{\mathbb{R}}e^{-ikB(w)t}\frac{\Gamma_{k}(v,w)}{B(v)-B(w)}\big{(}1-\Phi^{\ast}(v-w)\big{)}\partial_{w}B(w)\,dw.\end{split}$
(3.3)

Recalling the relation $r=e^{v},v\in\mathbb{R}$, we then set for $t>1,r>0$,

$f_{k\rm loc}(t,r):=F_{k\rm loc}(t,v),\quad f_{k\rm nloc}(t,r):=F_{k\rm
nloc}(t,v).$ (3.4)

It suffices to prove the for $v\in\mathbb{R},j\in[0,\infty)$, we have the
bounds for $t\geq 1$,

$\big{|}\partial_{v}^{j}F_{k\rm
loc}(t,v)\big{|}\lesssim_{j,k}M_{k}\frac{e^{\sqrt{k^{2}+8}\,v}}{1+e^{(|k|+8+\sqrt{k^{2}+8})v}},$
(3.5)

and

$\begin{split}&\big{|}\partial_{v}^{j}F_{k\rm
nloc}(t,v)\big{|}\lesssim_{j,k}M_{k}\,e^{2|v|}t^{-1}\min\Big{\\{}e^{2\sqrt{k^{2}+8}|v|}t^{-\sqrt{k^{2}+8}},\,1\Big{\\}}\frac{e^{\sqrt{k^{2}+8}\,v}}{1+e^{(|k|+8+\sqrt{k^{2}+8})v}}.\end{split}$
(3.6)

For (3.6), using the bounds (2.32)-(2.23), and the observation that for
$j_{1},j_{2}\in[0,\infty)$ and $|v-w|\geq 1$,

$\Big{|}\partial_{v}^{j_{1}}\partial_{w}^{j_{2}}\frac{1}{B(v)-B(w)}\Big{|}\lesssim_{j_{1},j_{2}}\Big{|}\frac{1}{B(v)-B(w)}\Big{|},$
(3.7)

we see that the proof for $j\geq 1$ is similar to the case $j=0$. A useful
inequality we can use is the following. Suppose $h(v,w)$ is smooth and
compactly supported on $(j,j+1)$ satisfying for $\alpha\geq 0$

$\big{|}\partial_{w}^{\alpha}h(v,w)\big{|}\lesssim_{\alpha}e^{-2\mu|w|}$ (3.8)

for some $j\in\mathbb{Z}$, and $\mu>0$, then for $t\geq 1$,

$\Big{|}\int_{\mathbb{R}}h(v,w)e^{-ikB(w)t}\,dw\Big{|}\lesssim_{\mu}\min\Big{\\{}\frac{1}{(|k|t)^{\mu}},e^{-2\mu|j|}\Big{\\}}.$
(3.9)

(3.9) can be proved by a simple integration by parts argument. The desired
bound (3.6) then follow from (2.32) and (3.9) by a lengthy but straightforward
calculation.

To establish the bound (3.5), it suffices to consider the first term on the
right hand side of (3.2). We follow the argument as in the proof of Theorem
1.5 in [9]. Assume that $|v-j|\leq 1$ for some $j\in\mathbb{Z}$, we define the
function $\zeta:[-4,4]\to\mathbb{R}$ such that we have the following
identities for $|\rho-j|\leq 5$,

$\nu:=\frac{B(\rho)-B(j)}{MB^{\prime}(j)},\qquad\rho:=\zeta(\nu)+j.$ (3.10)

In the above $M>1$ is chosen sufficiently large so that $\nu\in[-5,5]$ when
$\rho\in[j-5,j+5]$. $\zeta$ then satisfies the assumption in (iv) of Theorem
2.3. For $|v-j|\leq 1,|w-j|\leq 5$, denote

$\nu_{1}:=\frac{B(v)-B(j)}{MB^{\prime}(j)},\quad\nu_{2}:=\frac{B(w)-B(j)}{MB^{\prime}(j)}.$
(3.11)

Then we can compute

$\begin{split}&\int_{\mathbb{R}}e^{-ik(B(w)-B(v))t}\,{\rm
P.V.}\frac{\Theta_{k}(v-w,w)}{B(v)-B(w)}\Phi^{\ast}(v-w)\partial_{w}B(w)\,dw\\\
&=\int_{\mathbb{R}}e^{-ik(\nu_{2}-\nu_{1})MB^{\prime}(j)t}\,{\rm
P.V.}\frac{\Theta_{k}(\zeta(\nu_{1})-\zeta(\nu_{2}),\zeta(\nu_{2})+j)}{\nu_{1}-\nu_{2}}\Phi^{\ast}(\zeta(\nu_{1})-\zeta(\nu_{2}))\,d\nu_{2}\\\
&=\int_{\mathbb{R}}e^{-ik\rho MB^{\prime}(j)t}\,{\rm
P.V.}\frac{\Theta_{k}(\zeta(\nu_{1})-\zeta(\nu_{1}+\rho),\zeta(\nu_{1}+\rho)+j)}{-\rho}\Phi^{\ast}(\zeta(\nu_{1})-\zeta(\nu_{1}+\rho))\,d\rho.\end{split}$
(3.12)

The desired bounds (3.5) then follow from the smoothness of $\nu(v)$ in $v$,
(2.37) and smoothness of $\zeta$.

We now turn to the proof of (1.12) using (1.10)-(1.11). This is similar to the
argument in [1]. We first note that for $\eta\in C^{\infty}(0,\infty)$
satisfying for all $j\in[0,\infty)$,

$\eta\equiv 0\,\,{\rm for}\,\,r\gg
1,\quad|(r\partial_{r})^{j}\eta(r)|\lesssim_{j,k}r^{k+2},$ (3.13)

the bounds for $t\geq 1$ and $k\geq 2$,

$\Big{|}\int_{\mathbb{R}}\omega_{k}(t,r)\eta(r)\,rdr\Big{|}\lesssim_{j,k}\frac{1}{t^{1+\sqrt{k^{2}+8}}}$
(3.14)

follows from (1.9)-(1.11). For $k=1$, estimating the integral

$\int_{\mathbb{R}}\omega_{k}(t,r)\eta(r)\,rdr$

directly using (1.9)-(1.11) seems to lead to a logarithmic loss in $t$.
Similar issues also appeared when proving the bounds (1.9)-(1.11) in [9].
Remarkably, as observed already in [1] and later also in [3], for $k=1$ the
linearized equation (1.5) is explicitly solvable. Using the formula (2.38)
from [9], the desired bounds (1.12) can be proved from direct computations.

By the asymptotic expansion of $\eta_{k}(r)$ as $r\to 0+$, with leading order
given by $r^{k}$ and $r^{k+2}$, it suffices to show for $t\geq 1$,

$\Big{|}\int_{\mathbb{R}}r^{k}\omega_{k}(t,r)\varrho(r)\,rdr\Big{|}\lesssim_{k}\frac{1}{t^{1+\sqrt{k^{2}+8}}},$
(3.15)

if $\varrho\in C_{0}^{\infty}[0,\infty)$ with $\varrho\equiv 1$ for $r\leq
1/8$. Using

$\begin{split}\int_{\mathbb{R}}r^{k}\omega_{k}(t,r)\varrho(r)\,rdr&=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\omega(t,r,\theta)\,\,\overline{r^{k}e^{ik\theta}\varrho(r)}\,rdrd\theta\\\
&=\frac{1}{2\pi}\int_{\mathbb{R}^{2}}\psi(t,r,\theta)\Delta\,\Big{[}\,\overline{r^{k}e^{ik\theta}\varrho(r)}\,\Big{]}\,rdrd\theta.\end{split}$
(3.16)

Since $r^{k}e^{ik\theta}$ is harmonic on $\mathbb{R}^{2}$, the function

$H_{k}(r,\theta):=\Delta\,\Big{[}\,\overline{r^{k}e^{ik\theta}\varrho(r)}\,\Big{]}$
(3.17)

is compactly supported away from $r=0$. (3.15) follows from the bounds for
$t\geq 1$ and $\varphi\in C_{0}^{\infty}(\mathbb{R})$,

$\Big{|}\int_{\mathbb{R}}\phi_{k}(t,v)\varphi(v)\,dv\Big{|}\lesssim_{k}\frac{1}{t^{1+\sqrt{k^{2}+8}}}.$
(3.18)

(3.18) can now be proved by decomposing $\phi_{k}(t,v)$, similar to
$f_{k}(t,v)$, to the local and nonlocal part, and using bounds similar to
(1.10)-(1.12) for the local and nonlocal parts respectively. Theorem 1.2 is
then proved.

Acknowledgement
We thank Rajendra Beekie for valuable discussions on the paper [1].

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# Classical and quantum speed limits

Katarzyna<EMAIL_ADDRESS>Joanna
<EMAIL_ADDRESS>Piotr
<EMAIL_ADDRESS>
(1Department of Statistical Methods, Faculty of Economics and Sociology
University of Lodz, 41/43 Rewolucji 1905 St., 90-214 Lodz, Poland
2Department of Computer Science, Faculty of Physics and Applied Informatics
University of Lodz, 149/153 Pomorska St., 90-236 Lodz, Poland )

###### Abstract

The new bound on quantum speed limit in terms of relative purity is derived by
applying the original Mandelstam-Tamm one to the evolution in the space of
Hilbert-Schmidt operators acting in the initial space of states. It is shown
that it provides the quantum counterpart of the classical speed limit derived
in Phys. Rev. Lett. 120 (2018), 070402 and the $\hbar\rightarrow 0$ limit of
the former yields the latter. The existence of classical limit is related to
the degree of mixing of the quantum state.

##  I Introduction

There exist two seminal results concerning the bounds on the speed of quantum
evolution and related ability to distinguish quantum states connected via time
evolution. The first one, due to Mandelstam and Tamm, is expressed in terms of
energy dispersion of initial state [1]. Quite unexpectably, Margolus and
Levitin [2] established an independent bound based on expectation value of
excitation energy. Unifying both results one obtains the following constraint
on orthogonalization time [3]

$t_{\perp}\geq\max\left(\frac{\pi\hbar}{2(\left<E\right>-E_{0})},\frac{\pi\hbar}{2\Delta
E}\right)$ (1)

with $E_{0}$ being the ground state energy while $\Delta E$ is the energy
dispersion. This result has been further analyzed, extended in various
directions and applied in different context in numerous papers [4, 5, 6, 7, 8,
9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28].

Quite recently an interesting question has been raised whether there exists a
classical counterpart of quantum speed limit [29, 30]. Consider the
$\hbar\rightarrow 0$ limit of the bound (1). Typically, both the excitation
energy and its dispersion behave as $O(1)$ with $\hbar\rightarrow 0$. In fact,
denoting generically the ”principal” quantum number by $n$ one finds that
energy level has an expansion of the form
$a_{0}(n\hbar)+a_{1}(n\hbar)\hbar+a_{2}(n\hbar)\hbar^{2}+\ldots$. By
correspondence principle the classical limit is obtained letting
$\hbar\rightarrow 0$, $n\rightarrow\infty$, $n\hbar=O(1)$. Therefore, the
right hand side of eq. (1) vanishes in this limit. This is quite reasonable.
Consider the quantum pure states which saturate the Heisenberg uncertainty
principle. The $\hbar\rightarrow 0$ limit yields the pure classical state
$\rho_{0}$ with delta-like probability distribution. Clearly, the overlap
between $\rho$ and $\rho_{t}$ vanishes for any $t\neq 0$ (except some ”static”
states). However, the question becomes more subtle if mixed states are taken
into account.

In what follows we consider the Mandelstam-Tamm bound (the Margolus-Levitin
one seems to be less interesting in the classical limit [30]). It follows from
the following inequality

$\left|\left<\Psi(0)|\Psi(t)\right>\right|\geq\cos\left(\frac{(\Delta
E)_{0}}{\hbar}t\right).$ (2)

Both eqs. (1) and (2) show that the trouble with $\hbar\rightarrow 0$ limit
results from the fact that $\frac{\Delta E}{\hbar}\rightarrow\infty$. So the
question is whether the bound (2) can be replaced by the one not involving the
troublesome expression $\frac{\Delta E}{\hbar}$. It
$\hat{\rho}=\left|\Psi\right>\left<\Psi\right|$ is a pure state one can
immediately derive the identity

$\frac{2}{\hbar^{2}}\left(\Delta
E\right)^{2}_{\rho}=-\frac{1}{\hbar^{2}}Tr[\hat{H},\hat{\rho}]^{2}.$ (3)

The right hand side involves the commutator. Therefore, one can expect that is
possesses well-defined classical limit for ”reasonable” class of density
operators $\hat{\rho}$. This class obviously does not encompass the set of
pure states since the left-hand side is not well-defined in the limit
$\hbar\rightarrow 0$. However, the above identity is valid only for pure
states; the more mixed is the state $\hat{\rho}$ the more the right-hand side
of (3) deviate from the left one. Therefore, one can expect that for
$\hat{\rho}$ describing sufficiently ”regular” mixed states the right hand
side behaves reasonably for $\hbar\rightarrow 0$. Consequently, the idea to
modify the Mandelstam-Tamm bound to get sensible $\hbar\rightarrow 0$ limit by
replacing somehow $(\Delta E)^{2}_{\rho}$ by $Tr[\hat{H},\hat{\rho}]^{2}$ is
appealing.

On the classical level the bound on the speed of Hamiltonian evolution in
phase space has been derived by Okuyama and Ohzeki [30]. They considered the
Hilbert space of square integrable functions on phase space. By applying some
Hilbert space techniques they derived a bound valid for square integrable
classical probability distributions. They suggested that this bound is
specific for classical dynamics since the generator of dynamics in phase space
(Liouvillian) is a first order differential operator, contrary to the
Schrödinger one. The bound derived by Okuyama and Ohzeki involves the Poisson
bracket of Hamiltonian and probability density distribution. This suggests
that it is the classical limit of some quantum bound involving the relevant
commutator instead of $(\Delta E)_{\rho}$.

In the present paper we show that that the relevant quantum bound can be
readily obtained from the Mandelstam-Tamm relation applied to the pure states
in the Hilbert space of Hilbert-Schmidt operators acting in the original space
of states. The new bound is expressed in terms of relative purity and is
tighter than those encountered in the literature. Moreover, it is the quantum
counterpart of Okuyama-Ohzeki classical bound. As it has been mentioned above
their bound is valid for square integrable probability distributions. We show
that this assumption implies that they are obtained from density operators
which describe states which become more and more mixed as $\hbar\rightarrow
0$.

The paper is organized as follows. In Sec. I we rederive the Okuyama-Ohzeki
bound showing that it is a direct consequence of Mandelstam-Tamm one given by
eq. (2). Then in Sec. III we derive the quantum bound by using (2) in the
context of the Hilbert space of Hilbert-Schmidt operators. We show also why
one cannot expect that the bound to be saturated. Sec. IV is devoted to the
Wigner function formalism applied to speed limit. It is shown there that the
classical limit of the bound derived here yields Okuyama-Ohzeki bound.
Finally, Sec. V contains short summary.

##  II Speed limit in classical phase space

Let us rederive the bound on classical speed limit (CSL) obtained in Ref.
[30]. Consider a classical dynamical system described by some $2f$ dimensional
phase space $\Gamma$ and a Hamiltonian $H(\underline{q},\underline{p})$; for
simplicity we assume that $H$ is time independent but the generalization is
straightforward. Let $\rho(q,p,t)$ be a probability density of classical
states. The classical Hamiltonian dynamics is encoded in Liouville equation

$\frac{\partial\rho(\underline{q},\underline{p},t)}{\partial
t}+\left\\{\rho(\underline{q},\underline{p},t),H(\underline{q},\underline{p})\right\\}=0.$
(4)

Eq. (4) is simply the conservation law for
$\rho(\underline{q},\underline{p},t)$ along trajectories. Therefore, any
differentiable function $G(\rho)$ obey eq. (4) as well.

Eq. (4) may be rewritten in the form

$\frac{i\partial\rho(\underline{q},\underline{p},t)}{\partial
t}=(\hat{L}\rho)(\underline{q},\underline{p},t)$ (5)

where the Liouvillian $\hat{L}$ is defined by

$(\hat{L}\rho)(\underline{q},\underline{p},t)\equiv
i\left\\{H,\rho\right\\}(\underline{q},\underline{p},t)=i\sum_{k=1}^{f}\left(\frac{\partial
H}{\partial q_{k}}\frac{\partial\rho}{\partial p_{k}}-\frac{\partial
H}{\partial p_{k}}\frac{\partial\rho}{\partial
q_{k}}\right)(\underline{q},\underline{p},t)$ (6)

i.e,

$\hat{L}\equiv i\sum_{k=1}^{f}\left(\frac{\partial H}{\partial
q_{k}}\frac{\partial}{\partial p_{k}}-\frac{\partial H}{\partial
p_{k}}\frac{\partial}{\partial q_{k}}\right)$ (7)

is a first order differential operator.

Eq. (5) has the form of Schrödinger equation with $\hat{L}$ playing the role
of Hamiltonian. Therefore, we consider the Hilbert space of functions
$f(\underline{q},\underline{p})$, square integrable over the phase space
$\Gamma$; the relevant scalar product is defined by

$(f,g)\equiv\int\limits_{\Gamma}\mathrm{d}\underline{q}\,\mathrm{d}\underline{p}\,\overline{f(\underline{q},\underline{p})}\,g(\underline{q},\underline{p}).$
(8)

It is easy to see that $\hat{L}$ is (at least formally) selfadjoint with
respect to the above scalar product. Assuming that our probability
distribution $\rho(\underline{q},\underline{p},0)$ is square integrable we can
consider eq. (5) as describing an unitary evolution in our Hilbert space.
Therefore, the original Mandelstam-Tamm derivation remains valid.

Let $\|\rho_{t}\|$ denotes the norm in our Hilbert space,

$\|\rho_{t}\|^{2}\equiv\int\limits_{\Gamma}\mathrm{d}\underline{q}\,\mathrm{d}\underline{p}\,\rho^{2}(\underline{q},\underline{p},t);$
(9)

then $\frac{1}{\|\rho\|}\rho(\underline{q},\underline{p},t)$ is the normalized
vector. The Mandelstam-Tamm bound, applied in this context, yields (cf. [5]
for convenient form of MT bound)

$\frac{(\rho_{0},\rho_{t})}{\|\rho_{0}\|^{2}}\geq\cos\left((\Delta
L)_{0}t\right).$ (10)

Note that due to

$(\rho_{0},\rho_{t})\equiv\int\limits_{\Gamma}\mathrm{d}\underline{q}\,\mathrm{d}\underline{p}\,\rho(\underline{q},\underline{p},0)\,\rho(\underline{q},\underline{p},t)$
(11)

$\frac{1}{\|\rho_{0}\|^{2}}(\rho_{0},\rho_{t})$ can be viewed as classical
counterpart of relative purity [31], [32]. The dispersion $(\Delta L)_{0}$ is
defined as usual,

$(\Delta
L)_{0}^{2}\equiv\frac{1}{\|\rho_{0}\|^{2}}\left((\rho_{0},\hat{L}^{2}\rho_{0})-(\rho_{0},\hat{L}\rho_{0})^{2}\right).$
(12)

Now, let $K$ be the antiunitary operator of complex conjugation. Then, from
eq. (6) one finds

$K\hat{L}K=-\hat{L}.$ (13)

Moreover, $\rho$ is real, $K\rho=\rho$, and

$(\rho,\hat{L}\rho)=(K\rho,\hat{L}K\rho)=-(K\rho,K\hat{L}\rho)=-(\hat{L}\rho,\rho)=-(\rho,\hat{L}\rho)$
(14)

and eq. (12) simplifies to

$(\Delta
L)_{0}^{2}=\left<\hat{L}^{2}\right>_{0}\equiv\frac{(\rho_{0},\hat{L}^{2}\rho_{0})}{\|\rho_{0}\|^{2}}.$
(15)

So, eq. (10) can be rewritten as

$\frac{(\rho_{0},\rho_{t})}{(\rho_{0},\rho_{0})}\geq\cos\left(\frac{\sqrt{(\rho_{0},\hat{L}^{2}\rho_{0})}}{\|\rho_{0}\|}\,t\right).$
(16)

As it has been noted above, any function $G(\rho)$ of $\rho$ obeys Liouville
equation and one can repeat the above reasoning. In particular, taking
$G(\rho)=\rho^{\alpha}$ and assuming that $\rho^{\alpha}$ is square integrable
one obtains

$\frac{(\rho_{0}^{\alpha},\rho_{t}^{\alpha})}{(\rho_{0}^{\alpha},\rho_{0}^{\alpha})}\geq\cos\left(\sqrt{\frac{(\rho_{0}^{\alpha},\hat{L}^{2}\rho_{0}^{\alpha})}{(\rho_{0}^{\alpha},\rho_{0}^{\alpha})}}\,t\right)$
(17)

which coincides with eq. (18) of Ref. [30]. Note further that, by virtue of
eq. (6),

$(\rho_{0},\hat{L}^{2}\rho_{0})=(\hat{L}\rho_{0},\hat{L}\rho_{0})=\int\limits_{\Gamma}\mathrm{d}\underline{q}\,\mathrm{d}\underline{p}\left\\{H,\rho_{0}\right\\}^{2}.$
(18)

##  III Quantum speed limit

We shall now find the quantum counterpart of the bounds derived above. Let
$\mathcal{H}$ be the Hilbert space of states of the quantum system under
consideration. Any physical state is described by a density matrix
$\hat{\rho}(t)$ obeying

$i\hbar\dot{\hat{\rho}}(t)=\left[\hat{H},\hat{\rho}(t)\right]$ (19)

with $H$ being the Hamiltonian of the system under consideration.

Consider the Hilbert space $\mathcal{H}_{HS}$ of Hilbert-Schmidt operators
acting in $\mathcal{H}$, equipped with the scalar product

$(A,B)\equiv Tr(A^{+}B).$ (20)

Given the Hamiltonian $\hat{H}$ we define the operator $\tilde{H}$, acting in
$\mathcal{H}_{HS}$, by

$\tilde{H}A\equiv\left[\hat{H},A\right]$ (21)

$\tilde{H}$ is selfajoint with respect to the scalar product (20). Eq. (19)
may be rewritten as follows

$i\hbar\dot{\hat{\rho}}_{t}=\tilde{H}\hat{\rho}_{t}.$ (22)

Taking into account that $\frac{1}{\|\hat{\rho}_{t}\|}\hat{\rho}_{t}$
($\|\rho_{t}\|\equiv\sqrt{Tr(\hat{\rho}^{2}_{0})}$) is a unit vector in
$\mathcal{H}_{HS}$ one can again apply Mandelstam-Tamm inequality to find

$\frac{Tr(\hat{\rho}_{0}\hat{\rho}_{t})}{Tr(\hat{\rho}_{0}^{2})}\geq\cos\left(\frac{(\Delta\tilde{H})_{0}}{\hbar}\,t\right)$
(23)

with

$(\Delta\tilde{H})^{2}_{0}=\frac{1}{\|\hat{\rho}_{t}\|^{2}}\left((\hat{\rho}_{t},\tilde{H}^{2}\hat{\rho}_{t})-(\hat{\rho}_{t},\tilde{H}\hat{\rho}_{t})^{2}\right).$
(24)

Following the reasoning similar to that in previous section we define an
antiunitary operator $K$, acting in $\mathcal{H}_{HS}$:

$KA=A^{+}.$ (25)

Then

$K\tilde{H}K=-\tilde{H}$ (26)

and

$K\hat{\rho}_{t}=\hat{\rho}_{t}.$ (27)

Consequently

$(\hat{\rho}_{0},\tilde{H}\hat{\rho}_{0})=(K\hat{\rho}_{0},\tilde{H}K\hat{\rho}_{0})=-(K\hat{\rho}_{0},K\tilde{H}\hat{\rho}_{0})=-(\tilde{H}\hat{\rho}_{0},\hat{\rho}_{0})=-(\hat{\rho}_{0},\tilde{H}\hat{\rho}_{0})$
(28)

implying

$(\Delta\tilde{H})_{0}^{2}=\frac{1}{\|\hat{\rho}_{0}\|^{2}}(\hat{\rho}_{0},\tilde{H}^{2}\hat{\rho}_{0})=\frac{1}{\|\hat{\rho}_{0}\|^{2}}(\tilde{H}\hat{\rho}_{0},\tilde{H}\hat{\rho}_{0})=\frac{-Tr([\hat{H},\hat{\rho}_{0}]^{2})}{Tr(\hat{\rho}_{0}^{2})}.$
(29)

Eq. (23) yields then

$\frac{Tr(\hat{\rho}_{0}\hat{\rho}_{t})}{Tr(\hat{\rho}_{0}^{2})}\geq\cos\left(\sqrt{\frac{-Tr([\hat{H},\hat{\rho}_{0}]^{2})}{Tr(\hat{\rho}_{0}^{2})\hbar^{2}}}\,t\right).$
(30)

By comparying (30) with eqs. (16) and (18) we conclude that it provides the
quantum couterpart of (16).

Defining $\hat{\rho}^{\alpha}$, $\alpha\geq 0$, with the help of the spectral
decomposition of $\hat{\rho}$ we find that it obeys the equation of motion
(19). Therefore, assuming that $\hat{\rho}^{\alpha}$ is again a Hilbert-
Schmidt operator we arrive at the quantum counterpart of eq. (17). Concluding,
let us note that the bound (30) is stronger that the ones obtained in [18] and
[29] for the case of unitary evolution.

One can pose the question whether the bound (30) is attainable. If
$\hat{\rho}=\left|\Psi\right>\left<\Psi\right|$ is a pure state, eq. (30)
implies

$\left|\left<\Psi(0)|\Psi(t)\right>\right|^{2}\geq\cos\left(\frac{\sqrt{2}(\Delta
E)_{0}t}{\hbar}\right)$ (31)

while the original Mandelstam-Tamm bound yields

$\left|\left<\Psi(0)|\Psi(t)\right>\right|^{2}\geq\cos^{2}\left(\frac{(\Delta
E)_{0}t}{\hbar}\right)$ (32)

which is stronger. So (31) cannot be saturated. Now, what about genuinely
mixed states?

Within our framework based on $\mathcal{H}_{HS}$, the normalized density
matrix $\frac{1}{\|\hat{\rho}\|}\hat{\rho}$ is always viewed as a pure state.
It is known that for pure states the Mandelstam-Tamm bound is attainable in
the sense that, given two states,
$\left|\Psi\right>\equiv\left|\Psi(0)\right>$,
$\left|\phi\right>\equiv\left|\Psi(t)\right>$, one can find the Hamiltonian
saturating the inequality (32). It should be selected in such a way as to
generate the arc of the great circle connecting $\left|\Psi(0)\right>$ and
$\left|\Psi(t)\right>$ on $S^{2}$ [33]. It reads

$\hat{H}=\omega(\left|\Psi\right>\langle\tilde{\Psi}|+|\tilde{\Psi}\rangle\left<\Psi\right|)$
(33)

where

$|\tilde{\Psi}\rangle=i\left(\frac{\left|\phi\right>-\left|\Psi\right>\left<\Psi|\phi\right>}{\sqrt{1-\left|\left<\phi|\Psi\right>\right|^{2}}}\right)$
(34)

i.e. $\langle\Psi|\tilde{\Psi}\rangle=0$,
$\langle\tilde{\Psi}|\tilde{\Psi}\rangle=1$.

However, the trouble is that in our case the set of admissible Hamiltonians is
restricted to those satisfying eq. (21); this condition cannot be, in general,
satisfied by the Hamiltonians of the form described by eqs. (33) and (34), at
least in the finitedimensional case. In fact eq. (33) implies

$\hat{H}\left|\Psi\right>=\omega|\tilde{\Psi}\rangle,\quad\hat{H}|\tilde{\Psi}\rangle=\omega\left|\Psi\right>.$
(35)

Since in the derivation of eq. (30) $\frac{1}{\|\hat{\rho}\|}\hat{\rho}$ is
treated as a pure state in $\mathcal{H}_{HS}$, the Hamiltonian $\tilde{H}$,
acting in $\mathcal{H}_{HS}$, is given by eqs. (33) and (34) with the
substitution
$\left|\Psi\right>\rightarrow\frac{1}{\|\hat{\rho}_{0}\|}\hat{\rho}_{0}$,
$\left|\phi\right>\rightarrow\frac{1}{\|\hat{\rho}_{t}\|}\hat{\rho}_{t}$.
Using eqs. (21) and (35) we find the following relation for the initial
Hamiltonian $\hat{H}$ acting in $\mathcal{H}$,

$\left[\hat{H},\hat{\rho}_{0}\right]=i\omega\left(\frac{\|\hat{\rho}_{0}\|^{2}\hat{\rho}_{t}-Tr(\hat{\rho}_{0}\hat{\rho}_{t})\hat{\rho}_{0}}{\sqrt{\|\hat{\rho}_{0}\|^{2}\|\hat{\rho}_{t}\|^{2}-Tr^{2}(\hat{\rho}_{0}\hat{\rho}_{t})}}\right)$
(36)
$\left[\hat{H},\frac{\|\hat{\rho}_{0}\|^{2}\hat{\rho}_{t}-Tr(\hat{\rho}_{0}\hat{\rho}_{t})\hat{\rho}_{0}}{\sqrt{\|\hat{\rho}_{0}\|^{2}\|\hat{\rho}_{t}\|^{2}-Tr^{2}(\hat{\rho}_{0}\hat{\rho}_{t})}}\right]=-i\omega\hat{\rho}_{0}.$
(37)

Taking the trace of both sides we conclude that eqs. (36), (37) can be
satisfied only provided $\hat{\rho}_{0}=\hat{\rho}_{t}$.

##  IV Speed limit in terms of Wigner’s functions

As we have discussed in the Introduction one can expect that the smooth
classical limit for the speed bound exists rather for strongly mixed states
than pure ones. This is best seen if one uses the Wigner function formalism
(the description of quantum speed limit in the framework of Wigner’s function
has been discussed by a number of authors cf., e.g., [28],[29]). Assume our
space of states $\mathcal{H}$ describes the quantum system obtained by
quantizing some classical Hamiltonian dynamics (for simplicity, we assume one
degree of freedom). It is convenient to introduce a specific basis in the
space $\mathcal{H}_{HS}$. Following Mukunda [34] we define the operators
$\omega(\hat{q})$ and $\omega(\hat{p})$, acting in $\mathcal{H}_{HS}$, by

$\omega(\hat{q})\hat{A}=\frac{1}{2}(\hat{q}\hat{A}+\hat{A}\hat{q})$ (38)
$\omega(\hat{p})\hat{A}=\frac{1}{2}(\hat{p}\hat{A}+\hat{A}\hat{p}).$ (39)

They are selfadjoint with respect to the scalar product (20). Moreover,

$\left[\omega(\hat{q}),\omega(\hat{p})\right]=0.$ (40)

Therefore, their common eigenvectors $\hat{V}(q,p)$,

$\omega(\hat{q})\hat{V}(q,p)=q\hat{V}(q,p)$ (41)
$\omega(\hat{p})\hat{V}(q,p)=p\hat{V}(q,p)$ (42)

span the (generalized) basis in $\mathcal{H}_{HS}$. The solution to (41), (42)
reads

$\hat{V}(q,p)=\sqrt{\frac{2}{\pi\hbar}}\,e^{\frac{2i}{\hbar}(p\hat{q}-q\hat{p})}P$
(43)

where $P$ is the parity operator,

$P\left|q\right>=\left|-q\right>$ (44) $P\left|p\right>=\left|-p\right>.$ (45)

Moreover,

$\left(\hat{V}(q,p),\hat{V}(q^{\prime},p^{\prime})\right)\equiv
Tr\left(\hat{V}^{+}(q,p)\hat{V}(q^{\prime},p^{\prime})\right)=\delta(q-q^{\prime})\delta(p-p^{\prime}).$
(46)

For any $\hat{A}\in\mathcal{H}_{HS}$ one has the eigenfunctions expansion

$\hat{A}=\int\mathrm{d}q\,\mathrm{d}p\,\tilde{a}(q,p)\hat{V}(q,p).$ (47)

In particular,

$Tr(\hat{A}^{+}\hat{B})=\int\mathrm{d}q\,\mathrm{d}p\,\overline{\tilde{a}(q,p)}\,\tilde{b}(q,p).$
(48)

and

$\tilde{a}(q,p)=Tr(\hat{V}^{+}(q,p)\hat{A}).$ (49)

Computing the trace yields

$\tilde{a}(q,p)=\frac{1}{\sqrt{2\pi\hbar}}\int\limits_{-\infty}^{\infty}\mathrm{d}q^{\prime}\,\left<q+\frac{q^{\prime}}{2}\right|\hat{A}\left|q-\frac{q^{\prime}}{2}\right>e^{-\frac{ipq^{\prime}}{\hbar}}.$
(50)

The Wigner function is defined by

$W(q,p)\equiv\frac{1}{\sqrt{2\pi\hbar}}\,\tilde{\rho}(q,p)=\frac{1}{2\pi\hbar}\int\limits_{-\infty}^{\infty}\mathrm{d}q^{\prime}\,\left<q+\frac{q^{\prime}}{2}\right|\hat{\rho}\left|q-\frac{q^{\prime}}{2}\right>e^{-\frac{ipq^{\prime}}{\hbar}}.$
(51)

Eqs. (48) and (51) imply

$Tr(\hat{\rho}_{0}\hat{\rho}_{t})=2\pi\hbar\int\mathrm{d}q\,\mathrm{d}p\,W(q,p,0)\,W(q,p,t).$
(52)

One readily concludes from eq. (50) that

$a_{W}(q,p)=\sqrt{2\pi\hbar}\,\tilde{a}(q,p)$ (53)

is the Weyl symbol of $\hat{A}$. In particular, the Wigner function is the
Weyl symbol of density matrix $\hat{\rho}$ divided by $2\pi\hbar$. Therefore,
puting $\hat{B}=\hat{\rho}$ in eq. (48) we find

$\langle\hat{A}\rangle_{\rho}\equiv
Tr(\hat{A}\hat{\rho})=\int\mathrm{d}q\,\mathrm{d}p\,W(q,p)\,a_{W}(q,p).$ (54)

The normalization condition reads

$1=Tr\hat{\rho}=\int\mathrm{d}q\,\mathrm{d}p\,W(q,p)$ (55)

while

$1\geq Tr\hat{\rho}^{2}=2\pi\hbar\int\mathrm{d}q\,\mathrm{d}p\,W^{2}(q,p).$
(56)

Taking into account that in the limit $\hbar\rightarrow 0$ the Weyl symbol
becomes the corresponding classical dynamical variable we conclude from eq.
(54) that the Wigner function should be in this limit identified with
classical probability density on phase space, $\rho(q,p)$. Let us remind that
the derivation of classical speed bound (16) is based on the assumption that
the classical probability distribution $\rho(q,p)$ is square integrable over
phase space. Now, taking into account that the latter is given by the
$\hbar\rightarrow 0$ limit of Wigner’s function, one concludes from eq. (56)
that with $\hbar\rightarrow 0$ the state under consideration becomes more and
more mixed. This conclusion supports our general discussion in Section I.

In order to rewrite the right hand side of eq. (30) in terms of Wigner’s
function let us remaind that

$\left(\frac{1}{i\hbar}[\hat{A},\hat{B}]\right)_{W}=\left\\{\left\\{a_{W},b_{W}\right\\}\right\\}$
(57)

where on the left hand side $(\ldots)_{W}$ denotes the Weyl symbol of the
expression in bracket while $\\{\\{\;,\;\\}\\}$ on the right hand side denotes
the Moyal bracket [35], [36]. Using eqs. (48), (51) and (53) we find

$-\frac{1}{\hbar^{2}}Tr\left([\hat{H},\hat{\rho}_{0}]^{2}\right)=2\pi\hbar\int\mathrm{d}q\,\mathrm{d}p\left\\{\left\\{H_{W},W_{0}\right\\}\right\\}^{2}$
(58)

where $W_{0}\equiv W(q,p,0)$ and $H_{W}$ denotes the Weyl symbol of the
Hamiltonian $\hat{H}$. Finally, taking into account eq. (52) one can rewrite
(30) as

$\frac{\int\mathrm{d}q\,\mathrm{d}p\,W(q,p,0)\,W(q,p,t)}{\int\mathrm{d}q\,\mathrm{d}p\,W(q,p,0)^{2}}\geq\cos\left(\sqrt{\frac{\int\mathrm{d}q\,\mathrm{d}p\,\left\\{\left\\{H_{W},W_{0}\right\\}\right\\}^{2}}{\int\mathrm{d}q\,\mathrm{d}p\,W_{0}^{2}}}\,\cdot
t\right).$ (59)

Note that

$\left\\{\left\\{.\,,.\right\\}\right\\}=\left\\{.\,,.\right\\}+O(\hbar^{2})$
(60)

where $\left\\{.\,,.\right\\}$ denotes the classical Poisson bracket. By
comparying eqs. (59) and (60) with (16) and (18) we conclude that eq. (16)
coincides with $\hbar\rightarrow 0$ limit of eq. (59).

##  V Summary

We have derived the quantum bound on relative purity. It is given by eqs. (30)
and is tighter than some encountered in the literature. Its classical limit
coincides with the bound derived by Okuyama and Ohzeki. The existence of
classical limit is related to the degree of mixing of the state under
consideration: it becomes more and more mixed as $\hbar\rightarrow 0$.

In fact, as in the classical case, we have the whole family of bounds. One can
replace $\hat{\rho}$ by $\hat{\rho}_{\alpha}$, $\alpha\in\mathbbm{R}_{+}$,
provided the latter is well defined (unnormalized) density operator.

### Acknowledgement

We are grateful to Profs. Krzysztof Andrzejewski, Cezary Gonera and Paweł
Maślanka for fruitful discussion.

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