On 2nd-order fully-nonlinear equations with links to 3rd-order fully-nonlinear
equations

Marianna Euler and Norbert Euler

International Society of Nonlinear Mathematical Physics, Auf der Hardt 27,
56130 Bad Ems, Germany
and
Centro Internacional de Ciencias, Av. Universidad s/n, Colonia Chamilpa,
62210 Cuernavaca, Morelos, Mexico

Abstract: We derive the general conditions for fully-nonlinear symmetry-
integrable second-order evolution equations and their first-order recursion
operators. We then apply the established Propositions to find links between a
class of fully-nonlinear third-order symmetry-integrable evolution equations
and fully-nonlinear second-order symmetry-integrable evolution equations.

##  1 Introduction

We recently reported a class of third-order fully-nonlinear symmetry-
integrable evolution equations in 1+1 dimensions with rational nonlinearities
in their highest derivative [2]. For this class of equations we have
furthermore reported all the potentialisations and some multi-
potentialisations in [3]. It is interesting to note that some of these
potential equations are in fact members of hierarchies of fully-nonlinear
second-order evolution equations. In the current article we identify all those
potential equations and give the explicit recursion operators that generate
those hierarchies. In particular, the fully-nonlinear third-order symmetry-
integrable evolution equations that are relevant here are

$\displaystyle
u_{t}=\frac{u_{xx}^{3}}{u_{xxx}^{2}}\left(\lambda_{1}+\lambda_{2}u_{xx}\right)^{3}$
and $\displaystyle u_{t}=\frac{1}{u_{xxx}^{2}},$

where $\lambda_{1}$ and $\lambda_{2}$ are arbitrary constants. For more
details on symmetry-integrable hierarchies and recursion operators, we refer
to [4] and [2], and the references therein.

This article is organised as follows: In Section 2 we provide a Proposition to
identify second-order fully-nonlinear equations that admit third-order Lie-
Bäcklund symmetries (see Proposition 1). Furthermore a second Proposition then
gives the conditions for recursion operators of those second-order fully-
nonlinear equations (see Proposition 2). In Section 3 we apply the mentioned
Propositions to establish links to the two fully-nonlinear equations given
above. Finally in Section 4 we make our conclusions and propose some further
studies that could be of interest.

##  2 On fully-nonlinear second-order equations

We consider a second-order equation of the form

$\displaystyle u_{t}=F(u,u_{x},u_{xx})$ (2.1)

where the non-constant function $F$ is to be determined such that (2.1) is
fully nonlinear in $u_{xx}$ and admits a third-order Lie-Bäcklund symmetry
generated by $\displaystyle{Z=\eta[u]\frac{\partial\ }{\partial u}}$; hence a
recursion operator $R$ that generates this third-order symmetry and
consequently an infinite number of higher-order Lie-Bäcklund symmetries. The
hierarchy of symmetry-integrable evolution equations is then

$\displaystyle u_{t_{m}}=R^{m}[u]\,u_{t},\qquad m=0,1,2,\ldots\ .$ (2.2)

Our first step is to find the general form of $F$ and $\eta$ to satisfy this
requirement as a necessary condition. This is given by

######  Proposition 1.

Consider the class of second-order evolution equations of the form (2.1) viz

$\displaystyle u_{t}=F(u,u_{x},u_{xx}).$

Under the assumption that (2.1) is symmetry-integrable, the following
statements must be true:

  1. 1.

The function $F$ in (2.1) satisfies the condition

$\displaystyle\frac{\partial^{3}F}{\partial u_{xx}^{3}}\left(\frac{\partial
F}{\partial u_{xx}}\right)^{-1}-\frac{3}{2}\left(\frac{\partial^{2}F}{\partial
u_{xx}^{2}}\right)^{2}\left(\frac{\partial F}{\partial u_{xx}}\right)^{-2}=0$
(2.3)

with general solution

$\displaystyle
F(u,u_{x},u_{xx})=-\frac{A(u,u_{x})}{u_{xx}+B(u,u_{x})}+C(u,u_{x}),$ (2.4)

where $A,\ B$ and $C$ are arbitrary functions of their arguments. The quasi-
linear and semi-linear form of (2.1) follow from the singular solution of
(2.3), namely the solution for which $\partial^{2}F/\partial u_{xx}^{2}=0$.
[Note that (2.3) is the Schwarzian derivative in $F$.]

  2. 2.

Any third-order Lie-Bäcklund symmetry generator

$\displaystyle Z:=\eta(x,u,u_{x},u_{xx},u_{xxx})\frac{\partial\ }{\partial u}$
(2.5)

for (2.4) is of the general form

$\displaystyle\eta(x,u,u_{x},u_{xx},u_{xxx})=\frac{c_{0}A^{3/2}u_{xxx}}{(u_{xx}+B)^{3}}-\frac{c_{0}A^{3/2}}{(u_{xx}+B)^{3}}\left(B\frac{\partial
B}{\partial u_{x}}-u_{x}\frac{\partial B}{\partial u}\right)$
$\displaystyle\qquad+\frac{3c_{0}A^{1/2}}{2(u_{xx}+B)^{2}}\left(B\frac{\partial
A}{\partial u_{x}}-u_{x}\frac{\partial A}{\partial
u}\right)-\frac{f_{1}}{u_{xx}+B}+f_{2},$ (2.6)

whereby $c_{0}$ denotes an arbitrary but non-zero constant, and the functions
$A=A(u,u_{x})$, $B=B(u,u_{x})$, $C=C(u,u_{x})$, $f_{1}=f_{1}(x,u,u_{x})$ and
$f_{2}=f_{2}(x,u,u_{x})$ need to be determined such that the invariance
condition

$\displaystyle\left.\vphantom{\frac{DA}{DB}}L_{E}\eta(x,u,u_{x},u_{xx},u_{xxx})\right|_{E=0}=0$
(2.7)

is satisfied. Here $E:=u_{t}-F(u,u_{x},u_{xx})$ and $L_{E}$ denotes the linear
operator

$\displaystyle L_{E}[u]:=\frac{\partial E}{\partial u}+\frac{\partial
E}{\partial u_{t}}D_{t}+\frac{\partial E}{\partial u_{x}}D_{x}+\frac{\partial
E}{\partial u_{xx}}D_{x}^{2}+\frac{\partial E}{\partial u_{xxx}}D_{x}^{3}.$
(2.8)

We now seek 1st-order recursion operators $R$ for every equation that admits a
third-order Lie-Bäcklund symmetry, which, according to Proposition 1, are
equations of the general form (2.4) We consider both differential recursion
operators

$\displaystyle R[u]=G_{1}(u,u_{x},u_{xx})D_{x}+G_{0}(u,u_{x})$ (2.9)

as well as integro-differential recursion operators

$\displaystyle
R[u]=G_{1}(u,u_{x},u_{xx})D_{x}+G_{0}(u,u_{x})+I(u,u_{x},u_{xx})D_{x}^{-1}\circ\Lambda(u,u_{x},u_{xx}).$
(2.10)

Applying the standard condition for $R$ (see for example [1]), namely

$\displaystyle\left.\vphantom{\frac{DA}{DB}}\left[L_{E},\
R[u]\right]=D_{t}R\right|_{E=0}$ (2.11)

we obtain the following

######  Proposition 2.

For finding recursion operators of (2.4) viz.

$\displaystyle u_{t}=-\frac{A(u,u_{x})}{u_{xx}+B(u,u_{x})}+C(u,u_{x}).$

we distinguish between the following four cases:

  1. 1.

With the assumption $A=F_{1}(u_{x})$ and $B(u,u_{x})=0$ in (2.4), it follows
that

$\displaystyle
u_{t}=-\frac{F_{1}(u_{x})}{u_{xx}}+u\left(\frac{a_{1}\sqrt{F_{1}(u_{x})}}{u_{x}^{2}P(u_{x})}-\frac{F_{1}(u_{x})}{u_{x}P(u_{x})}\frac{dP}{du_{x}}+\frac{F_{1}(u_{x})}{u_{x}^{2}}-\frac{1}{2u_{x}}\frac{dF_{1}}{du_{x}}\right)$
$\displaystyle\qquad+\frac{a_{0}\sqrt{F_{1}(u_{x})}}{P(u_{x})}+H(u_{x})$
(2.12)

admits a recursion operator of the form (2.10) with

$\displaystyle G_{1}(u_{x},u_{xx})=\frac{k_{1}\sqrt{F_{1}(u_{x})}}{u_{xx}}$
(2.13a) $\displaystyle
G_{0}(u_{x})=\frac{k_{1}\sqrt{F_{1}(u_{x})}}{2P(u_{x})}\frac{dP}{du_{x}}-\frac{k_{1}\sqrt{F_{1}(u_{x})}}{2u_{x}}-\frac{a_{1}k_{1}}{2u_{x}P(u_{x})}+k_{0}$
(2.13b) $\displaystyle I(u_{x})=\alpha u_{x}$ (2.13c)
$\displaystyle\Lambda(u_{x},u_{xx})=\frac{P(u_{x})u_{xx}}{\sqrt{F_{1}(u_{x})}},$
(2.13d)

where $a_{0},\ a_{1}$ and $k_{0}$ are arbitrary constants, $k_{1}$ and
$\alpha$ are an arbitrary non-zero constants, $H$ is an arbitrary function of
$u_{x}$ and, furthermore, $F_{1}$ and $P$ must satisfy the following
condition:

$\displaystyle
0=k_{1}u_{x}^{2}P^{2}\left(u_{x}\frac{dP}{du_{x}}+P\right)\frac{d^{2}F_{1}}{du_{x}^{2}}-k_{1}u_{x}P\left[-3u_{x}^{2}P\frac{d^{2}P}{du_{x}^{2}}+2u_{x}^{2}\left(\frac{dP}{du_{x}}\right)^{2}-2u_{x}P\frac{dP}{du_{x}}\right.$
$\displaystyle\left.\vphantom{\frac{DA}{DB}}+2P^{2}\right]\frac{dF_{1}}{du_{x}}+2k_{1}\left[u_{x}^{3}P^{2}\frac{d^{3}P}{du_{x}^{3}}-2u_{x}^{3}P\frac{dP}{du_{x}}\frac{d^{2}P}{du_{x}^{2}}+u_{x}^{2}P^{2}\frac{d^{2}P}{du_{x}^{2}}+u_{x}^{3}\left(\frac{dP}{du_{x}}\right)^{3}\right.$
$\displaystyle\left.-u_{x}^{2}P\left(\frac{dP}{du_{x}}\right)^{2}-u_{x}P^{2}\frac{dP}{du_{x}}+P^{3}\right]F_{1}+2\left(4\alpha
u_{x}^{3}P^{3}-k_{1}a_{1}^{2}\right)\left(u_{x}\frac{dP}{du_{x}}+P\right).$
(2.14)

  2. 2.

With the assumption $A=F_{1}(u_{x})$ and $B=B(u_{x})$ in (2.4), it follows
that

$\displaystyle
u_{t}=-\frac{F_{1}(u_{x})}{u_{xx}+(b_{1}+b_{2}u_{x})\sqrt{F_{1}(u_{x})}}+\frac{\sqrt{F_{1}(u_{x})}}{b_{1}+b_{2}u_{x}}$
(2.15)

admits a recursion operator of the form (2.10) with

$\displaystyle
G_{1}(u_{x},u_{xx})=\frac{k_{1}\sqrt{F_{1}(u_{x})}}{u_{xx}+(b_{1}+b_{2}u_{x})\sqrt{F_{1}(u_{x})}}$
(2.16a) $\displaystyle
G_{0}(u_{x})=\frac{k_{1}b_{2}F_{1}(u_{x})}{u_{xx}+(b_{1}+b_{2}u_{x})\sqrt{F_{1}(u_{x})}}-\frac{k_{1}b_{2}\sqrt{F_{1}(u_{x})}}{b_{1}+b_{2}u_{x}}+k_{0}$
(2.16b) $\displaystyle I(u_{x})=u_{x}+\frac{b_{1}}{b_{2}}$ (2.16c)
$\displaystyle\Lambda(u_{x},u_{xx})=\alpha
b_{2}\left(\frac{u_{xx}+(b_{1}+b_{2}u_{x})\sqrt{F_{1}(u_{x})}}{(b_{1}+b_{2}u_{x})\sqrt{F_{1}(u_{x})}}\right),$
(2.16d)

where $b_{1}$ and $\alpha$ are arbitrary constants, whereas $b_{2}$ and
$k_{1}$ are arbitrary non-zero constants.

  3. 3.

With the assumption $A=F_{1}(u_{x})$, $B=0$ and $C=C(u_{x})$ in (2.4), it
follows that

$\displaystyle u_{t}=-\frac{F_{1}(u_{x})}{u_{xx}}+C(u_{x}).$ (2.17)

This leads to two subcases:

3.1. Equation (2.17) admits the recursion operator of the form (2.10) with

$\displaystyle G_{1}(u_{x},u_{xx})=\frac{k_{1}\sqrt{F_{1}(u_{x})}}{u_{xx}}$
(2.18a) $\displaystyle
G_{0}(u_{x})=-\frac{k_{1}}{4\sqrt{F_{1}(u_{x})}}\frac{dF_{1}}{du_{x}}+k_{0}$
(2.18b) $\displaystyle I(u_{x})=\alpha$ (2.18c)
$\displaystyle\Lambda(u_{x},u_{xx})=\frac{u_{xx}}{F_{1}(u_{x})},$ (2.18d)

where $k_{0}$ is an arbitrary constant, $k_{1}$ and $\alpha$ are arbitrary
non-zero constants, $C$ is an arbitrary function of $u_{x}$, and $F_{1}$ must
satisfy the following condition:

$\displaystyle
k_{1}\left(4F_{1}^{4}\frac{d^{3}F_{1}}{du_{x}^{3}}-6F_{1}^{3}\frac{dF_{1}}{du_{x}}\frac{d^{2}F_{1}}{du_{x}^{2}}+3F_{1}^{2}\left(\frac{dF_{1}}{du_{x}}\right)^{3}\right)+16\alpha
F_{1}^{5/2}\frac{dF_{1}}{du_{x}}=0.$ (2.19)

3.2. Equation (2.17) admits the recursion operator of the form (2.10) with

$\displaystyle G_{1}(u_{x},u_{xx})=\frac{k_{1}\sqrt{F_{1}(u_{x})}}{u_{xx}}$
(2.20a) $\displaystyle
G_{0}(u_{x})=-\frac{k_{1}}{4\sqrt{F_{1}(u_{x})}}\frac{dF_{1}}{du_{x}}+k_{0}$
(2.20b) $\displaystyle I(u_{x})=\alpha u_{x}$ (2.20c)
$\displaystyle\Lambda(u_{x},u_{xx})=\frac{u_{x}u_{xx}}{F_{1}(u_{x})},$ (2.20d)

where $k_{0}$ is an arbitrary constant, $k_{1}$ and $\alpha$ are arbitrary
non-zero constants, $C$ is an arbitrary function of $u_{x}$, and $F_{1}$ must
satisfy the following condition:

$\displaystyle
k_{1}\left(4F_{1}^{4}\frac{d^{3}F_{1}}{du_{x}^{3}}-6F_{1}^{3}\frac{dF_{1}}{du_{x}}\frac{d^{2}F_{1}}{du_{x}^{2}}+3F_{1}^{2}\left(\frac{dF_{1}}{du_{x}}\right)^{3}\right)$
$\displaystyle+16\alpha
F_{1}^{5/2}u_{x}\left(u_{x}\frac{dF_{1}}{du_{x}}-4F_{1}\right)=0.$ (2.21)

  4. 4.

The equation

$\displaystyle
u_{t}=-\frac{F_{1}(u_{x})}{u_{xx}+(b_{1}+b_{2}u_{x})\sqrt{F_{1}(u_{x})}}+F_{2}(u_{x})$
(2.22)

admits a recursion operator of the form (2.9) with

$\displaystyle
G_{1}(u_{x})=\frac{k_{1}\sqrt{F_{1}}}{u_{xx}+(b_{1}+u_{x}b_{2})\sqrt{F_{1}}}$
(2.23a) $\displaystyle
G_{0}(u_{x})=\frac{k_{1}b_{2}F_{1}}{u_{xx}+(b_{1}+b_{2}u_{x})\sqrt{F_{1}}}-\frac{k_{1}}{4\sqrt{F_{1}}}\frac{dF_{1}}{du_{x}}+\frac{k_{1}}{2}(b_{1}+b_{2}u_{x})\frac{dF_{2}}{du_{x}}$
$\displaystyle\qquad-\frac{k_{1}b_{2}}{2}F_{2}+k_{0},$ (2.23b)

where $b_{1}$ and $b_{2}$ are arbitrary constants, $k_{1}$ is an arbitrary
non-zero constant, and $F_{1}$ and $F_{2}$ must satisfy the following
condition:

$\displaystyle
4F_{1}^{2}\frac{d^{3}F_{1}}{du_{x}^{3}}-6F_{1}\frac{dF_{1}}{du_{x}}\frac{d^{2}F_{1}}{du_{x}^{2}}+3\left(\frac{dF_{1}}{du_{x}}\right)^{3}-8F_{1}^{5/2}\left(\left(b_{1}+b_{2}u_{x}\right)\frac{d^{3}F_{2}}{du_{x}^{3}}\right.$
$\displaystyle\left.+3b_{2}\frac{d^{2}F_{2}}{du_{x}^{2}}\right)=0.$ (2.24)

##  3 Fully-nonlinear second-order equations related to fully-non-
linear third-order equations

We now apply Proposition 1 and Proposition 2 to identify fully-nonlinear
second-order equations that are related to fully-nonlinear third-order
equations. For this purpose we consider two main cases:

Case 1: Consider the fully-nonlinear symmetry-integrable equation [2]

$\displaystyle
u_{t}=\frac{u_{xx}^{3}}{u_{xxx}^{2}}\left(\lambda_{1}+\lambda_{2}u_{xx}\right)^{3},$
(3.1)

where $\lambda_{1}$ and $\lambda_{2}$ are arbitrary constants. This leads to
the following subcases.

Subcase 1.1: Consider equation (3.1), with $\lambda_{1}\neq 0$ and
$\lambda_{2}\neq 0$. Then (3.1) admits the zero-order potentialisation [3]

$\displaystyle
v_{t}=\frac{\lambda_{1}^{3}}{4}\left(\vphantom{\frac{A}{B}}1+\lambda_{1}^{2}\lambda_{2}v_{x}^{2}\right)^{3/2}\left(\frac{v_{x}^{3}v_{xxx}}{v_{xx}^{3}}-\frac{3v_{x}^{2}}{v_{xx}}\right),$
(3.2)

where

$\displaystyle
v_{x}=-\frac{2}{\lambda_{1}^{2}}\left[u_{xx}(\lambda_{1}+\lambda_{2}u_{xx})\right]^{1/2}.$
(3.3)

Applying Proposition 1 we find that the fully-nonlinear second-order equation

$\displaystyle\boxed{\vphantom{\frac{DA}{DB}}v_{t}=-\left(\vphantom{\frac{A}{B}}1+\lambda_{1}^{2}\lambda_{2}v_{x}^{2}\right)\frac{v_{x}^{2}}{v_{xx}}-v}$
(3.4)

admits the Lie-Bäcklund symmetry generator
$\displaystyle{Z=\eta[v]\frac{\partial\ }{\partial v}}$ for which (3.2) is the
third-order flow, i.e.

$\displaystyle\eta(x,v,v_{x},v_{xx},v_{xxx})=\frac{\lambda_{1}^{3}}{4}\left(\vphantom{\frac{A}{B}}1+\lambda_{1}^{2}\lambda_{2}v_{x}^{2}\right)^{3/2}\left(\frac{v_{x}^{3}v_{xxx}}{v_{xx}^{3}}-\frac{3v_{x}^{2}}{v_{xx}}\right).$
(3.5)

Applying now Proposition 2 we find that equation (3.4) admits the recursion
operator $R_{11}[v]$ of the form (2.10), where

$\displaystyle
G_{1}=\frac{\lambda_{1}^{3}}{4}\left(\vphantom{\frac{A}{B}}1+\lambda_{1}^{2}\lambda_{2}v_{x}^{2}\right)^{1/2}\frac{v_{x}}{v_{xx}}$
(3.6a) $\displaystyle
G_{0}=-\frac{\lambda_{1}^{5}\lambda_{2}}{4}\left(\vphantom{\frac{A}{B}}1+\lambda_{1}^{2}\lambda_{2}v_{x}^{2}\right)^{-1/2}v_{x}^{2}$
(3.6b) $\displaystyle I=v_{x}$ (3.6c)
$\displaystyle\Lambda=\frac{\lambda_{1}^{5}\lambda_{2}}{4}\left(\vphantom{\frac{A}{B}}1+\lambda_{1}^{2}\lambda_{2}v_{x}^{2}\right)^{-3/2}v_{xx}$
(3.6d)

for which (3.2) is the second member $v_{t_{1}}$ in the hierarchy
$\displaystyle{v_{t_{m}}=R_{11}^{m}[v]v_{t}}$, $m=0,1,2,\ldots$ and $v_{t}$ is
the equation (3.4) .

Subcase 1.2: Consider equation (3.1), with $\lambda_{1}=0$ and
$\lambda_{2}=1$, i.e.

$\displaystyle u_{t}=\frac{u_{xx}^{6}}{u_{xxx}^{2}}$ (3.7)

Equation (3.7) admits the potentialisation [3]

$\displaystyle
v_{t}=\frac{v_{x}^{6}v_{xxx}}{v_{xx}^{3}}-\frac{3v_{x}^{5}}{v_{xx}},$ (3.8)

where

$\displaystyle v_{x}=-2^{1/3}u_{xx}.$ (3.9)

Applying Proposition 1 we find that the fully-nonlinear second-order equation

$\displaystyle\boxed{\vphantom{\frac{DA}{DB}}v_{t}=-\frac{v_{x}^{4}}{v_{xx}}}$
(3.10)

admits the Lie-Bäcklund symmetry generator
$\displaystyle{Z=\eta[v]\frac{\partial\ }{\partial v}}$ for which (3.8) is the
third-order flow, i.e.

$\displaystyle\eta(x,v,v_{x},v_{xx},v_{xxx})=\frac{v_{x}^{6}v_{xxx}}{v_{xx}^{3}}-\frac{3v_{x}^{5}}{v_{xx}}.$
(3.11)

Applying now Proposition 2 we find that equation (3.10) admits the recursion
operator $R_{12}[v]$ of the form (2.9), where

$\displaystyle G_{1}=\frac{v_{x}^{2}}{v_{xx}}$ (3.12a) $\displaystyle
G_{0}=-v_{x}$ (3.12b)

for which (3.8) is the second member $v_{t_{1}}$ in the hierarchy
$\displaystyle{v_{t_{m}}=R_{12}^{m}[v]v_{t}}$, $m=0,1,2,\ldots$ and $v_{t}$ is
the equation (3.10) .

Subcase 1.3: Consider equation (3.1), with $\lambda_{1}=-1$ and
$\lambda_{2}=0$, i.e.

$\displaystyle u_{t}=-\frac{u_{xx}^{3}}{u_{xxx}^{2}}$ (3.13)

For this equation we have established several potentialisations in [3]. We
treat the relevant cases below:

  * 1.3a:

Equation (3.13) admits the potentialisation [3]

$\displaystyle
v_{t}=\frac{2v_{x}^{3}v_{xxx}}{v_{xx}^{3}}-\frac{3v_{x}^{2}}{v_{xx}},$ (3.14)

where

$\displaystyle v_{x}=-u_{xx}.$ (3.15)

Applying Proposition 1 we find that the fully-nonlinear second-order equation

$\displaystyle\boxed{\vphantom{\frac{DA}{DB}}v_{t}=-\frac{v_{x}^{2}}{v_{xx}}}$
(3.16)

admits the Lie-Bäcklund symmetry generator
$\displaystyle{Z=\eta[v]\frac{\partial\ }{\partial v}}$ for which (3.14) is
the third-order flow, i.e.

$\displaystyle\eta(x,v,v_{x},v_{xx},v_{xxx})=\frac{2v_{x}^{3}v_{xxx}}{v_{xx}^{3}}-\frac{3v_{x}^{2}}{v_{xx}}.$
(3.17)

Applying now Proposition 2 we find that equation (3.16) admits the recursion
operator $R_{13}[v]$ of the form (2.9), where

$\displaystyle G_{1}=\frac{2v_{x}}{v_{xx}}$ (3.18a) $\displaystyle G_{0}=-1$
(3.18b)

for which (3.14) is the second member $v_{t_{1}}$ in the hierarchy
$\displaystyle{v_{t_{m}}=R_{13}^{m}[v]v_{t}}$, $m=0,1,2,\ldots$ and $v_{t}$ is
the equation (3.16).

  * 1.3b:

Furthermore we know from the results reported in [3] that a zero-order
potentialisation of equation (3.14) is

$\displaystyle V_{t}=\frac{V_{xxx}}{V_{xx}^{3}}-3\cdot
2^{-2/3}\frac{1}{V_{xx}}-2^{-1/3}\,x,$ (3.19)

where

$\displaystyle V_{x}=2^{-1/2}\ln(v_{x})$ (3.20)

gives the relation to (3.14) and

$\displaystyle V_{x}=2^{-1/2}\ln|-u_{xx}|$ (3.21)

the relation to (3.13). Applying Proposition 1 for (3.19) we find that the
fully-nonlinear second-order equation

$\displaystyle\boxed{\vphantom{\frac{DA}{DB}}V_{t}=-\frac{1}{V_{xx}}}$ (3.22)

admits the Lie-Bäcklund symmetry generator
$\displaystyle{Z=\eta[V]\frac{\partial\ }{\partial V}}$ for which (3.19) is
the third-order flow, i.e.

$\displaystyle\eta(x,V,V_{x},V_{xx},V_{xxx})=\frac{V_{xxx}}{V_{xx}^{3}}-3\cdot
2^{-2/3}\frac{1}{V_{xx}}-2^{-1/3}\,x.$ (3.23)

Applying now Proposition 2 we find that equation (3.22) admits the recursion
operator $R_{131}[V]$ of the form (2.10), where

$\displaystyle G_{1}=\frac{1}{V_{xx}}$ (3.24a) $\displaystyle G_{0}=3\cdot
2^{-2/3}$ (3.24b) $\displaystyle I=1$ (3.24c)
$\displaystyle\Lambda=2^{-1/3}V_{xx}$ (3.24d)

for which (3.19) is the second member $V_{t_{1}}$ in the hierarchy
$\displaystyle{V_{t_{m}}=R_{131}^{m}[V]V_{t}}$, $m=0,1,2,\ldots$ and $V_{t}$
is the equation (3.22).

  * 1.3c:

In [3] it was shown that a multi-potentialisation of (3.13) leads to

$\displaystyle w_{t}=\frac{w_{xxx}}{w_{xx}^{3}}-2^{-2/3}\frac{3}{w_{xx}},$
(3.25)

where

$\displaystyle w_{xx}=2^{-1/3}\frac{u_{xxx}}{u_{xx}}$ (3.26)

gives the relation to (3.13). Applying Proposition 1 for (3.25) we find that
the fully-nonlinear second-order equation

$\displaystyle\boxed{\vphantom{\frac{DA}{DB}}w_{t}=-\frac{1}{w_{xx}}}$ (3.27)

admits the Lie-Bäcklund symmetry generator
$\displaystyle{Z=\eta[w]\frac{\partial\ }{\partial w}}$ for which (3.25) is
the third-order flow, i.e.

$\displaystyle\eta(x,w,w_{x},w_{xx},w_{xxx})=\frac{w_{xxx}}{w_{xx}^{3}}-2^{-2/3}\frac{3}{w_{xx}}.$
(3.28)

Applying now Proposition 2 we find that equation (3.27) admits the recursion
operator $R_{132}[w]$ of the form (2.9), where

$\displaystyle G_{1}=\frac{1}{w_{xx}}$ (3.29a) $\displaystyle G_{0}=3\cdot
2^{-2/3}$ (3.29b)

for which (3.25) is the second member $w_{t_{1}}$ in the hierarchy
$\displaystyle{w_{t_{m}}=R_{132}^{m}[w]w_{t}}$, $m=0,1,2,\ldots$ and $w_{t}$
is the equation (3.27).

Case 2: Consider the fully-nonlinear symmetry-integrable equation [2]

$\displaystyle u_{t}=\frac{1}{u_{xxx}^{2}}.$ (3.30)

A multi-potentialisation of (3.30) leads the quasi-linear equation [3]

$\displaystyle v_{t}=\frac{v_{xxx}}{v_{xx}^{3}},$ (3.31)

where

$\displaystyle v_{x}=-2^{-1/3}u_{xx}.$ (3.32)

Applying Proposition 1 we find that the fully-nonlinear second-order equation

$\displaystyle\boxed{\vphantom{\frac{DA}{DB}}v_{t}=-\frac{1}{v_{xx}}}$ (3.33)

admits the Lie-Bäcklund symmetry generator
$\displaystyle{Z=\eta[v]\frac{\partial\ }{\partial v}}$ for which (3.31) is
the third-order flow, i.e.

$\displaystyle\eta(x,v,v_{x},v_{xx},v_{xxx})=\frac{v_{xxx}}{v_{xx}^{3}}.$
(3.34)

Applying now Proposition 2 we find that equation (3.33) admits the recursion
operator $R_{14}[v]$ of the form (2.9), where

$\displaystyle G_{1}=-\frac{1}{v_{xx}}$ (3.35a) $\displaystyle G_{0}=0$
(3.35b)

for which (3.31) is the second member $v_{t_{1}}$ in the hierarchy
$\displaystyle{v_{t_{m}}=R_{14}^{m}[v]v_{t}}$, $m=0,1,2,\ldots$ and $v_{t}$ is
the equation (3.33).

##  4 Concluding remarks

In this article we have provided examples of third-order fully-nonlinear
symmetry-integrable equations that are linked to second-order fully nonlinear
symmetrey-integrable equations. In particular, the examples of Case 1 and Case
2 show that certain potentialisations of considered third-order fully-
nonlinear equations are in fact the third-order flows of the Lie-Bäcklund
symmetries of particular fully-nonlinear second-order equations. Hence these
potential equations, which are quasi-linear third-order equations, are members
of hierarchies of symmetry-integrable equations for which the seed equation is
a second-order fully-nonlinear equation.

We should point out that not every fully-nonlinear third-order symmetry-
integrable equation can be linked in this way to a second-order equation.
Moreover, not every potential equation that results from third-order fully-
nonlinear symmetry-integrable equation belongs to a symmetry-integrable
hierarchy of second-order equations, even if it does so for other
potentialisations. For example, the fully-nonlinear third-order symmetry
integrable equation (3.7), viz

$\displaystyle u_{t}=\frac{u_{xx}^{6}}{u_{xxx}^{2}}$

is related to the fully-nonlinear second-order equation

$\displaystyle v_{t}=-\frac{v_{x}^{4}}{v_{xx}}$

via the potential equation

$\displaystyle
v_{t}=\frac{v_{x}^{6}v_{xxx}}{v_{xx}^{3}}-\frac{3v_{x}^{5}}{v_{xx}}$

as shown in Subcase 1.2. However, as reported in [3], equation (3.7) also
admits the zero-order potentialisation

$\displaystyle v_{t}=\frac{v_{xxx}}{v_{xx}^{3}}+\frac{3}{v_{x}v_{xx}},$ (4.1)

where

$\displaystyle v_{x}=\frac{1}{2^{1/3}u_{xx}}.$

Using Proposition 1 it is easy to shown that there exists no second-order
equation that admits the Lie-Bäcklund symmetry generator

$\displaystyle
Z=\left(\frac{v_{xxx}}{v_{xx}^{3}}+\frac{3}{v_{x}v_{xx}}\right)\frac{\partial\
}{\partial v}.$

We conclude that the potential equation (4.1) does not provide a link between
a second-order equation and the third-order fully-nonlinear equation (3.7).
Furthermore, an example of a fully-nonlinear third-order symmetry-integrable
equation that does not admit a potentialisation which links this equation to a
second-order symmetry-integrable equation is [3]

$\displaystyle
u_{t}=\frac{4u_{x}^{5}}{(2bu_{x}^{2}-2u_{x}u_{xxx}+3u_{xx}^{2})^{2}},$

for any constant $b$.

Finally, we give an example of a more general second-order fully-nonlinear
symmetry-integrable evolution than those that are known to use to be linked to
a fully-nonlinear third-order equation from the results reported in [3]:
Applying Proposition 1 and Proposition 2 we find that

$\displaystyle
v_{t}=-\frac{v_{x}^{2}}{v_{xx}}+c_{3}v+c_{4}v_{x}\ln(v_{x})+c_{4}+c_{5}v_{x}$
(4.2)

admits the recursion operator

$\displaystyle
R[v]=\frac{27}{4}\left(\frac{v_{x}}{v_{xx}}\right)D_{x}-\frac{27}{4}+\frac{27c_{3}}{4}+\alpha
v_{x}D_{x}^{-1}\circ\frac{v_{xx}}{v_{x}^{2}},$ (4.3)

where $\alpha$, $c_{3},\ c_{4}$ and $c_{5}$ are arbitrary constants. The
third-order equation in the hierarchy $v_{t_{m}}=R^{m}[v]v_{t}$ is then

$\displaystyle
v_{t_{1}}=\frac{27}{4}\frac{v_{x}^{3}v_{xxx}}{v_{xx}^{3}}-\frac{27}{4}\frac{v_{x}^{2}}{v_{xx}}+\alpha(c_{3}-1)xv_{x}+c_{3}\left(\frac{27c_{3}}{4}-\frac{27}{4}-\alpha\right)v+\frac{27}{4}(c_{4}+c_{3}c_{5})v_{x}$
$\displaystyle\qquad+\left(\frac{27c_{3}c_{4}}{4}+\alpha
c_{5}\right)v_{x}\ln(v_{x})+\frac{\alpha
c_{4}}{2}v_{x}\ln^{2}(v_{x})+c_{4}\left(\frac{27}{4}(c_{3}-1)-\alpha\right).$
(4.4)

By letting $c_{3}=c_{4}=c_{5}=\alpha=0$ we obtain

$\displaystyle
v_{t_{1}}=\frac{27}{4}\frac{v_{x}^{3}v_{xxx}}{v_{xx}^{3}}-\frac{27}{4}\frac{v_{x}^{2}}{v_{xx}}$
(4.5)

which is an equation that was obtained in [3] by a multi-potentialisation of
(3.13), viz

$\displaystyle u_{t}=-\frac{u_{xx}^{3}}{u_{xxx}^{2}}.$

The relation between (3.13) and (4.5) is given my

$\displaystyle\frac{v_{xx}}{v_{x}}=\frac{3}{2}\frac{u_{xxx}}{u_{xx}}.$ (4.6)

In view of this example one may ask whether this method could be applied to
construct further fully-nonlinear third-order symmetry-integrable evolution
equations different from those that have been reported in [2]. Furthermore,
the method proposed here could be exploited to the construct higher-order
fully-nonlinear symmetry-integrable equations.

## References

  * [1] Euler M and Euler N, Nonlocal invariance of the multipotentialisations of the Kupershmidt equation and its higher-order hierarchies In: Nonlinear Systems and Their Remarkable Mathematical Structures, N Euler (ed), CRC Press, Boca Raton, 317–351, 2018.
  * [2] Euler M and Euler N, On fully-nonlinear symmetry-integrable equations with rational functions in their highest derivative: Recursion operators, Open Communications in Nonlinear Mathematical Physics, 2, 216–228, 2022.
  * [3] Euler M and Euler N, Potentialisations of a class of fully-nonlinear symmetry-integrable evolution equations, Open Communications in Nonlinear Mathematical Physics, 4, 44–78, 2024.
  * [4] Fokas A S and Fuchssteiner B, On the structure of symplectic operators and hereditary symmetries, Lettere al Nuovo Cimento 28, 299–303, 1980.


# Quantumness Speeds up Quantum Thermodynamics Processes

Ming-Xing Luo 1,2,3,∗ 1School of Information Science and Technology, Southwest
Jiaotong University, Chengdu 610031, China
2CAS Center for Excellence in Quantum Information and Quantum Physics, Hefei,
230026, China  3Lead contact: Further information and requests for resources
should be directed to the lead contact (M.X.Luo<EMAIL_ADDRESS>
###### Abstract

Quantum thermodynamic process involves manipulating and controlling quantum
states to extract energy or perform computational tasks with high efficiency.
There is still no efficient general method to theoretically quantify the
effect of the quantumness of coherence and entanglement in work extraction. In
this work, we propose a thermodynamics speed to quantify the extracting work.
We show that the coherence of quantum systems can speed up work extracting
with respect to some cyclic evolution beyond all incoherent states. We further
show the genuine entanglement of quantum systems may speed up work extracting
beyond any bi-separable states. This provides a new thermodynamic method to
witness entangled systems without state tomography.

††journal: iScience

[table]capposition=top capbtabboxtable[][]

## Introduction

Quantum thermodynamics provides a bridge to explore energy transfer and
conversion of two systems at the microscopic level. By incorporating quantum
effects into thermodynamic systems, it can gain insights into the fundamental
limits of energy extraction and charging behavior of small-scale devices1-4.
So far, quantum thermodynamics has intrigued great improvements in energy
storage and transfer, such as quantum heat engines5-7 and quantum
refrigerators7-9. Quantum thermodynamics sheds light on the fundamental
principles governing the behavior of quantum systems, paving the way for
quantum computing and quantum information processing10,11.

Exploiting the quantum features of the coherence or entanglement beyond the
classical counterparts is one of the most important tasks in quantum
thermodynamics7,12,13. This can trace back to a basic problem from the birth
of thermodynamics in 1824, i.e., what criteria is useful to compare different
states with respect to their energies. In the thermodynamic process, a quantum
system can provide energy for the other systems or be charged by others. This
allows building battery-like quantum devices14. Lots of potential examples are
proposed from qubits15, spins16-18, flywheels19,20, to collision model21-23.
In general, it require auxiliary systems to control the batteries24-28.
Moreover, with the coherence some quantum batteries show interesting features
of faster and higher-power charging capability than the classical
counterparts14,25-31. This is recently extended for entangled quantum
systems32-40. Specially, most battery systems will generate entanglement under
global evolution, and then show an advantage over those with only local
operations32,37,41,42. These results show new quantum effects of the coherence
and entanglement in the thermodynamic processes of quantum batteries.

Recently, most quantum work extracting protocols focus on exploiting the
optimal final energy that can be exchanged for a given quantum battery under
cyclic control3. This intrigues to define the maximum of the amount of
extracted work as so-called ergotropy. As for the charging process, the
maximum of the amount of charged work as so-called antiergotropy. Combining
both quantities allows characterizing a given quantum battery using a
fundamental quantity of quantum battery capacity43. Dividing these quantities
by the evolution time period provides bare Hamiltonian-based ways to
characterize the energy transferring or charging processes27.

Figure 1: Schematic thermodynamic process of battery. A classical battery
system undergoes a cyclic controlling with the probing Hamiltonian within a
time period in the up protocol. The quantum battery system undergoes a cyclic
controlling with the same probing Hamiltonian within the same time period in
the down protocol. The final extracted energy is defined according to a given
bare Hamiltonian. The goal is to characterize the difference of the maximal
energy-exchange speed over time-independent probing Hamiltonians.

In this paper, we define a _quantum energy-exchange speed_ as a figure of the
thermodynamic energy transferring that will link to quantum features of the
coherence44 and entanglement47. We define an energy-exchange speed as the
operational ratio of the Hellinger distance51 of two final extractable
energies of one battery in two complement thermodynamic procedures. This
quantity is independent of the governor Hamiltonian and shows an operational
relationship to the statistical speed of quantum Fisher information52,53. We
further show the present quantity is the evolution speed of energies on a unit
geometric sphere. We propose a novel approach to exploit the energy extraction
of coherent or entangled battery systems based on the thermodynamics speed, as
shown in [fig-1]. We prove that quantum coherence can speed up the quantum
thermodynamic process beyond all incoherent states. We then show the entangled
battery systems can provide a speed up beyond separable states, even quadratic
speeding up in an Ising model. This allows witnessing both coherence and
entanglement with new thermodynamic features.

## Results

### Quantum energy-exchange speed

Consider a single-particle quantum $B$ which can exchange energy with external
sources. Define the evolution of $B$ is generated by a Hamiltonian
$\hat{H}(t)$ which accounts for energy exchanging. Assume $B$ is in a cyclic
process, in which originally isolated is coupled at the time $t=0$ to external
sources, and decouples from them at the time $t$ 2. The bare Hamiltonian is
defined by $H_{0}$. The battery is thermally isolated but may involve energy
exchanges between its parts, i.e., it does not involve heat exchange with a
thermal environment. The initial extractable energy is defined by

$\displaystyle E_{0}={\rm Tr}(\rho_{0}H_{0})$ (1)

with an initial state $\rho_{0}$ 2. The final extractable energy is then given
by $E_{t}={\rm Tr}(\rho(t)H_{0})$, where the final state $\rho(t)$ is defined
according to the Liouville-Neumann equation

$\displaystyle i\hbar\dot{\rho}=[H(t),\rho(t)].$ (2)

For continuous systems the optimal energy depends on von Neumann entropy of
final Gibbsian state 3. But this is not always right for finite systems.

In what follows, we consider a finite-dimensional single-particle battery
system. More precisely, define the evolution of $B$ under a unitary operation
$U$ as $\rho(t)=U\rho_{0}U^{\dagger}$. The bare Hamiltonian $H_{0}$ is
decomposed into

$\displaystyle
H_{0}=\sum_{j=0}^{d-1}\lambda_{j}E|\lambda_{j}\rangle\langle\lambda_{j}|,$ (3)

where $E$ denotes the unit energy and eigenenergies $\lambda_{i}$ satisfy
$\lambda_{0}\leq\cdots\leq\lambda_{d-1}$. Without loss of generality, we
associate zero energy to the lowest energetic state $|\lambda_{0}\rangle$.
When the quantum battery goes a cyclic evolution $U_{t}=e^{-it\hat{H}}$ with a
time-independent Hamiltonian $\hat{H}$, the final extractable energy is given
by55,56:

$\displaystyle E_{U_{t}}(\rho_{0};H_{0})\equiv{\rm
Tr}(U_{t}\rho_{0}U_{t}^{\dagger}H_{0}).$ (4)

We extend the result to multiparticle quantum batteries. Suppose an isolated
$N$-particle quantum battery is in the initial state $\rho_{0}$ on Hilbert
space. Intead of the interaction-free Hamiltonian
$\hat{H}(t)=\sum_{i=1}^{N}H_{i}$ with Hamiltonian $H_{i}$ of the $i$-th
particle, we consider the battery equipped with a total global Hamiltonian
$\hat{H}$. Suppose the quantum battery is subjected to a cyclic evolution with
a $k$-body correlated time-independent Hamiltonian $\hat{H}$ given by

$\displaystyle\hat{H}=\frac{1}{d}\sum_{i=1}^{N}\alpha_{i}\sigma_{\mathbf{v}_{i}}^{(i)}+\frac{\gamma}{d^{k}}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{k}=1\end{subarray}}^{N}\beta_{i_{1}\cdots
i_{k}}\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\cdots\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})},$
(5)

where $\alpha_{i}\in[0,1]$ account for the strength of local operations, and
$\beta_{i_{1}\cdots{}i_{k}}$ denote $k$-body correlating parameters satisfying
the symmetry condition of
$\beta_{i_{1}\cdots{}i_{k}}=\beta_{\tau(i_{1}\cdots{}i_{k})}$ for any
different $i_{1},\cdots,i_{k}$ and permutation $\tau$ in the permutation group
$S_{k}$. $\gamma$ is an arbitrary real number. Here,
$\sigma^{(i_{j})}_{\mathbf{u}_{i_{j}}}\equiv\vec{\sigma}^{(i_{j})}\cdot\mathbf{u}_{i_{j}}$
are $d\times d$ generalized Pauli matrices of the $j$-th particle, where
$\vec{\sigma}$ denotes the $d$-dimensional Gell-Mann matrix vector and
$\mathbf{u}_{i_{j}}$ is a unit vector on the Gell-Mann sphere54. After the
evolution period $t$, the final state is given by
$\rho=e^{-it\hat{H}}\rho_{0}e^{it\hat{H}}$. The final extractable energy has
the same form (4). This intrigues us to identify the difference of extractable
energies between a given evolution period. Specially, when $E_{U_{0}}\geq
E_{U_{t}}$, the difference of the extractable energy gives as $W\equiv
E_{U_{0}}-E_{U_{t}}$, which characterizes the work extracted from the battery.
On the other hand, it exhibits the amount of work being charged into the
battery from other systems when $E_{U_{0}}\leq E_{U_{t}}$. The corresponding
maximal works under any unitary evolutions are defined as the _ergotropy_ and
the _antiergotropy_ , respectively3,43.

Our goal in what follows is to characterize the extractable energy (4) in
terms of the quantum features of battery systems. The main idea is using the
thermodynamic speed to characterize the extractable energy that can be charged
or extracted from the battery systems within any cyclic control protocols, as
shown in Fig.1. Suppose that a given quantum battery in the state $\rho$ will
be probed by applying a global transformation $U_{t}\equiv e^{-it\hat{H}}$
with an evolution time $t$. Define an operational measure to quantify the
difference of extractable energies $E_{U_{t}}$ and $E_{U_{t^{\prime}}}$ during
the evolution period $[t,t^{\prime}]$ as

$\displaystyle\mathcal{D}_{E}(t,t^{\prime})$ $\displaystyle=$
$\displaystyle\left(\frac{1}{{\rm
Tr}H_{0}}\left(\sqrt{E_{U_{t}}(\rho;H_{0})}-\sqrt{E_{U_{t^{\prime}}}(\rho,H_{0})}\right)^{2}\right.$
(6)
$\displaystyle+\left.\left(\sqrt{E_{U_{t}}(\rho;\overline{H}_{0})}-\sqrt{E_{U_{t^{\prime}}}(\rho,\overline{H}_{0})}\right)^{2}\right)^{1/2},$

where $\overline{H}\equiv\mathbbm{1}-H_{0}/{\rm Tr}H_{0}$ defines the
complement Hamiltonian of $H_{0}$, with the corresponding final extractable
energy $E_{U_{t}}(\rho;\overline{H}_{0})={\rm Tr}(U_{t}\rho
U_{t}^{\dagger}\overline{H}_{0})$. Both quantities satisfy the operational
relationship of

$\displaystyle E_{U_{t}}(\rho,H_{0})+{\rm
Tr}{}H_{0}E_{U_{t}}(\rho,\overline{H}_{0})={\rm Tr}{}H_{0}$ (7)

for any $U_{t},\rho$ and $H_{0}$. The present distance (6) is zero, if and
only if two final states are same. The maximal difference is $\sqrt{2}E$ if
one final extractable energy is zero and the other is $E/2$. The metric (6)
can be regarded as a Hellinger distance of extractable energies52,53. This
allows us to define a thermodynamic speed to figure out how fast can the
extractable energy be exchanged under the cyclic control of the given battery
system as:

$\displaystyle v\equiv v(t)=\frac{d\mathcal{D}_{E}}{dt}\Big{|}_{t},$ (8)

i.e., the ratio at which $\mathcal{D}_{W}$ changes with $t^{\prime}$ around
the reference time $t$. From the Taylor expansion it follows that

$\displaystyle v^{2}$ $\displaystyle=$ $\displaystyle\frac{1}{4W_{U_{t}}{\rm
Tr}{}H_{0}}\dot{W}_{U_{t}}^{2}+\frac{1}{4W_{U_{t}}(\rho;\overline{H}_{0})}\dot{W}_{U_{t}}(\rho;\overline{H}_{0})^{2}$
(9) $\displaystyle=$ $\displaystyle\frac{1}{4W_{U_{t}}({\rm
Tr}{}H_{0}-W_{U_{t}})}\dot{W}_{U_{t}}^{2}$

with $\dot{W}_{U_{t}}\equiv\frac{dW_{U_{t}}}{dt}$. The present speed will be
used to explore the capability of extracting energies.

Define a maximal quantum energy-exchange speed as

$\displaystyle v_{w}\equiv\max_{\hat{H}}v,$ (10)

where the maximum is over all equipped Hamiltonians in the unitary evolution
protocols. This means the present quantity $v_{w}$ is independent of the
probing Hamiltonian and provides a general feature of the controlling protocol
while the known definition of quantum speed depends on the controlling
Hamiltonian8,27,35.

Inspired by the classical Fisher information53, the present quantity (10) can
be evaluated according to the symmetric logarithmic derivative as

$\displaystyle v^{2}_{w}={\rm Tr}(\rho\mathcal{W}^{2}),$ (11)

where the symmetric logarithmic derivative $\mathcal{W}$ is uniquely defined
on the support of $\rho$ via the relation
$\dot{\rho}=\tfrac{1}{4\sqrt{E}}(\mathcal{W}\rho+\rho\mathcal{W})$. This
implies the quantity $v^{2}_{w}$ is convex from the convexity of the Fisher
information57.

From Eq.(9) it follows that

$\displaystyle v_{w}=\frac{1}{2\sqrt{W_{U_{t}}({\rm
Tr}{}H_{0}-W_{U_{t}})}}\dot{W}_{U_{t}}$ (12)

if $\dot{W}_{U_{t}}\geq 0$. By integral over time period $[0,t]$ this implies
a new formula of the operational charging work as (Method):

$\displaystyle\mathcal{E}(\rho,H_{0})\equiv\int_{0}^{t}v_{w}dt=\arcsin\sqrt{\frac{W_{U_{t}}}{{\rm
Tr}{}H_{0}}}-\arcsin\sqrt{\frac{W_{U_{0}}}{{\rm Tr}{}H_{0}}}$ (13)

under a cyclic charging evolution $U_{t}$. Instead, if $\dot{W}_{U_{t}}\leq 0$
we get a new form of the operational extracting work as

$\displaystyle\mathcal{A}(\rho,H_{0})=\arcsin\sqrt{\frac{W_{U_{0}}}{{\rm
Tr}{}H_{0}}}-\arcsin\sqrt{\frac{W_{U_{t}}}{{\rm Tr}{}H_{0}}}$ (14)

under a cyclic discharging evolution $U_{t}$. Both quantity $\mathcal{E}$ and
$\mathcal{A}$ provide different metrics of quantum work procedure from
previous definitions43, as shown in Fig.2. This further intrigues to define
the maximal works of _ergotropy_ and _antiergotropy_ with the present metric,
which is valuable for further exploration.

Figure 2: Operational work extracting. For an $n$-particle quantum state
$\rho$ the normalized extracting work $W_{U_{t}}(\rho,H_{0})$ and dual work
$W_{U_{t}}(\rho,\overline{H}_{0})$ consist of a unit cycle. When the state
undergoes a unitary transformation $U_{t}$, the present quantity
$\mathcal{D}_{W}$ denotes the quantum work distance (green line) while
$\mathcal{A}(\rho,H_{0})$ denotes the evolution distance (orange arc).

Now, we present the main result to show the energy-exchange speed of quantum
battery systems in the superposition states44 and multipartite entangled
states47. Eq.(10) implies this quantity is independent of all bare
Hamiltonians. Given two batteries in the states $\rho$ and $\varrho$ on the
same Hilbert space, suppose both are under the same control with Hamiltonian
$\hat{H}$. From Eq.(11), if $v_{w}(\rho)^{2}\geq v_{w}(\varrho)^{2}$, it means
the battery in the state $\rho$ has a larger maximum energy-exchange speed
beyond the battery in the state $\varrho$. Informally, as for any isolated
quantum battery systems, its coherence can speed up the fastest energy-
exchange over all the incoherent systems. This provides a new state-
independent supremacy in quantum thermodynamic over all classical
counterparts. As for multiparticle quantum batteries, the optimal work
extraction may not generate multipartite entanglement33. However, we will show
the genuine entanglement63 may show a state-dependent supremacy to speed up
the fastest energy-exchange over biseparable systems. Here, the genuine
entanglement means it cannot be decomposed into a mixture of any biseparable
state $\rho_{S_{1}}\otimes\rho_{S_{2}}$, where $S_{1}$ and $S_{2}$ denotes a
bipartition of all particles.

Theorem 1. Given an $n$-particle quantum battery in the pure state
$|\phi\rangle$, the following results hold:

  * (i)

there exists a time-independent probing Hamiltonian $\hat{H}$ such that
$v_{w}(|\phi\rangle)^{2}>v_{w}(\rho_{ic})^{2}$ for any incoherent states
$\rho_{ic}$ if $|\phi\rangle$ is a coherent state;

  * (ii)

there exists a time-independent probing Hamiltonian $\hat{H}$ such that
$v_{w}(|\phi\rangle)^{2}\geq v_{w}(\rho_{bs})^{2}$ for any bi-separable state
$\rho_{bs}$ if $|\phi\rangle$ is genuinely entangled state.

The proof is inspired by recent methods53,58. For any pure states and a
control protocol with the time-independent probing Hamiltonian $\hat{H}$, the
quantum energy-exchange speed can be evaluated as

$\displaystyle v_{w}^{2}=E\langle(\Delta\hat{H})^{2}\rangle.$ (15)

To show the effect of quantum coherence, from the convexity of quantum energy-
exchange speed, it is sufficient to prove the result for all the isolated
systems, i.e., the pure states. A simple fact is that any two different
incoherent states are orthogonal. This allows finding a simple Hamiltonian for
each coherent state such that the average energy-exchange speed is larger than
its of any incoherent states. The detailed proof is shown in Method. For
multiparticle entangled batteries, the quantum energy-exchange speed depends
on specific Hamiltonian $H$ and its state, but is no less than its of any bi-
separable batteries.

### Coherent quantum batteries

Quantum coherence refers to the state of a quantum system where its
constituent particles are in a superposition44,45. Quantum coherent states are
characterized by their ability to exhibit interference effects, making them
valuable for applications such as quantum computing, quantum communication,
and quantum metrology. We quantify the maximal energy-exchange speed of a
given coherent battery. We estimate the largest energy-exchange speed over all
classically correlated states. From the convexity, we obtain the following
inequality

$\displaystyle v^{2}\leq v^{2}_{ic}\equiv\max_{|\psi_{ic}\rangle}v_{w}^{2},$
(16)

where the maximum is over all pure product incoherent states
$|\psi_{ic}\rangle\in\\{\otimes_{i=1}^{N}|i_{j}\rangle,i_{j}=0,\cdots,d-1\\}$.
Specially, for a given quantum battery the maximum energy-exchange speed can
be saturated by optimal Hamiltonians53.

As its derived in Method, we prove that the maximum energy-exchange of
incoherent state $|\psi_{ic}\rangle$ under the cyclic evolution with the
probing Hamiltonian (5) is given by

$\displaystyle
v_{ic}^{2}=\nu_{1}^{2}+\nu_{2}^{2}\gamma+\nu_{3}^{2}\gamma^{2},$ (17)

where $\nu_{1}=(\Delta\hat{H}_{1})^{2}$,
$\nu_{2}=\langle\\{\hat{H}_{1},\hat{H}_{2}\\}\rangle$ and
$\nu_{3}=(\Delta\hat{H}_{2})^{2}$ with local probing Hamiltonians
$\hat{H}_{1}$ and nonlocal probing Hamiltonians $\hat{H}_{2}$, i.e.,
$\hat{H}=\hat{H}_{1}+\hat{H}_{2}$. All the details of these correlations are
shown in Method. As $v_{ic}$ bounds the energy-exchange speed over all probing
Hamiltonians and all incoherent states, the coherent battery system has a
larger maximum energy-exchange speed than incoherent systems if the battery
states violate the inequality (16).

Example 1. Consider a multiple-qubit battery with a constant Hamiltonian,
i.e., $\gamma=0$. From Eq.(17) we obtain the maximum energy-exchange speed as

$\displaystyle
v_{ic}^{2}=\frac{E}{8}\sum_{\sigma^{(i)}_{\mathbf{v}_{i}}\not=\sigma_{z}}\alpha_{i}^{2}+\frac{E}{4}\sum_{\sigma^{(i)}_{\mathbf{v}_{i}}=\sigma_{z}}\alpha_{i}^{2}.$
(18)

This implies that for a product coherent pure state $|+\rangle^{\otimes N}$,
with the homogeneous Hamiltonian of $\alpha_{i}=a>0$, from Eq.(15) the local
probing Hamiltonian $\hat{H}=\sigma_{z}^{\otimes N}$ gives the maximum energy-
exchange speed satisfies $v_{Q}^{2}=N^{2}a^{2}/4$, which provides a quadratic
speed up beyond the maximum energy-exchange speed $v_{ic}^{2}=Na^{2}/4$ for
all incoherent states. For general coherent batteries, the maximum energy-
exchange speed depends on nonlocal probing Hamiltonians.

### Entangled batteries

Entanglement is a fundamental concept in quantum mechanics46-48. For an
entangled two particles measuring one particle can instantaneously affect the
state of the other46. This phenomenon cannot be described by a classical
understanding of cause and effect49,50. Here, we quantify the maximum energy-
exchange speed for an entangled battery. From the convexity of the maximum
energy-exchange speed we obtain the following inequality

$\displaystyle v^{2}\leq v^{2}_{fs}\equiv\max_{|\psi_{s}\rangle}v^{2}_{w},$
(19)

where the maximum is over all product pure states
$|\psi_{fs}\rangle\in\\{\otimes_{i=1}^{N}|\psi^{(i)}\rangle\\}$ or all bi-
separable product states
$|\psi_{bs}\rangle\in\\{|\psi\rangle_{S_{1}}|\psi\rangle_{S_{2}},\forall
S_{1}\cup S_{2}=\\{1,\cdots,N\\},S_{1}\cap S_{2}=\emptyset\\}$ for witnessing
entangled or genuinely entangled batteries, respectively63. Similar to the
proof for coherent systems in Method, the maximum energy-exchange speed of
$|\psi_{s}\rangle$ with the probing Hamiltonian $\hat{H}$ has the form (17)
with respect to separable states. The entangled battery may violate the
inequality (19). The general case depends on the probing Hamiltonians.

Example 2. Consider an Ising model with the $k$-paired nearest-neighbor
interaction59, i.e., $V_{ij}=\frac{1}{2k}\sum_{\forall j,0<|j-i|\leq
k}\delta(j,i)$ with Dirac delta function $\delta$ and
$\mathbf{v}\cdot\mathbf{u}=1$. $\gamma$ denotes the ferromagnetic coupling
factor. For the homogeneous case of $\alpha_{i}=a$ and
$\mathbf{v}=\mathbf{u}$, the optimal fully separable state for maximizing Eq.
(17) is given by $\otimes_{i=1}^{N}|\psi_{i}\rangle$, where $|\psi_{i}\rangle$
is a superposition state in terms of two eigenstates of $\sigma_{\mathbf{u}}$.
We obtain the maximum energy-exchange speed from Eq. (17) satisfy (Method):

$\displaystyle
v_{fs}^{2}=\frac{NE}{4}\left\\{\begin{split}&a^{2}+a_{0}\gamma^{2}+O(\gamma^{4}),\gamma\leq\gamma_{c};\\\
&\frac{a^{2}}{k}+a\gamma+(\frac{1}{k}+\frac{k}{N})\gamma^{2},\gamma\geq\gamma_{c};\end{split}\right.$
(20)

where $a_{0}=(8(N-k+1)ka-Nka^{2}+N+k^{2})/(4Nk)$. The critical value
$\gamma_{c}$ is defined with equal of two speeds in Eq.(20). Numerical
evaluations are shown in Fig.3.

Figure 3: Witness of entangled battery with the maximum energy-exchange speed.
Maximum energy-exchange speed of fully separable states, $v_{\max}$ (dots),
probed by the Ising Hamiltonian. The present Hamiltonian $\hat{H}$ provides a
method to witness entanglement if the battery has larger energy-exchange speed
than $v_{\max}$.

Under local probing Hamiltonians, the maximum energy-exchange speed of special
entangled states may be larger than its of fully separable batteries. For
exploring the quantum supremacy of general entangled battery, it requires both
local and nonlocal probing Hamiltonians.

Example 3. Consider the Dicke state $|N,k\rangle$ with $k$ number of excited
spins60. If the local probing Hamiltonian $\hat{H}_{1}$ satisfies
$\mathbf{v}\cdot\mathbf{u}=0$ and $k=2$, from Eq.(15) its maximum energy-
exchange speed is given by $v_{Q}^{2}=E((n+2)N-n^{2})/2$. This is larger than
the maximum energy-exchange speed $v_{fs}^{2}=EN/4$ for all fully separable
states when $n<N/2$.

Consider the maximally entangled Greenberger-Horne-Zeilinger (GHZ) state61, it
is easy to show the maximum energy-exchange speed $N^{2}E/4$ from Eq.(15),
which shows a quadratic speed up over all fully separable states with nonlocal
nearest-neighbor Hamiltonians.

## Discussion

In applications, if there are some averaged Hamiltonians $\langle
W\rangle_{t}=\sum_{i}p_{i}W_{i}(t)$, it is useful to extend the Hellinger
distance and energy-exchange speed to the work distribution $\overline{W}(t)$,
where $w_{0}=\frac{1}{m}\sum_{i=1}^{m}p_{i}W_{i}$ and $W_{1},\cdots,W_{m}$ are
extracted works from different bare Hamiltonians $H_{1},\cdots,H_{m}$. This
allows to characterize the statistical mixture of quantum work speed. But we
can not obtain the operational standard forms (13) and (14) of extracting
work. This intrigues new problems to explore the relationships between the
probing Hamiltonians and average energy-exchange speed.

The present energy-exchange speed reveals the quantum features of quantum
batteries in a unified manner. This requires to probe a battery system with a
generic multi-particle Hamiltonian. The present method shares several
important properties. One is that the quantum features of the coherence and
entanglement allow quantum supremacy of the maximum energy-exchange speed even
quadratic speed up. This shows the importance of quantum features in quantum-
size batteries for extracting or charging energies. The second is that the
present quantity provides a new way to witness the quantum properties of
coherence and entanglement hold in given quantum batteries. Beside the
possibility to witness a larger class of entangled states, the extracting work
speed allows to take in account the residual coupling among neighboring
particles62.

To conclude, the method discussed in this manuscript allows the experimental
characterization of a larger class of quantum states. The present results show
that entanglement can be detected even when the probing Hamiltonian is
nonlinear and therefore generates entanglement. This opens the way to study
entanglement near quantum phase transition points by quenching the parameters
of the governing many-body Hamiltonian.

## Limitations of Study

This paper aims to show the quantum properties of coherence and entanglement
can speed up quantum thermodynamics processes. The main limitation of the
proposed method is from the Fish information.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China
(Nos. 62172341,61772437), Sichuan Natural Science Foundation (No.
2023NSFSC0447), and Interdisciplinary Research of Southwest Jiaotong
University China (No.2682022KJ004).

## Author Contributions

M.X. conceived the study and wrote the manuscript.

## Declaration of Interests

The authors declare no competing interests.

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## STAR*Methods

### Resource Availability

#### Lead Contact

Further information and requests for resources should be directed to the lead
contact Ming-Xing Luo (mxluo@swjtu.edu.cn).

#### Materials Availability

This study did not generate new materials.

#### Data and Code Availability

This study has no data and code available.

### Key resources table

This study is no key resources table.

## Method details

### Proof of Eq.(13)

Since ${W_{U_{t}}}\leq{{\rm Tr}H_{0}}$ for any state $\rho$ and Hamiltonian
$H_{0}$, define $\sin\theta\equiv\sqrt{W_{U_{t}}/{\rm Tr}H_{0}}$ and
$\cos\theta\equiv\sqrt{1-W_{U_{t}}/{\rm Tr}H_{0}}$. This implies that

$\displaystyle\Delta\mathcal{D}_{w}$ $\displaystyle\equiv$
$\displaystyle\int_{0}^{t}vdt$ (21) $\displaystyle=$
$\displaystyle\int_{0}^{t}\frac{1}{2\sqrt{W_{U_{t}}({\rm
Tr}H_{0}-W_{U_{t}})}}\dot{W}_{U_{t}}dt$ $\displaystyle=$
$\displaystyle\int_{\arcsin(W_{U_{t}}/{\rm Tr}H_{0})}^{\arcsin(W_{U_{0}}/{\rm
Tr}H_{0})}\frac{1}{2\sin\theta\cos\theta}d\sin^{2}\theta$ $\displaystyle=$
$\displaystyle\arcsin\sqrt{\frac{W_{U_{t}}}{{\rm
Tr}H_{0}}}-\arcsin\sqrt{\frac{W_{U_{0}}}{{\rm Tr}H_{0}}}.$

### Proof of Theorem 1

The proof is inspired by recent methods53,58. Given a quantum battery in the
pure state $|\phi\rangle$, and a unitary transformation $e^{-i\hat{H}t}$ under
the time-independent probing Hamiltonian $\hat{H}$, the energy-exchange speed
is given by

$\displaystyle v_{w}^{2}(|\phi\rangle)=E\Delta\hat{H}^{2}.$ (22)

We firstly prove the result for quantum batteries in the coherent states. For
a given quantum battery in the coherent state $|\phi\rangle$, let the probing
Hamiltonian be $\hat{H}=|\psi_{ic}\rangle\langle\psi_{ic}|^{\otimes n}$, where
$|\psi_{ic}\rangle$ satisfies
$|\langle\psi_{ic}|\phi\rangle|^{2}\equiv\lambda\not=1/2$. This implies from
Eq.(22) that

$\displaystyle v(|\phi\rangle)^{2}$ $\displaystyle=$ $\displaystyle
E\langle\Delta\hat{H}^{2}\rangle_{|\phi\rangle}$ (23) $\displaystyle=$
$\displaystyle
E\langle\hat{H}-\langle\hat{H}\rangle^{2}\rangle_{|\phi\rangle}$
$\displaystyle=$ $\displaystyle E(\lambda-\lambda^{2})>0.$

But for any incoherent state $|\phi_{ic}\rangle$, we obtain that

$\displaystyle v(|\phi_{ic}\rangle)^{2}$ $\displaystyle=$ $\displaystyle
E\langle\Delta\hat{H}^{2}\rangle_{|\phi_{ic}\rangle}$ (24) $\displaystyle=$
$\displaystyle
E\langle\hat{H}-\langle\hat{H}\rangle^{2}\rangle_{|\phi_{ic}\rangle}$
$\displaystyle=$ $\displaystyle 0,$

where we have used the fact
$\langle\hat{H}\rangle_{|\phi_{ic}\rangle}\in\\{0,1\\}$ for incoherent states
$|\phi_{ic}\rangle$. This implies that
$v(|\phi\rangle)^{2}>\max_{|\phi_{ic}\rangle}v(|\phi_{ic}\rangle)^{2}$.

Now, we show the result for genuinely entangled batteries. Consider a given
isolated battery system in the entangled pure state $|\phi\rangle$. If one
makes use of the special entanglement witness operator47 of
$\mathcal{W}_{\phi}\equiv|\phi\rangle\langle\phi|-\lambda\mathbbm{1}$ as the
probing Hamiltonian $\hat{H}$, it follows that $v(|\phi\rangle)^{2}=0$ and
$v(|\phi_{ic}\rangle)^{2}\geq 0$, where $\lambda=\max_{\rho_{ic}}{\rm
Tr}(\rho_{ic}|\phi\rangle\langle\phi|)$. This means the well-known
entanglement witness is useless to design the probing Hamiltonian. Instead, we
use the following fact, i.e., there is a biseparable pure state
$|\phi_{bs}\rangle$ such that $|\langle\phi_{bs}|\phi\rangle|^{2}=1/2$. This
can be proved by using Schmidt decomposition of $|\phi\rangle$ in terms of any
bipartition of all particles. With this state, we define the probing
Hamiltonian as $\hat{H}=|\phi_{bs}\rangle\langle\phi_{bs}|$. It follows that

$\displaystyle v(|\phi\rangle)^{2}$ $\displaystyle=$ $\displaystyle
E\langle\Delta\hat{H}^{2}\rangle_{|\phi\rangle}$ (25) $\displaystyle=$
$\displaystyle
E\langle\hat{H}-\langle\hat{H}\rangle^{2}\rangle_{|\phi\rangle}=\frac{E}{4}.$

For any biseperable pure state $|\psi_{bs}\rangle$, we obtain that

$\displaystyle v(|\psi_{bs}\rangle)^{2}$ $\displaystyle=$ $\displaystyle
E\langle\Delta\hat{H}^{2}\rangle_{|\psi_{bs}\rangle}$ (26) $\displaystyle=$
$\displaystyle
E(\langle\hat{H}\rangle_{|\psi_{bs}\rangle}-\langle\hat{H}\rangle^{2}_{|\psi_{bs}\rangle})\leq\frac{E}{4}.$

This has proved the result for genuinely entangled batteries.

Similar proof holds for entangled batteries by ruling out all fully separable
batteries if there is a fully separable state $|\phi_{fs}\rangle$ such that
$|\langle\phi_{fs}|\phi\rangle|^{2}=1/2$.

### Proof of Eq. (17)

In this section, we evaluate the energy-exchange speed of incoherent systems
as

$\displaystyle
v_{ic}^{2}=\nu_{1}^{2}+\nu_{2}^{2}\gamma+\nu_{3}^{2}\gamma^{2},$ (27)

where $\nu_{i}$ are given by

$\displaystyle\nu_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{E}{d^{2}}\sum_{i=1}^{N}\alpha_{i}^{2}(1-s_{i}^{2}),$
$\displaystyle\nu_{2}^{2}$ $\displaystyle=$
$\displaystyle\frac{4E}{d^{k+1}}\sum_{\bf j}\alpha_{i}\beta_{\bf
j}(\mathbf{v}_{j_{t}}^{(j_{t})}\cdot{}\mathbf{u}_{j_{t}}^{(j_{t})}-(s_{j_{t}}^{(j_{t})})^{2})\prod_{\ell\not=t}s_{\mathbf{u}_{j_{\ell}}}^{(j_{\ell})},$
$\displaystyle\nu_{3}^{2}$ $\displaystyle=$
$\displaystyle\frac{E}{d^{2k}}\left[\sum_{\bf i}\beta^{2}_{\bf
i}(1-(s_{\mathbf{u}_{i_{1}}}^{(i_{1})}\cdots
s_{\mathbf{u}_{i_{k}}}^{(i_{k})})^{2})+\sum_{{\bf i,j},i_{t}\neq j_{t},\exists
t}\beta_{\bf i}\beta_{\bf
j}(1-\prod_{i_{t}=j_{\ell}}(s_{\mathbf{u}_{i_{t}}}^{(i_{t})})^{2})\prod_{i_{t}\not=j_{t}}s_{\mathbf{u}_{i_{t}}}^{(i_{t})}\right]$

with $s_{i}=\langle\sigma_{\mathbf{v}_{i}}^{(i)}\rangle$,
$s_{\mathbf{u}_{i}}^{(i)}=\langle\sigma_{\mathbf{u}_{i}}^{(i)}\rangle$,
$\mathbf{i}=i_{1}\cdots{}i_{k}$ and $\mathbf{j}=j_{1}\cdots{}j_{k}$.

Inspired by recent methods53,58, for pure states and unitary transformation
$e^{-i\hat{H}t}$, the energy-exchange speed is given by
$v_{w}^{2}(\gamma)=E\Delta\hat{H}^{2}$. Taking
$\hat{H}=\hat{H}_{1}+\gamma\hat{H}_{2}$, this equality implies that

$\displaystyle\nu_{1}^{2}$ $\displaystyle=$ $\displaystyle
E\Delta\hat{H}_{1}^{2},$ (28) $\displaystyle\nu_{2}^{2}$ $\displaystyle=$
$\displaystyle
E(\langle\\{\hat{H}_{1},\hat{H}_{2}\\}\rangle-2\langle\hat{H}_{1}\rangle\langle\hat{H}_{2}\rangle),$
(29) $\displaystyle\nu_{3}^{2}$ $\displaystyle=$ $\displaystyle
E\Delta\hat{H}_{2}^{2}.$ (30)

From the convexity of the energy-exchange speed, it is sufficient to show
$v_{\max}$ for all product incoherent pure states.

We first evaluate $\nu_{1}^{2}$ with respect to product incoherent pure
states. For the single-particle Hamiltonian $\hat{H}_{1}$ given by
$\hat{H}_{1}=\frac{1}{d}\sum_{i=1}^{N}\alpha_{i}\sigma_{\mathbf{v}_{i}}^{(i)}$,
we obtain that

$\displaystyle\langle\hat{H}_{1}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{d^{2}}\sum_{i,j=1}^{N}\alpha_{i}\alpha_{j}\langle\sigma_{\mathbf{v}_{i}}^{(i)}\sigma_{\mathbf{v}_{j}}^{(j)}\rangle$
(31) $\displaystyle:=$
$\displaystyle\frac{1}{d^{2}}\sum_{i,j=1}^{N}\alpha_{i}\alpha_{j}s_{ij},$

where
$s_{ij}=\langle\sigma_{\mathbf{v}_{i}}^{(i)}\sigma_{\mathbf{v}_{j}}^{(j)}\rangle$.
If all vectors $\mathbf{v}_{j}$ are the same, it follows a special case of
$s_{ij}=1$ for $i=j$; and $0$ for other cases. Moreover, we obtain that

$\displaystyle\langle\hat{H}_{1}\rangle^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{d^{2}}\sum_{i,j=1}^{N}\alpha_{i}\alpha_{j}\langle\sigma_{\mathbf{v}_{i}}^{(i)}\rangle\langle\sigma_{\mathbf{v}_{j}}^{(j)}\rangle$
(32) $\displaystyle:=$
$\displaystyle\frac{1}{d^{2}}\sum_{i,j=1}^{N}\alpha_{i}\alpha_{j}s_{i}s_{j},$

where $s_{i}=\langle\sigma_{\mathbf{v}_{i}}^{(i)}\rangle$. This implies from
the equality (28) that

$\displaystyle\nu_{1}^{2}$ $\displaystyle=$ $\displaystyle
E(\langle\hat{H}_{1}^{2}\rangle-\langle\hat{H}_{1}\rangle^{2})$ (33)
$\displaystyle=$
$\displaystyle\frac{E}{d^{2}}\sum_{i,j=1}^{N}\alpha_{i}\alpha_{j}(s_{ij}-s_{i}s_{j})$
$\displaystyle=$
$\displaystyle\frac{E}{d^{2}}\sum_{i=1}^{N}\alpha_{i}^{2}(1-s_{i}^{2})$

as $s_{ij}=s_{i}s_{j}$ for product incherent pure states.

Now, we evaluate $\nu_{2}^{2}$ with respect to product incoherent pure states.
Let $\hat{H}_{2}$ be given by
$\hat{H}_{2}=\frac{1}{d^{k}}\sum_{\begin{subarray}{c}i_{1},\cdots,i_{k}=1\end{subarray}}^{N}\beta_{i_{1}\cdots
i_{k}}\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\cdots\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}$.
We obtain that

$\displaystyle\langle\\{\hat{H}_{1},\hat{H}_{2}\\}\rangle$ $\displaystyle=$
$\displaystyle\langle\hat{H}_{1}\hat{H}_{2}\rangle+\langle\hat{H}_{2}\hat{H}_{1}\rangle$
(34) $\displaystyle=$ $\displaystyle\frac{1}{d^{k+1}}\left[\sum_{i,{\bf
j}}\alpha_{i}\beta_{\bf
j}\langle\sigma_{\mathbf{v}_{i}}^{(i)}\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle+\sum_{i,{\bf
j}}\alpha_{i}\beta_{\bf
j}\langle\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\sigma_{\mathbf{v}_{i}}^{(i)}\rangle\right]$
$\displaystyle=$ $\displaystyle\frac{2}{d^{k+1}}\left[\sum_{\bf
j}\alpha_{i}\beta_{\bf
j}(\mathbf{v}_{j_{t}}^{(j_{t})}\cdot{}\mathbf{u}_{j_{t}}^{(j_{t})}\prod_{\ell\not=t}s_{\mathbf{u}_{j_{\ell}}}^{(j_{\ell})}+\sum_{j_{1},\cdots,j_{k+1}}\alpha_{j_{k+1}}\beta_{\bf
j}s_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots
s_{\mathbf{u}_{j_{k+1}}}^{(j_{k+1})}\right]$

where $s_{\mathbf{u}_{i}}^{(i)}=\langle\sigma_{\mathbf{u}_{i}}^{(i)}\rangle$.
Moreover, we obtain that

$\displaystyle\langle\hat{H}_{1}\rangle\langle\hat{H}_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\hat{H}_{1}\hat{H}_{2}\rangle+\langle\hat{H}_{2}\hat{H}_{1}\rangle$
(35) $\displaystyle=$ $\displaystyle\frac{1}{d^{k+1}}\sum_{i,{\bf
j}}\alpha_{i}\beta_{\bf
j}\langle\sigma_{\mathbf{v}_{i}}^{(i)}\rangle\langle\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\rangle\cdots\langle\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{d^{k+1}}\left[\sum_{\bf
j}\alpha_{i}\beta_{\bf
j}(s_{j_{t}}^{(j_{t})}s_{j_{t}}^{(j_{t})}\prod_{\ell\not=t}s_{\mathbf{u}_{j_{\ell}}}^{(j_{\ell})}+\sum_{j_{1},\cdots,j_{k+1}}\alpha_{j_{k+1}}\beta_{\bf
j}s_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots
s_{\mathbf{u}_{j_{k+1}}}^{(j_{k+1})}\right]$

This implies from the equality (28) that

$\displaystyle\nu_{2}^{2}$ $\displaystyle=$ $\displaystyle
E(\langle\\{\hat{H}_{1},\hat{H}_{2}\\}\rangle-2\langle\hat{H}_{1}\rangle\langle\hat{H}_{2}\rangle)$
(36) $\displaystyle=$ $\displaystyle\frac{4E}{d^{k+1}}\sum_{\bf
j}\alpha_{i}\beta_{\bf
j}(\mathbf{v}_{j_{t}}^{(j_{t})}\cdot{}\mathbf{u}_{j_{t}}^{(j_{t})}-s_{j_{t}}^{(j_{t})}s_{j_{t}}^{(j_{t})})\prod_{\ell\not=t}s_{\mathbf{u}_{j_{\ell}}}^{(j_{\ell})}.$

Now, we evaluate $\nu_{3}^{2}$ with respect to product incherent pure states.
We obtain that

$\displaystyle\langle\hat{H}_{2}^{2}\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{d^{2k}}\sum_{\bf i,j}\beta_{\bf i}\beta_{\bf
j}\langle\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle,$
(37)

where ${\bf i}=i_{1}\cdots i_{k}$. Moreover, we obtain that

$\displaystyle\langle\hat{H}_{2}\rangle^{2}=\frac{1}{d^{2k}}\sum_{\bf
i,j}\beta_{\bf i}\beta_{\bf
j}\langle\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\rangle\langle\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\rangle\cdots\langle\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}\rangle\langle\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle.$
(38)

This implies from the equality (30) that

$\displaystyle\nu_{3}^{2}$ $\displaystyle=$ $\displaystyle
E(\langle\hat{H}_{2}^{2}\rangle-\langle\hat{H}_{2}\rangle^{2})$ (39)
$\displaystyle=$ $\displaystyle\frac{E}{d^{2k}}\sum_{\bf i,j}\beta_{\bf
i}\beta_{\bf
j}(\langle\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle-\langle\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\rangle\langle\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\rangle\cdots\langle\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}\rangle\langle\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle).$

Given a product incoherent state, for $i_{t}\not=j_{t}$ with any
$t=1,\cdots,k$, we obtain that

$\displaystyle\langle\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle=\langle\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\rangle\langle\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\rangle\cdots\langle\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}\rangle\langle\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle.$
(40)

If ${\bf i}={\bf j}$, we obtain that

$\displaystyle\langle\sigma_{\mathbf{u}_{i_{1}}}^{(i_{1})}\sigma_{\mathbf{u}_{j_{1}}}^{(j_{1})}\cdots\sigma_{\mathbf{u}_{i_{k}}}^{(i_{k})}\sigma_{\mathbf{u}_{j_{k}}}^{(j_{k})}\rangle=1.$
(41)

This implies that

$\displaystyle\nu_{3}^{2}$ $\displaystyle=$ $\displaystyle
E(\langle\hat{H}_{2}^{2}\rangle-\langle\hat{H}_{2}\rangle^{2})$ (42)
$\displaystyle=$ $\displaystyle\frac{E}{d^{2k}}\sum_{\bf i}\beta^{2}_{\bf
i}(1-(s_{\mathbf{u}_{i_{1}}}^{(i_{1})}\cdots
s_{\mathbf{u}_{j_{k}}}^{(j_{k})})^{2})+\sum_{\begin{subarray}{c}\bf i,j\\\
i_{t}\neq j_{t},\exists t\end{subarray}}\beta_{\bf i}\beta_{\bf
j}(1-\prod_{i_{t}=j_{\ell}}(s_{\mathbf{u}_{i_{t}}}^{(i_{t})})^{2})\prod_{i_{x}\not=b_{y}}s_{\mathbf{u}_{i_{t}}}^{(i_{t})}.$

### Ising model

Let us consider an Ising model with the $k$-pair nearest-neighbor interaction,
i.e., $V_{ij}=\frac{1}{2k}\sum_{\forall j,0<|j-i|\leq k}\delta(j,i)$ with
Dirac delta function $\delta$ and $\mathbf{v}\cdot\mathbf{u}=1$. For $\gamma$
smaller than a critical value $\gamma_{c}$, the maximum speed $v_{fs}$ is
achieved when $\langle\sigma^{(i)}_{\mathbf{u}}\rangle$ are all equal to $s$.
From Eq. (17), with forward tedious evaluations the energy-exchange speed for
fully separable states is given by

$\displaystyle
v_{fs}^{2}=\nu_{1}^{2}+\nu_{2}^{2}\gamma+\nu_{3}^{2}\gamma^{2},$ (43)

where $\nu_{i}$ are given by

$\displaystyle\nu_{1}^{2}$ $\displaystyle=$
$\displaystyle\frac{NE}{4}a^{2}(1-s^{2}),$ $\displaystyle\nu_{2}^{2}$
$\displaystyle=$ $\displaystyle\frac{E}{2}a(N-k+1)(s-s^{3}),$
$\displaystyle\nu_{3}^{2}$ $\displaystyle=$
$\displaystyle\frac{E}{4}[\frac{1}{4k}(N+k^{2})+((1-\frac{1}{2k})N-\frac{5k}{6}-1)s^{2}-((1-\frac{1}{4k})N-\frac{7k}{12}-\frac{9}{8})s^{4}].$

For a large $N$, $\gamma\ll 1$ and $0\leq a\leq 1$, we find
$s=\gamma+O(\gamma^{3})$ for optimizing $v_{fs}^{2}$ as

$\displaystyle v_{fs}^{2}$ $\displaystyle=$
$\displaystyle\frac{NE}{4}\left[a^{2}(1-s^{2})+2a(s-s^{3})(1-\frac{k-1}{N})\gamma+g(s)\gamma^{2}\right]$
(44) $\displaystyle=$
$\displaystyle\frac{NE}{4}[a^{2}+a_{0}\gamma^{2}+O(\gamma^{4})],$

where $a_{0}=(8(N-k+1)ka-Nka^{2}+N+k^{2})/(4Nk)$.

For $\gamma>\gamma_{c}$, we obtain the optimized energy-exchange speed when
$\langle\sigma_{\mathbf{v}}^{(i)}\rangle=0$ for all $i$, and
$\langle\sigma_{\mathbf{u}}^{(kj+1)}\rangle=1$ and
$\langle\sigma_{\mathbf{u}}^{(kj)}\rangle=\cdots=\langle\sigma_{\mathbf{u}}^{(kj+k-1)}\rangle=0$
for all $j$. This implies that $\nu_{1}^{2}=Na^{2}E/4k,\nu_{2}^{2}=(N-1)aE/4$
and $\nu_{3}^{2}=(N+k^{2})E/4k$, i.e.,

$v_{fs}^{2}(\beta)=\frac{NE}{4}\left[\frac{a^{2}}{k}+a\gamma+(\frac{1}{k}+\frac{k}{N})\gamma^{2}\right].$
(45)

Indeed, in the limit $\gamma\gg 1$, Eq. (45) goes as $NE\gamma^{2}/4k$ and
thus is larger than Eq. (43) for any $k$, which goes as
$Nk(4k-1+4N)\gamma^{2}E/(16N(4k-1))$. The critical value of $\gamma$ is
obtained setting Eq. (43) is equal to the maximum over $s$ of Eq. (45), and
given by

$\displaystyle\gamma_{c}=-\frac{c_{2}}{2c_{1}}+\frac{1}{2c_{1}}\sqrt{c_{2}^{2}-4c_{1}c_{3}},$
(46)

where $c_{i}$ are defined by

$\displaystyle c_{1}$ $\displaystyle=$
$\displaystyle-\frac{3}{4k}-\frac{3k}{4N}+\frac{1}{N}((1-\frac{1}{2k})N-\frac{5k}{6}-1)s^{2}$
$\displaystyle-\frac{1}{N}((1-\frac{1}{4k})N-\frac{7k}{12}-\frac{9}{8})s^{4}-\frac{1}{k}-\frac{k}{N},$
$\displaystyle c_{2}$ $\displaystyle=$ $\displaystyle a(2(s-s^{3})(N-k+1)-1),$
$\displaystyle c_{3}$ $\displaystyle=$ $\displaystyle
a^{2}(\frac{k-1}{k}-s^{2}).$

The upped bound of $v^{2}_{\max}$ for the inhomogeneous case ($\alpha_{i}\neq
0$) can be obtained by maximizing each term in Eq. (17) separately. This gives

$\displaystyle
v^{2}_{\max}\leq\frac{E}{4}\sum_{i=1}^{N}\alpha_{i}^{2}+\frac{E\alpha_{0}}{4}\gamma+\frac{NE}{8}\gamma^{2}$
(47)

with $\alpha_{0}=\max\\{\sum_{{\rm odd}\,i}\alpha_{i},\sum_{{\rm
even}\,i}\alpha_{i}\\}$.