# Decay-angular-distribution correlated $C\\!P$ violation in heavy hadron
cascade decays

Yu-Jie Zhao  Zhen-Hua Zhang  School of Nuclear Science and Technology,
University of South China, Hengyang, 421001, Hunan, China  Xin-Heng Guo
College of Nuclear Science and Technology, Beijing Normal University, Beijing
100875, China

###### Abstract

$C\\!P$ violation in baryon decay processes is still undiscovered to date. We
present a general analysis of the decay-angular-distributions and the
corresponding $C\\!P$ asymmetries in cascade decays of the type $\mathbb{H}\to
R(\to ab)c$, where $\mathbb{H}$ is a heavy hadron that decays through weak
interactions $\mathbb{H}\to Rc$, and the resonance $R$ decays strongly via
$R\to ab$. Based on the analysis, we propose to search for $C\\!P$ violation
in the decay-angular-distributions in the cascade decay processes
${\mathbb{B}}\to\mathcal{B}M$, with $\mathcal{B}$ or $M$ subsequently decaying
through strong interactions, where ${\mathbb{B}}$ is the mother baryon,
$\mathcal{B}$ or $M$ are the daughter baryon and meson, respectively, and $M$
has to be spin-nonzero. We also present some typical decay channels in which
the search for such kinds of $C\\!P$ asymmetries can be performed.

##  I introduction

In the Observable Universe, the density of baryons is by far larger than that
of anti-baryons [1], which is a clear evidence of Charge-Parity ($C\\!P$)
violation and Charge-conjugate ($C$) violation, according to Sakharov’s
criteria for the generation of the Baryon Asymmetry of the Universe [2].
$C\\!P$ violation (CPV) has been observed in the decays of $K$ [3], $D$ [4],
$B$ [5, 6, 7] and $B_{s}$ [8] mesons, all of which are consistent with the
description of the Standard Model of particle physics, in which a single
$C$-violating weak phase in the Cabibbo–Kobayashi–Maskawa (CKM) matrix is the
origin of CPV [9, 10]. Nevertheless, CPV in baryon decay processes is still
undiscovered in laboratories to date, despite the fact that many efforts have
been made.

In baryon decay processes, the non-zero baryon spin provides us with more
freedom to construct CPV observables. The $C\\!P$ asymmetries (CPAs) induced
by the decay asymmetry parameters in the hyperon weak decay transitions
${\frac{1}{2}}^{+}\to{\frac{1}{2}}^{+}+0^{-}$ such as $\Xi\to\Lambda\pi$ and
$\Lambda\to p\pi$ are one kind of typical examples. The decay asymmetry
parameters of transitions ${\frac{1}{2}}^{+}\to{\frac{1}{2}}^{+}+0^{-}$, are
induced by the interference of the parity-even S-wave and the parity-odd
P-wave amplitudes [11]. They can be measured from the decay-angular-
distribution if (1) the mother baryon is polarized, and/or (2) the daughter
baryon decays subsequently via weak interactions. The decay asymmetry
parameter induced CPAs in hyperon decays, which have been investigated
extensively on the theoretical side [12, 13, 14, 15], are expected to be
larger than those induced by the branching ratios in the aforementioned
hyperon decay channels. Recently, experimental studies on the hyperon decay
asymmetry parameter induced CPAs were performed by BESIII and Belle, with the
precision at about one percent level [16, 17]. Moreover, decay asymmetry
parameter induced CPAs in various charmed baryon decay channels such as
$\Lambda_{c}^{+}\to(\Lambda,\Sigma^{0})h^{+}$ and
$\Sigma_{c}^{0}\to\Sigma^{-}\pi^{+}$ were also investigated by Belle [17, 18].
However, they are still too small to be confirmed experimentally at the
current stage for both the hyperon and the charmed baryon decays. Theoretical
analyses of CPA induced by the decay parameters were also made in bottom
baryon decays such as $\Lambda_{b}^{0}\to\Lambda D$ [19, 20], while the
corresponding experimental study is still absent currently in bottom baryon
decays.

Just like the case of the aforementioned decay asymmetry parameter induced
CPAs, the mechanism of generating CPV via the interference between different
canonical amplitudes is a good idea for decay processes with particles of non-
zero spin involved, and is complementary to CPAs corresponding to the partial
decay width. It should be pointed out, however, that the interference between
the parity-even and -odd amplitudes can only show up in the angular-
distributions of the final particles when the subsequent decay is also a weak
one. In the situation that the corresponding subsequent decay is through
strong or electromagnetic interactions which respect parity symmetry, the
interference between parity-even and -odd amplitudes would be simply absent
111The interference between parity-even and -odd amplitudes will show up when
there are two resonances with similar masses but opposite parities [21, 22]..
Interestingly, for some decay processes, when more than one parity-even and/or
-odd amplitudes enter in the decay, the interference between the amplitudes
with the same parity can then show up in the decay-angular-distributions, even
when the subsequent decay is a strong one. We will analyse this in more detail
in this paper.

This paper is organized as follows. In Sec. II, we present the general
analysis of the decay-angular-distribution correlated CPAs. In Sec. III, by
expressing the decay-angular-distribution in terms of the canonical decay
amplitudes, the interfering behaviour between different canonical decay
amplitudes can be seen in a more transparent way. In Sec. IV, we present some
suggested decay channels in which the decay-angular-distribution correlated
CPAs are suitable for searching for. In the last section, we make our
conclusion.

##  II  General analysis of the Decay-angular-distributions and the correlated
CPAs

In general, we will consider a mother hadron $\mathbb{H}$ decaying via a weak
decay process $\mathbb{H}\to Rc$ with the intermediate resonance $R$ decaying
through strong interactions $R\to ab$. The differential decay width of the
aforementioned cascade decay process $\mathbb{H}\to R(\to ab)c$ for
unpolarized ${\mathbb{H}}$ can be expressed as [8] 222One reason why we only
consider the unpolarized case is that the polarizations of the heavy baryons
produced on colliders are still too small to be detected [23, 24, 25].,

$\frac{d\Gamma_{\mathbb{H}\to R(\to
ab)c}}{ds_{ab}ds_{ac}}=\frac{1}{(2\pi)^{3}}\frac{1}{32m_{\mathbb{H}}^{3}}\overline{\left|\mathcal{M}\right|^{2}},$
(1)

where $s_{ij}$ is the invariant mass squared of the particle $i$ and particle
$j$ system, $m_{\mathbb{H}}$ is the mass of $\mathbb{H}$, and
$\overline{\left|\mathcal{M}\right|^{2}}$ is the decay amplitude squared of
the cascade decay $\mathbb{H}\to R(\to ab)c$, which is defined as
$\overline{\left|\mathcal{M}\right|^{2}}\equiv\frac{1}{2s_{\mathbb{H}}+1}\sum_{m_{z},\lambda_{a},\lambda_{b},\lambda_{c}}\overline{\left|\mathcal{M}^{s_{\mathbb{H}},m_{z}}_{\lambda_{a}\lambda_{b}\lambda_{c}}\right|^{2}}$,
where $\mathcal{M}^{s_{\mathbb{H}},m_{z}}_{\lambda_{a}\lambda_{b}\lambda_{c}}$
is the covariant decay amplitude for the cascade decay $\mathbb{H}\to R(\to
ab)c$, $s_{\mathbb{H}}$ is the spin of $\mathbb{H}$ and $m_{z}$ is its
$z$-component (the direction of $z$ is irrelevant here since we are dealing
with unpolarized $\mathbb{H}$), $\lambda_{a}$, $\lambda_{b}$, and
$\lambda_{c}$ are the helicities of $a$, $b$, and $c$, respectively. The decay
amplitude squared can be further expressed as [22]

$\overline{\left|\mathcal{M}\right|^{2}}=\sum_{\stackrel{{\scriptstyle 0\leq
j\leq 2{s_{R}}}}{{j~{}\text{even}}}}w^{(j)}P_{j}\left(c_{\theta}\right),$ (2)

where $P_{j}$ is the $j$-th Legendre polynomial, $c_{\theta}\equiv\cos\theta$
with $\theta$ being the helicity angle of particle $a$ with respect to $c$ (or
equivalently, to $\mathbb{H}$) in the center-of-mass frame of the $a\&b$
system, $s_{R}$ is the spin quantum number for resonance $R$. Note that $j$
can only take even values (from 0 to $2s_{R}$), because the decay of $R$ is
through strong interactions which respect the parity symmetry [22].

Obviously, all the weights $w^{(j)}$ descibe the decay-angular-distributions,
i.e., the angular distributions of the final particles. The weight $w^{(j)}$
for the $j$-th Legendre polynomial in Eq. (2) can be expressed as

$\displaystyle w^{(j)}=\frac{{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}^{2}}{\left|s_{ab}-m_{R}^{2}+im_{R}\Gamma_{R}\right|^{2}}\mathcal{W}^{(j)}\mathcal{S}^{(j)},$
(3)

where $\mathcal{W}^{(j)}$ and $\mathcal{S}^{(j)}$ contains the information for
the decay $\mathbb{H}\to Rc$ and $R\to ab$, respectively, and can be expressed
in terms of the helicity decay amplitudes as

$\displaystyle\mathcal{W}^{(j)}$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\frac{(-)^{\sigma-s_{R}}\langle
s_{R},-\sigma;s_{R},\sigma|s_{R}s_{R}j0\rangle}{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}\sum_{\lambda_{c}}\left|\mathcal{F}^{s_{\mathbb{H}}}_{\sigma\lambda_{c}}\right|^{2},$
(4)

and

$\displaystyle\mathcal{S}^{(j)}$ $\displaystyle=$
$\displaystyle\left.\sum_{\lambda_{a}\lambda_{b}}\frac{(-)^{-\lambda+s_{R}}\langle
s_{R},-\\!\lambda;s_{R},\lambda|s_{R}s_{R}j0\rangle}{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}\left|\mathcal{G}^{s_{R}}_{\lambda_{a}\lambda_{b}}\right|^{2}\right|_{\lambda=\lambda_{a}-\lambda_{b}},$
(5)

with $\mathcal{F}^{s_{\mathbb{H}}}_{\sigma\lambda_{c}}$ and
$\mathcal{G}^{s_{R}}_{\lambda_{a}\lambda_{b}}$ being the helicity decay
amplitudes of ${\mathbb{H}}\to Rc$ and $R\to ab$, respectively, $\sigma$ being
the helicity of $R$, “$\langle\cdots|\cdots\rangle$” being the notation for
Clebsch-Gordan coefficients. Our definitions of $\mathcal{W}^{(j)}$ and
$\mathcal{S}^{(j)}$ are slightly different from those in previous works such
as Ref. [22], i.e., with an extra factor $\frac{1}{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}$. The motivation for the
introduction of this factor is to guarantee that
$\mathcal{W}^{(0)}=\sum_{\sigma,\lambda_{c}}\left|\mathcal{F}^{s_{\mathbb{H}}}_{\sigma\lambda_{c}}\right|^{2}$
and
$\mathcal{S}^{(0)}=\sum_{\lambda_{a}\lambda_{b}}\left|\mathcal{G}^{s_{R}}_{\lambda_{a}\lambda_{b}}\right|^{2}$.
Moreover, one can see that all the coefficients in front of the helicity
amplitudes squared in $\mathcal{W}^{(j)}$ and $\mathcal{S}^{(j)}$ are now
simple rational numbers. One can relax the constraint of $j$ taking even
values in Eq. (2), because $\mathcal{S}^{(j)}$ automatically equals to 0 for
odd $j$ due to the parity symmetry in $R\to ab$ [22].

We will focus on the decay-angular-distributions with respect to the helicity
angle $\theta$. To this end, we need to integrate out $s_{ab}$. In the narrow
width approximation of $R$, the differential decay width of the cascade decay
$\mathbb{H}\to R(\to ab)c$ can then be expressed as

$\displaystyle\frac{1}{\Gamma_{\mathbb{H}\to R(\to
ab)c}}\frac{d\Gamma_{\mathbb{H}\to R(\to
ab)c}}{dc_{\theta}}=\frac{1}{2}\sum_{\stackrel{{\scriptstyle 0\leq j\leq
2{s_{R}}}}{{j~{}\text{even}}}}\gamma_{\mathbb{H}\to R(\to
ab)c}^{(j)}P_{j}(c_{\theta}),$ (6)

where $\gamma_{\mathbb{H}\to R(\to ab)c}^{(j)}$ can be further parameterized
as $\gamma_{\mathbb{H}\to R(\to ab)c}^{(j)}={\lambda}^{(j)}_{\mathbb{H}\to
Rc}\alpha^{(j)}_{R\to ab}$, with ${\lambda}^{(j)}_{\mathbb{H}\to Rc}$ and
$\alpha^{(j)}_{R\to ab}$ being defined as

${\lambda}^{(j)}_{\mathbb{H}\to Rc}\equiv\frac{{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}}{{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}00\rangle}}\frac{\mathcal{W}^{(j)}}{\mathcal{W}^{(0)}},$
(7)

and

$\alpha^{(j)}_{R\to ab}=\frac{{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}}{{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}00\rangle}}\frac{\mathcal{S}^{(j)}}{\mathcal{S}^{(0)}},$
(8)

respectively. Here, $\alpha^{(j)}_{R\to ab}$ are a set of decay parameters of
$R\to ab$. On the other hand, ${\lambda}^{(j)}_{\mathbb{H}\to Rc}$
($j=1,\cdots,2s_{R}$) describe the polarization of $R$, as can be seen from
Eq. (4) 333 We can further parameterize ${\lambda}^{(j)}_{\mathbb{H}\to Rc}$
as ${\lambda}^{(j)}_{\mathbb{H}\to
Rc}=\mathcal{P}^{(j)}_{R}\kappa^{(j)}_{\mathbb{H}\to Rc}$. where
$\mathcal{P}^{(j)}_{R}$ are the normalized polarization parameters which are
defined as
$\mathcal{P}^{(j)}_{R}\equiv\frac{\mathcal{W}^{(j)}}{\mathcal{W}^{(j)}_{\text{abs}}}$,
and $\kappa^{(j)}_{\mathbb{H}\to Rc}$ can be viewed as production
asymmetry/ratio and are defined through $\kappa^{(j)}_{\mathbb{H}\to
Rc}\equiv\frac{\mathcal{W}^{(j)}_{\text{abs}}}{\mathcal{W}^{(0)}}$, where
$\mathcal{W}^{(j)}_{\text{abs}}\equiv\sum_{\sigma,\lambda_{c}}\left|\frac{(-)^{\sigma-
s_{R}}\langle s_{R},-\sigma;s_{R},\sigma|s_{R}s_{R}j0\rangle}{\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}\right|\left|\mathcal{F}^{s_{\mathbb{H}}}_{\sigma\lambda_{c}}\right|^{2}$.
. They can be viewed as the generalization of the polarization from spin-half
particles to particles of any spins produced in weak decay processes. We say
that $R$ is unpolarized if ${\lambda}^{(j)}_{\mathbb{H}\to Rc}=0$, for all
$j=1,2,\cdots,2s_{R}$.

The differences between $\lambda_{\mathbb{H}\to Rc}^{(j)}$ and their $CP$
correspondence
$\lambda_{\overline{\mathbb{H}}\to\overline{R}\overline{c}}^{(j)}$, are
measures of $CP$V. One can define the decay-angular-distribution correlated
CPAs as [26, 27, 28]

${A}_{CP}^{(j)}\equiv\frac{1}{2}\left(\lambda_{\mathbb{H}\to
Rc}^{(j)}-\lambda_{\overline{\mathbb{H}}\to\overline{R}\overline{c}}^{(j)}\right).$
(9)

Note that ${A}_{CP}^{(0)}\equiv 0$ by definition. From the experimental side,
the direct observables are

$\tilde{A}_{CP}^{(j)}={A}_{CP}^{(j)}\alpha^{(j)}_{R\to ab},$ (10)

from which one can see that the observability of ${A}_{CP}^{(j)}$ are
restricted by the values of $\alpha^{(j)}_{R\to ab}$. In other words,
$\tilde{A}_{CP}^{(j)}$ can be observed only when both ${A}_{CP}^{(j)}$ and
$\alpha^{(j)}_{R\to ab}$ are not too small.

In practice, the following definition of CPA observables is easier accessible
experimentally. The null points of $P_{j}(c_{\theta})$ devide the interval of
$c_{\theta}\in[-1,1]$ into $j+1$ sub-intervals, which can be ordered from
right ($c_{\theta}=+1$) to left ($c_{\theta}=-1$) as the 1st, 2nd, $\cdots$,
$k$-th, $\cdots$, $(j+1)$-th sub-intervals. Denoting the invent yields of each
sub-interval as $N^{(j)}_{k}$ and $\overline{N^{(j)}_{k}}$ for
$k=1,\cdots,j+1$, the CPA observables can be defined as

$\hat{A}_{CP}^{(j)}\equiv\frac{\sum_{k=1}^{j+1}(-)^{k}\left[N_{k}^{(j)}-\overline{N_{k}^{(j)}}\right]}{\sum_{k}\left[N_{k}^{(j)}+\overline{N_{k}^{(j)}}\right]}.$
(11)

It can be easily seen that $\hat{A}_{CP}^{(0)}$ is in fact the conventionally
defined CPA corresponding to the branching ratios, ${A}_{CP}(\mathbb{H}\to
R(\to ab)c)$.

##  III  Interference behaviour of canonical decay amplitudes in the Decay-
angular-distributions

Interference between amplitudes with different weak phases may generate large
a $CP$ asymmetry parameter, provided that there is also a large strong phase
difference. Potential final state interaction may generate such non-
perturbative strong phase difference in different canonical partial wave
amplitudes [29]. Besides, the interference behaviour is quite obscure in Eq.
(4) since there are no interfering terms between different helicity amplitudes
for the current situation. Consequently, it will be more transparent to see
the interference pattern if we re-express $\mathcal{W}^{(j)}$ in terms of
canonical amplitudes, which reads

$\displaystyle\mathcal{W}^{(j)}$ $\displaystyle=$
$\displaystyle\sum_{ls,l^{\prime}s^{\prime}}\rho_{ls,l^{\prime}s^{\prime}}^{(j)}a^{s_{\mathbb{H}}}_{ls}a^{s_{\mathbb{H}}\ast}_{l^{\prime}s^{\prime}},$
(12)

where $a^{s_{\mathbb{H}}}_{ls}$’s are the canonical decay amplitudes of
${\mathbb{H}}\to Rc$ and are related to the helicity amplitudes through [30,
31]

$\mathcal{F}^{s_{\mathbb{H}}}_{\sigma\lambda_{3}}=\sum_{ls}\left(\frac{2l+1}{2s_{\mathbb{H}}+1}\right)^{\frac{1}{2}}\langle
s_{R},\sigma;s_{c},-\\!\lambda_{c}|s_{R}s_{c}s(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle\langle
l,0;s,\sigma\\!\\!-\\!\\!\lambda_{c}|lss_{\mathbb{H}}(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle
a^{s_{\mathbb{H}}}_{ls},$ (13)

or inversely,

$a^{s_{\mathbb{H}}}_{ls}=\sum_{\sigma\lambda_{c}}\left(\frac{2l+1}{2s_{\mathbb{H}}+1}\right)^{\frac{1}{2}}\langle
s_{R},\sigma;s_{c},-\\!\lambda_{c}|s_{R}s_{c}s(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle\langle
l,0;S,\sigma\\!\\!-\\!\\!\lambda_{c}|lss_{\mathbb{H}}(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle\mathcal{F}^{s_{\mathbb{H}}}_{\sigma\lambda_{c}},$
(14)

and the rotational-invariant coefficient
$\rho_{ls,l^{\prime}s^{\prime}}^{(j)}$ reads

$\displaystyle\rho_{ls,l^{\prime}s^{\prime}}^{(j)}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{(2l+1)(2l^{\prime}+1)}}{(2s_{\mathbb{H}}+1)\langle
s_{R},-s_{R};s_{R},s_{R}|s_{R}s_{R}j0\rangle}\sum_{\sigma\lambda_{c}}(-)^{\sigma-
s_{R}}\langle s_{R},-\\!\sigma;s_{R},\sigma|s_{R}s_{R}j0\rangle$ (15)
$\displaystyle\times$ $\displaystyle\langle
l,0;s,{\sigma\\!\\!-\\!\\!\lambda_{c}}|lss_{\mathbb{H}}(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle\langle
s_{R},\sigma;s_{c},-\\!\lambda_{c}|s_{R}s_{c}s(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle$
$\displaystyle\times$ $\displaystyle\langle
l^{\prime},0;s^{\prime},(\sigma\\!\\!-\\!\\!\lambda_{c})|l^{\prime}s^{\prime}s_{\mathbb{H}}(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle\langle
s_{R},\sigma;s_{c},-\\!\lambda_{c}|s_{R}s_{c}s^{\prime}(\sigma\\!\\!-\\!\\!\lambda_{c})\rangle.$

In general, the presence of the interference between two canonical decay
amplitudes in certain $\mathcal{W}^{(j)}$ is constrained by the properties of
the coefficients $\rho_{ls,l^{\prime}s^{\prime}}^{(j)}$. The following two
properties of the coefficients $\rho_{ls,l^{\prime}s^{\prime}}^{(j)}$ , which
can be proven with the aid of the properties of the Clabsch-Gordan
coefficients, turn out to be very important for our analysis here:

  1. 1.

For a given value of $j$ satisfying $0\leq j\leq 2s_{R}$, the indices of the
nonzero elements of $\rho_{ls,l^{\prime}s^{\prime}}^{(j)}$ fulfill the
triangle inequality (necessary condition):

$|l-l^{\prime}|\leq j\leq l+l^{\prime},$ (16) $|s-s^{\prime}|\leq j\leq
s+s^{\prime}.$ (17)

  2. 2.

Zero elements:

$\rho_{ls,l^{\prime}s^{\prime}}^{j}=0,~{}\text{if}\left\\{\begin{array}[]{l}\text{$j$
is even, one of $l$ and $l^{\prime}$ is even and the other is odd;}\\\
\text{$j$ is odd, both $l$ and $l^{\prime}$ are even or
odd.}\end{array}\right.$ (18)

The first consequence of Property 2 is that —since $j$ can only take even
values in Eq. (2)— the interference between parity-even and -odd amplitudes is
absent in the decay amplitude squared of Eq. (2) when the subsequential decay
of $R$ is strong, where the parity of the canonical decay amplitude
$a^{s_{\mathbb{H}}}_{ls}$ is determined according to
$\Pi_{a^{s_{\mathbb{H}}}_{ls}}=(-)^{l}\Pi_{\mathbb{H}}\Pi_{R}\Pi_{c}$, with
$\Pi$ representing parity. This means that there can be only interference of
$a^{s_{\mathbb{H}}}_{ls}$ and $a^{s_{\mathbb{H}}}_{l^{\prime}s^{\prime}}$ in
$\mathcal{W}^{(j)}$ of Eq. (2) when both $l$ and $l^{\prime}$ are even or odd
simultaneously for even $j$ (so that $a^{s_{\mathbb{H}}}_{ls}$ and
$a^{s_{\mathbb{H}}}_{l^{\prime}s^{\prime}}$ have the same parity).

Of course, Property 1 can also set constraints on the presence of the
interfering terms in $\mathcal{W}^{(j)}$. For example, there will be no
interfering terms between different canonical amplitudes in
$\mathcal{W}^{(0)}$, because according to Property 1, $l=l^{\prime}$ and
$s=s^{\prime}$ for $j=0$. Hence one has
$\rho_{ls,l^{\prime}s^{\prime}}^{0}=\delta_{ll^{\prime}}\delta_{ss^{\prime}}$
when $j=0$, so that
$\mathcal{W}^{(0)}=\sum_{\sigma\lambda_{c}}\left|\mathcal{F}_{\sigma\lambda_{c}}^{s_{\mathbb{H}}}\right|^{2}=\sum_{ls}\left|a_{ls}^{s_{\mathbb{H}}}\right|^{2}$,
from which one can see that there is no interference between different
canonical amplitudes in $w^{(0)}$, as expected.

For the weak process $\mathbb{H}\to Rc$, it can be proved that the number of
the independent canonical decay amplitudes, which is of course the same as
that of the independent helicity decay amplitudes, are in total

$N_{c.a.}=(2s_{1}+1)(2s_{2}+1)-\kappa(\kappa+1),$ (19)

where $\kappa\equiv\max\left\\{s_{1}+s_{2}-s_{3},0\right\\}$. Here, $s_{1}$,
$s_{2}$, and $s_{3}$ represent the spins of the particles involved in the weak
decay, $\mathbb{H}$, $R$, and $c$, which are ordered according to $s_{1}\leq
s_{2}\leq s_{3}$. Moreover, it can be proved that the number of parity-even
and -odd canonical amplitudes will be either the same (when $N_{c.a.}$ is a
even number), or with a difference of 1 (when $N_{c.a.}$ is an odd number).
Consequently, in order for the interferences between the canonical decay
amplitudes with the same parity in $w^{(j)}$ Eq. (2) for certain $j$ to
happen, the number of the independent canonical decay amplitudes should be no
less than 3, so that at least two of them share the same parity. The first few
combinations of $(s_{1},s_{2},s_{3})$ that fulfill this requirement are
$(0,1,1)$, $(0,1,2)$, $(0,\frac{3}{2},\frac{3}{2})$, $(0,2,2)$,
$(\frac{1}{2},\frac{1}{2},1)$, $(\frac{1}{2},\frac{1}{2},2)$,
$(\frac{1}{2},1,\frac{3}{2})$, and $(\frac{1}{2},\frac{3}{2},2)$, with the
number of independent canonical decay amplitudes of 3, 3, 4, 5, 4, 4, 6, and
8, respectively. Phenomenologically, if we confine ourself only on discussions
of unstable $\mathbb{H}$ that is dominated by weak decays, the first four
combinations will be the cases of pseudo-scalar heavy meson decays, while the
last four combinations will be the cases of heavy baryon decays.

Take the typical weak transition of the type $\mathbb{B}\to\mathcal{B}V$
(${\frac{1}{2}}^{+}\to{\frac{1}{2}}^{+}+1^{-}$), such as
$\Lambda_{b}\to\Lambda\rho^{0}$, as an example, where the vector meson decays
subsequently to two pseudo-scalar mesons through strong interactions. It can
be easily seen that there are four independent weak decay canonical amplitudes
in total, according to Eq. (19). There are in total three
$\mathcal{W}^{(j)}$’s for $j=0,1$, and $2$, which can be respectively
expressed in terms of the helicity amplitudes as well as the canonical ones as

$\displaystyle\mathcal{W}^{(0)}$ $\displaystyle=$
$\displaystyle|\mathcal{F}_{1\frac{1}{2}}|^{2}+|\mathcal{F}_{\\!-1-\frac{1}{2}}|^{2}+|\mathcal{F}_{0\frac{1}{2}}|^{2}+|\mathcal{F}_{0~{}-\frac{1}{2}}|^{2},$
(20) $\displaystyle\mathcal{W}^{(1)}$ $\displaystyle=$
$\displaystyle|\mathcal{F}_{1\frac{1}{2}}|^{2}-|\mathcal{F}_{\\!-1-\frac{1}{2}}|^{2},$
(21) $\displaystyle\mathcal{W}^{(2)}$ $\displaystyle=$
$\displaystyle|\mathcal{F}_{1\frac{1}{2}}|^{2}+|\mathcal{F}_{\\!-1-\frac{1}{2}}|^{2}-2\left(|\mathcal{F}_{0\frac{1}{2}}|^{2}+|\mathcal{F}_{0~{}-\frac{1}{2}}|^{2}\right),$
(22)

and

$\displaystyle\mathcal{W}^{(0)}$ $\displaystyle=$
$\displaystyle\left(|a_{0\frac{1}{2}}|^{2}+|a_{1\frac{1}{2}}|^{2}+|a_{1\frac{3}{2}}|^{2}+|a_{2\frac{3}{2}}|^{2}\right),$
(23) $\displaystyle\mathcal{W}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{3}}\left[\sqrt{2}\Re\left(a_{0\frac{1}{2}}a_{1\frac{1}{2}}^{\ast}\right)+\Re\left(a_{0\frac{1}{2}}a_{1\frac{3}{2}}^{\ast}\right)+\Re\left(a_{1\frac{1}{2}}a_{2\frac{3}{2}}^{\ast}\right)+\sqrt{\frac{1}{2}}\Re\left(a_{1\frac{3}{2}}a_{2\frac{3}{2}}^{\ast}\right)\right],$
(24) $\displaystyle\mathcal{W}^{(2)}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\left[-\left(|a_{1\frac{3}{2}}|^{2}+|a_{2\frac{3}{2}}|^{2}\right)+2\sqrt{2}\Re\left(a_{0\frac{1}{2}}a_{2\frac{3}{2}}^{\ast}+a_{1\frac{1}{2}}a_{1\frac{3}{2}}^{\ast}\right)\right].$
(25)

From the helicity forms of $\mathcal{W}^{(1)}$ and $\mathcal{W}^{(2)}$ one can
clearly see that they describe the polarization of the vector meson $V$:
$\mathcal{W}^{(1)}$ represents the asymmetry between helicity $+1$ and $-1$ of
$V$; while $\mathcal{W}^{(2)}$ represents the asymmetry of transverse and
longitudinal polarizations. The factor 2 in front of the longitudinal
polarization parts is important, and also understandable. It reflects the
simple fact that the degrees of freedom for the longitudinal polarizations
($\sigma=\pm 1$) and the transverse parts ($\sigma=0$) are different. On the
other hand, the interference behaviour between different amplitudes is easier
to be seen from the canonical forms of $\mathcal{W}^{(1)}$ and
$\mathcal{W}^{(2)}$.

The differential decay width can then be expressed as 444The behaviour in the
expression seems to be different from the one used in the literature such as
[32]. One can see from Eq. (13) of Ref. [32] that there are interference terms
of S- and D-waves in the integrated decay width $\Gamma$, while there is no
such kind of term (interference between $a_{0\frac{1}{2}}$ and
$a_{2\frac{3}{2}}$ in our notation) in Eq. (24). This is because the
definitions of the canonical amplitudes are different. In the current paper,
$a_{ls}$ are the canonical amplitudes in the sense that they transform
irreducibly under SO(3), while the $S$\- and $D$\- waves in Ref. [32] are not.

$\frac{1}{\Gamma}\frac{d\Gamma}{d\cos\theta}=\frac{1}{2}+\frac{\mathcal{W}^{(2)}{\mathcal{S}}^{(2)}}{2\mathcal{W}^{(0)}{\mathcal{S}}^{(0)}}P_{2}(c_{\theta})=\frac{1}{2}-\frac{1}{\sqrt{2}}\lambda^{(2)}_{\mathbb{B}\to\mathcal{B}V}P_{2}(c_{\theta}).$
(26)

Note that $\mathcal{S}^{(0)}$ and $\mathcal{S}^{(2)}$ are correlated according
to $\mathcal{S}^{(2)}/\mathcal{S}^{(0)}=-\sqrt{2}$. The reason for the
correlation is understandable. Since the vector meson decays into two
pesudoscalar ones, there can only be one independent helicity amplitude. Both
of $\mathcal{S}^{(0)}$ and $\mathcal{S}^{(2)}$ must proportional to the square
of this helicity amplitude, hence they are correlated. If the relative strong
phase differences between $a_{0\frac{1}{2}}$ and $a_{2\frac{3}{2}}$ and/or
$a_{1\frac{1}{2}}$ and $a_{1\frac{3}{2}}$ are not small, the contributions of
the corresponding interfering terms to the CPA associated with the decay-
angular-distributions $A_{CP}^{(2)}$ could be large.

To see the CPA behaviour in more detail, let us first parameterize the
canonical decay amplitudes $a_{ls}^{s_{\mathbb{H}}}$ and their $C\\!P$
counterparts $\overline{a_{ls}^{s_{\mathbb{H}}}}$ as

$a_{ls}^{s_{\mathbb{H}}}=\left(a_{ls}^{T}+a_{ls}^{P}e^{i(\phi_{\text{w}}+\delta_{ls}^{PT})}\right)e^{i\delta_{ls}},$
(27)
$\overline{a_{ls}^{s_{\mathbb{H}}}}=\left(a_{ls}^{T}+a_{ls}^{P}e^{i(-\phi_{\text{w}}+\delta_{ls}^{PT})}\right)e^{i\delta_{ls}},$
(28)

where $a_{ls}^{T}$ and $a_{ls}^{P}$ are the tree and penguin parts of the
canonical decay amplitudes, respectively, $\phi_{\text{w}}$ and
$\delta_{ls}^{PT}$ are respectively the weak and the strong phase difference
between the tree and penguin parts of the same canonical decay amplitude
$a_{ls}$, and $\delta_{ls}$ is the overall strong phase of the canonical
amplitudes $a_{ls}^{s_{\mathbb{H}}}$. The CPA parameters will behave as

$A^{(j)}_{CP}\sim
2\sin\phi_{\text{w}}\sum_{ls,l^{\prime}s^{\prime}}\rho^{(j)}_{ls,l^{\prime}s^{\prime}}\left\\{a^{T}_{ls}a^{P}_{l^{\prime}s^{\prime}}\sin\left[\left(\delta_{ls}-\delta_{l^{\prime}s^{\prime}}\right)-\delta_{l^{\prime}s^{\prime}}^{PT}\right]-a^{P}_{ls}a^{T}_{l^{\prime}s^{\prime}}\sin\left[\left(\delta_{ls}-\delta_{l^{\prime}s^{\prime}}\right)+\delta_{ls}^{PT}\right]\right\\}.$
(29)

It is naturally expected (although this is only a possibility) that the
relative strong phases between different canonical decay amplitudes could be
relatively larger, comparing to the strong phase difference of the tree and
the penguin parts with different weak phases within the same canonical decay
amplitude,

$\delta_{ls}-\delta_{l^{\prime}s^{\prime}}>\delta_{ls}^{PT}\sim\delta_{l^{\prime}s^{\prime}}^{PT}.$
(30)

This means that the CPA parameter(s) $A^{(j)}_{CP}$ will have a good chance to
be dominated by the interference terms between different canonical decay
amplitudes:

$A^{(j)}_{CP}\sim 2\sin\phi_{\text{w}}\sum_{ls\neq
l^{\prime}s^{\prime}}\rho^{(j)}_{ls,l^{\prime}s^{\prime}}\left(a^{T}_{ls}a^{P}_{l^{\prime}s^{\prime}}-a^{P}_{ls}a^{T}_{l^{\prime}s^{\prime}}\right)\sin\left(\delta_{ls}-\delta_{l^{\prime}s^{\prime}}\right).$
(31)

Since there is no interference between different canonical decay amplitudes in
$\hat{A}^{(0)}_{CP}$, it is probably that $A^{(j)}_{CP}$ for $j\neq 0$ will be
larger than $\hat{A}^{(0)}_{CP}$.

While the presence of the strong phase difference between different partial-
wave amplitudes is an important necessary condition for large CPA
corresponding to the decay-angular distributions, it should be pointed out
that this is not a sufficient condition. To see this, let us use again the
decay $\mathbb{B}\to\mathcal{B}V$ as an example to illustrate. Theoretical
analyses of this type of decays have been performed via Generalized
Factorization Approach (GFA) for bottom baryon decays [33, 34, 35]. Based on
helicity decay amplitudes obtained from GFA, it can be shown that the
canonical decay amplitudes, and their $C\\!P$ conjugates, take the forms
$a_{ls}=\mathcal{C}_{V}K_{ls}$, and
$\overline{a_{ls}}=\overline{\mathcal{C}_{V}}K_{ls}$, where $K_{ls}$ is the
kinematical part which depends on $l$ and $s$, and the parameters
$\mathcal{C}_{V}$ and $\overline{\mathcal{C}_{V}}$ contain the CKM matrix
elements and are universal for all the $a_{ls}$. Especially, $\mathcal{C}_{V}$
and $\overline{\mathcal{C}_{V}}$ are independent of $l$ and $s$. While it can
generate a non-zero direct CPA corresponding to the branching ratios, the CPAs
corresponding to the decay-angular-distributions, which are defined in Eq.
(11), are predicted to be exactly zero by GFA, because the universal
$\mathcal{C}_{V}$ ($\overline{\mathcal{C}_{V}}$) is cancelled out in
$\lambda_{\mathbb{H}\to Rc}^{(j)}$
($\lambda_{\overline{\mathbb{H}}\to\overline{R}\overline{c}}^{(j)}$), so that
$\lambda_{\mathbb{H}\to
Rc}^{(j)}=\lambda_{\overline{\mathbb{H}}\to\overline{R}\overline{c}}^{(j)}$.
In other words, there can be only one CPA in the approach of GFA, i.e., CPA
corresponding to the branching ratios, $\tilde{A}_{CP}^{(0)}$. Consequently,
it is crucially important to go beyond GFA for the predictions of CPA
corresponding to the decay-angular distributions.

##  IV suggested channels for searching for decay-angular-distributions
correlated CPAs in baryon decays

It is very hard to make a concrete prediction of any decay-distribution-
correlated CPAs in heavy baryon cascade decays. Nevertheless, our analysis do
provide some guidelines for searching for such kind of CPAs in certain decay
processes. According to the above analysis, two of the guidelines are
especially important. The first one is that the number of independent
canonical decay amplitudes $N_{c.a.}$ should be no less than $3$, which will
constrain the spins of the particle involved according to Eq. (19). The second
one is the that $j$ can only take even values from $0$ to $2s_{R}$, which
implies that $s_{R}$ should be no less than 1 in order for the non-trivial
decay-angular-distributions ($j\geq 2$) to appear. Of course, the properties
of the coefficients $\rho^{(j)}_{ls,l^{\prime}s^{\prime}}$ can provide us more
detailed information on the interference of different canonical decay
amplitudes in $\mathcal{W}^{(j)}$.

The above guidelines suggest that we can study the decay-angular-distributions
and the corresponding $CP$As in the following two immediate situations. The
first one is that $\mathbb{H}$ is a pseudo-scalar heavy meson, which will be
denoted as $\mathbb{M}$ for this situation. From Eq. (19) one can see in this
situation that the necessary condition for $N_{c.a.}\geq 3$ is that the spins
of both $R$ and $c$ should be non-zero and larger than half. As
aforementioned, typical examples for the spin combinations
$(s_{1},s_{2},s_{3})$ of $\mathbb{M}$, $R$ and $c$ that fulfill this
requirement are (0, 1, 1), (0, 1, 2), $(0,\frac{3}{2},\frac{3}{2})$, (0, 2,
2). The simplest decays which can be used to perform the search of decay-
angular-distribution correlated CPAs are 1) $\mathbb{M}\to V_{1}V_{2}$, with
$V_{1}$ or $V_{2}$ decaying strongly to two pseudoscalar mesons; 2)
$\mathbb{M}\to B_{1}^{\ast}B_{2}^{\ast}$, with $B_{1}^{\ast}$ and
$B_{2}^{\ast}$ being spin-one-and-a-half baryons and one of them decaying via
strong interactions. Since our main concern is heavy baryon decay processes,
we will not go through the heavy meson decay processes any further [36].

The second situation, which is our main concern in this paper, is that
$\mathbb{H}$ represents a spin-half heavy baryon, which will be denoted as
$\mathbb{B}$ in what follows. In this case, since either $R$ or $c$ is a
baryon, in order for $N_{c.a.}\geq 3$, Eq. (19) indicates that the other
particle must be a spin-non-zero meson. In view of the above constraints, we
propose to search for decay-distribution-correlated CPV in cascade decays of
the following types: 1) ${\mathbb{B}}\to{\mathcal{B}}M$, $M\to M_{1}M_{2}$,
with the spin of the meson $M$ being nonzero; 2)
${\mathbb{B}}\to{\mathcal{B}}M$,
${\mathcal{B}}\to{\mathcal{B}}^{\prime}M^{\prime}$, with the spin of the
baryon resonance $\mathcal{B}$ being larger than $\frac{1}{2}$, and the spin
of $M$ being nonzero. Here, $\mathcal{B}$ and $\mathcal{B}^{\prime}$ represent
light baryons, and $M$, $M^{\prime}$, $M_{1}$ and $M_{2}$ represent light
mesons.

The first type seems to be more common and more applicable, among which the
most relevant decay type is of the form ${\mathbb{B}}\to{\mathcal{B}}V$, with
a subsequent strong decay of the vector resonance $V\to P_{1}P_{2}$. Typical
decays of the form ${\mathbb{B}}\to{\mathcal{B}}V$ include 1) $b\to
du\overline{u}$ transitions: $\Lambda_{b}^{0}\to p\rho(770)^{+}$,
$\Lambda_{b}^{0}\to N(1520)^{\ast}\rho(770)^{+}$; 2) $b\to su\overline{u}$
transitions: $\Lambda_{b}^{0}\to\Lambda\rho(770)^{0}$, $\Lambda_{b}^{0}\to
p{K^{\ast}}(892)^{-}$, $\Lambda_{b}^{0}\to N(1520)K^{\ast}$; 3) $c\to
ud\overline{d}$ transitions: $\Lambda_{c}^{+}\to p\rho(770)^{0}$,
$\Xi_{c}^{+}\to p\overline{K^{\ast}}(892)^{0}$; 4) $c\to us\overline{s}$
transitions: $\Lambda_{c}^{+}\to p\phi$,
$\Lambda_{c}^{+}\to\Sigma^{+}{K^{\ast}}(892)^{0}$.

For the second type, ${\mathbb{B}}\to\mathcal{B}M$,
$\mathcal{B}\to\mathcal{B}^{\prime}M^{\prime}$, where the spin of the
intermediate baryon resonance is no less than $\frac{3}{2}$, and the spin of
$M$ is no less than 1, typical decays include 1) $c\to ud\overline{d}$
transitions: $\Lambda_{c}^{+}\to N(1520)^{\ast}\rho(770)^{0}$, $\Xi_{c}^{+}\to
N(1520)^{\ast}\overline{K^{\ast}}(892)^{0}$; 2) $c\to us\overline{s}$
transitions: $\Lambda_{c}^{+}\to N(1520)^{\ast}\phi$,
$\Lambda_{c}^{+}\to\Sigma^{+}{K^{\ast}}(892)^{0}$; 3) $b\to du\overline{u}$
transition: $\Lambda_{b}^{0}\to N(1520)^{\ast}\rho(770)^{+}$; 4) $b\to
su\overline{u}$ transition: $\Lambda_{b}^{0}\to N(1520)K^{\ast}$. For most of
the cases, the study of $A^{(j)}$ and/or $\hat{A}^{(j)}$ for $j=2$ will be
enough since all the spins of the resonances mentioned in the above examples
are less than 2.

##  V Summary and Conclusion

$CP$V in baryon decay processes has not been observed yet. In this paper, the
decay-angular-distribution correlated CPAs for cascade decays of unpolarized
heavy hadrons are analyzed. By expressing the differential decay width in
terms of the canonical decay amplitudes, we analyse the general condition for
the presence of the interfering terms between different canonical decay
amplitudes. The presence of the interfering terms are important for the
generation of large decay-angular-distribution correlated CPAs.

We focus mainly on one typical type of decays, in which the heavy baryon
decays weakly into two daughter hadrons, with one of them decaying strongly
into two granddaughter hadrons. The analysis indicates that when the two
daughter hadrons are both spin-nonzero, there will be at least two parity-even
and -odd canonical amplitudes. The interference between the canonical
amplitudes with the same parity properties will be present in the angular
distributions of the final particles. With a possible large strong phase
between different canonical amplitudes, a large CPA corresponding to the
decay-angular-distributions may be generated. We also present some typical
decay channels in which the search for such kind of CPAs can be performed.

###### Acknowledgements.

We thank Prof. Hai-Yang Cheng, Prof. Hsiang-nan Li, and Dr. Chia-Wei Liu for
valuable discussions. This work was supported by National Natural Science
Foundation of China under Grants Nos. 12192261 and 12275024, Natural Science
Foundation of Hunan Province under Grants No. 2022JJ30483, and Scientific
Research Fund of Hunan Provincial Education Department under Grants No.
22A0319.

## References

  * Aghanim  _et al._ [2020] N. Aghanim _et al._ (Planck), Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641, A6 (2020), [Erratum: Astron.Astrophys. 652, C4 (2021)], arXiv:1807.06209 [astro-ph.CO] .
  * Sakharov [1967] A. D. Sakharov, Violation of CP Invariance, C asymmetry, and baryon asymmetry of the universe, Pisma Zh. Eksp. Teor. Fiz. 5, 32 (1967).
  * Christenson  _et al._ [1964] J. H. Christenson, J. W. Cronin, V. L. Fitch, and R. Turlay, Evidence for the $2\pi$ Decay of the $K_{2}^{0}$ Meson, Phys. Rev. Lett. 13, 138 (1964).
  * Aaij  _et al._ [2019] R. Aaij _et al._ (LHCb), Observation of CP Violation in Charm Decays, Phys. Rev. Lett. 122, 211803 (2019), arXiv:1903.08726 [hep-ex] .
  * Aubert  _et al._ [2001] B. Aubert _et al._ (BaBar), Measurement of CP violating asymmetries in $B^{0}$ decays to CP eigenstates, Phys. Rev. Lett. 86, 2515 (2001), arXiv:hep-ex/0102030 .
  * Abe  _et al._ [2001] K. Abe _et al._ (Belle), Observation of large CP violation in the neutral $B$ meson system, Phys. Rev. Lett. 87, 091802 (2001), arXiv:hep-ex/0107061 .
  * Aaij  _et al._ [2013a] R. Aaij _et al._ (LHCb), First observation of $CP$ violation in the decays of $B^{0}_{s}$ mesons, Phys. Rev. Lett. 110, 221601 (2013a), arXiv:1304.6173 [hep-ex] .
  * Workman [2022] R. L. Workman (Particle Data Group), Review of Particle Physics, PTEP 2022, 083C01 (2022).
  * Cabibbo and Maksymowicz [1965] N. Cabibbo and A. Maksymowicz, Angular Correlations in $K_{e4}$ Decays and Determination of Low-Energy $\pi-\pi$ Phase Shifts, Phys. Rev. 137, B438 (1965), [Erratum: Phys.Rev. 168, 1926 (1968)].
  * Kobayashi and Maskawa [1973] M. Kobayashi and T. Maskawa, CP Violation in the Renormalizable Theory of Weak Interaction, Prog. Theor. Phys. 49, 652 (1973).
  * Lee and Yang [1957] T. D. Lee and C.-N. Yang, General Partial Wave Analysis of the Decay of a Hyperon of Spin 1/2, Phys. Rev. 108, 1645 (1957).
  * Donoghue and Pakvasa [1985] J. F. Donoghue and S. Pakvasa, Signals of CP Nonconservation in Hyperon Decay, Phys. Rev. Lett. 55, 162 (1985).
  * Donoghue  _et al._ [1986] J. F. Donoghue, X.-G. He, and S. Pakvasa, Hyperon Decays and CP Nonconservation, Phys. Rev. D 34, 833 (1986).
  * Bigi  _et al._ [2018] I. I. Bigi, X.-W. Kang, and H.-B. Li, CP Asymmetries in Strange Baryon Decays, Chin. Phys. C 42, 013101 (2018), arXiv:1704.04708 [hep-ph] .
  * Wang and Yu [2022] J.-P. Wang and F.-S. Yu, Probing hyperon CP violation from charmed baryon decays, (2022), arXiv:2208.01589 [hep-ph] .
  * Ablikim  _et al._ [2019] M. Ablikim _et al._ (BESIII), Polarization and Entanglement in Baryon-Antibaryon Pair Production in Electron-Positron Annihilation, Nature Phys. 15, 631 (2019), arXiv:1808.08917 [hep-ex] .
  * Li  _et al._ [2023] L. K. Li _et al._ (Belle), Search for CP violation and measurement of branching fractions and decay asymmetry parameters for $\Lambda_{c}^{+}\to\Lambda h^{+}$ and $\Lambda_{c}^{+}\to\Sigma^{0}h^{+}$ (h=$K$,$\pi$), Sci. Bull. 68, 583 (2023), arXiv:2208.08695 [hep-ex] .
  * Tang  _et al._ [2023] S.-S. Tang, L.-K. Li, X.-Y. Zhou, and C.-P. Shen, Recent Measurements of Decay Asymmetry Parameter and CP Asymmetry for Charmed Baryon Decays at Belle, Symmetry 15, 91 (2023).
  * Giri  _et al._ [2002] A. K. Giri, R. Mohanta, and M. P. Khanna, Possibility of extracting the weak phase gamma from $\Lambda_{b}\to\Lambda D^{0}$ decays, Phys. Rev. D 65, 073029 (2002), arXiv:hep-ph/0112220 .
  * Zhang  _et al._ [2021] S. Zhang, Y. Jiang, Z. Chen, and W. Qian, Sensitivity studies on the CKM angle $\gamma$ in $\Lambda_{b}^{0}\to D\Lambda$ decays, (2021), arXiv:2112.12954 [hep-ph] .
  * Zhang  _et al._ [2013] Z.-H. Zhang, X.-H. Guo, and Y.-D. Yang, CP violation in $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$ in the region with low invariant mass of one $\pi^{+}\pi^{-}$ pair, Phys. Rev. D 87, 076007 (2013), arXiv:1303.3676 [hep-ph] .
  * Zhang and Qi [2023] Z.-H. Zhang and J.-J. Qi, Analysis of angular distribution asymmetries and the associated $C\\!P$ asymmetries in three-body decays of bottom baryons, Eur. Phys. J. C 83, 133 (2023), arXiv:2208.13411 [hep-ph] .
  * Aaij  _et al._ [2013b] R. Aaij _et al._ (LHCb), Measurements of the $\Lambda_{b}^{0}\to J/\psi\Lambda$ decay amplitudes and the $\Lambda_{b}^{0}$ polarisation in $pp$ collisions at $\sqrt{s}=7$ TeV, Phys. Lett. B 724, 27 (2013b), arXiv:1302.5578 [hep-ex] .
  * Aaij  _et al._ [2020] R. Aaij _et al._ (LHCb), Measurement of the $\Lambda^{0}_{b}\rightarrow J/\psi\Lambda$ angular distribution and the $\Lambda^{0}_{b}$ polarisation in $pp$ collisions, JHEP 06, 110, arXiv:2004.10563 [hep-ex] .
  * Avery  _et al._ [1990] P. Avery _et al._ (CLEO), Measurement of the Lambda(c) decay asymmetry parameter, Phys. Rev. Lett. 65, 2842 (1990).
  * Zhang and Guo [2021] Z.-H. Zhang and X.-H. Guo, A novel strategy for searching for CP violations in the baryon sector, JHEP 07, 177, arXiv:2103.11335 [hep-ph] .
  * Wei and Zhang [2022] Y.-R. Wei and Z.-H. Zhang, The Forward-Backward Asymmetry induced $CP$ Asymmetry of $B^{\pm}\rightarrow\pi^{\pm}\pi^{+}\pi^{-}$, (2022), arXiv:2209.02348 [hep-ph] .
  * Zhang [2023] Z.-H. Zhang, Searching for CP violation through two-dimensional angular distributions in four-body decays of bottom and charmed baryons, Phys. Rev. D 107, L011301 (2023), arXiv:2209.13196 [hep-ph] .
  * Watson [1952] K. M. Watson, The Effect of final state interactions on reaction cross-sections, Phys. Rev. 88, 1163 (1952).
  * Jacob and Wick [1959] M. Jacob and G. C. Wick, On the General Theory of Collisions for Particles with Spin, Annals Phys. 7, 404 (1959).
  * Chung [1971] S. U. Chung, SPIN FORMALISMS 10.5170/CERN-1971-008 (1971).
  * Pakvasa  _et al._ [1990] S. Pakvasa, S. P. Rosen, and S. F. Tuan, Parity Violation and Flavor Selection Rules in Charmed Baryon Decays, Phys. Rev. D 42, 3746 (1990).
  * Leitner and Ajaltouni [2007] O. Leitner and Z. J. Ajaltouni, Testing CP and Time Reversal Symmetries with $\Lambda_{b}\to\Lambda V(1-)$ Decays, Nucl. Phys. B Proc. Suppl. 174, 169 (2007), arXiv:hep-ph/0610189 .
  * Geng  _et al._ [2020] C. Q. Geng, C.-W. Liu, and T.-H. Tsai, Charmed Baryon Weak Decays with Vector Mesons, Phys. Rev. D 101, 053002 (2020), arXiv:2001.05079 [hep-ph] .
  * Geng and Liu [2021] C.-Q. Geng and C.-W. Liu, Time-reversal asymmetries and angular distributions in $\Lambda$b → $\Lambda$V, JHEP 11, 104, arXiv:2109.09524 [hep-ph] .
  * Dunietz  _et al._ [1991] I. Dunietz, H. R. Quinn, A. Snyder, W. Toki, and H. J. Lipkin, How to extract CP violating asymmetries from angular correlations, Phys. Rev. D 43, 2193 (1991).


Thus, combining all of the above, we obtain inequality (183) and thus (165).

Now, we can prove inequality (166) by noticing

$\displaystyle\frac{2[A^{\mathsf{T}}\nabla_{r^{p}}^{2}A]}{R_{p}}=\frac{\sum_{n\sim_{A}m}(a_{n}-a_{m})^{2}r_{nm}^{p}}{\sum_{n\sim_{A}m}r_{nm}^{p}}\leq\|\nabla
A\|_{\infty}^{2}.$ (186)

###  C.2 Proof of inequality (167)

We can bound $R_{p}$ with state-independent quantity as

$\displaystyle R_{p}$
$\displaystyle\leq\sum_{n\sim_{A}m}\|\hat{H}_{nm}\|_{\mathrm{op}}\frac{p_{n}+p_{m}}{2}$
$\displaystyle=\sum_{n\in\mathcal{V}}p_{n}\sum_{m(\sim_{A}n)}\|\hat{H}_{nm}\|_{\mathrm{op}}$
$\displaystyle=\max_{n\in\mathcal{V}}\sum_{m(\sim_{A}n)}\|\hat{H}_{nm}\|_{\mathrm{op}}:=C_{H}.$
(187)

Thus, inequality (167) follows.

###  C.3 Proof of (169)

Inequality (169) is obtained in a manner similar to (165). We begin to show

$\displaystyle\sum_{n\sim_{A}m}|J^{q}_{nm}|\leq
2\sqrt{R^{2}-E_{\mathrm{trans}}^{2}},$ (188)

where $r_{nm}$ and $R$ are defined in Eqs. (170) and (172). For this purpose,
we employ the same parametrization of $J_{nm}^{q}$ and $Y_{nm}$ using
$\theta_{nm}$.

Now, Instead of inequality (176), we use the following inequality

$\displaystyle|\sin\theta|\leq\sqrt{1+b^{2}}-b|\cos\theta|$ (189)

for $b,\theta\in\mathbb{R}$. Then, we have

$\displaystyle\sum_{n\sim_{A}m}|J^{q}_{nm}|$
$\displaystyle\leq\sum_{n\sim_{A}m}2r_{nm}\sqrt{1+b^{2}}-b|Y_{nm}|$
$\displaystyle\leq 2\sqrt{1+b^{2}}R-2b|E_{\mathrm{trans}}|.$ (190)

The optimal upper bound is obtained for
$b=\frac{|E_{\mathrm{trans}}|/R}{\sqrt{1-(|E_{\mathrm{trans}}|/R)^{2}}}$, from
which we have the inequality (188). Thus, we have

$\displaystyle\mathcal{B}_{\mathrm{Lip}}\leq\|\nabla
A\|_{\infty}\sqrt{R^{2}-E_{\mathrm{trans}}^{2}}.$ (191)

Finally, as in the previous subsection, we use $R\leq R_{p}$ and obtain

$\displaystyle\mathcal{B}_{\mathrm{Lip}}\leq\|\nabla
A\|_{\infty}\sqrt{R_{p}^{2}-E_{\mathrm{trans}}^{2}},$ (192)

which is inequality (169).

##  Appendix D Derivation of the speed limit for acceleration

While many bounds in the main text are derived using inequalities (20) and
(21), we can find a speed limit for quantities that are not written as $F(p)$.
Specifically, we can show that the acceleration of the expectation value,
$\braket{\ddot{\hat{A}}}$, can be bounded above provided that $\hat{H}$ is
independent of time. This leads to speed limits of $\braket{\hat{A}}$ that is
distinct from $\mathcal{B}_{\mathrm{gL}}$ and $\mathcal{B}_{\mathrm{Lip}}$
discussed in the main text.

Indeed, we find, e.g.,

$\displaystyle|\braket{\ddot{\hat{A}}}|$
$\displaystyle\leq\max_{m,n(\sim_{A}m),l(\sim_{A}m)}|a_{n}+a_{l}-2a_{m}|\cdot{Q_{p}}$
$\displaystyle\leq\max_{m,n(\sim_{A}m),l(\sim_{A}m)}|a_{n}+a_{l}-2a_{m}|\cdot{C_{H}^{2}},$
(193)

where

$\displaystyle
Q_{p}=\sum_{m,n(\sim_{A}m),l(\sim_{A}m)}\sqrt{p_{n}p_{l}}\|\hat{H}_{nm}\|_{\mathrm{op}}\|\hat{H}_{ml}\|_{\mathrm{op}}.$
(194)

From this acceleration bound, we find a new type of the bound of velocity. If
we assume that $\braket{\dot{A}(0)}=0$, we find

$\displaystyle|\braket{\dot{A}(t)}|\leq\mathcal{B}_{\mathrm{acc}}=\max_{m,n(\sim_{A}m),l(\sim_{A}m)}|a_{n}+a_{l}-2a_{m}|\int_{0}^{t}d\tau{Q_{p}},$
(195)

which is bounded by the state-independent quantity
$\max_{m,n(\sim_{A}m),l(\sim_{A}m)}|a_{n}+a_{l}-2a_{m}|C_{H}^{2}t$. Note that
$\mathcal{B}_{\mathrm{acc}}$ can be better for short times than
$\mathcal{B}_{\mathrm{gL}}$ and $\mathcal{B}_{\mathrm{Lip}}$. Indeed, we can
see that $\mathcal{B}_{\mathrm{acc}}\rightarrow 0$ for $t\rightarrow 0$, while
$\mathcal{B}_{\mathrm{gL}}$ and $\mathcal{B}_{\mathrm{Lip}}$ may not be
necessarily zero even for $t\rightarrow 0$.

To prove the above inequalities, We first note that

$\displaystyle\braket{\ddot{\hat{A}}}$
$\displaystyle=-i\sum_{n\sim_{A}m}(a_{n}-a_{m})\mathrm{Tr}[\hat{H}_{nm}\dot{\hat{\rho}}_{mn}]$
$\displaystyle=-i\sum_{nml}(a_{n}-a_{m})\mathrm{Tr}[\hat{H}_{nm}(\hat{H}_{ml}\hat{\rho}_{ln}-\hat{\rho}_{ml}\hat{H}_{ln})]$
$\displaystyle=-i\sum_{nml}(a_{n}-a_{m})\mathrm{Tr}[\hat{H}_{nm}\hat{H}_{ml}\hat{\rho}_{ln}+\hat{\rho}_{nl}\hat{H}_{lm}\hat{H}_{mn}]$
$\displaystyle=-\frac{i}{2}\sum_{m,n(\sim_{A}m),l(\sim_{A}m)}(a_{n}+a_{l}-2a_{m})$
$\displaystyle\quad\quad\quad\quad\quad\times\mathrm{Tr}[\hat{H}_{nm}\hat{H}_{ml}\hat{\rho}_{ln}+\hat{\rho}_{nl}\hat{H}_{lm}\hat{H}_{mn}].$
(196)

We then have

$\displaystyle|\braket{\ddot{\hat{A}}}|$
$\displaystyle\leq\max_{m,n(\sim_{A}m),l(\sim_{A}m)}|a_{n}+a_{l}-2a_{m}|$
$\displaystyle\quad\quad\times\sum_{m,n(\sim_{A}m),l(\sim_{A}m)}|\mathrm{Tr}[\hat{H}_{nm}\hat{H}_{ml}\hat{\rho}_{ln}]|$
$\displaystyle\leq\max_{m,n(\sim_{A}m),l(\sim_{A}m)}|a_{n}+a_{l}-2a_{m}|\cdot
Q_{p},$ (197)

where

$\displaystyle
Q_{p}=\sum_{m,n(\sim_{A}m),l(\sim_{A}m)}\sqrt{p_{n}p_{l}}\|\hat{H}_{nm}\|_{\mathrm{op}}\|\hat{H}_{ml}\|_{\mathrm{op}}.$
(198)

We can also show the state-independent inequality

$\displaystyle|\braket{\ddot{\hat{A}}}|\leq\max_{m,n(\sim_{A}m),l(\sim_{A}m)}|a_{n}+a_{l}-2a_{m}|\cdot
C_{H}^{2},$ (199)

since

$\displaystyle Q_{p}$
$\displaystyle\leq\sum_{m,n(\sim_{A}m),l(\sim_{A}m)}\frac{p_{n}+p_{l}}{2}\|\hat{H}_{nm}\|_{\mathrm{op}}\|\hat{H}_{ml}\|_{\mathrm{op}}$
$\displaystyle=\sum_{n\in\mathcal{V}}p_{n}\sum_{m(\sim_{A}n)}\|\hat{H}_{nm}\|_{\mathrm{op}}\sum_{l(\sim_{A}m)}\|\hat{H}_{ml}\|_{\mathrm{op}}\leq
C_{H}^{2}.$ (200)

##  Appendix E Derivation of the speed limit for coherence for general states

Here, we derive the speed limit for macroscopic coherence for arbitrary
states. Using the assumption $c_{nm}=c_{mn}$, we have

$\displaystyle\frac{\mathrm{d}\mathcal{C}}{\mathrm{d}t}$
$\displaystyle=-i\sum_{nml}c^{nm}\mathrm{Tr}[(\hat{H}_{nl}\hat{\rho}_{lm}-\hat{\rho}_{nl}\hat{H}_{lm})\hat{\rho}_{mn}]$
$\displaystyle\quad\quad-i\sum_{nml}c^{nm}\mathrm{Tr}[\hat{\rho}_{nm}(\hat{H}_{ml}\hat{\rho}_{ln}-\hat{\rho}_{ml}\hat{H}_{ln})]$
$\displaystyle=-2i\sum_{nml}c^{nm}\mathrm{Tr}[\hat{H}_{nl}\hat{\rho}_{lm}\hat{\rho}_{mn}-\hat{\rho}_{ml}\hat{H}_{ln}\hat{\rho}_{nm}]$
$\displaystyle=-i\sum_{nml}\left(c^{nm}-c^{lm}\right)\operatorname{Tr}\left[\hat{H}_{nl}\hat{\rho}_{lm}\hat{\rho}_{mn}-\hat{H}_{ln}\hat{\rho}_{nm}\hat{\rho}_{ml}\right]$
$\displaystyle=\sum_{n\sim_{\mathcal{C}}l,m}\left(c^{nm}-c^{ml}\right)\mathcal{J}_{nlm},$
(201)

where

$\displaystyle\mathcal{J}_{nlm}=-i\operatorname{Tr}\left[\hat{H}_{nl}\hat{\rho}_{lm}\hat{\rho}_{mn}-\hat{H}_{ln}\hat{\rho}_{nm}\hat{\rho}_{ml}\right].$
(202)

Thus, we have, e.g.,

$\displaystyle|\dot{\mathcal{C}}|$
$\displaystyle\leq\max_{n\sim_{\mathcal{C}}l,m}|c^{nm}-c^{ml}|\sum_{n\sim_{\mathcal{C}}l,m}|\mathcal{J}_{nlm}|$
(203)

While we focus on this inequality here, inequalities using $r_{nm}$ are
obtained similarly.

Now, as in inequality (188) to bound $\sum_{nm}|J_{nm}|$, we can show that

$\displaystyle\sum_{n\sim_{\mathcal{C}}l,m}|\mathcal{J}_{nlm}|\leq
2\sqrt{\mathtt{R}^{2}-\mathtt{E}^{2}},$ (204)

where

$\displaystyle\mathtt{R}=\sum_{nlm}|\mathrm{Tr}[\hat{H}_{nl}\hat{\rho}_{lm}\hat{\rho}_{mn}]|$
(205)

and

$\displaystyle\mathtt{E}=\sum_{nlm}\mathrm{Tr}[\hat{H}_{nl}\hat{\rho}_{lm}\hat{\rho}_{mn}]=\mathrm{Tr}\left[\sum_{nl}\hat{H}_{nl}\hat{\rho}^{2}\right].$
(206)

We can further show that

$\displaystyle\mathtt{R}\leq\sum_{nlm}\|\hat{H}_{nl}\|_{\mathrm{op}}\|\hat{\rho}_{lm}\hat{\rho}_{mn}\|_{1}.$
(207)

The trace-1 norm is bounded using the representation
$\hat{\rho}=\sum_{k}\lambda_{k}\ket{\lambda_{k}}\bra{\lambda_{k}}$:

$\displaystyle\|\hat{\rho}_{lm}\hat{\rho}_{mn}\|_{1}$
$\displaystyle\leq\sum_{kq}\lambda_{k}\lambda_{q}\|\hat{P}_{l}\ket{\lambda_{k}}\bra{\lambda_{k}}\hat{P}_{m}\ket{\lambda_{q}}\bra{\lambda_{q}}\hat{P}_{n}\|_{1}$
$\displaystyle=\sum_{kq}\lambda_{k}\lambda_{q}\sqrt{\braket{\lambda_{k}}{\hat{P}_{l}}{\lambda_{k}}\braket{\lambda_{q}}{\hat{P}_{n}}{\lambda_{q}}}|\braket{\lambda_{k}}{\hat{P}_{m}}{\lambda_{q}}|$
$\displaystyle\leq\sqrt{\sum_{kq}\lambda_{k}\lambda_{q}\braket{\lambda_{k}}{\hat{P}_{l}}{\lambda_{k}}\braket{\lambda_{q}}{\hat{P}_{n}}{\lambda_{q}}}$
$\displaystyle\quad\quad\times\sqrt{\sum_{kq}\lambda_{k}\lambda_{q}|\braket{\lambda_{k}}{\hat{P}_{m}}{\lambda_{q}}|^{2}}$
$\displaystyle=\sqrt{p_{l}p_{n}\mathrm{Tr}[(\hat{\rho}_{mm})^{2}]}.$ (208)

Thus, we have

$\displaystyle|\dot{\mathcal{C}}|$ $\displaystyle\leq
2\max_{n\sim_{\mathcal{C}}l,m}|c^{nm}-c^{ml}|\cdot\sqrt{\mathtt{\tilde{R}}^{2}-\mathtt{E}^{2}},$
(209)

where

$\displaystyle\mathtt{\tilde{R}}=R_{p}\sum_{m}\sqrt{\mathrm{Tr}[(\hat{\rho}_{mm})^{2}]}.$
(210)

When $c_{nm}$ is assumed to be a distance between $n$ and $m$, which satisfies
the triangle inequality $|c^{nm}-c^{ml}|\leq c^{nl}$, we have

$\displaystyle|\dot{\mathcal{C}}|$
$\displaystyle\leq\max_{n\sim_{\mathcal{C}}l}(c^{nl})\cdot\sum_{n\sim_{\mathcal{C}}l,m}|\mathcal{J}_{nlm}|,$
(211)

For pure states, we recover inequality (III.3) since
$\mathtt{\tilde{R}}=R_{p}$ and $\mathtt{E}=E_{\mathrm{trans}}$ in this case.

##  Appendix F Derivation of the inequality (56)

We have, using the Cauchy-Schwarz inequality,

$\displaystyle|\dot{\mathbb{V}}[\hat{A}]|\leq$ $\displaystyle
2\cdot\frac{\|\nabla
A\|_{\infty}}{2}\sqrt{\sum_{n\sim_{A}m}(a_{n}-\braket{A})^{2}p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}$
$\displaystyle\times\sqrt{\sum_{n\sim_{A}m}\frac{|J_{nm}^{q}|^{2}}{p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}}.$
(212)

Here,

$\displaystyle\sum_{n\sim_{A}m}(a_{n}-\braket{A})^{2}$ $\displaystyle
p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}$ $\displaystyle=$
$\displaystyle\sum_{n}(a_{n}-\braket{A})^{2}p_{n}\sum_{m(\sim_{A}n)}\|\hat{H}_{nm}\|_{\mathrm{op}}$
$\displaystyle\leq$ $\displaystyle C_{H}{\mathbb{V}}[\hat{A}]$ (213)

and

$\displaystyle{\sum_{n\sim_{A}m}\frac{|J_{nm}^{q}|^{2}}{p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}}$
$\displaystyle\leq\sum_{n\sim_{A}m}\frac{4r_{nm}^{2}\sin^{2}\theta_{nm}}{p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}$
$\displaystyle\leq\sum_{n\sim_{A}m}\frac{4r_{nm}^{2}}{p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}\left(1+\frac{d^{2}}{4}-d|\cos\theta_{nm}|\right)$
$\displaystyle\leq\sum_{n\sim_{A}m}\frac{4r_{nm}^{2}}{p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}\left(1+\frac{d^{2}}{4}-d|\cos\theta_{nm}|\right)$
$\displaystyle\leq\sum_{n\sim_{A}m}[(4+d^{2})p_{m}\|\hat{H}_{nm}\|_{\mathrm{op}}$
$\displaystyle\quad\quad\quad-4dp_{m}\|\hat{H}_{nm}\|_{\mathrm{op}}|\cos\theta_{nm}|]$
$\displaystyle\leq(4+d^{2})C_{H}$
$\displaystyle\quad\quad\quad-\sum_{n\sim_{A}m}2d(p_{m}+p_{n})\|\hat{H}_{nm}\|_{\mathrm{op}}|\cos\theta_{nm}|$
$\displaystyle\leq(4+d^{2})C_{H}-\sum_{n\sim_{A}m}4dr_{nm}\cos\theta_{nm}|$
$\displaystyle\leq(4+d^{2})C_{H}-4d|E_{\mathrm{trans}}|$

with $d\in\mathbb{R}$, where we have used inequality (176) and
$r_{nm}\leq\sqrt{p_{n}p_{m}}\|\hat{H}_{nm}\|_{\mathrm{op}}\leq\frac{p_{n}+p_{m}}{2}\|\hat{H}_{nm}\|_{\mathrm{op}}$.
Optimization is carried out by taking $d=2|E_{\mathrm{trans}}|/C_{H}$, which
leads to

$\displaystyle{\sum_{n\sim_{A}m}\frac{|J_{nm}^{q}|^{2}}{p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}}\leq
4C_{H}-\frac{4|E_{\mathrm{trans}}|^{2}}{C_{H}}.$ (214)

Thus

$\displaystyle|\dot{\mathbb{V}}[\hat{A}]|\leq 2{\|\nabla
A\|_{\infty}}\sqrt{{\mathbb{V}}[\hat{A}](C_{H}^{2}-|E_{\mathrm{trans}}|^{2})}$
(215)

and

$\displaystyle\left|\frac{\mathrm{d}\Delta A}{\mathrm{d}t}\right|\leq{\|\nabla
A\|_{\infty}}\sqrt{C_{H}^{2}-E_{\mathrm{trans}}^{2}}.$ (216)

Then, integration from $t=0$ to $t=T$ leads to

$\displaystyle\left|\Delta A(T)-\Delta A(0)\right|$
$\displaystyle\leq{\|\nabla
A\|_{\infty}}\int_{0}^{T}dt\sqrt{C_{H}^{2}-E_{\mathrm{trans}}^{2}.}$
$\displaystyle={\|\nabla
A\|_{\infty}}T\overline{\sqrt{C_{H}^{2}-E_{\mathrm{trans}}^{2}}}$ (217)

##  Appendix G Derivation of the speed limit for Shannon entropy

We consider the speed limit of the Shannon entropy

$\displaystyle S(p)=-\sum_{n}p_{n}\ln p_{n}.$ (218)

We have

$\displaystyle|\dot{S}|$ $\displaystyle=\left|\sum_{n}\dot{p}_{n}\ln
p_{n}\right|$ $\displaystyle\leq\left|\frac{1}{2}\sum_{n\sim
m}J_{nm}^{q}\ln\frac{p_{n}}{p_{m}}\right|$
$\displaystyle\leq\frac{1}{2}\sqrt{\sum_{n\sim
m}r_{nm}\left(\ln\frac{p_{n}}{p_{m}}\right)^{2}\sum_{n\sim
m}\frac{|J_{nm}^{q}|^{2}}{r_{nm}}}$ $\displaystyle\leq\sqrt{\sum_{n\sim
m}r^{p}_{nm}\left(\ln\frac{p_{n}}{p_{m}}\right)^{2}}\sqrt{R_{p}-\frac{E_{\mathrm{trans}}^{2}}{R_{p}}}.$
(219)

To proceed, we notice the following inequality

$\displaystyle\sqrt{p_{n}p_{m}}$
$\displaystyle=\frac{p_{n}+p_{m}}{2}-\frac{(\sqrt{p_{n}}-\sqrt{p_{m}})^{2}}{2}$
$\displaystyle\leq\frac{p_{n}+p_{m}}{2}-\frac{\sqrt{p_{n}p_{m}}}{8}\left(\ln\frac{p_{n}}{p_{m}}\right)^{2},$
(220)

where we have used

$\displaystyle(\sqrt{a}-\sqrt{b})^{2}\geq\frac{\sqrt{ab}}{4}\left(\ln\frac{a}{b}\right)^{2}$
(221)

for $a,b>0$. Then,

$\displaystyle\sum_{n\sim
m}r_{nm}^{p}\left(\ln\frac{p_{n}}{p_{m}}\right)^{2}\leq 8(C_{H}-R_{p})$ (222)

and

$\displaystyle|\dot{S}|\leq\sqrt{8\left(\frac{C_{H}}{R_{p}}-1\right)}\sqrt{{R_{p}^{2}-{E_{\mathrm{trans}}^{2}}}}$
(223)

By replacing $E_{\mathrm{trans}}^{2}/R_{p}$ with
$E_{\mathrm{trans}}^{2}/C_{H}$ and completing the square, we have

$\displaystyle|\dot{S}|\leq\sqrt{2}\left(C_{H}-\frac{E_{\mathrm{trans}}^{2}}{C_{H}}\right).$
(224)

We note that a different type of inequality holds by noting

$\displaystyle R_{p}\leq C_{H}-\frac{1}{8}\sum_{n\sim
m}r_{nm}^{p}\left(\ln\frac{p_{n}}{p_{m}}\right)^{2}.$ (225)

We introduce the quantity similar to the symmetric version of discrete Fisher
information

$\displaystyle\mathcal{I}(p)=\frac{1}{2}\sum_{n\sim
m}r_{nm}^{p}\left(\ln\frac{p_{n}}{p_{m}}\right)^{2},$ (226)

and obtain

$\displaystyle|\dot{S}|\leq\sqrt{2\mathcal{I}(p)\left(C_{H}-\frac{E_{\mathrm{trans}}^{2}}{C_{H}}-\frac{\mathcal{I}(p)}{4}\right)}.$
(227)

Completing the square again leads to inequality (224).

##  Appendix H Tightness of the bounds for simple systems

As discussed in Sec. IV, for a single-particle one-dimensional system with
$W_{l}=0$, the initial state $\ket{\psi_{1}}$ leads to

$\displaystyle\braket{\dot{\hat{x}}}=\mathcal{B}_{\mathrm{gL}}=\mathcal{B}_{\mathrm{Lip}}=\mathcal{B}_{p}$
(228)

for short times $t\lesssim 2$, indicating the tightness of these bounds. We
here discuss why this tightness holds true in this situation.

We first show $\mathcal{B}_{\mathrm{Lip}}=\mathcal{B}_{p}$. The left-hand side
can be written as

$\displaystyle\mathcal{B}_{\mathrm{Lip}}={K}\sum_{l}|\braket{\psi(t)}{\hat{a}_{l+1}^{\dagger}\hat{a}_{l}-\hat{a}_{l}^{\dagger}\hat{a}_{l+1}}{\psi(t)}|$
(229)

because $\|\nabla x\|_{\infty}=1$. To proceed, we show that

$\displaystyle\braket{\psi(t)}{\hat{a}_{l+1}^{\dagger}\hat{a}_{l}}{\psi(t)}=-\braket{\psi(t)}{\hat{a}_{l}^{\dagger}\hat{a}_{l+1}}{\psi(t)}$
(230)

for $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ with any $l$ and $W_{l}=0$. To see
this, we introduce the following anti-unitary symmetry

$\displaystyle\hat{\mathcal{T}}=\hat{\mathcal{K}}\bigotimes_{l=1}^{L/2}(2\hat{n}_{2l}-1),$
(231)

where $\hat{\mathcal{K}}$ denotes the complex conjugation operator. We then
find, assuming the commutation relation for the hardcore boson 101010Our
discussion can be similarly made for fermions.

$\displaystyle\hat{\mathcal{T}}\ket{\psi_{1}}\propto\ket{\psi_{1}},\>\hat{\mathcal{T}}\ket{\psi_{2}}\propto\ket{\psi_{2}},$
(232)
$\displaystyle\hat{\mathcal{T}}e^{-i\hat{H}t}\hat{\mathcal{T}}^{-1}=e^{-i\hat{H}t},$
(233)

and

$\displaystyle\hat{\mathcal{T}}\hat{a}_{l+1}^{\dagger}\hat{a}_{l}\hat{\mathcal{T}}^{-1}=-\hat{a}_{l+1}^{\dagger}\hat{a}_{l}.$
(234)

Then, we have

$\displaystyle\braket{\psi(t)}{\hat{a}_{l+1}^{\dagger}\hat{a}_{l}}{\psi(t)}$
$\displaystyle=(\ket{\psi},e^{i\hat{H}t}\hat{a}_{l+1}^{\dagger}\hat{a}_{l}e^{-i\hat{H}t}\ket{\psi})$
$\displaystyle=(\hat{\mathcal{T}}\ket{\psi},\hat{\mathcal{T}}e^{i\hat{H}t}\hat{a}_{l+1}^{\dagger}\hat{a}_{l}e^{-i\hat{H}t}\ket{\psi})^{*}$
$\displaystyle=(\ket{\psi},-e^{i\hat{H}t}\hat{a}_{l+1}^{\dagger}\hat{a}_{l}e^{-i\hat{H}t}\ket{\psi})^{*}$
$\displaystyle=-\braket{\psi(t)}{\hat{a}_{l}^{\dagger}\hat{a}_{l+1}}{\psi(t)}.$
(235)

From this relation, we find

$\displaystyle\mathcal{B}_{\mathrm{Lip}}$
$\displaystyle=2{K}\sum_{l}|\braket{\psi(t)}{\hat{a}_{l+1}^{\dagger}\hat{a}_{l}}{\psi(t)}|$
$\displaystyle=2K\sum_{l}\sqrt{p_{l}p_{l+1}}=R_{p}.$ (236)

On the other hand, Eq. (230) also leads to $E_{\mathrm{trans}}=0$. Since we
consider pure states and
$\sqrt{2[x^{\mathsf{T}}\nabla_{r^{p}}^{2}x]/R_{p}}=1$, we then find

$\displaystyle\mathcal{B}_{p}=R_{p},$ (237)

which is equal to $\mathcal{B}_{\mathrm{Lip}}$, Similarly, we have
$\mathcal{B}_{\mathrm{gL}}=\mathcal{B}_{\mathrm{Lip}}$.

We next discuss the condition for
$\braket{\dot{\hat{x}}}=\mathcal{B}_{\mathrm{Lip}}$. To see this, we notice
that the equality condition of the inequality (11) is obtained when the signs
of $(a_{n}-a_{m})J_{mn}$ are equal (or zero) for all $n\sim_{A}m$. For the
current situation, this reduces to the condition that the signs of $J_{l,l+1}$
are equal (or zero) for all $l$. For short times from the initial state
$\ket{\psi_{1}}$ we find that this condition is indeed satisfied with
$J_{l+1,l}\geq 0$. Note that this is consistent with the intuition that the
positive current indicates the transfer of the particle to the right.

##  Appendix I Derivation of the speed limits for classical stochastic systems

Here, we prove the speed limits for classical stochastic systems. We first
derive the bound using the entropy production, which is done with our general
framework and the inequalities used in Refs. [75, 41]. Then we derive the
bound using the Hatano-Sasa entropy production for general Markovian systems.

###  I.1 Speed limit using the entropy production rate

We apply the inequality (10) with $r_{nm}=W_{nm}p_{m}+W_{mn}p_{n}$. We have

$\displaystyle|\braket{\dot{A}}|\leq\sqrt{\frac{\Theta_{A}}{2}\sum_{n\sim_{A}m}\frac{(W_{nm}p_{m}-W_{mn}p_{n})^{2}}{W_{nm}p_{m}+W_{mn}p_{n}}},$
(238)

where

$\displaystyle\Theta_{A}$ $\displaystyle=\frac{1}{2}\sum_{n\sim
m}(a_{n}-a_{m})^{2}(W_{nm}p_{m}+W_{mn}p_{n})$ $\displaystyle=\sum_{n\sim
m}(a_{n}-a_{m})^{2}W_{nm}p_{m}$ (239)

is the second moment of the transition speed. Now, using
$\frac{2(a-b)^{2}}{a+b}\leq(a-b)\log\frac{a}{b}$ for $a,b>0$ [75, 41], we have

$\displaystyle\sum_{n\sim_{A}m}\frac{(W_{nm}p_{m}-W_{mn}p_{n})^{2}}{W_{nm}p_{m}+W_{mn}p_{n}}$
$\displaystyle\leq\sum_{n\sim_{A}m}W_{nm}p_{m}\ln\frac{W_{nm}p_{m}}{W_{mn}p_{n}}$
$\displaystyle:=\dot{\Sigma}_{\mathrm{cg}}.$ (240)

We then obtain

$\displaystyle|\braket{\dot{A}}|\leq\sqrt{\frac{\Theta_{A}\dot{\Sigma}_{\mathrm{cg}}}{2}}\leq\|\nabla
A\|_{\infty}\sqrt{\frac{\mathcal{A}_{\mathrm{cg}}\dot{\Sigma}_{\mathrm{cg}}}{2}},$
(241)

where $\mathcal{A}_{\mathrm{cg}}$ is the coarse-grained dynamical activity.

The coarse-grained variables above are bounded from above by the non-coarse-
grained variables. In fact, we have

$\displaystyle\mathcal{A}_{\mathrm{cg}}=\sum_{x\in\mathcal{S}_{n}}\sum_{y\in\mathcal{S}_{m}}\mathcal{W}_{xy}\rho_{y}\leq\mathcal{A}$
(242)

and

$\displaystyle\dot{\Sigma}_{\mathrm{cg}}$ $\displaystyle\leq\sum_{n\neq
m}\sum_{x\in\mathcal{S}_{n}}\sum_{y\in\mathcal{S}_{m}}\mathcal{W}_{xy}\rho_{y}\ln\frac{\sum_{x\in\mathcal{S}_{n}}\sum_{y\in\mathcal{S}_{m}}\mathcal{W}_{xy}\rho_{y}}{\sum_{y\in\mathcal{S}_{m}}\sum_{x\in\mathcal{S}_{n}}\mathcal{W}_{yx}\rho_{x}}$
$\displaystyle\leq\sum_{n\neq
m}\sum_{x\in\mathcal{S}_{n}}\sum_{y\in\mathcal{S}_{m}}\mathcal{W}_{xy}\rho_{y}\ln\frac{\mathcal{W}_{xy}\rho_{y}}{\mathcal{W}_{yx}\rho_{x}}$
$\displaystyle\leq\sum_{x\neq
y}\mathcal{W}_{xy}\rho_{y}\ln\frac{\mathcal{W}_{xy}\rho_{y}}{\mathcal{W}_{yx}\rho_{x}}=\dot{\Sigma},$
(243)

where we have used the log-sum inequality [84].

Instead of the expectation values of observables, we can consider the speed
limit for ${\Sigma}_{\mathrm{cg}}$ itself. Indeed, we have

$\displaystyle\dot{\Sigma}_{\mathrm{cg}}$
$\displaystyle=\frac{1}{2}\sum_{n\sim_{A}m}J^{c}_{nm}\ln\frac{W_{nm}p_{m}}{W_{mn}p_{n}}$
$\displaystyle\leq\sqrt{\frac{1}{2}\sum_{n\sim_{A}m}W_{nm}p_{m}\left(\ln\frac{W_{nm}p_{m}}{W_{mn}p_{n}}\right)^{2}\dot{\Sigma}_{\mathrm{cg}}}.$
(244)

Thus,

$\displaystyle\dot{\Sigma}_{\mathrm{cg}}$
$\displaystyle\leq\frac{1}{2}\sum_{n\sim_{A}m}W_{nm}p_{m}\left(\ln\frac{W_{nm}p_{m}}{W_{mn}p_{n}}\right)^{2}:=\frac{\Theta_{\Sigma}}{2},$
(245)

where $\Theta_{\Sigma}$ is the second moment of $\dot{\Sigma}_{\mathrm{cg}}$.
We then obtain

$\displaystyle|\braket{\dot{A}}|\leq\frac{\sqrt{\Theta_{A}\Theta_{\Sigma}}}{2}.$
(246)

When the detailed balance condition is satisfied and the non-coarse-grained
version is considered, inequality (245) is further bounded as

$\displaystyle\dot{\Sigma}\leq{\max_{x\sim_{A}y}\mathcal{W}_{xy}}\cdot\tilde{\mathcal{F}}(\rho||\rho^{\mathrm{ss}}),$
(247)

where

$\displaystyle\tilde{\mathcal{F}}(\rho||\rho^{\mathrm{ss}})=\frac{1}{2}\sum_{x\sim
y}\rho_{x}\left(\ln\frac{\rho_{x}}{\rho_{y}}-\ln\frac{\rho^{\mathrm{ss}}_{x}}{\rho_{y}^{\mathrm{ss}}}\right)^{2}$
(248)

is the discrete version of the Fisher divergence [103].

###  I.2 Speed limit using the Hatano-Sasa entropy production rate

We next derive the speed limit with the Hatano-Sasa entropy production rate.
For this purpose, we first notice that

$\displaystyle-
J_{mn}^{c}=W_{nm}p_{m}-\tilde{W}_{mn}p_{n}-J_{mn}^{c,\mathrm{ss}}\frac{p_{n}}{p_{n}^{\mathrm{ss}}},$
(249)

where

$\displaystyle
J_{mn}^{c,\mathrm{ss}}=-W_{nm}p_{m}^{\mathrm{ss}}+W_{mn}p_{n}^{\mathrm{ss}}$
(250)

is the stationary current.

From this, we have

$\displaystyle\braket{\dot{A}}=$
$\displaystyle-\frac{1}{2}\sum_{n\sim_{A}m}(a_{n}-a_{m})J_{mn}^{c,\mathrm{ss}}\frac{p_{n}}{p_{n}^{\mathrm{ss}}}$
(251)
$\displaystyle+\frac{1}{2}\sum_{n\sim_{A}m}(a_{n}-a_{m})(W_{nm}p_{m}-\tilde{W}_{mn}p_{n}).$
(252)

Then, introducing

$\displaystyle V_{A}$
$\displaystyle=-\frac{1}{2}\sum_{n\sim_{A}m}(a_{n}-a_{m})J_{mn}^{c,\mathrm{ss}}\frac{p_{n}}{p_{n}^{\mathrm{ss}}}$
(253)
$\displaystyle=-\frac{1}{2}\sum_{n\sim_{A}m}a_{n}J_{mn}^{c,\mathrm{ss}}\left(\frac{p_{n}}{p_{n}^{\mathrm{ss}}}+\frac{p_{m}}{p_{m}^{\mathrm{ss}}}\right),$
(254)

whose value becomes a stationary current of ${A}$ for the stationary state, we
have

$\displaystyle|\braket{\dot{A}}-V_{A}|=\left|\frac{1}{2}\sum_{n\sim_{A}m}(a_{n}-a_{m})(W_{nm}p_{m}-\tilde{W}_{mn}p_{n})\right|.$
(255)

To relate the right-hand side with the Hatano-Sasa entropy production, we use
the following deformation of the inequality:

$\displaystyle|\braket{\dot{A}}-V_{A}|^{2}$
$\displaystyle=\frac{1}{2}\int_{0}^{1}dzz\left|\sum_{n\sim_{A}m}(a_{n}-a_{m})(W_{nm}p_{m}-\tilde{W}_{mn}p_{n})\right|^{2}$
$\displaystyle\leq\frac{1}{2}\int_{0}^{1}dz{\sum_{n\sim_{A}m}(a_{n}-a_{m})^{2}(zW_{nm}p_{m}+(1-z)\tilde{W}_{mn}p_{n})}$
$\displaystyle\quad\quad\times{\sum_{n\sim_{A}m}\frac{z(W_{nm}p_{m}-\tilde{W}_{mn}p_{n})^{2}}{zW_{nm}p_{m}+(1-z)\tilde{W}_{mn}p_{n}}}$
$\displaystyle\leq\frac{1}{2}\max_{0\leq z\leq
1}\left[\sum_{n\sim_{A}m}(a_{n}-a_{m})^{2}(zW_{nm}p_{m}+(1-z)\tilde{W}_{mn}p_{n})\right]$
$\displaystyle\quad\quad\times\int_{0}^{1}dz{\sum_{n\sim_{A}m}\frac{z(W_{nm}p_{m}-\tilde{W}_{mn}p_{n})^{2}}{zW_{nm}p_{m}+(1-z)\tilde{W}_{mn}p_{n}}}$
$\displaystyle\leq{\frac{\tilde{\Theta}_{A}\dot{\Sigma}_{\mathrm{cg}}^{\mathrm{HS}}}{2}},$
(256)

where

$\displaystyle\tilde{\Theta}_{A}$ $\displaystyle=\max_{0\leq z\leq
1}\left[\sum_{n\sim_{A}m}(a_{n}-a_{m})^{2}(zW_{nm}p_{m}+(1-z)\tilde{W}_{mn}p_{n})\right]$
$\displaystyle=\max\left[\sum_{n\sim_{A}m}(a_{n}-a_{m})^{2}W_{nm}p_{m},\sum_{n\sim_{A}m}(a_{n}-a_{m})^{2}\tilde{W}_{nm}p_{m}\right]$
(257)

and we have used
$\int_{0}^{1}dz\frac{z(a-b)^{2}}{za+(1-z)b}=a\ln\frac{a}{b}-a+b$ [161]. Since

$\displaystyle\tilde{\Theta}_{A}\leq\|\nabla
A\|_{\infty}^{2}\mathcal{A}_{\mathrm{cg}},$ (258)

we have

$\displaystyle|\braket{\dot{A}}-V_{A}|\leq\sqrt{{\frac{\tilde{\Theta}_{A}\dot{\Sigma}_{\mathrm{cg}}^{\mathrm{HS}}}{2}}}\leq\|\nabla
A\|_{\infty}\sqrt{\frac{\mathcal{A}_{\mathrm{cg}}\dot{\Sigma}_{\mathrm{cg}}^{\mathrm{HS}}}{2}}.$
(259)

##  Appendix J Derivation of inequality (126)

We here show the inequality (126). We first note that

$\displaystyle\sum_{n}a_{n}K_{n}^{\eta}=\sum_{nml}\left(a_{n}-\frac{a_{m}+a_{l}}{2}\right)\mathrm{Tr}[(\hat{L}_{\eta})_{nm}\hat{\rho}_{ml}(\hat{L}_{\eta}^{\dagger})_{ln}].$
(260)

As mentioned in the main text, we impose one assumption about the jump
operator: we require that each jump $\eta$ moves a state in the subspace
$\mathcal{H}_{m}$ to that in $\mathcal{H}_{n=f_{\eta}(m)}$ with an injective
function $f_{\eta}$. Put differently, when $(\hat{L}_{\eta})_{nm}$ is nonzero
for some $n$ and $m$, $(\hat{L}_{\eta})_{n^{\prime}m}\>(n^{\prime}\neq n)$ and
$(\hat{L}_{\eta})_{nm^{\prime}}\>(m^{\prime}\neq m)$ should be zero. Under
this assumption, we can set $m=l$ in Eq. (260) and have

$\displaystyle\sum_{\eta}\sum_{n}a_{n}K_{n}^{\eta}$
$\displaystyle=\sum_{nm}(a_{n}-a_{m})\sum_{\eta}\mathrm{Tr}[(\hat{L}_{\eta}^{\dagger})_{mn}(\hat{L}_{\eta})_{nm}\hat{\rho}_{mm}]$
$\displaystyle=\frac{1}{2}\sum_{nm}(a_{n}-a_{m})\left(T_{m\rightarrow
n}^{\mathrm{cg}}-T_{n\rightarrow m}^{\mathrm{cg}}\right)$ (261)

with $T_{m\rightarrow
n}^{\mathrm{cg}}=\sum_{\eta}\mathrm{Tr}[(\hat{L}_{\eta}^{\dagger})_{mn}(\hat{L}_{\eta})_{nm}\hat{\rho}_{mm}].$

We use the same technique discussed in Appendix I to obtain inequality (126):

$\displaystyle\left|\sum_{\eta}\sum_{n}a_{n}K_{n}^{\eta}\right|$
$\displaystyle\leq\frac{1}{2}\max_{(n,m)\in\mathcal{E}_{\mathrm{D}}}|a_{n}-a_{m}|$
$\displaystyle\times\sqrt{\sum_{nm}(T_{m\rightarrow
n}^{\mathrm{cg}}+T_{n\rightarrow
m}^{\mathrm{cg}})\sum_{nm}\frac{(T_{m\rightarrow
n}^{\mathrm{cg}}-T_{n\rightarrow m}^{\mathrm{cg}})^{2}}{T_{m\rightarrow
n}^{\mathrm{cg}}+T_{n\rightarrow m}^{\mathrm{cg}}}}$
$\displaystyle\leq\max_{(n,m)\in\mathcal{E}_{\mathrm{D}}}|a_{n}-a_{m}|\sqrt{\frac{\mathcal{A}_{\mathrm{cg}}^{d}\dot{\Sigma}_{\mathrm{cg}}^{d}}{2}},$
(262)

where we have defined

$\displaystyle\mathcal{A}_{\mathrm{cg}}^{d}=\sum_{n\neq m}T_{m\rightarrow
n}^{\mathrm{cg}},$ (263)

and

$\displaystyle\dot{\Sigma}_{\mathrm{cg}}^{d}=\sum_{n,m}T_{m\rightarrow
n}^{\mathrm{cg}}\ln\frac{T_{m\rightarrow n}^{\mathrm{cg}}}{T_{n\rightarrow
m}^{\mathrm{cg}}}\geq 0.$ (264)

##  Appendix K Derivation of inequality (133)

We start from evaluating the speed of $\braket{e^{\pm\lambda\hat{A}}}$ with
$\lambda>0$:

$\displaystyle\left|\frac{\mathrm{d}\braket{e^{\pm\lambda\hat{A}}}}{\mathrm{d}t}\right|$
$\displaystyle\leq\left|\sum_{n}e^{\pm\lambda a_{n}}\dot{p}_{n}\right|$
$\displaystyle\leq\frac{1}{2}\left|\sum_{n\sim_{A}m}(e^{\pm\lambda
a_{n}}-e^{\pm\lambda a_{m}})J_{nm}^{q}\right|$
$\displaystyle\leq\left|\sum_{n\sim_{A}m}(e^{\pm\lambda a_{n}}-e^{\pm\lambda
a_{m}})\sqrt{p_{n}p_{m}}\|\hat{H}_{nm}\|_{\mathrm{op}}\right|.$ (265)

Now, we use the fact that, for $0<\lambda\leq\lambda_{M}:=\frac{2}{\|\nabla
A\|_{\infty}}\log\frac{\sqrt{5}-1}{2}$, we have

$\displaystyle e^{\pm\lambda a_{n}}-e^{\pm\lambda a_{m}}\leq
e^{\pm\frac{\lambda a_{n}}{2}}e^{\pm\frac{\lambda a_{m}}{2}}$ (266)

for arbitrary $n\sim m$. In fact, inequality (266) is equivalent to
$y^{2}-y-1\leq 0$ with $y=e^{\pm\frac{\lambda}{2}(a_{n}-a_{m})}$. This is
satisfied when $\pm\lambda(a_{n}-a_{m})\leq 2\log\frac{\sqrt{5}-1}{2}$. Since
the maximum value of $\pm(a_{n}-a_{m})$ is bounded by $\|\nabla A\|_{\infty}$,
$0<\lambda\leq\lambda_{M}$ ensures inequality (266).

Then, we have

$\displaystyle\left|\frac{\mathrm{d}\braket{e^{\pm\lambda\hat{A}}}}{\mathrm{d}t}\right|$
$\displaystyle\leq\left|\sum_{n\sim_{A}m}(e^{\pm\frac{\lambda
a_{n}}{2}}e^{\pm\frac{\lambda
a_{m}}{2}})\sqrt{p_{n}p_{m}}\|\hat{H}_{nm}\|_{\mathrm{op}}\right|$
$\displaystyle\leq\sqrt{\sum_{n\sim_{A}m}e^{\pm\lambda
a_{n}}p_{n}\|\hat{H}_{nm}\|_{\mathrm{op}}}\sqrt{\sum_{n\sim_{A}m}e^{\pm\lambda
a_{m}}p_{m}\|\hat{H}_{nm}\|_{\mathrm{op}}}$ $\displaystyle\leq
C_{H}\braket{e^{\pm\lambda\hat{A}}}.$ (267)

Integration from time 0 to $t$ leads to

$\displaystyle\braket{e^{\pm\lambda\hat{A}}}\leq\braket{e^{\pm\lambda\hat{A}}}_{0}e^{\overline{C_{H}}t},$
(268)

where $\braket{\cdots}_{0}$ is the expectation value with respect to
$\hat{\rho}(0)$.

Then, the Markov inequality indicates that

$\displaystyle\mathbb{P}_{t}\left[\left|\frac{A-A^{\prime}}{\|\nabla
A\|_{\infty}}\right|>\epsilon\right]$
$\displaystyle=\mathbb{P}_{t}\left[{{A-A^{\prime}}>{\|\nabla
A\|_{\infty}}}\epsilon\right]+\mathbb{P}_{t}\left[{A^{\prime}-A>{\|\nabla
A\|_{\infty}}}\epsilon\right]$
$\displaystyle\leq\frac{\braket{e^{\lambda(\hat{A}-A^{\prime})}}}{e^{\lambda\|\nabla
A\|_{\infty}\epsilon}}+\frac{\braket{e^{-\lambda(\hat{A}-A^{\prime})}}}{e^{\lambda\|\nabla
A\|_{\infty}\epsilon}}$
$\displaystyle\leq\braket{e^{\lambda(\hat{A}-A^{\prime})}}_{0}e^{\overline{C_{H}}t-\lambda\|\nabla
A\|_{\infty}\epsilon}+\braket{e^{-\lambda(\hat{A}-A^{\prime})}}_{0}e^{\overline{C_{H}}t-\lambda\|\nabla
A\|_{\infty}\epsilon}.$ (269)

Setting $\lambda$ as $\lambda\|\nabla A\|_{\infty}$, we have

$\displaystyle\mathbb{P}_{t}\left[\left|\frac{A-A^{\prime}}{\|\nabla
A\|_{\infty}}\right|>\epsilon\right]\leq
2\left\langle\cosh\frac{\lambda(\hat{A}-A^{\prime})}{\|\nabla
A\|_{\infty}}\right\rangle_{0}e^{\overline{C_{H}}t-\lambda\epsilon}$ (270)

for $0<\lambda\leq 2\log\frac{\sqrt{5}-1}{2}$.

## References

  * Mandelstam and Tamm [1945] L. Mandelstam and I. Tamm, The energy–time uncertainty relation in non-relativistic quantum mechanics, J. Phys. 9, 249 (1945).
  * Deffner and Campbell [2017] S. Deffner and S. Campbell, Quantum speed limits: from heisenberg’s uncertainty principle to optimal quantum control, Journal of Physics A: Mathematical and Theoretical 50, 453001 (2017).
  * Margolus and Levitin [1998] N. Margolus and L. B. Levitin, The maximum speed of dynamical evolution, Physica D: Nonlinear Phenomena 120, 188 (1998).
  * Giovannetti  _et al._ [2003] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum limits to dynamical evolution, Phys. Rev. A 67, 052109 (2003).
  * Levitin and Toffoli [2009] L. B. Levitin and T. Toffoli, Fundamental limit on the rate of quantum dynamics: The unified bound is tight, Phys. Rev. Lett. 103, 160502 (2009).
  * Fleming [1973] G. N. Fleming, A unitarity bound on the evolution of nonstationary states, Il Nuovo Cimento A (1965-1970) 16, 232 (1973).
  * Bhattacharyya [1983] K. Bhattacharyya, Quantum decay and the mandelstam-tamm-energy inequality, Journal of Physics A: Mathematical and General 16, 2993 (1983).
  * Anandan and Aharonov [1990] J. Anandan and Y. Aharonov, Geometry of quantum evolution, Phys. Rev. Lett. 65, 1697 (1990).
  * Pfeifer [1993] P. Pfeifer, How fast can a quantum state change with time?, Phys. Rev. Lett. 70, 3365 (1993).
  * Taddei  _et al._ [2013] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos Filho, Quantum speed limit for physical processes, Phys. Rev. Lett. 110, 050402 (2013).
  * Deffner and Lutz [2013] S. Deffner and E. Lutz, Quantum speed limit for non-markovian dynamics, Phys. Rev. Lett. 111, 010402 (2013).
  * del Campo  _et al._ [2013] A. del Campo, I. L. Egusquiza, M. B. Plenio, and S. F. Huelga, Quantum speed limits in open system dynamics, Phys. Rev. Lett. 110, 050403 (2013).
  * Xu  _et al._ [2014] Z.-Y. Xu, S. Luo, W. L. Yang, C. Liu, and S. Zhu, Quantum speedup in a memory environment, Phys. Rev. A 89, 012307 (2014).
  * Zhang  _et al._ [2014] Y.-J. Zhang, W. Han, Y.-J. Xia, J.-P. Cao, and H. Fan, Quantum speed limit for arbitrary initial states, Scientific reports 4, 1 (2014).
  * Mirkin  _et al._ [2016] N. Mirkin, F. Toscano, and D. A. Wisniacki, Quantum-speed-limit bounds in an open quantum evolution, Phys. Rev. A 94, 052125 (2016).
  * Mondal  _et al._ [2016] D. Mondal, C. Datta, and S. Sazim, Quantum coherence sets the quantum speed limit for mixed states, Physics Letters A 380, 689 (2016).
  * Marvian  _et al._ [2016] I. Marvian, R. W. Spekkens, and P. Zanardi, Quantum speed limits, coherence, and asymmetry, Phys. Rev. A 93, 052331 (2016).
  * Campaioli  _et al._ [2018] F. Campaioli, F. A. Pollock, F. C. Binder, and K. Modi, Tightening quantum speed limits for almost all states, Phys. Rev. Lett. 120, 060409 (2018).
  * Il’in and Lychkovskiy [2021] N. Il’in and O. Lychkovskiy, Quantum speed limit for thermal states, Phys. Rev. A 103, 062204 (2021).
  * Liu  _et al._ [2019] J. Liu, H. Yuan, X.-M. Lu, and X. Wang, Quantum fisher information matrix and multiparameter estimation, Journal of Physics A: Mathematical and Theoretical 53, 023001 (2019).
  * Jones and Kok [2010] P. J. Jones and P. Kok, Geometric derivation of the quantum speed limit, Phys. Rev. A 82, 022107 (2010).
  * Pires  _et al._ [2016] D. P. Pires, M. Cianciaruso, L. C. Céleri, G. Adesso, and D. O. Soares-Pinto, Generalized geometric quantum speed limits, Phys. Rev. X 6, 021031 (2016).
  * García-Pintos  _et al._ [2021] L. P. García-Pintos, S. Nicholson, J. R. Green, A. del Campo, and A. V. Gorshkov, Unifying quantum and classical speed limits on observables, arXiv preprint arXiv:2108.04261 (2021).
  * Yung [2006] M.-H. Yung, Quantum speed limit for perfect state transfer in one dimension, Phys. Rev. A 74, 030303 (2006).
  * Caneva  _et al._ [2009] T. Caneva, M. Murphy, T. Calarco, R. Fazio, S. Montangero, V. Giovannetti, and G. E. Santoro, Optimal control at the quantum speed limit, Phys. Rev. Lett. 103, 240501 (2009).
  * Murphy  _et al._ [2010] M. Murphy, S. Montangero, V. Giovannetti, and T. Calarco, Communication at the quantum speed limit along a spin chain, Phys. Rev. A 82, 022318 (2010).
  * Ashhab  _et al._ [2012] S. Ashhab, P. C. de Groot, and F. Nori, Speed limits for quantum gates in multiqubit systems, Phys. Rev. A 85, 052327 (2012).
  * Bukov  _et al._ [2019] M. Bukov, D. Sels, and A. Polkovnikov, Geometric speed limit of accessible many-body state preparation, Phys. Rev. X 9, 011034 (2019).
  * Lam  _et al._ [2021] M. R. Lam, N. Peter, T. Groh, W. Alt, C. Robens, D. Meschede, A. Negretti, S. Montangero, T. Calarco, and A. Alberti, Demonstration of quantum brachistochrones between distant states of an atom, Phys. Rev. X 11, 011035 (2021).
  * Campbell and Deffner [2017] S. Campbell and S. Deffner, Trade-off between speed and cost in shortcuts to adiabaticity, Phys. Rev. Lett. 118, 100601 (2017).
  * Funo  _et al._ [2017] K. Funo, J.-N. Zhang, C. Chatou, K. Kim, M. Ueda, and A. del Campo, Universal work fluctuations during shortcuts to adiabaticity by counterdiabatic driving, Phys. Rev. Lett. 118, 100602 (2017).
  * Guéry-Odelin  _et al._ [2019] D. Guéry-Odelin, A. Ruschhaupt, A. Kiely, E. Torrontegui, S. Martínez-Garaot, and J. G. Muga, Shortcuts to adiabaticity: Concepts, methods, and applications, Rev. Mod. Phys. 91, 045001 (2019).
  * Giovannetti  _et al._ [2001] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced positioning and clock synchronization, Nature 412, 417 (2001).
  * Giovannetti  _et al._ [2004] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced measurements: beating the standard quantum limit, Science 306, 1330 (2004).
  * Tóth and Apellaniz [2014] G. Tóth and I. Apellaniz, Quantum metrology from a quantum information science perspective, Journal of Physics A: Mathematical and Theoretical 47, 424006 (2014).
  * Okuyama and Ohzeki [2018] M. Okuyama and M. Ohzeki, Quantum speed limit is not quantum, Phys. Rev. Lett. 120, 070402 (2018).
  * Shanahan  _et al._ [2018] B. Shanahan, A. Chenu, N. Margolus, and A. del Campo, Quantum speed limits across the quantum-to-classical transition, Phys. Rev. Lett. 120, 070401 (2018).
  * Ito [2018] S. Ito, Stochastic thermodynamic interpretation of information geometry, Phys. Rev. Lett. 121, 030605 (2018).
  * Ito and Dechant [2020] S. Ito and A. Dechant, Stochastic time evolution, information geometry, and the cramér-rao bound, Phys. Rev. X 10, 021056 (2020).
  * Nicholson  _et al._ [2020] S. B. Nicholson, L. P. Garcia-Pintos, A. del Campo, and J. R. Green, Time–information uncertainty relations in thermodynamics, Nature Physics 16, 1211 (2020).
  * Shiraishi  _et al._ [2018] N. Shiraishi, K. Funo, and K. Saito, Speed limit for classical stochastic processes, Phys. Rev. Lett. 121, 070601 (2018).
  * Falasco and Esposito [2020] G. Falasco and M. Esposito, Dissipation-time uncertainty relation, Phys. Rev. Lett. 125, 120604 (2020).
  * Note [1] Here, $a=1$ for a ballistic and $a=2$ for a diffusive process.
  * Note [2] In a typical situation, $b=1$ for a ballistic and $b=1/2$ for a diffusive process. For long times, $\mathcal{B}_{\mathrm{ur}}$ saturates to a value $\mathrm{O}(L)$.
  * Polkovnikov  _et al._ [2011] A. Polkovnikov, K. Sengupta, A. Silva, and M. Vengalattore, Colloquium: Nonequilibrium dynamics of closed interacting quantum systems, Rev. Mod. Phys. 83, 863 (2011).
  * Yukalov [2011] V. Yukalov, Equilibration and thermalization in finite quantum systems, Laser Physics Letters 8, 485 (2011).
  * Eisert  _et al._ [2015] J. Eisert, M. Friesdorf, and C. Gogolin, Quantum many-body systems out of equilibrium, Nature Physics 11, 124 (2015).
  * D’Alessio  _et al._ [2016] L. D’Alessio, Y. Kafri, A. Polkovnikov, and M. Rigol, From quantum chaos and eigenstate thermalization to statistical mechanics and thermodynamics, Advances in Physics 65, 239 (2016).
  * Bloch  _et al._ [2012] I. Bloch, J. Dalibard, and S. Nascimbene, Quantum simulations with ultracold quantum gases, Nature Physics 8, 267 (2012).
  * Gross and Bloch [2017] C. Gross and I. Bloch, Quantum simulations with ultracold atoms in optical lattices, Science 357, 995 (2017).
  * Goldstein  _et al._ [2013] S. Goldstein, T. Hara, and H. Tasaki, Time scales in the approach to equilibrium of macroscopic quantum systems, Phys. Rev. Lett. 111, 140401 (2013).
  * Goldstein  _et al._ [2015] S. Goldstein, T. Hara, and H. Tasaki, Extremely quick thermalization in a macroscopic quantum system for a typical nonequilibrium subspace, New Journal of Physics 17, 045002 (2015).
  * Reimann [2016] P. Reimann, Typical fast thermalization processes in closed many-body systems, Nature communications 7, 1 (2016).
  * de Oliveira  _et al._ [2018] T. R. de Oliveira, C. Charalambous, D. Jonathan, M. Lewenstein, and A. Riera, Equilibration time scales in closed many-body quantum systems, New Journal of Physics 20, 033032 (2018).
  * Bose [2003] S. Bose, Quantum communication through an unmodulated spin chain, Phys. Rev. Lett. 91, 207901 (2003).
  * Kay [2010] A. Kay, Perfect, efficient, state transfer and its application as a constructive tool, International Journal of Quantum Information 8, 641 (2010).
  * Deffner [2017] S. Deffner, Geometric quantum speed limits: a case for wigner phase space, New Journal of Physics 19, 103018 (2017).
  * Funo  _et al._ [2019] K. Funo, N. Shiraishi, and K. Saito, Speed limit for open quantum systems, New Journal of Physics 21, 013006 (2019).
  * Villani [2009] C. Villani,  _Optimal transport: old and new_ , Vol. 338 (Springer, 2009).
  * Dechant and Sakurai [2019] A. Dechant and Y. Sakurai, Thermodynamic interpretation of wasserstein distance, arXiv preprint arXiv:1912.08405 (2019).
  * Van Vu and Hasegawa [2021] T. Van Vu and Y. Hasegawa, Geometrical bounds of the irreversibility in markovian systems, Phys. Rev. Lett. 126, 010601 (2021).
  * Nakazato and Ito [2021] M. Nakazato and S. Ito, Geometrical aspects of entropy production in stochastic thermodynamics based on wasserstein distance, arXiv preprint arXiv:2103.00503 (2021).
  * Dechant  _et al._ [2021] A. Dechant, S.-i. Sasa, and S. Ito, Geometric decomposition of entropy production in out-of-equilibrium systems, arXiv preprint arXiv:2109.12817 (2021).
  * Dechant [2021] A. Dechant, Minimum entropy production, detailed balance and wasserstein distance for continuous-time markov processes, arXiv preprint arXiv:2110.01141 (2021).
  * Carlen and Maas [2014] E. A. Carlen and J. Maas, An analog of the 2-wasserstein metric in non-commutative probability under which the fermionic fokker–planck equation is gradient flow for the entropy, Communications in mathematical physics 331, 887 (2014).
  * Datta and Rouzé [2020] N. Datta and C. Rouzé, Relating relative entropy, optimal transport and fisher information: a quantum hwi inequality, in  _Annales Henri Poincaré_ , Vol. 21 (Springer, 2020) pp. 2115–2150.
  * Chow  _et al._ [2019] S.-N. Chow, W. Li, and H. Zhou, A discrete schrödinger equation via optimal transport on graphs, Journal of Functional Analysis 276, 2440 (2019).
  * De Palma  _et al._ [2021] G. De Palma, M. Marvian, D. Trevisan, and S. Lloyd, The quantum wasserstein distance of order 1, IEEE Transactions on Information Theory (2021).
  * Hu  _et al._ [2020] X. Hu, S. Sun, and Y. Zheng, Quantum speed limit via the trajectory ensemble, Phys. Rev. A 101, 042107 (2020).
  * Lieb and Robinson [1972] E. H. Lieb and D. W. Robinson, The finite group velocity of quantum spin systems, in  _Statistical mechanics_ (Springer, 1972) pp. 425–431.
  * Dechant and Sasa [2018a] A. Dechant and S.-i. Sasa, Entropic bounds on currents in langevin systems, Phys. Rev. E 97, 062101 (2018a).
  * Bakr  _et al._ [2009] W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, A quantum gas microscope for detecting single atoms in a hubbard-regime optical lattice, Nature 462, 74 (2009).
  * Deffner and Lutz [2010] S. Deffner and E. Lutz, Generalized clausius inequality for nonequilibrium quantum processes, Phys. Rev. Lett. 105, 170402 (2010).
  * Deffner and Lutz [2011] S. Deffner and E. Lutz, Nonequilibrium entropy production for open quantum systems, Phys. Rev. Lett. 107, 140404 (2011).
  * Shiraishi  _et al._ [2016] N. Shiraishi, K. Saito, and H. Tasaki, Universal trade-off relation between power and efficiency for heat engines, Phys. Rev. Lett. 117, 190601 (2016).
  * Barato and Seifert [2015] A. C. Barato and U. Seifert, Thermodynamic uncertainty relation for biomolecular processes, Phys. Rev. Lett. 114, 158101 (2015).
  * Gingrich  _et al._ [2016] T. R. Gingrich, J. M. Horowitz, N. Perunov, and J. L. England, Dissipation bounds all steady-state current fluctuations, Phys. Rev. Lett. 116, 120601 (2016).
  * Horowitz and Gingrich [2020] J. M. Horowitz and T. R. Gingrich, Thermodynamic uncertainty relations constrain non-equilibrium fluctuations, Nature Physics 16, 15 (2020).
  * Dechant and Sasa [2018b] A. Dechant and S.-i. Sasa, Current fluctuations and transport efficiency for general langevin systems, Journal of Statistical Mechanics: Theory and Experiment 2018, 063209 (2018b).
  * Hatano and Sasa [2001] T. Hatano and S.-i. Sasa, Steady-state thermodynamics of langevin systems, Phys. Rev. Lett. 86, 3463 (2001).
  * Gorini  _et al._ [1976] V. Gorini, A. Kossakowski, and E. C. G. Sudarshan, Completely positive dynamical semigroups of n-level systems, Journal of Mathematical Physics 17, 821 (1976).
  * Lindblad [1976] G. Lindblad, On the generators of quantum dynamical semigroups, Communications in Mathematical Physics 48, 119 (1976).
  * Merris [1994] R. Merris, Laplacian matrices of graphs: a survey, Linear algebra and its applications 197, 143 (1994).
  * Cover [1999] T. M. Cover,  _Elements of information theory_ (John Wiley & Sons, 1999).
  * Strasberg and Winter [2021] P. Strasberg and A. Winter, First and second law of quantum thermodynamics: A consistent derivation based on a microscopic definition of entropy, PRX Quantum 2, 030202 (2021).
  * Note [3] Assuming that the current fluctuates randomly in time with zero mean, we obtain $\gamma=1/2$.
  * Note [4] When we consider a pure state and take the entire basis $\\{\mathinner{|{i}\rangle}\\}$ of the Hilbert space to describe each subspace, $\theta_{ij}$ indeed becomes the standard phase difference, $\theta_{j}-\theta_{i}$. Here, we take $\mathinner{\langle{i|\psi}\rangle}=|\mathinner{\langle{i|\psi}\rangle}|e^{i\theta_{i}}$.
  * Nachtergaele and Sims [2006] B. Nachtergaele and R. Sims, Lieb-robinson bounds and the exponential clustering theorem, Communications in mathematical physics 265, 119 (2006).
  * Bravyi  _et al._ [2006] S. Bravyi, M. B. Hastings, and F. Verstraete, Lieb-robinson bounds and the generation of correlations and topological quantum order, Phys. Rev. Lett. 97, 050401 (2006).
  * Lashkari  _et al._ [2013] N. Lashkari, D. Stanford, M. Hastings, T. Osborne, and P. Hayden, Towards the fast scrambling conjecture, Journal of High Energy Physics 2013, 1 (2013).
  * Abanin  _et al._ [2015] D. A. Abanin, W. De Roeck, and F. m. c. Huveneers, Exponentially slow heating in periodically driven many-body systems, Phys. Rev. Lett. 115, 256803 (2015).
  * Kuwahara  _et al._ [2016] T. Kuwahara, T. Mori, and K. Saito, Floquet–magnus theory and generic transient dynamics in periodically driven many-body quantum systems, Annals of Physics 367, 96 (2016).
  * Abanin  _et al._ [2017] D. Abanin, W. De Roeck, W. W. Ho, and F. Huveneers, A rigorous theory of many-body prethermalization for periodically driven and closed quantum systems, Communications in Mathematical Physics 354, 809 (2017).
  * Kukuljan  _et al._ [2017] I. Kukuljan, S. c. v. Grozdanov, and T. c. v. Prosen, Weak quantum chaos, Phys. Rev. B 96, 060301 (2017).
  * Gong  _et al._ [2020a] Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, Universal error bound for constrained quantum dynamics, Phys. Rev. Lett. 124, 210606 (2020a).
  * Gong  _et al._ [2020b] Z. Gong, N. Yoshioka, N. Shibata, and R. Hamazaki, Error bounds for constrained dynamics in gapped quantum systems: Rigorous results and generalizations, Phys. Rev. A 101, 052122 (2020b).
  * Baumgratz  _et al._ [2014] T. Baumgratz, M. Cramer, and M. B. Plenio, Quantifying coherence, Phys. Rev. Lett. 113, 140401 (2014).
  * Shimizu and Morimae [2005] A. Shimizu and T. Morimae, Detection of macroscopic entanglement by correlation of local observables, Phys. Rev. Lett. 95, 090401 (2005).
  * Yadin and Vedral [2016] B. Yadin and V. Vedral, General framework for quantum macroscopicity in terms of coherence, Phys. Rev. A 93, 022122 (2016).
  * Streltsov  _et al._ [2017] A. Streltsov, G. Adesso, and M. B. Plenio, Colloquium: Quantum coherence as a resource, Rev. Mod. Phys. 89, 041003 (2017).
  * Fröwis  _et al._ [2018] F. Fröwis, P. Sekatski, W. Dür, N. Gisin, and N. Sangouard, Macroscopic quantum states: Measures, fragility, and implementations, Rev. Mod. Phys. 90, 025004 (2018).
  * Note [5] To be precise, $\mathcal{C}$ obtained from this specific choice of $c_{nm}=|n-m|$ may not satisfy some criteria [97] for quantum coherence. Nevertheless, the measure obtained by this choice can be useful for discussing the speed of certain transitions between states with different macroscopic coherence. We thus use this choice in this work as well as the coherence measure based on the variance.
  * Lyu [2012] S. Lyu, Interpretation and generalization of score matching, arXiv preprint arXiv:1205.2629 (2012).
  * Deffner [2020] S. Deffner, Quantum speed limits and the maximal rate of information production, Phys. Rev. Research 2, 013161 (2020).
  * Bremermann [1967] H. J. Bremermann, Quantum noise and information, in  _Proceedings of the fifth berkeley symposium on mathematical statistics and probability_ , Vol. 4 (University of California Press Berkeley, CA, 1967) pp. 15–20.
  * Bekenstein [1981] J. D. Bekenstein, Energy cost of information transfer, Phys. Rev. Lett. 46, 623 (1981).
  * Dalfovo  _et al._ [1999] F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of bose-einstein condensation in trapped gases, Rev. Mod. Phys. 71, 463 (1999).
  * Note [6] This divergence for the speed is different from the divergence of the transition time discussed in [15, 162], which is caused by the vanishing averaged speed near the stationary state.
  * Schneider  _et al._ [2012] U. Schneider, L. Hackermüller, J. P. Ronzheimer, S. Will, S. Braun, T. Best, I. Bloch, E. Demler, S. Mandt, D. Rasch, _et al._ , Fermionic transport and out-of-equilibrium dynamics in a homogeneous hubbard model with ultracold atoms, Nature Physics 8, 213 (2012).
  * Ronzheimer  _et al._ [2013] J. P. Ronzheimer, M. Schreiber, S. Braun, S. S. Hodgman, S. Langer, I. P. McCulloch, F. Heidrich-Meisner, I. Bloch, and U. Schneider, Expansion dynamics of interacting bosons in homogeneous lattices in one and two dimensions, Phys. Rev. Lett. 110, 205301 (2013).
  * Schuch  _et al._ [2011] N. Schuch, S. K. Harrison, T. J. Osborne, and J. Eisert, Information propagation for interacting-particle systems, Phys. Rev. A 84, 032309 (2011).
  * Wang and Hazzard [2020] Z. Wang and K. R. Hazzard, Tightening the lieb-robinson bound in locally interacting systems, PRX Quantum 1, 010303 (2020).
  * Kuwahara and Saito [2021] T. Kuwahara and K. Saito, Lieb-robinson bound and almost-linear light cone in interacting boson systems, Phys. Rev. Lett. 127, 070403 (2021).
  * Faupin  _et al._ [2021] J. Faupin, M. Lemm, and I. M. Sigal, Maximal speed for macroscopic particle transport in the bose-hubbard model, arXiv preprint arXiv:2110.04313 (2021).
  * Note [7] We have $\sum@\slimits@_{l}\sqrt{m_{l,i}(m_{l\pm 1,i}+1)}=\sum@\slimits@_{l:m_{l,i}\neq 0}\sqrt{m_{l,i}(m_{l\pm 1,i}+1)}$. Using the Cauchy-Schwarz inequality, this is further bounded by $C_{H}\lesssim 2K\sqrt{\sum@\slimits@_{l:m_{l,i}\neq 0}{m_{l,i}}\sum@\slimits@_{l:m_{l,i}\neq 0}{(m_{l\pm 1,i}+1)}}\leq 2\sqrt{2}KM$. Thus, $C_{M}/M$ does not grow with $M$.
  * Pietzonka  _et al._ [2016] P. Pietzonka, A. C. Barato, and U. Seifert, Universal bounds on current fluctuations, Phys. Rev. E 93, 052145 (2016).
  * Horowitz and Gingrich [2017] J. M. Horowitz and T. R. Gingrich, Proof of the finite-time thermodynamic uncertainty relation for steady-state currents, Phys. Rev. E 96, 020103 (2017).
  * Proesmans and Van den Broeck [2017] K. Proesmans and C. Van den Broeck, Discrete-time thermodynamic uncertainty relation, EPL (Europhysics Letters) 119, 20001 (2017).
  * Pigolotti  _et al._ [2017] S. Pigolotti, I. Neri, E. Roldán, and F. Jülicher, Generic properties of stochastic entropy production, Phys. Rev. Lett. 119, 140604 (2017).
  * Hasegawa and Van Vu [2019a] Y. Hasegawa and T. Van Vu, Fluctuation theorem uncertainty relation, Phys. Rev. Lett. 123, 110602 (2019a).
  * Brandner  _et al._ [2018] K. Brandner, T. Hanazato, and K. Saito, Thermodynamic bounds on precision in ballistic multiterminal transport, Phys. Rev. Lett. 120, 090601 (2018).
  * Dechant and Sasa [2020] A. Dechant and S.-i. Sasa, Fluctuation–response inequality out of equilibrium, Proceedings of the National Academy of Sciences 117, 6430 (2020).
  * Hasegawa and Van Vu [2019b] Y. Hasegawa and T. Van Vu, Uncertainty relations in stochastic processes: An information inequality approach, Phys. Rev. E 99, 062126 (2019b).
  * Liu  _et al._ [2020] K. Liu, Z. Gong, and M. Ueda, Thermodynamic uncertainty relation for arbitrary initial states, Phys. Rev. Lett. 125, 140602 (2020).
  * Esposito [2012] M. Esposito, Stochastic thermodynamics under coarse graining, Phys. Rev. E 85, 041125 (2012).
  * Seifert [2012] U. Seifert, Stochastic thermodynamics, fluctuation theorems and molecular machines, Reports on progress in physics 75, 126001 (2012).
  * Otsubo  _et al._ [2020] S. Otsubo, S. Ito, A. Dechant, and T. Sagawa, Estimating entropy production by machine learning of short-time fluctuating currents, Phys. Rev. E 101, 062106 (2020).
  * Note [8] This bound may be further tightened by using the recently found inequality for the Wasserstein distance [64].
  * Derrida [1998] B. Derrida, An exactly soluble non-equilibrium system: the asymmetric simple exclusion process, Physics Reports 301, 65 (1998).
  * Eisler [2011] V. Eisler, Crossover between ballistic and diffusive transport: the quantum exclusion process, Journal of Statistical Mechanics: Theory and Experiment 2011, P06007 (2011).
  * Temme  _et al._ [2012] K. Temme, M. M. Wolf, and F. Verstraete, Stochastic exclusion processes versus coherent transport, New Journal of Physics 14, 075004 (2012).
  * Yusipov  _et al._ [2017] I. Yusipov, T. Laptyeva, S. Denisov, and M. Ivanchenko, Localization in open quantum systems, Phys. Rev. Lett. 118, 070402 (2017).
  * Haga  _et al._ [2021] T. Haga, M. Nakagawa, R. Hamazaki, and M. Ueda, Liouvillian skin effect: Slowing down of relaxation processes without gap closing, Phys. Rev. Lett. 127, 070402 (2021).
  * Diehl  _et al._ [2008] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. Büchler, and P. Zoller, Quantum states and phases in driven open quantum systems with cold atoms, Nature Physics 4, 878 (2008).
  * Note [9] As an example, if we consider the system with $M$ particles in Sec. IV and take $n$ as the sum of the particle positions, $t_{n}$ for $n\sim l_{\mathrm{ave}}M$ (i.e., $l_{\mathrm{ave}}$ denotes the average position of particles) is $t_{n}\sim l_{\mathrm{ave}}/(2K\lambda_{M})$.
  * Pai  _et al._ [2019] S. Pai, M. Pretko, and R. M. Nandkishore, Localization in fractonic random circuits, Phys. Rev. X 9, 021003 (2019).
  * Sala  _et al._ [2020] P. Sala, T. Rakovszky, R. Verresen, M. Knap, and F. Pollmann, Ergodicity breaking arising from hilbert space fragmentation in dipole-conserving hamiltonians, Phys. Rev. X 10, 011047 (2020).
  * Castro-Alvaredo  _et al._ [2016] O. A. Castro-Alvaredo, B. Doyon, and T. Yoshimura, Emergent hydrodynamics in integrable quantum systems out of equilibrium, Phys. Rev. X 6, 041065 (2016).
  * Bulchandani  _et al._ [2017] V. B. Bulchandani, R. Vasseur, C. Karrasch, and J. E. Moore, Solvable hydrodynamics of quantum integrable systems, Phys. Rev. Lett. 119, 220604 (2017).
  * Spohn [2012] H. Spohn,  _Large scale dynamics of interacting particles_ (Springer Science & Business Media, 2012).
  * Tsubota  _et al._ [2013] M. Tsubota, M. Kobayashi, and H. Takeuchi, Quantum hydrodynamics, Physics Reports 522, 191 (2013).
  * Schiulaz  _et al._ [2019] M. Schiulaz, E. J. Torres-Herrera, and L. F. Santos, Thouless and relaxation time scales in many-body quantum systems, Phys. Rev. B 99, 174313 (2019).
  * García-Pintos  _et al._ [2017] L. P. García-Pintos, N. Linden, A. S. L. Malabarba, A. J. Short, and A. Winter, Equilibration time scales of physically relevant observables, Phys. Rev. X 7, 031027 (2017).
  * Jensen and Shankar [1985] R. V. Jensen and R. Shankar, Statistical behavior in deterministic quantum systems with few degrees of freedom, Phys. Rev. Lett. 54, 1879 (1985).
  * Deutsch [1991] J. M. Deutsch, Quantum statistical mechanics in a closed system, Phys. Rev. A 43, 2046 (1991).
  * Srednicki [1994] M. Srednicki, Chaos and quantum thermalization, Phys. Rev. E 50, 888 (1994).
  * Rigol  _et al._ [2008] M. Rigol, V. Dunjko, and M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems, Nature 452, 854 (2008).
  * Nahum  _et al._ [2017] A. Nahum, J. Ruhman, S. Vijay, and J. Haah, Quantum entanglement growth under random unitary dynamics, Phys. Rev. X 7, 031016 (2017).
  * Nahum  _et al._ [2018] A. Nahum, S. Vijay, and J. Haah, Operator spreading in random unitary circuits, Phys. Rev. X 8, 021014 (2018).
  * von Keyserlingk  _et al._ [2018] C. W. von Keyserlingk, T. Rakovszky, F. Pollmann, and S. L. Sondhi, Operator hydrodynamics, otocs, and entanglement growth in systems without conservation laws, Phys. Rev. X 8, 021013 (2018).
  * Khemani  _et al._ [2018] V. Khemani, A. Vishwanath, and D. A. Huse, Operator spreading and the emergence of dissipative hydrodynamics under unitary evolution with conservation laws, Phys. Rev. X 8, 031057 (2018).
  * Rakovszky  _et al._ [2018] T. Rakovszky, F. Pollmann, and C. W. von Keyserlingk, Diffusive hydrodynamics of out-of-time-ordered correlators with charge conservation, Phys. Rev. X 8, 031058 (2018).
  * Hamazaki and Ueda [2018] R. Hamazaki and M. Ueda, Atypicality of most few-body observables, Phys. Rev. Lett. 120, 080603 (2018).
  * Sugimoto  _et al._ [2021] S. Sugimoto, R. Hamazaki, and M. Ueda, Test of the eigenstate thermalization hypothesis based on local random matrix theory, Phys. Rev. Lett. 126, 120602 (2021).
  * Boucheron  _et al._ [2013] S. Boucheron, G. Lugosi, and P. Massart,  _Concentration inequalities: A nonasymptotic theory of independence_ (Oxford university press, 2013).
  * Van Handel [2014] R. Van Handel,  _Probability in high dimension_ , Tech. Rep. (PRINCETON UNIV NJ, 2014).
  * Chung and Graham [1997] F. R. Chung and F. C. Graham,  _Spectral graph theory_ , 92 (American Mathematical Soc., 1997).
  * Weinberg and Bukov [2017] P. Weinberg and M. Bukov, QuSpin: a Python Package for Dynamics and Exact Diagonalisation of Quantum Many Body Systems part I: spin chains, SciPost Phys. 2, 003 (2017).
  * Weinberg and Bukov [2019] P. Weinberg and M. Bukov, Quspin: a python package for dynamics and exact diagonalisation of quantum many body systems. part ii: bosons, fermions and higher spins, SciPost Phys 7, 97 (2019).
  * Note [10] Our discussion can be similarly made for fermions.
  * Vo  _et al._ [2020] V. T. Vo, T. Van Vu, and Y. Hasegawa, Unified approach to classical speed limit and thermodynamic uncertainty relation, Phys. Rev. E 102, 062132 (2020).
  * O’Connor  _et al._ [2021] E. O’Connor, G. Guarnieri, and S. Campbell, Action quantum speed limits, Phys. Rev. A 103, 022210 (2021).