# Disturbance Enhanced Uncertainty Relations

Liang-Liang Sun  Hefei National Laboratory for Physical Sciences at Microscale
and Department of Modern Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China  Kishor Bharti  Centre for Quantum
Technologies, National University of Singapore,3 Science Drive 2, Singapore
117543, Singapore.  Ya-Li Mao  Shenzhen Institute for Quantum Science and
Engineering and Department of Physics, Southern University of Science and
Technology, Shenzhen, 518055, China.  Xiang Zhou  Hefei National Laboratory
for Physical Sciences at Microscale and Department of Modern Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Leong-Chuan Kwek 222email<EMAIL_ADDRESS>Centre for Quantum
Technologies, National University of Singapore,3 Science Drive 2, Singapore
117543, Singapore.  MajuLab, CNRS-UNS-NUS-NTU International Joint Research
Unit, UMI 3654, Singapore  National Institute of Education, Nanyang
Technological University, 1 Nanyang Walk, Singapore 637616  School of
Electrical and Electronic Engineering Block S2.1, 50 Nanyang Avenue, Singapore
639798  Jingyun Fan 222email<EMAIL_ADDRESS>Shenzhen Institute for
Quantum Science and Engineering and Department of Physics, Southern University
of Science and Technology, Shenzhen, 518055, China  Guangdong Provincial Key
Laboratory of Quantum Science and Engineering,Southern University of Science
and Technology, Shenzhen, 518055, China,  Sixia Yu 333email:
<EMAIL_ADDRESS>Hefei National Laboratory for Physical Sciences at
Microscale and Department of Modern Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China

###### Abstract

Uncertainty and disturbance are two most fundamental properties of a quantum
measurement and they are usually separately studied in terms of the
preparation and the measurement uncertainty relations. Here we shall establish
an intimate connection between them that goes beyond the above mentioned two
kinds of uncertainty relations. Our basic observation is that the disturbance
of one measurement to a subsequent measurement, which can be quantified based
on observed data, sets lower-bounds to uncertainty. This idea can be
universally applied to various measures of uncertainty and disturbance, with
the help of data processing inequality. The obtained relations, referred to as
_disturbance enhanced uncertainty relations_ , immediately find various
applications in the field of quantum information. They ensure preparation
uncertainty relation such as novel entropic uncertainty relations independent
of the Maassen and Uffink relation. And they also result in a simple protocol
to estimate coherence. We anticipate that this new twist on uncertainty
principle may shed new light on quantum foundations and may also inspire
further applications in the field of quantum information.

##  I I. Introduction

Uncertainty principle, one of fundamental traits of quantum mechanics, is
quantified by two kinds of uncertainty relations, namely, for preparation and
for measurement. They are commonly investigated in terms of the uncertainties
Kennard (1927); Heisenberg (1927); Robertson (1929); Beckner (1975); Coles et
al. (2017); Maassen and Uffink (1988) and the error-disturbance Ozawa (1991);
Busch et al. (2014a); Busch and Shilladay (2006); Ozawa (2003a); Paul (2007);
Branciard (2013); Busch et al. (2014b), respectively. Till now, these two
fundamental properties have gained a relative well understanding and have been
harnessed for applications such as the security proof of quantum communication
Fuchs and Peres (1996); Barrett et al. (2005); Wehner and Winter (2010); Coles
et al. (2017), witnessing entanglement Hofmann and Takeuchi (2003); Gühne and
Lewenstein (2004), bounding quantum correlations Steeg and Wehner (2008);
Oppenheim and Wehner (2010); Girolami et al. (2013), and quantum metrology
Braunstein and Caves (1994); Braunstein et al. (1996); Luo (2003); Giovannetti
et al. (2011, 2012). As two basic properties, the disturbance and uncertainty
present themselves in a quantum measurement process simultaneously.
Frequently, they are studied separately within one of the two types of
uncertainty relations.

A physical state, which may be prepared in a coherent superposition of
classically distinguishable ones, undergoes a sudden random collapse after a
measurement. The post-measurement state and subsequent measurements are
therefore disturbed. Intuitively, with suitable measures, the less uncertainty
a measurement exhibits, the less it will disturb quantum state and the
subsequent measurements. Therefore the measurement uncertainty should impose
some constraints on its disturbance effect, and vise versa. For a given
outcome of measurement, Winter’s gentle measurement lemma states that the
post-measurement state is disturbed slightly if the associated outcome happens
with a high probability (exhibiting small uncertainty) Winter (2014) and this
special link has already led to some important applications such as channel
coding theorem Wilde and Renes (2012); Ogawa and Nagaoka (2007). By using a
specific distance measure to quantify disturbance, a trade-off between
uncertainty and disturbance has been established, which, though dealing with
local properties, can be used to bound nonlocality Sun et al. (2018). These
results suggest an intrinsic link between uncertainty and disturbance, which,
if existing, shall bring a deeper insight on uncertainty principle and may
find novel applications in the fields of quantum information sciences.

In this paper, we present such a link that is referred to as _disturbance
enhanced uncertainty relation_ , exploiting the data processing inequality and
the fact that the disturbance caused by measurement to the state can be
quantified by physical measures based on observed data. We establish a
universal relation between uncertainty and disturbance, valid for all
disturbance measures that are induced by state distance measures with suitable
properties (see Eq.(1) below). Specifically we also derive corresponding
uncertainty-disturbance trade-offs for some widely used distance measures in
various quantum informational scenarios. As applications, our relations, which
impose constraints on the intrinsic “spreads” of incompatible quantities, can
be seen as preparation uncertainty relations, giving rise to novel entropic
uncertainty relations that are independent of the Maassen and Uffink’s Maassen
and Uffink (1988). Moreover, our disturbance enhanced uncertainty relations
are suitable for coherence detection, providing operational upper and lower
bounds for coherence measures based on distance. Therefore we anticipate this
new perspective of uncertainty principle may promise far-reaching impact on
quantum information science and quantum foundations.

##  II II. Uncertainty-Disturbance Relations

We consider a sequential measurement $\hat{A}\rightarrow\hat{B}$ of two
incompatible observables $\hat{A}=\\{|A_{i}\rangle\langle
A_{i}|=\Pi^{A}_{i}\\}$ and $\hat{B}=\\{|B_{j}\rangle\langle
B_{j}|=\Pi^{B}_{j}\\}$, where $\\{|A_{i}\rangle\\}$ and $\\{|B_{j}\rangle\\}$
are two sets of orthonormal bases. The intrinsic distributions of $\hat{A}$
and $\hat{B}$ on quantum state $\rho$ are given by probability vectors
$\mathbf{p}=\\{p_{i}=\operatorname{tr}(\rho\Pi^{A}_{i})\\}$ and
$\mathbf{q}=\\{q_{j}=\operatorname{tr}(\rho\Pi^{B}_{j})\\}$. Measuring
$\hat{A}$ on the state $\rho$ leaves a disturbed state
$\rho_{A}=\Phi_{A}(\rho)=\sum_{i}p_{i}\Pi^{A}_{i}$, where
$\Phi_{A}(\cdot)\equiv\sum_{i}\Pi^{A}_{i}\cdot\Pi^{A}_{i}$ and similarly
$\Phi_{B}(\cdot)=\sum_{j}\Pi^{B}_{j}\cdot\Pi^{B}_{j}$ are two complete-
positive-trace-preserving (CPTP) operations. A subsequent measurement
$\hat{B}$ on the state $\rho_{A}$ would yield a disturbed distribution
$\mathbf{q}^{\prime}=\\{q^{\prime}_{j}\\}$ with
$q^{\prime}_{j}=\operatorname{tr}(\rho_{A}\Pi_{j}^{B})=\sum_{i}c_{ij}p_{i}=[[\mathcal{C}\cdot\mathbf{p}]]_{j},$
where the overlaps of eigenstates $\\{c_{ij}=|\langle
A_{i}|B_{j}\rangle|^{2}\\}$ form a unitary stochastic matrix Pakonski et al.
(2001); Życzkowski et al. (2003), denoted by $\mathcal{C}$, which depends only
on the choices of $\hat{A}$ and $\hat{B}$ and characterizes their
incompatibility.

Here we shall quantify the uncertainty of $\hat{A}$ by some measure
$\delta_{A}$, which is expected to respect Shur concavity as a function of
probability distribution and may depend on which state distance that is used
to quantify the distinguishability of states (as shown below). We denote by
$\operatorname{D}_{A\to B}$ the disturbance introduced by $\hat{A}$ to
$\hat{B}$ which is quantified by the distance between $\mathbf{q}$ and
$\mathbf{q}^{\prime}$ and depends also on the state distance we choose. We
note that the disturbance is estimated directly from experimental data, i.e.,
the undisturbed and disturbed statistics of $\hat{B}$.

Two key tools developed in the field of quantum information turn out to be
very useful to explore the relation between $\delta_{A}$ and
$\operatorname{D}_{A\to B}$. One tool is the state distance, specified by
$\operatorname{D}(\cdot,\cdot)$, which quantifies how distinguishable two
quantum states are. Widely used measures of state distance are trace distance
Nielsen and Chuang (2011), Rényi divergence Müller-Lennert et al. (2013);
Leditzky et al. (2016), Tsallis relative entropy Abe (2003), and infidelity.
The state distance we choose should be non-increasing under at least CPTP
operations $\Phi_{B}$. The other tool is the data processing inequality, i.e.,
two quantum states become less distinguishable after a general CPTP operation
$\operatorname{D}(\rho_{1},\rho_{2})\geq\operatorname{D}[\Phi(\rho_{1}),\Phi(\rho_{2})].$
The data processing inequality has been used to derive entropic uncertainty
relations Berta et al. (2009); Coles et al. (2012); Coles and Piani (2014).
Here, we shall see that it can lead to more fruitful results. In what follows
we shall present two kinds of characterization of uncertainty and disturbance
trade-off, a universal one and a distance measure dependent one.

For the purpose of a universal relation we shall make two reasonable
assumptions about the state distance measure we choose besides its
monotonicity under CPTP operations. First, we assume that the distance between
two pure states, say $|\phi\rangle$ and $|\phi_{A}\rangle$, is an increasing
function 111For some distance measure, e.g., relative entropy, such a function
is not well-defined.

$\operatorname{D}(\phi,\phi_{A})=\mathcal{G}_{D}\Big{(}\operatorname{IF}(\phi,\phi_{A})\Big{)},$
(1)

of their infidelity $\operatorname{IF}(\phi,\phi_{A})$, where
$\mathcal{G}_{D}(x)$ is an increasing single variable function for $x\geq 0$,
depending on the distance measure we choose, and the infidelity for two
general density matrices is
$\operatorname{IF}(\rho_{1},\rho_{2})=\sqrt{1-\operatorname{F}(\rho_{1},\rho_{2})^{2}}$
with
$\operatorname{F}(\rho_{1},\rho_{2})=\operatorname{tr}\large(\textstyle\sqrt{\operatorname{\sqrt{\rho_{2}}\rho_{1}\sqrt{\rho_{2}}}}\large)$
being the quantum fidelity. For two pure states, their infidelity becomes
simply their non-overlap
$\operatorname{IF}(\phi,\phi_{A})=\sqrt{1-|\langle\phi|\phi_{A}\rangle|^{2}}.$
This assumption is reasonable since the more distinguishable the states the
larger the infidelity and therefore the larger the distance should be. For
each distance measure satisfying above property Eq.(1), we can introduce a
gauged distance measure

$\tilde{\operatorname{D}}(\rho_{1},\rho_{2})=\mathcal{G}_{\operatorname{D}}^{-1}\Big{(}{\operatorname{D}}(\rho_{1},\rho_{2})\Big{)}$
(2)

which also satisfies the data processing inequality as function
$\mathcal{G}_{D}$ is monotonous. We shall refer to such kinds of distance
measures as gaugeable. As can be seen in Table I, almost all commonly used
distance measures are gaugeable. Second, we assume the state distance is
unitary invariant, i.e.,
$\operatorname{D}(\rho_{1},\rho_{2})=\operatorname{D}({U}\rho_{1}{U^{\dagger}},{U}\rho_{2}{U^{\dagger}})$,
which is satisfied by all distance measures considered below. As a result we
can write
${\operatorname{D}}(\Phi_{B}(\rho),\Phi_{B}(\rho_{A}))={\operatorname{D}}(\mathbf{q},\mathbf{q}^{\prime})$
since these two density matrices are diagonal in the same basis. For the state
distance measure assuming above two assumptions, we have an universal
disturbance-uncertainty relation (see Methods).

_Theorem 1_ (_Uncertainty-Disturbance_ relation). Performing sharp
measurements in order $\hat{A}\to\hat{B}$ on a system in state $\rho$,
resulting in data $\mathbf{p},\mathbf{q}$ and $\mathbf{q}^{\prime}$, it holds

$\delta_{A}(\rho)\geq=\max_{\operatorname{D}}\tilde{\operatorname{D}}(\mathbf{q},\mathbf{q}^{\prime}):=\tilde{\rm
D}_{A\to B}(\rho).$ (3)

where the maximization is taken over all monotonous gaugeable distance
measures.

In general, if we can bound a given specific distance measure
${\operatorname{D}}(\rho,\rho_{A})$ from above by a Schur concave function
$\delta_{A}^{\operatorname{D}}$ of $\mathbf{p}$, the data processing
inequality will lead to a specific pairwise definitions of uncertainty
$\delta_{A}^{\operatorname{D}}$ and disturbance
$\operatorname{D}_{A\rightarrow B}$ via a distance-measure dependent
uncertainty-disturbance relation

$\delta^{\operatorname{D}}_{A}(\rho)\geq{\operatorname{D}}(\rho,\rho_{A})\geq{\operatorname{D}}[\Phi_{B}(\rho),\Phi_{B}(\rho_{A})]=\operatorname{D}_{A\rightarrow
B}(\rho),$ (4)

where $\operatorname{D}(\rho,\rho_{A})$ specifies disturbance in $\rho$, and
$\operatorname{D}_{A\rightarrow B}$ specifies disturbance in measurement. For
a given distance measure, the specific uncertainty-disturbance relation Eq.(4)
may be different from the universal one, i.e., Eq(3).

In Table 1, we have summarized uncertainty-disturbance relations, universal
Eq.(3) or distance-measure dependent Eq.(4), arising from some widely used
distance measures with proofs present in Methods. These distance measures
include infidelity (if), trace distance (tr), Tsallis relative entropy (ts),
Rényi divergence (rd), relative entropy (re), and Hilbert-Schimdt (hs)
distance from first line to the last, respectively, in Table 1. Some remarks
are in order. First, $\operatorname{U}_{\rm if}$ is equivalent to
$\operatorname{U}^{0.5}_{\rm rd}$ and ${\rm U}_{\rm tr}$ and ${\rm U}_{\rm
tr}^{\prime}$ arise from the same distance measure, namely, the trace
distance, while their uncertainties are quantified by different measures. We
note that these two relations are independent. Second, in spite of the fact
that relative entropy is not gaugeable, a specific uncertainty-disturbance
relation still exists. Third, we also include Hilbert-Schmidt distance measure
because, even though it is monotonous under dephasing operation but not a
general CPTP operation Coles et al. (2019), a uncertainty-disturbance relation
can be obtained. Fourth, when considering the measurement sequence
$\hat{B}\rightarrow\hat{A}$, one obtains a dual of the corresponding
uncertainty-disturbance relations. Lastly, for other monotonous distance
measures we can obtain similar distance-measure dependent uncertainty-
disturbance relations.

D | U${}_{\rm D}\ $ (measure-dependent) |  $\tilde{\rm U}_{\rm D}$ (uinversal)
---|---|---
IF$(\rho,\rho_{A})$ |  $\rm U_{\rm if}:\ $ $\operatorname{H}_{\frac{1}{2}}(\mathbf{p})\geq{\operatorname{D}}_{2}(\mathbf{q}\|\mathbf{q}^{\prime})$ |  $\tilde{\rm U}_{\rm if}:\ $$\delta_{A}(\rho)\geq\sqrt{\textstyle 1-\left(\sum_{i}\sqrt{q_{i}{q^{\prime}}_{i}}\right)^{2}}$
$\frac{1}{2}\operatorname{tr}|\rho-\rho_{A}|$ |  $\rm U_{\rm tr}:\ $ $\delta_{A}(\rho)\geq|\mathbf{q}-\mathbf{q}^{\prime}|$ |  $\rm\tilde{U}_{\rm tr}:$ $\delta_{A}(\rho)\geq|\mathbf{q}-\mathbf{q}^{\prime}|$
$\frac{1}{2}\operatorname{tr}|\rho-\rho_{A}|$ |  $\rm U_{\rm tr}^{\prime}:\ $ $\frac{1}{2}\left({\|\mathbf{p}\|_{\frac{1}{2}}-1}\right)\geq|\mathbf{q}-\mathbf{q}^{\prime}|$ |  $\tilde{\rm U}_{\rm tr}^{\prime}:\ $$\delta_{A}(\rho)\geq|\mathbf{q}-\mathbf{q}^{\prime}|$
$\frac{1}{\bar{\alpha}}[1-\operatorname{Tr}(\rho^{\alpha}\rho^{\bar{\alpha}}_{A})]$ |  $\rm U_{\rm ts}:\ $ $\frac{1}{2-\alpha}\operatorname{H}_{2-\alpha}(\mathbf{p})\geq{\operatorname{D}}_{\alpha}(\mathbf{q}\|\mathbf{q}^{\prime})$ |  $\tilde{\rm U}_{\rm ts}:$ $\delta_{A}(\rho)\geq\sqrt{\textstyle 1-\left(\sum_{i}q_{i}^{\alpha}{q^{\prime}}_{i}^{\bar{\alpha}}\right)^{\frac{1}{\alpha}}}$
$\frac{-1}{\bar{\alpha}}\log\large\operatorname{tr}(\rho^{\frac{\bar{\alpha}}{2\alpha}}_{A}\rho^{\frac{1}{\alpha}}\rho^{\frac{\bar{\alpha}}{2\alpha}}_{A})^{\alpha}\large$ |  $\rm U_{\rm rd}\ :$ $\operatorname{H}_{\frac{1}{\alpha}}(\mathbf{p})\geq{\operatorname{D}}_{\alpha}(\mathbf{q}\|\mathbf{q}^{\prime})$ |  $\tilde{\rm U}_{\rm rd}:$ $\delta_{A}(\rho)\geq\sqrt{\textstyle 1-\left(\sum_{i}q_{i}^{\alpha}{q^{\prime}}_{i}^{\bar{\alpha}}\right)^{\frac{1}{\alpha}}}$
$S(\rho||\rho_{A})$ |  $\rm U_{\rm re}:\ $ $\operatorname{H}(\mathbf{p})\geq\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime})$ | —
$\sqrt{\rm Tr(\rho-\rho_{A})^{2}}$ |  $\rm U_{\rm hs}:\ $ $\delta_{A}(\rho)\geq\sqrt{\textstyle\sum_{i}(q_{i}-q_{i}^{\prime})^{2}}$ | —
Table 1: A list of universal and distance-measure dependent uncertainty-
disturbance relations arising from commonly used distance measures. For
convenience we have denoted $\bar{\alpha}=1-\alpha$ with
$\frac{1}{2}\leq\alpha<1$ Rényi divergence and $0\leq\alpha<1$ for Tsallis
relative entropy. Moreover,
$|\mathbf{q}-\mathbf{q^{\prime}}|=\frac{1}{2}\sum_{i}|q_{i}-q^{\prime}_{i}|$,
$\operatorname{H}_{\alpha}(\mathbf{p})=\frac{\alpha}{\bar{\alpha}}\log\|\mathbf{p}\|_{\alpha}$
for Rényi entropy, and
${\operatorname{D}}_{\alpha}(\mathbf{q}\|\mathbf{q}^{\prime})=\frac{-1}{\bar{\alpha}}\log\sum_{i}q^{\alpha}_{i}{q^{\prime}}_{i}^{1-\alpha}$
for classical Rényi divergence. For universal trade-offs we list only the
disturbance arising from gauged distance measures in last column.

From discussion above, $\delta^{\operatorname{D}}_{A}$ and
$\operatorname{D}_{A\to B}$ can be pairwisely defined from a given state
distance measure, where uncertainty measures are Shur concave and are equal or
equivalent to $\alpha-$Rényi entropy. It is worthy of pointing out that these
uncertainty measures, such as Shannon entropy $\operatorname{H}(\mathbf{p})$
(employed in $\operatorname{U}_{\rm re}$), are intimately related to
information gain. Previous researches from error-disturbance relations, in
which a lower bound on disturbance is placed by the error allowed in a
previous measurement Busch et al. (2014a, b); Busch and Shilladay (2006);
Ozawa (2003b), reveal an aspect of uncertainty principle that _information
gain implies disturbance_. Here, our uncertainty-disturbance relations,
especially $\operatorname{U}_{re}$, uncover another facet that _disturbance of
a measurement lower-bounds its information gain_.

##  III III. Uncertainty Disturbance trade-off as Preparation Uncertainty
Relations

In general, for two measurements $\hat{A}$ and $\hat{B}$, any relation
reflecting the fact that probability distributions $\mathbf{q}$ and
$\mathbf{p}$ cannot be simultaneously sharp, is an uncertainty relation. The
_disturbance enhanced uncertainty_ relations listed in the third column are
formulated in terms of $\mathbf{p}$, $\mathbf{q}$, and
$\mathbf{q}^{\prime}=\mathcal{C}\cdot\mathbf{p}$ with $\mathcal{C}$ being the
matrix formed by the overlaps of eigenstates of $\hat{A}$ and $\hat{B}$.
Provided $\mathcal{C}$, the relations, which are actually formulated in terms
of intrinsic distributions $\mathbf{p}$ and $\mathbf{q}$, can constrain their
uncertainties and capture the spirit of preparation uncertainty.

As an example, we shall derive a novel entropic uncertainty relation from our
uncertainty-disturbance relation. From uncertainty-disturbance relation
$\operatorname{U}^{\alpha}_{\rm ts}$ arising form Tsallis relative entropy, it
follows the following novel entropic uncertainty relation

$\displaystyle\frac{1}{2-\alpha}\operatorname{H}_{2-\alpha}(\mathbf{q})+\operatorname{H}_{\alpha}(\mathbf{p})\geq-\log
c,$ (5)

where $0\leq\alpha<1$. This is due to the fact that
$q^{\prime}_{i}=\sum_{j}c_{ij}p_{j}\leq\max_{ij}c_{ij}\equiv c$ so that

$\frac{1}{2-\alpha}\operatorname{H}_{2-\alpha}(\mathbf{p})\geq\frac{1}{\alpha-1}\log\sum_{i}q^{\alpha}_{i}c^{1-\alpha}=-\log
c-\operatorname{H}_{\alpha}(\mathbf{p}).$

A dual of Eq.(5) can be readily obtained by swapping $\mathbf{p}$ and
$\mathbf{q}$. In comparison, the well-known Maassen-Uffink (MU) entropic
uncertainty relation reads

$\displaystyle\operatorname{H}_{\alpha}(\mathbf{p})+\operatorname{H}_{\beta}(\mathbf{q})\geq-\text{log}c,$
(6)

where $\alpha,\beta\geq\frac{1}{2}$ satisfying
$\frac{1}{\alpha}+\frac{1}{\beta}=2$ are referred to as conjugated indices.
The MU relation have found numerous applications in many information tasks and
have been generalized in various scenarios Coles et al. (2017); Koashi (2006);
Schneeloch et al. (2013); Berta et al. (2009); Coles and Piani (2014).
Generalizing entropic uncertainty relation to the cases of nonconjugated
indices is a topic that has sustained interests Zozor et al. (2014).

We shall see that uncertainty relations Eq.(5) and Eq.(6) are independent. In
fact, the factor $(2-\alpha)^{-1}\leq 1$ in Eq.(5) allows a possible
strengthening over MU uncertainty Eq.(6). To see this we assume that
$\mathbf{p}$ is uniform so that $\operatorname{H}_{\alpha}(\mathbf{p})=\log d$
for arbitrary $\alpha$ and the indices pair for Eq.(5) is
$\\{\beta,2-\beta\\}$, then and $\log
d+\frac{1}{\beta}\operatorname{H}_{\beta}(\mathbf{q})\leq\log
d+\operatorname{H}_{\beta}(\mathbf{q})$ when $1\leq\beta$. Therefore, Eq.(5)
indicates an alternative approach of generalization, namely, by introducing
modifying fractions, which is made possible by considering disturbance.

Figure 1: Geometrical visualization of the constraints of uncertainty
relations imposed on the observed data. In qubit case, the distributions from
measuring $\hat{A}$ and $\hat{B}$ can be specified respectively by $p_{0}$ and
$q_{0}$, and their incompability is characterized by $c_{00}$, which are
constrained by various uncertainty relations. Plots (a-e) correspond to
$\operatorname{U}_{tr1}$, $\operatorname{U}_{tr2}$,
$\operatorname{U}^{0.5}_{re}$, $\operatorname{U}^{0.5}_{ts}$,
$\operatorname{U}_{re}$, and plot (f) for the The MU relation
($\alpha=\beta=1$).

In MU entropic uncertainty relations, the incompatibility is characterized by
a single number $c$ instead of the whole set of $\\{c_{ij}\\}$ or the
transition matrix $\mathcal{C}$. This simplification leads to neat
representations of uncertainty relations while sacrifices their tightness. In
contrast, our uncertainty-disturbance trade-offs include all $\\{c_{ij}\\}$
and therefore are tighter, justifying the name disturbance enhanced
uncertainty relations. For a visualized comparison, we present two case
studies of the _uncertainty-disturbance_ type uncertainty relations regarding
to the tightness in the case of qubit ($d=2$) and qutrit ($d=3$) as
illustrated in Table 2.

Relations | Volume(d=2) | Volume(d=3)
---|---|---
$\operatorname{U}_{\rm tr}$ | 0.930 | 0.94675
$\operatorname{U}_{\rm tr}^{\prime}$ | 0.705 | 0.94682
$\operatorname{U}^{0.5}_{\rm rd}$ | 0.787 | 0.917
$\operatorname{U}_{\rm re}$ | 0.770 | 0.905
$\operatorname{U}^{0.5}_{\rm ts}$ | 0.814 | 0.937
$\operatorname{U}_{\rm hs}$ | 0.705 | 0.887
Maassen-Uffink | 0.974 | 0.999
Table 2: Computed _volumes_ according to different uncertainty-disturbance
relations and and their duals in pure states. The difference between the
computed _volumes_ via $\operatorname{U}_{\rm tr}$, $\operatorname{U}_{\rm
tr}^{\prime}$ for qutrit states are small but nonvanishing.

For the qubit case, $\mathbf{p}$, $\mathbf{q}$, and $\mathcal{C}$ are
determined by three independent parameters,
$\\{p_{0},q_{0},c_{00}\\}\in[0,1]$. Tightness means how strong the uncertainty
relations can bound $p_{0}$ and $q_{0}$ under a given $c_{00}$. Geometrically
one may visualize that each one of the inequalities in $\operatorname{U}_{\rm
tr}$, $\operatorname{U}_{\rm tr}^{\prime}$, $\operatorname{U}^{\alpha}_{\rm
ts}$, $\operatorname{U}^{\alpha}_{\rm rd}$, $\operatorname{U}_{\rm re}$ and
the MU relation for measurements $\hat{A}\to\hat{B}$ together with their dual
inequalities for measurements $\hat{B}\to\hat{A}$ enclose a region shown in
Fig.1 and their _volumes_ are computed in Table 2. Hence, the diagrams and the
_volumes_ present a direct comparison regarding the tightness of these
uncertainty relations. The smaller the _volume_ , the tighter the constraint.
The volume of parameter space is normalized to 1 if without any other
constraints. Under constraint of our disturbance enhanced uncertainty
relations, the volumes are in the range [0.705, 0.930], which are
significantly smaller than the _volume_(=0.974) for the MU relation
($\alpha=1$).

For the qutrit case, the hypersurfaces corresponding to the _uncertainty-
disturbance_ relations enclose some regions in the parameter space spanned by
$\mathbf{p}$, $\mathbf{q}$, and $\mathcal{C}$. These conditions respectively
yield the same computed _volumes_ (see Table 2). The corresponding volumes are
in the range [0.887, 0.947], which are also significantly smaller than the
_volume_(=0.999) for the MU relation (see Table 2). Therefore, we have shown
that the _uncertainty-disturbance_ relations impose stronger constraints than
the MU relation on the observed data, since in our cases we have used all
elements of matrix $\mathcal{C}$ in the analysis.

##  IV IV. Detecting Coherence with Uncertainty-Disturbance Relation.

Coherence is a fundamental quantum feature that finds many applications
ranging from quantum information science and quantum foundations Streltsov et
al. (2017) to quantum biologyEngel et al. (2007); Gauger et al. (2011). In
practice detecting coherence is a crucial but complicated task. As another
application, our disturbance enhanced uncertainty relation can provide an
efficient detection of coherence.

Coherence is characterized with respect to some computation basis, for
example, $\\{|A_{i}\rangle\\}$ and a coherence measure quantifies how much
coherence contained in a quantum state Baumgratz et al. (2014). A widely used
measure of coherence Liu et al. (2017); Winter and Yang (2016); Streltsov et
al. (2017) is defined by the minimum distance
$C_{\operatorname{D}}=\min_{\sigma}\operatorname{D}(\rho,\sigma)$ to the set
of all incoherent states ($\sigma$), which are diagonal states in the given
basis. Here $\operatorname{D}$ is some suitable state distance measure.
Naturally, an operational upper-bound
$C_{\operatorname{D}}\leq\operatorname{D}(\rho,\rho_{A})\leq\delta^{\operatorname{D}}_{A}(\rho)$
in terms of the uncertainty follows from uncertainty disturbance trade-off
Eq.(4) since $\rho_{A}$ is an incoherent state. We note that both uncertainty
and coherence are quantities defined with respect to computation basis, and
the former relates to the diagonal terms of density matrix while the latter
relates to the off-diagonal terms. The upper bound above provide a connection
between them. For those distance measures whose nearest incoherent state is
$\rho_{A}$, our uncertainty disturbance trade-off Eq.(4) also provides an
operational lower-bound $\operatorname{D}_{A\to B}\leq C_{\operatorname{D}}$,
in terms of disturbance.

As a case illustration, we take the relative entropy as distance measure and
we have relative entropy measure of coherence
$C_{r}(\rho)\equiv\mathcal{S}(\rho\|\rho_{A})$, which quantifies the
distillable coherence from a state $\rho$ as well as extractable quantum
randomness Liu et al. (2018); Yuan et al. (2019). In this case the nearest
incoherent state to $\rho$ is exactly $\rho_{A}$. By the above analysis, one
immediate has operational bounds as

$\displaystyle\operatorname{H}(\mathbf{p})\geq
C_{r}(\rho)\geq\operatorname{H}(\mathbf{q}\|\mathbf{q^{\prime}}).$ (7)

One needs to independently measure $\hat{A}\to\hat{B}$ and observable
$\hat{B}$, which yield distributions $\mathbf{p}$, $\mathbf{q}$ and
$\mathbf{q}^{\prime}=\mathcal{C}\cdot\mathbf{p}$. When
$\mathbf{q}^{\prime}=\mathbf{q}$, the observable $\hat{B}$ yields a trivial
estimation and these failing cases compose a set of measure zero. Therefore,
it is almost impossible to yield a trivial estimation even only one additional
measurement is employed. For comparison, in a previous approach based on the
theory of majorization Yu and Gühne (2019); Ding et al. (2021), where a
nontrivial estimation of the spectrum of $\rho$ require that the distribution
from a test measurement majorize the diagonal parts of the state, one may need
several measurements so as to obtain such a distribution.

##  V V. Discussions

To summarize, we have established a quantitative link, a universal one
(Theorem 1) and a distance-measure dependent one (for commonly used distance
measures), between disturbance and uncertainty in sequential sharp
measurements, uncovering a facet of uncertainty principle that has not been
covered by previous approaches via preparation and the measurement uncertainty
relations. Our uncertainty-disturbance trade-offs involve many basic concepts
such as $\alpha-$Rényi entropy, Tsallis relative entropy, _etc._ , which have
nice properties and are frequently employed in the field of quantum
information. Thus, the reported relations naturally find corresponding
applications such as in deriving novel uncertainty relations and detecting
coherence. This new twist on uncertainty principle promises potential further
applications in quantum information science and may shed new light in quantum
foundations. One next immediate task is to generalize the uncertainty
disturbance relations to general measurements find possible applications in
tasks where uncertainty relation plays a key role, e.g., the detection of
coherence quantified in other measures and the certification of quantum
randomness. Moreover our relations, as generalization of Winter’s gentle
measurement lemma, may inspire further applications in channel coding theorem.

References

## References

  * Kennard (1927) E. H. Kennard, Zeitschrift für Physik 44, 326 (1927).
  * Heisenberg (1927) W. Heisenberg, Zeitschrift für Physik 43, 172 (1927).
  * Robertson (1929) H. P. Robertson, Phys. Rev. 34, 163 (1929).
  * Beckner (1975) W. Beckner, Ann. Math. 102, 159 (1975).
  * Coles et al. (2017) P. J. Coles, M. Berta, M. Tomamichel, and S. Wehner, Rev. Mod. Phys. 89, 015002 (2017).
  * Maassen and Uffink (1988) H. Maassen and J. B. M. Uffink, Phys. Rev. Lett. 60, 1103 (1988).
  * Ozawa (1991) M. Ozawa, Lecture Notes in Physics 378, 1 (1991).
  * Busch et al. (2014a) P. Busch, P. Lahti, and R. F. Werner, Rev. Mod. Phys. 86, 1261 (2014a).
  * Busch and Shilladay (2006) P. Busch and C. Shilladay, Phys. Rep. 435, 1 (2006).
  * Ozawa (2003a) M. Ozawa, Phys. Lett. A 318, 21 (2003a).
  * Paul (2007) B. Paul, _“no information without disturbance”: Quantum limitations of measurement_ (2007), eprint quant-ph/0706.3526.
  * Branciard (2013) C. Branciard, PNAS 110, 6742 (2013).
  * Busch et al. (2014b) P. Busch, P. Lahti, and R. F. Werner, J. Math. Phys. 55, 172 (2014b).
  * Fuchs and Peres (1996) C. A. Fuchs and A. Peres, Phys. Rev. A 53, 2038 (1996).
  * Barrett et al. (2005) J. Barrett, L. Hardy, and A. Kent, Phys. Rev. Lett. 95, 010503 (2005).
  * Wehner and Winter (2010) S. Wehner and A. Winter, New J. Phys. 12, 025009 (2010).
  * Hofmann and Takeuchi (2003) H. F. Hofmann and S. Takeuchi, Phys. Rev. A 68, 032103 (2003).
  * Gühne and Lewenstein (2004) O. Gühne and M. Lewenstein, Phys. Rev. A 70, 022316 (2004).
  * Steeg and Wehner (2008) G. V. Steeg and S. Wehner, Quantum Information and Computation 9, 801 (2008).
  * Oppenheim and Wehner (2010) J. Oppenheim and S. Wehner, Science 330, 1072 (2010).
  * Girolami et al. (2013) D. Girolami, T. Tufarelli, and G. Adesso, Phys. Rev. Lett. 110, 240402 (2013).
  * Braunstein and Caves (1994) S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72, 3439 (1994).
  * Braunstein et al. (1996) L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. 247, 135 (1996).
  * Luo (2003) S. Luo, Phys. Rev. Lett. 91, 180403 (2003).
  * Giovannetti et al. (2011) V. Giovannetti, S. Lloyd, and L. Maccone, Nat. Photon. 5, 222 (2011).
  * Giovannetti et al. (2012) V. Giovannetti, S. Lloyd, and L. Maccone, Phys. Rev. Lett. 108, 260405 (2012).
  * Winter (2014) A. Winter, IEEE Trans. Inf. Theory 45, 2481 (2014).
  * Wilde and Renes (2012) M. M. Wilde and J. M. Renes, 2012 IEEE International Symposium on Information Theory Proceedings (2012).
  * Ogawa and Nagaoka (2007) T. Ogawa and H. Nagaoka, IEEE Trans. Inf. Theory 53, 2261 (2007).
  * Sun et al. (2018) L.-L. Sun, S. Yu, and Z.-B. Chen, _Uncertainty-complementarity balance as a general constraint on non-locality_ (2018), eprint quant-ph/1808.06416.
  * Pakonski et al. (2001) P. Pakonski, K. Życzkowski, and M. Kus, J. Phys. A: Gen. Phys. 34, 9303 (2001).
  * Życzkowski et al. (2003) K. Życzkowski, M. Kus, W. Słomczyński, and H. Sommers, J. Phys. A: Gen. Phys. 36, 3425 (2003).
  * Nielsen and Chuang (2011) M. A. Nielsen and I. L. Chuang, _Quantum Computation and Quantum Information: 10th Anniversary Edition_ (Cambridge University Press, USA, 2011), 10th ed., ISBN 1107002176.
  * Müller-Lennert et al. (2013) M. Müller-Lennert, F. Dupuis, O. Szehr, S. Fehr, and M. Tomamichel, J. Math. Phys. 54, 122203 (2013).
  * Leditzky et al. (2016) F. Leditzky, C. Rouzé, and N. Datta, Lett. Math. Phys. 107, 61 (2016).
  * Abe (2003) S. Abe, Phys. Rev. A 312, 336 (2003), ISSN 0375-9601.
  * Berta et al. (2009) M. Berta, M. Christandl, R. Colbeck, J. M. Renes, and R. Renner, Nat. Phys. 6, 757 (2009).
  * Coles et al. (2012) P. J. Coles, R. Colbeck, L. Yu, and M. Zwolak, Phys. Rev. Lett. 108, 210405 (2012).
  * Coles and Piani (2014) P. J. Coles and M. Piani, Phys. Rev. A 89, 022112 (2014).
  * Coles et al. (2019) P. J. Coles, M. Cerezo, and L. Cincio, Phys. Rev. A 100, 022103 (2019).
  * Ozawa (2003b) M. Ozawa, Phys. Rev. A 67, 042105 (2003b).
  * Koashi (2006) M. Koashi, J. Phys. Conf. Ser. 36, 98 (2006).
  * Schneeloch et al. (2013) J. Schneeloch, C. J. Broadbent, S. P. Walborn, E. G. Cavalcanti, and J. C. Howell, Phys. Rev. A 87, 062103 (2013).
  * Zozor et al. (2014) S. Zozor, G. M. Bosyk, and M. Portesi, J. Phys. A: Math. Theor. 47, 495302 (2014).
  * Streltsov et al. (2017) A. Streltsov, G. Adesso, and M. B. Plenio, Rev. Mod. Phys. 89, 041003 (2017).
  * Engel et al. (2007) G. S. Engel, T. R. Calhoun, E. L. Read, T. K. Ahn, T. Manal, Y. C. Cheng, R. E. Blankenship, and G. R. Fleming, Nature 446 (2007).
  * Gauger et al. (2011) E. M. Gauger, E. Rieper, J. J. L. Morton, S. C. Benjamin, and V. Vedral, Phys. Rev. Lett. 106, 040503 (2011).
  * Baumgratz et al. (2014) T. Baumgratz, M. Cramer, and M. B. Plenio, Phys. Rev. Lett. 113, 140401 (2014).
  * Liu et al. (2017) Z.-W. Liu, X. Hu, and S. Lloyd, Phys. Rev. Lett. 118, 060502 (2017).
  * Winter and Yang (2016) A. Winter and D. Yang, Phys. Rev. Lett. 116, 120404 (2016).
  * Liu et al. (2018) Y. Liu, Q. Zhao, and X. Yuan, J. Phys. A: Math. Theor. 51, 414018 (2018), ISSN 1751-8121.
  * Yuan et al. (2019) X. Yuan, Q. Zhao, D. Girolami, and X. Ma, Advanced Quantum Technologies 2, 1900053 (2019), ISSN 2511-9044, URL http://dx.doi.org/10.1002/qute.201900053.
  * Yu and Gühne (2019) X.-D. Yu and O. Gühne, Phys. Rev. A 99, 062310 (2019).
  * Ding et al. (2021) Q.-M. Ding, X.-X. Fang, X. Yuan, T. Zhang, and H. Lu, Phys. Rev. Research 3, 023228 (2021).
  * Miszczak et al. (2009) J. A. Miszczak, Z. Puchała, P. Horodecki, A. Uhlmann, and K. Zyczkowski, Quantum Info. Comput. 9 (2009), ISSN 1533-7146.

##  VI Supplementary Materials

###  VI.1 Proof of Theorem 1.

Our main tool is the data processing inequality. Consider an arbitrary
monotonous (under CPTP) and gaugeable distance (satisfying assumption Eq.(1)),
we have

$\displaystyle\delta_{A}(\rho)$ $\displaystyle:=$
$\displaystyle\textstyle\sqrt{1-\|\mathbf{p}\|^{2}_{2}}=\sqrt{1-\operatorname{tr}(\rho\rho_{A})}$
$\displaystyle\geq$
$\displaystyle\operatorname{IF}(\rho,\rho_{A})=\operatorname{IF}(\phi,\phi_{A})$
$\displaystyle=$
$\displaystyle\tilde{\operatorname{D}}(\phi,\phi_{A})\geq\tilde{\operatorname{D}}(\rho,\rho_{A})$
$\displaystyle\geq$
$\displaystyle\tilde{\operatorname{D}}(\Phi_{B}(\rho),\Phi_{B}(\rho_{A}))=\tilde{\operatorname{D}}(\mathbf{q},\mathbf{q}^{\prime})$

In the first line above we have denoted
$\|\mathbf{p}\|_{\alpha}=(\sum_{i}p^{\alpha}_{i})^{\frac{1}{\alpha}}$ and
$\sqrt{1-\|\mathbf{p}\|^{2}_{2}}$ is Shur concave and therefore a well-defined
uncertainty measure. In the second line the inequality is due to the fact
$\operatorname{F}(\rho_{1},\rho_{2})^{2}\geq\operatorname{tr}(\rho_{1}\cdot\rho_{2})$
Miszczak et al. (2009) for two general density matrices $\rho_{1}$ and
$\rho_{2}$ and the equality is becuause for two mixed states, say $\rho$ and
$\rho_{A}$, there are different purifications and the optimal ones
$|\phi\rangle$ and $|\phi_{A}\rangle$ give the quantum fidelity
$|\langle\phi|\phi_{A}\rangle|=\operatorname{F}(\rho,\rho_{A})$. In the third
line the equality is due to the definition of gauged distance measure and the
inequality is due to data processing inequality applied to $\phi,\phi_{A}$
under partial trace. In the fourth line data processing inequality is employed
again for $\rho,\rho_{A}$ under $\Phi_{B}$. In the last line that we have
noted that $\Phi_{B}(\rho)$ and $\Phi_{B}(\rho_{A})$ are diagonal states in
the same basis and the distance is basis-independent due to the unitary
invariance, therefore the distance is the function of diagonal terms, namely,
statistics arising from measuring $B$ on $\rho$ and $\rho_{A}$.

###  VI.2 Proof of uncertainty-disturbance $\rm U_{\rm tr}$ and $\rm U_{\rm
tr}^{\prime}$

As the first example we employ the trace distance $\operatorname{D}_{\rm
tr}(\rho,\rho_{A})=\frac{1}{2}\operatorname{tr}|\rho-\rho_{A}|$ as the
distance measure where
$\operatorname{tr}|N|=\operatorname{tr}\sqrt{NN^{\dagger}}$. For two pure
states we have $\operatorname{D}_{\rm
tr}(\phi,\phi_{A})={\operatorname{IF}(\phi,\phi_{A})}$ and therefore
$\mathcal{G}_{\rm tr}(x)=x$, i.e., the gauged distance is identical to the
original one. Similarly by using data processing inequality as above we have

$\delta_{A}(\rho):=\sqrt{1-\|\mathbf{p}\|_{2}^{2}}\geq\frac{1}{2}\operatorname{tr}|\rho-\rho_{A}|\geq\frac{1}{2}\sum_{i}|q_{i}-q^{\prime}_{i}|:=\operatorname{D}_{A\to
B}^{\rm tr}(\rho),$ (8)

where $\operatorname{D}_{A\to B}^{\rm
tr}(\rho)\equiv\frac{1}{2}\sum_{i}|q_{i}-q^{\prime}_{i}|$ is $l_{1}$ or the
Kolmogorov distance commonly used disturbance measure. From an alternative
upper bound of trace distance, i.e.,

$\frac{1}{2}({\|\mathbf{p}\|_{\frac{1}{2}}-1})=\sum_{i>j}\sqrt{p_{i}p_{j}}\geq\sum_{i>j}|\rho_{ij}|\geq\frac{1}{2}\operatorname{tr}|\rho-\rho_{A}|,$

where $\\{\rho_{ij}\\}$ are elements of density matrix $\rho$ we obtain $\rm
U_{\rm tr}^{\prime}$ which was first derived in a previous work Sun et al.
(2018).

###  VI.3 Proof of uncertainty-disturbance relations $\rm U_{\rm
rd}^{\alpha}$, $\rm\tilde{U}_{\rm rd}^{\alpha}$, $\rm U_{\rm if}$,
$\rm\tilde{U}_{\rm if}$, and $\rm U_{\rm re}$

We present below the variants associated with the distance measures of
$\alpha-$Rényi divergence and relative entropy. Employ Rényi divergence

$\operatorname{D}_{\alpha}(\rho\|\rho_{A})=\frac{1}{\alpha-1}\log\large\operatorname{tr}(\rho^{\frac{1-\alpha}{2\alpha}}_{A}\rho\rho^{\frac{1-\alpha}{2\alpha}}_{A})^{\alpha},$

with $1/2\leq\alpha<1$, as state distance measure. It is obviously invariant
under unitaries and satisfying data processing inequality. For two pure states
we have
$\operatorname{D}_{\alpha}(\phi\|\phi_{A})=\frac{\alpha}{1-\alpha}\log(1-\operatorname{IF}(\phi,\phi_{A})^{2})$
and $\mathcal{G}_{D}(x)=\frac{\alpha}{1-\alpha}\log(1-x^{2})$ which is a
reversible and monotonously increasing function for $1/2\leq\alpha<1$. The
corresponding gauged distance measure reads

$\tilde{\operatorname{D}}_{\alpha}(\rho||\rho_{A})=\textstyle\sqrt{1-\left(\operatorname{tr}(\rho^{\frac{1-\alpha}{2\alpha}}_{A}\rho\rho^{\frac{1-\alpha}{2\alpha}}_{A})^{\alpha}\right)^{\frac{1}{\alpha}}}$

giving the corresponding trade-off

$\delta_{A}(\rho)\geq\sqrt{\textstyle
1-\left(\sum_{i}q_{i}^{\alpha}{q^{\prime}}_{i}^{1-\alpha}\right)^{\frac{1}{\alpha}}}.$
(9)

There are also other ways of bounding the Rényi divergence. We note that for a
positive Hermitian operator $O$ it holds
$\operatorname{Tr}O^{\alpha}\geq(\operatorname{Tr}O)^{{}^{\alpha}}$ for
$\alpha<1$. As a result we have

$\operatorname{D}_{\alpha}(\rho||\rho_{A})\leq\frac{\alpha}{\alpha-1}\log\large\operatorname{tr}(\rho^{\frac{1-\alpha}{2\alpha}}_{A}\rho\rho^{\frac{1-\alpha}{2\alpha}}_{A})=\frac{\alpha}{\alpha-1}\log\|\mathbf{p}\|_{\frac{1}{\alpha}}$

which gives rise to trade-off

$\displaystyle\operatorname{H}_{\frac{1}{\alpha}}(\mathbf{p})\geq{\operatorname{D}}_{\alpha}(\mathbf{q}\|\mathbf{q}^{\prime}):=\frac{1}{\alpha-1}\log\sum_{i}q^{\alpha}_{i}{q^{\prime}}_{i}^{1-\alpha},$
(10)

with R$\operatorname{\acute{e}}$nyi entropy defined as

$\operatorname{H}_{\alpha}(\mathbf{p})=\frac{\alpha}{1-\alpha}\log\|\mathbf{p}\|_{\alpha}$

for the uncertainty measure. We note that the uncertainty in the first variant
Eq.(8) is equivalent to the collision entropy
$\operatorname{H}_{2}(\mathbf{p})=-\log(1-{\delta^{2}_{A}(\rho)}).$

Specifically, the $\alpha-$Rényi divergence becomes relative entropy when
$\alpha$ tends to 1, namely, $\operatorname{D}_{\alpha\to
1}(\rho\|\rho_{A})=S(\rho\|\rho_{A})=\operatorname{tr}(\rho\log\rho)-\operatorname{tr}(\rho\log\rho_{A})$.
We note that in this case the distance measure does not satisfy condition
Eq.(1), i.e., $\mathcal{G}_{D}$ is not well-defined. However, we still have
similar uncertainty-disturbance trade-off following from the data processing
inequality

$\displaystyle\operatorname{H}(\mathbf{p})\geq
S(\rho||\rho_{A})\geq\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime}),$ (11)

where the Shannon entropy
$\operatorname{H}(\mathbf{p})\equiv-\sum_{i}p_{i}\log p_{i}$ and classical
relative entropy
$\operatorname{H}(\mathbf{q}\|\mathbf{q}^{\prime})\equiv\sum_{i}q_{i}\log\frac{q_{i}}{q^{\prime}_{i}}$
quantify uncertainty and disturbance, respectively.

Take infidelity ${\rm IF}(\rho,\rho_{A})$ as the distance measure, we have
$\rm U_{\rm if}$ as a special case of Eq.(10) for $\alpha=\frac{1}{2}$, i.e.,

$\displaystyle\operatorname{H}_{2}(\mathbf{p})\geq\operatorname{D}_{\frac{1}{2}}(\mathbf{q}\|\mathbf{q}^{\prime}),$
(12)

where $\delta_{A}(\rho)=\operatorname{H}_{2}(\mathbf{p})$ and
$\operatorname{D}_{A\to
B}=\operatorname{D}_{\frac{1}{2}}(\mathbf{q}\|\mathbf{q}^{\prime})$. It turns
out that it is equivalent to $\tilde{\rm}U_{\rm if}$ and is tighter than the
first variant Eq.(8).

###  VI.4 Proof of $\rm U_{\rm ts}^{\alpha}$ and $\rm\tilde{U}_{\rm
ts}^{\alpha}$

. Let us now employ Tsallis relative entropy as the state distance measure

$\operatorname{T}_{\alpha}(\rho,\rho_{A})=\frac{1}{1-\alpha}[1-\operatorname{Tr}(\rho^{\alpha}\rho^{1-\alpha}_{A})],$

with $0\leq\alpha<1$. The assumption Eq.(1) is satisfied as
$\operatorname{T}_{\alpha}(\phi,\phi_{A})=\frac{1}{1-\alpha}\operatorname{IF}(\phi,\phi_{A})^{2}$
and we have a monotonous increasing function
$\mathcal{G}_{D}(x)=\frac{1}{1-\alpha}x^{2}$, which gives an identical
universal uncertainty disturbance trade-off as Eq.(9) from Rényi divergence.
As an alternative way of bounding the Tsallis distance we note first that

$\operatorname{tr}(\rho^{\alpha}\rho_{A}^{1-\alpha})\geq\operatorname{tr}(\rho\rho_{A}^{1-\alpha})=\|{\mathbf{p}}\|_{2-\alpha}^{2-\alpha}$

for $\alpha<1$ so that, by formulating in a similar way as uncertainty
Eq.(10), we have

$\displaystyle\frac{1}{2-\alpha}\operatorname{H}_{2-\alpha}(\mathbf{p})\geq{\operatorname{D}}_{\alpha}(\mathbf{q}\|\mathbf{q}^{\prime}),$
(13)

where $\frac{1}{2-\alpha}\operatorname{H}_{2-\alpha}(\mathbf{p})$ and
$\operatorname{D}_{\alpha}(\mathbf{q}\|\mathbf{q}^{\prime})$ quantify
uncertainty and disturbance, respectively. We note that uncertainty relation
above is stronger than Eq.(10) and also that from quantum Rényi divergence we
can also obtain this stronger version of trade-off by considering Araki-Lieb-
Thirring inequality, in the same manner.

###  VI.5 Proof of $\rm U_{\rm hs}$

In fact our method can be also slightly generalized to cover more distance
measures. As Eq.(4) only requires the monotonicity of distance measure under
the dephasing operation $\Phi_{B}(\cdot)$, the uncertainty-disturbance trade-
off applies also to the Hilbert-Schmidt distance
$\operatorname{D}_{\operatorname{HS}}(\rho,\rho_{A})\equiv\sqrt{\operatorname{tr}(\rho-\rho_{A})^{2}}$
which is monotonous under dephasing operation (but not a general under CPTP
Coles et al. (2019)). It follows from the data processing inequality that

$\displaystyle\delta_{A}(\rho)$ $\displaystyle=$
$\displaystyle\textstyle\sqrt{1-\|\mathbf{p}\|^{2}_{2}}\geq\textstyle\sqrt{\operatorname{tr}\rho^{2}-\|\mathbf{p}\|^{2}_{2}}$
(14) $\displaystyle=$
$\displaystyle{\operatorname{D}_{\operatorname{HS}}(\rho,\rho_{A})}\geq\operatorname{D}_{\operatorname{HS}}(\Phi_{B}(\rho),\Phi_{B}(\rho_{A}))$
$\displaystyle=$
$\displaystyle\sqrt{\textstyle\sum_{i}(q_{i}-q_{i}^{\prime})^{2}}:=\operatorname{D}^{\operatorname{HS}}_{A\to
B}(\rho).$


# The Primordial Black Hole Abundance: The Broader, the Better

A. Ianniccari  A.J. Iovino  A. Kehagias  D. Perrone  and A. Riotto

###### Abstract

We show that the abundance of primordial black holes, if formed through the
collapse of large fluctuations generated during inflation and unless the power
spectrum of the curvature perturbation is very peaked, is always dominated by
the broadest profile of the compaction function, even though statistically it
is not the most frequent. The corresponding threshold is therefore 2/5. This
result exacerbates the tension when combining the primordial black hole
abundance with the signal seen by pulsar timing arrays and originated from
gravitational waves induced by the same large primordial perturbations.

##  1 Introduction and Conclusions

Primordial Black Holes (PBHs) have emerged as one of the most interesting
topics in cosmology in the last years (see Ref. [1] for a recent review). PBHs
could explain both some of the signals from binary black hole mergers observed
in gravitational wave detectors [2] and be an important component of the dark
matter in the Universe.

One of the crucial parameter in PBHs physics is the relative abundance of PBHs
with respect to the dark matter component. This quantity is not easy to
calculate in the scenario in which PBHs are formed by the collapse of large
fluctuations generated during inflation upon horizon re-entry. Indeed, the
formation probability is very sensitive to tiny changes in the various
ingredients, such as the critical threshold of collapse, the non-Gaussian
nature of the fluctuations, the choice of the window function to define
smoothed observables (see again Ref. [1] for a nice discussion on such
issues), the nonlinear corrections entering in the calculation of the PBHs
abundance from the nonlinear radiation transfer function and the determination
of the true physical horizon crossing [3] and the appearance of an infinite
tower of local, non-local and higher-derivative operators upon dealing with
the nonlinear overdensity [4].

One intrinsic and therefore unavoidable source of uncertainty in calculating
the PBHs abundance arises from the inability to predict the value of a given
observable with zero uncertainty, e.g. the compaction function or its
curvature at its peak, in a given point or region. This is due to the fact
that the theory delivers only stochastic quantities, e.g. the curvature
perturbation, of which we know only the power spectrum and the higher-order
correlators. Therefore, we are allowed to calculate only ensemble averages and
typical values, which come with intrinsic uncertainties quantified by, for
example, root mean square deviations.

Since the critical PBHs abundance depends crucially on the curvature of the
compaction function at its peak, the natural question which arises is the
following: in order to calculate the PBHs abundance, which value of the
critical threshold should we use? In other words, which value of the curvature
should one adopt to derive the formation threshold?

A natural answer to this question might be to use the average profile of the
compaction function, and this is done routinely in the literature. After all,
most of the Hubble volumes are populated by peaks with such average profile at
horizon re-entry.

In this paper we wish to make a simple, but relevant observation: only if the
power spectrum of the curvature perturbation is very peaked, the critical
threshold for formation is determined by the average value of the curvature of
the compaction function at the peak; in the realistic cases in which the power
spectrum of the curvature perturbation is not peaked, the critical threshold
for formation is determined by the broadest possible compaction function. This
is because the abundance is dominated by the smallest critical threshold,
which corresponds to the broadest profile. In such a case, the threshold for
the compaction function is fixed to be 2/5.

This paper is organized as follows. In section 2 we briefly summarize the
properties of the compaction function; in sections 3 and 4 we prove our
observation, while in section 5 we make a comparison with the recent
literature and its implication with Pulsar Timing Arrays experiments. In
section 6 we provide some final comments.

##  2 The compaction function

The key starting object is the curvature perturbation $\zeta(\bf x)$ on
superhorizon scales which appears in the metric in the comoving uniform-energy
density gauge

${\rm d}s^{2}=-{\rm d}t^{2}+a^{2}(t)e^{2\zeta(\bf x)}{\rm d}{\bf x}^{2},$
(2.1)

where $a(t)$ is the scale factor in terms of cosmic time. Cosmological
perturbations may gravitationally collapse to form a PBH depending on the
amplitude measured at the peak of the compaction function, defined as the mass
excess compared to the background value within a given radius (see for
instance Ref. [5])

$C({\bf x})=2\frac{M({\bf x},t)-M_{\rm b}({\bf x},t)}{R({\bf x},t)},$ (2.2)

where $M({\bf x},t)$ is the Misner-Sharp mass and $M_{\rm b}({\bf x},t)$ its
background value. The Misner-Sharp mass gives the mass within a sphere of
areal radius

$R({\bf x},t)=a(t)re^{\zeta(\bf x)}$ (2.3)

with spherical coordinate radius $r$, centered around position ${\bf x}$ and
evaluated at time $t$. The compaction directly measures the overabundance of
mass in a region and is therefore better suited than the curvature
perturbation for determining when an overdensity collapses into a PBH.
Furthermore, the compaction has the advantage to be time-independent on
superhorizon scales. It can be written in terms of the density contrast as

$C({\bf x})=\frac{2\rho_{\rm b}}{R({\bf x},t)}\int{\rm d}^{3}{\bf
x}\,\delta({\bf x},t),$ (2.4)

where $\rho_{\rm b}$ is the background energy density. On superhorizon scales
the density contrast is related to the curvature perturbation in real space by
the nonlinear relation

$\delta({\bf x},t)=-\frac{4}{9}\frac{1}{a^{2}H^{2}}e^{-2\zeta(\bf
x)}\left(\nabla^{2}\zeta({\bf x})+\frac{1}{2}\left({\mathbf{\nabla}}\zeta({\bf
x})\right)^{2}\right).$ (2.5)

Assuming spherical symmetry and defining $\zeta^{\prime}={\rm d}\zeta/{\rm
d}r$, the compaction function becomes

$\displaystyle C(r)$ $\displaystyle=$ $\displaystyle\frac{8\pi\rho_{\rm
b}}{R(r,t)}\int_{0}^{R}{\rm
d}\widetilde{R}\,\widetilde{R}^{2}(r,t)\delta(r,t)=C_{\zeta}(r)-\frac{3}{8}C_{\zeta}^{2}(r),$
$\displaystyle C_{\zeta}(r)$ $\displaystyle=$
$\displaystyle-\frac{4}{3}r\zeta^{\prime}(r).$ (2.6)

Suppose now that there is peak in the curvature perturbation $\zeta({\bf x})$
with a given peak value $\zeta(0)$ and profile $\zeta(r)$ away from the
center, which we arbitrarily can set at the origin of the coordinates. The
corresponding compaction function will have a maximum at the distance $r_{m}$
from the origin of the peak. Since

$C^{\prime}(r_{m})=C^{\prime}_{\zeta}(r_{m})\left[1-\frac{3}{4}C_{\zeta}(r_{m})\right]=0,$
(2.7)

the extremum of the compaction function $C(r)$ coincides with the extremum of
$C_{\zeta}(r)$. Furthermore, since

$C^{\prime\prime}(r_{m})=C^{\prime\prime}_{\zeta}(r_{m})\left[1-\frac{3}{4}C_{\zeta}(r_{m})\right],$
(2.8)

the maximum of the compaction function $C(r)$ coincides with the maximum of
$C_{\zeta}(r)$ as long as $C_{\zeta}(r_{m})<4/3$ (the so-called type I case).
We will focus therefore mainly on this quantity. Notice that sometimes we will
call $C_{\zeta}(r)$ “linear” compaction function for simplicity, where the
term linear stems from the fact that its expression is linear in the curvature
perturbation $\zeta(r)$. However, $C_{\zeta}(r)$ is not necessarily Gaussian
if the curvature perturbation $\zeta(r)$ is not.

The maximum of the compaction function is fixed by the equation

$C^{\prime}(r_{m})=C^{\prime}_{\zeta}(r_{m})=0\,\,\,\,{\rm
or}\,\,\,\,\zeta^{\prime}(r_{m})+r_{m}\zeta^{\prime\prime}(r_{m})=0.$ (2.9)

Consider now a family of compaction functions which have in common the same
value of $r_{m}$, but a different curvature at the maximum parametrized by [6]

$q=-\frac{1}{4}\frac{r_{m}^{2}C^{\prime\prime}(r_{m})}{C(r_{m})},$ (2.10)

Numerically, it has been noticed that the critical threshold depends on the
curvature at the peak of the compaction function [6, 7, 8]

$C_{c}(q)=\frac{4}{15}e^{-1/q}\frac{q^{1-5/2q}}{\Gamma(5/2q)-\Gamma(5/2q,1/q)},$
(2.11)

such that $C_{c}(q\ll 1)\simeq 2/5$ and $C_{c}(q\gg 1)\simeq 2/3$. We also
notice that

$q=-\frac{1}{4}\frac{r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m})\left[1-\frac{3}{4}C_{\zeta}(r_{m})\right]}{C_{\zeta}(r_{m})\left[1-\frac{3}{8}C_{\zeta}(r_{m})\right]}\simeq-\frac{1}{4}\frac{r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m})}{C_{\zeta}(r_{m})}\left[1-\frac{3}{8}C_{\zeta}(r_{m})\right]\equiv
q_{\zeta}\left[1-\frac{3}{8}C_{\zeta}(r_{m})\right].$ (2.12)

###  2.1 The average profile

One question to pose is the following: which profile should one make use of to
calculate the critical value for PBH abundance, given that it depends on the
peak profile? The natural answer, routinely adopted in the literature, would
be the average profile of the compaction function with the constraint that
there is a peak of the curvature perturbation at the center of the coordinates
with value $\zeta(0)$. This is the most obvious answer as the average profile
is the most frequent, statistically speaking. Supposing for the moment that
$\zeta(r)$ is Gaussian, such an average profile would be

$\langle
C_{\zeta}(r)\rangle_{\zeta(0)}=-\frac{4}{3}r\langle\zeta^{\prime}(r)\rangle_{\zeta(0)}=-\frac{4}{3}r\langle\zeta(r)\rangle_{\zeta(0)}^{\prime}=-\frac{4}{3}r\frac{\xi^{\prime}(r)}{\xi(0)}\zeta_{0},$
(2.13)

where

$\xi(r)=\int\frac{{\rm d}k}{k}{\cal P}_{\zeta}(k)\frac{\sin kr}{kr}$ (2.14)

is the two-point correlation of the curvature perturbation. In such a case the
value of $r_{m}$ where the most likely compaction function has its maximum
would then be fixed by the equation

$\xi^{\prime}(r_{m})+r_{m}\xi^{\prime\prime}(r_{m})=0.$ (2.15)

A standard choice is therefore to calculate the curvature of the peak of the
compaction function as111This is clearly not correct as, for instance, the
average of the ratio of two stochastic variables is not the ratio of their
averages.

$q=-\frac{1}{4}\frac{r_{m}^{2}\langle
C^{\prime\prime}_{\zeta}(r_{m})\rangle_{\zeta(0)}}{\langle
C_{\zeta}(r_{m})\rangle_{\zeta(0)}}\left[1-\frac{3}{8}\langle
C_{\zeta}(r_{m})\rangle_{\zeta(0)}\right].$ (2.16)

The crucial point is that, the smaller the value of the curvature, the smaller
the value of the threshold. Since the PBHs abundance has an exponentially
strong dependence on the threshold, one expects that broad compaction
functions should be very relevant in the determination of the abundance of
PBHs even though they are more rare than the average profiles. This is what we
discuss next.

##  3  The relevance of Broadness: the Gaussian case

In this section we assume $C_{\zeta}(r_{m})$ and
$C_{\zeta}^{\prime\prime}(r_{m})$ to be Gaussian (and correlated) variables.
This will allow us to gain some analytical intuition. We define

$\sigma^{2}_{0}=\langle
C_{\zeta}^{2}(r_{m})\rangle,\,\,\,\sigma^{2}_{1}=-\frac{1}{4}r_{m}^{2}\langle
C_{\zeta}^{\prime\prime}(r_{m})C_{\zeta}(r_{m})\rangle,\,\,\,{\rm
and}\,\,\,\sigma^{2}_{2}=\frac{1}{16}r_{m}^{4}\langle{C_{\zeta}^{\prime\prime}(r_{m})}^{2}\rangle.$
(3.1)

Such correlations are easily computed knowing that the Fourier transform of
the linear compaction function reads

$C_{\zeta}({\bf k},r)=\frac{4}{9}k^{2}r^{2}W(kr)\zeta({\bf
k}),\,\,\,\,W(x)=3\frac{\sin x-x\cos x}{x^{3}},$ (3.2)

where $W(x)$ is the Fourier transform of the Heaviside window function in real
space. We will use the conservation of the probabilities

$\displaystyle P\left[C(r_{m}),C^{\prime\prime}(r_{m})\right]{\rm
d}C(r_{m}){\rm d}C^{\prime\prime}(r_{m})$ $\displaystyle=$ $\displaystyle{\cal
P}\left[C_{\zeta}(r_{m}),C_{\zeta}^{\prime\prime}(r_{m})\right]{\rm
d}C_{\zeta}(r_{m}){\rm d}C_{\zeta}^{\prime\prime}(r_{m})$ (3.3)
$\displaystyle=$ $\displaystyle\widetilde{{\cal
P}}\left[C_{\zeta}(r_{m}),q_{\zeta}\right]{\rm d}C_{\zeta}(r_{m}){\rm
d}q_{\zeta}$

where

$\displaystyle{\cal
P}\left[-\frac{1}{4}r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m}),C_{\zeta}(r_{m})\right]$
$\displaystyle=$ $\displaystyle\frac{1}{2\pi\sqrt{{\rm det}\,\Sigma}}\,{\rm
exp}\left(-\vec{V}^{T}\Sigma^{-1}\vec{V}/2\right),$ $\displaystyle\vec{V}^{T}$
$\displaystyle=$
$\displaystyle\left[-\frac{1}{4}r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m}),C_{\zeta}(r_{m})\right],$
$\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cc}\sigma^{2}_{2}&\sigma^{2}_{1}\\\
\sigma^{2}_{1}&\sigma^{2}_{0}\end{array}\right).$ (3.6)

We find it convenient to define the parameter

$\gamma=\frac{\sigma_{1}^{2}}{\sigma_{2}\sigma_{0}},$ (3.7)

which will play an important role in the following and indicates the broadness
of a given power spectrum of the curvature perturbation. The closer $\gamma$
is to unity, the more spiky is the peak of the curvature perturbation.

###  3.1 The average of the curvature

The average curvature of the linear compaction function $C_{\zeta}$ can be
computed by using the conditional probability to have a peak at $r_{m}$222In
fact we use threshold statistics rather than peak statistics to elaborate our
point. However, regions well above the corresponding square root of the
variance are very likely local maxima [9].

$\langle q_{\zeta}\rangle=\int_{0}^{\infty}{\rm
d}q_{\zeta}\,q_{\zeta}\,P[q|C_{\zeta}(r_{m})>C_{\zeta,c}(q_{\zeta})],$ (3.8)

with

$P[q|C_{\zeta}(r_{m})>C_{\zeta,c}(q_{\zeta})]=\frac{\widetilde{{\cal
P}}[q_{\zeta},C_{\zeta}(r_{m})>C_{\zeta,c}(q_{\zeta})]}{\widetilde{{\cal
P}}[C_{\zeta}(r_{m})>C_{\zeta,c}(q_{\zeta})]}.$ (3.9)

The conditional probability, in the limit of large thresholds, becomes

$\displaystyle P[q_{\zeta}|C_{\zeta}(r_{m})>C_{\zeta,c}(q_{\zeta})]$
$\displaystyle\simeq$
$\displaystyle\frac{\left(1-\gamma^{2}\right)^{1/2}\sigma_{2}C_{\zeta,c}(q_{\zeta})}{\sqrt{2\pi}\sigma_{0}\left[(q_{\zeta}-\gamma\sigma_{2}/\sigma_{0})^{2}+(1-\gamma^{2})\sigma_{2}^{2}/\sigma_{0}^{2}\right]^{1/2}}\cdot$
(3.10) $\displaystyle\cdot$
$\displaystyle\exp\left[-\frac{(q-\gamma\sigma_{2}/\sigma_{0})^{2}C^{2}_{\zeta,c}(q_{\zeta})}{2(1-\gamma^{2})\sigma_{2}^{2}}\right].$

For a monochromatic power spectrum of the curvature perturbation, that is
$\gamma\simeq 1$, we recognize the Dirac delta and the value of $q_{\zeta}$,
which minimizes the exponent and maximizes the PBHs abundance, is the average
value $\langle q_{\zeta}\rangle=\sigma_{2}/\sigma_{0}$.

Departing from $\gamma\simeq 1$, and integrating numerically, one discovers
departures from the value $\gamma\sigma_{2}/\sigma_{0}$ for the average of
$q_{\zeta}$, but not dramatically, and one has

$\langle q_{\zeta}\rangle\simeq\gamma\frac{\sigma_{2}}{\sigma_{0}}.$ (3.11)

###  3.2 The PBHs abundance

The PBHs abundance is given by333We do not account for the extra factor
counting the mass of the PBH with respect to the mass contained in the horizon
volume at re-entry as we give priority to getting analytical results. We will
reintegrate it in the next section.

$\beta=\int_{C_{c}(q)}^{\infty}{\rm d}C(r_{m})\int_{-\infty}^{0}{\rm
d}C^{\prime\prime}(r_{m})P\left[C(r_{m}),C^{\prime\prime}(r_{m})\right]=\int_{0}^{\infty}{\rm
d}q_{\zeta}\int_{C_{\zeta,c}(q_{\zeta})}^{\infty}{\rm
d}C_{\zeta}(r_{m})\widetilde{{\cal
P}}\left[C_{\zeta}(r_{m}),q_{\zeta}\right].$ (3.12)

Going back to the initial probability, it can be written as

$\displaystyle{\cal
P}\left[-\frac{1}{4}r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m}),C_{\zeta}(r_{m})\right]$
$\displaystyle=$
$\displaystyle\frac{1}{2\pi}\frac{1}{\sigma_{2}\sigma_{0}\sqrt{1-\gamma^{2}}}{\rm
exp}\left[-\frac{r_{m}^{4}{C_{\zeta}^{\prime\prime}(r_{m})}^{2}}{16\cdot
2\sigma_{2}^{2}}\right]\cdot$ (3.13) $\displaystyle\cdot$ $\displaystyle{\rm
exp}\left[-\frac{1}{2(1-\gamma^{2})}\left(\frac{C_{\zeta}(r_{m})}{\sigma_{0}}+\gamma\frac{r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m})}{4\sigma_{2}}\right)^{2}\right].$

For a monochromatic, very peaked, power spectrum of the curvature
perturbation, where $\gamma\simeq 1$, the probability reduces to

$\lim_{\gamma\rightarrow 1}{\cal
P}\left[-\frac{1}{4}r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m}),C_{\zeta}(r_{m})\right]=\frac{1}{\sqrt{2\pi}}\frac{1}{\sigma_{2}\sigma_{0}}{\rm
exp}\left(-\frac{r_{m}^{4}{C_{\zeta}^{\prime\prime}(r_{m})}^{2}}{16\cdot
2\sigma_{2}^{2}}\right)\delta_{D}\left(\frac{C_{\zeta}(r_{m})}{\sigma_{0}}+\frac{r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m})}{4\sigma_{2}}\right),$
(3.14)

which fixes

$q_{\zeta}=\frac{\sigma_{2}}{\sigma_{0}}=\frac{\sigma_{2}\sigma_{0}}{\sigma_{0}^{2}}=\frac{\sigma^{2}_{1}}{\sigma^{2}_{0}}=\langle
q_{\zeta}\rangle,$ (3.15)

and

$\displaystyle\beta$ $\displaystyle=$ $\displaystyle\int_{0}^{\infty}{\rm
d}q_{\zeta}\int_{C_{\zeta,c}(q_{\zeta})}^{4/3}{\rm d}C_{\zeta}(r_{m}){\cal
P}\left[-\frac{1}{4}r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m}),C_{\zeta}(r_{m})\right]$
(3.16) $\displaystyle=$ $\displaystyle\int_{C_{\zeta,c}(\langle
q_{\zeta}\rangle)}^{4/3}{\rm
d}C_{\zeta}(r_{m})\frac{1}{\sqrt{2\pi}\sigma_{0}}{\rm
exp}\left(-\frac{C^{2}_{\zeta}(r_{m})}{2\sigma_{0}^{2}}\right)=\frac{1}{2}{\rm
Erfc}\left[\frac{C_{\zeta,c}(\langle
q_{\zeta}\rangle)}{\sqrt{2}\sigma_{0}}\right],$

where

$C_{\zeta,c}(q_{\zeta})\simeq\frac{4}{3}\left(1-\sqrt{\frac{2-3C_{c}(q_{\zeta})}{2}}\right).$
(3.17)

Therefore for monochromatic spectra of the curvature perturbation the PBHs
abundance is fixed by the value of the threshold corresponding to the average
value of the curvature of the compaction function at its peak

$C^{\rm peaked}_{\zeta,c}=C_{\zeta,c}(\langle q_{\zeta}\rangle).$ (3.18)

For a generic power spectrum, we change the variables from
$(-r_{m}^{2}C_{\zeta}^{\prime\prime}(r_{m}),C_{\zeta}(r_{m}))$ to
$(q_{\zeta},C_{\zeta}(r_{m}))$ and making use of the conservation of the
probability we obtain

$\displaystyle\beta(q_{\zeta})$ $\displaystyle=$
$\displaystyle\int_{C_{\zeta,c}(q_{\zeta})}^{4/3}{\rm
d}C_{\zeta}(r_{m})|C_{\zeta}(r_{m})|\,{\cal
P}\left[qC_{\zeta}(r_{m}),C_{\zeta}(r_{m})\right]$ $\displaystyle\simeq$
$\displaystyle\frac{\sqrt{1-\gamma^{2}}\sigma_{2}}{2\pi\sigma_{0}\left[(q_{\zeta}-\gamma\sigma_{2}/\sigma_{0})^{2}+(1-\gamma^{2})\sigma^{2}_{2}/\sigma^{2}_{0}\right]}\cdot$
$\displaystyle\cdot$
$\displaystyle\exp\left[-\frac{\left[(q_{\zeta}-\gamma\sigma_{2}/\sigma_{0})^{2}+(1-\gamma^{2})\sigma^{2}_{2}/\sigma^{2}_{0}\right]C^{2}_{\zeta,c}(q_{\zeta})}{2(1-\gamma^{2})\sigma^{2}_{2}}\right].$

We see that the square of the critical threshold is replaced by an effective
squared critical threshold

$C^{2}_{\zeta,c}(q_{\zeta})\Big{|}_{\rm
eff}=\left[(q_{\zeta}-\gamma\sigma_{2}/\sigma_{0})^{2}+(1-\gamma^{2})\sigma_{2}^{2}/\sigma_{0}^{2}\right]C^{2}_{\zeta,c}(q_{\zeta}).$
(3.20)

Its minimum is determined by the equation

$(q_{\zeta}-\gamma\sigma_{2}/\sigma_{0})C_{\zeta,c}(q_{\zeta})+\left[(q_{\zeta}-\gamma\sigma_{2}/\sigma_{0})^{2}+(1-\gamma^{2})\sigma_{2}^{2}/\sigma_{0}^{2}\right]\frac{{\rm
d}C_{\zeta,c}(q_{\zeta})}{{\rm d}q_{\zeta}}=0.$ (3.21)

For peaked profiles where $\gamma\simeq 1$ we have

$(q_{\zeta}-\sigma_{2}/\sigma_{0})\left[C_{\zeta,c}(q_{\zeta})+(q_{\zeta}-\sigma_{2}/\sigma_{0})\frac{{\rm
d}C_{\zeta,c}(q_{\zeta})}{{\rm d}q_{\zeta}}\right]=0$ (3.22)

and the mimimum lies at the value of the average $q_{\zeta}=\langle
q_{\zeta}\rangle=\sigma_{2}/\sigma_{0}$. For broad spectra $\gamma\ll 1$, the
effective threshold is minimized for $q_{\zeta}\simeq 0$ as it reduces to

$C^{2}_{\zeta,c}(q_{\zeta})\Big{|}_{\rm
eff}=\left[q_{\zeta}^{2}+\sigma_{2}^{2}/\sigma_{0}^{2}\right]C^{2}_{\zeta,c}(q_{\zeta}).$
(3.23)

and the threshold $C_{\zeta,c}(q_{\zeta})$ is also minimized for small
$q_{\zeta}$. There is in general a critical value of $q_{\zeta}$ for which the
abundance is always dominated by the broad spectra. We can see this behaviour
by plotting the curve $(q_{\zeta\;\rm{min}},\gamma)$ obtain from the Eq.
(3.21), as shown in Fig. 2. As we start decreasing from $\gamma=1$ where the
minimum is in $\sigma_{2}/\sigma_{0}$, also the value of
$q_{\zeta,\;\rm{min}}$ decreases, up until a critical value $\gamma_{\rm
crit}$. For values of $\gamma$ below this point the function
$C_{\zeta,c}(q_{\zeta})|_{\rm{eff}}$ does not have a minimum, but is
monotonically increasing with $q_{\zeta}$, hence the minimum lies at the
boundary of the interval, i.e. $q_{\zeta}=0$. The transition is therefore very
sharp after the critical value.
It is also possible to evaluate the position of this minimum for different
values of the parameter $\sigma_{2}/\sigma_{0}$, as shown in Fig. 2. We can
understand the behaviour because having larger values of this parameter the
transition happens for larger values of $\gamma$, being easier to enter in the
regime of Eq. (3.23), where $\sigma_{2}/\sigma_{0}$ dominates.

Figure 1: Plot of $q_{\zeta\;\rm{\min}}$ as a function of $\gamma$ for
$\sigma_{2}/\sigma_{0}=2$.

Figure 2: Critical value of $\gamma$ as a function of $\sigma_{2}/\sigma_{0}$

To show this explicitly, in Fig. 3 we plot the formation probability for three
different values of $\gamma$. It demonstrates that the abundance is dominated
by the broadest profiles when the curvature perturbation is not very spiky and
not by the average value of $q_{\zeta}$. The corresponding critical value
needed to be used is therefore

$C_{\zeta,c}(q_{\zeta}\simeq 0)\simeq\frac{4}{3}\left(1-\sqrt{\frac{2-3\cdot
2/5}{2}}\right)\simeq 0.49.$ (3.24)

Figure 3: The PBHs formation probability as a function of $q_{\zeta}$ for
$\sigma_{0}=\sigma_{2}/2=0.05$ and three different values of
$\gamma=(0.3,0.5,0.8)$ for which $\langle q_{\zeta}\rangle=(0.6,1,1.6)$ for
the Gaussian case.

##  4  The relevance of Broadness: the non-Gaussian case

As a matter of fact, the curvature perturbation generated in models producing
large overdensities is typically non-Gaussian. Non-Gaussianity among the modes
interested in the growth of the curvature perturbation is generated either by
their self-interaction during the ultra slow-roll phase [10] or after Hubble
radius exit when the curvature perturbation is sourced by a curvaton-like
field [11, 12, 13].

We proceed therefore by assuming that the initial curvature perturbation is
non-Gaussian, but a function of a Gaussian component

$\zeta(r)=F[\zeta_{\rm g}(r)].$ (4.1)

In such a case the compaction function is still given by Eq. (2), where

$C_{\zeta}(r)=F_{1}(\zeta_{\rm g})C_{\rm g}(r),\,\,\,\,C_{\rm
g}(r)=-\frac{4}{3}r\zeta^{\prime}_{\rm g}(r),$ (4.2)

and we have indicated the derivatives of $F$ with respect to $\zeta_{\rm g}$
by $F_{n}={\rm d}F(\zeta_{\rm g})/{\rm d}\zeta_{\rm g}$. The maximum of the
compaction function can be found solving the equation

$C_{\zeta}^{\prime}(r_{m})=F_{1}(\zeta_{\rm g})C^{\prime}_{{\rm
g}}(r_{m})+C_{{\rm g}}(r_{m})\zeta_{\rm g}^{\prime}(r_{m})F_{2}(\zeta_{\rm
g})=0,$ (4.3)

as long as $C_{\zeta}(r_{m})<4/3$. The next step is to define the following
Gaussian and correlated variables

$x_{0}=\zeta_{\rm g},\quad x_{1}=r\zeta^{\prime}_{\rm g},\quad
x_{2}=r^{2}\zeta^{\prime\prime}_{\rm g},\quad
x_{3}=r^{3}\zeta^{\prime\prime\prime}_{\rm g},$ (4.4)

for which the condition of the maximum becomes

$x_{2}=-x_{1}\left(1+x_{1}\frac{F_{2}(x_{0})}{F_{1}(x_{0})}\right).$ (4.5)

One can construct the corresponding probability distribution as

$P(x_{0},x_{1},x_{2},x_{3})=\frac{1}{(2\pi)^{2}\sqrt{{\rm det}\,\Sigma}}\,{\rm
exp}\left(-\vec{V}^{T}\Sigma^{-1}\vec{V}/2\right),$ (4.6)

where

$\vec{V}^{T}=\left[x_{0},x_{1},x_{2},x_{3}\right],$

and

$\displaystyle\Sigma$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{cccc}\sigma^{2}_{0}&\gamma_{01}\sigma_{1}\sigma_{0}&\gamma_{02}\sigma_{2}\sigma_{0}&\gamma_{03}\sigma_{3}\sigma_{0}\\\
\gamma_{01}\sigma_{1}\sigma_{0}&\sigma_{1}^{2}&\gamma_{12}\sigma_{2}\sigma_{1}&\gamma_{13}\sigma_{1}\sigma_{3}\\\
\gamma_{02}\sigma_{2}\sigma_{0}&\gamma_{12}\sigma_{2}\sigma_{1}&\sigma^{2}_{2}&\gamma_{23}\sigma_{2}\sigma_{3}\\\
\gamma_{03}\sigma_{3}\sigma_{0}&\gamma_{13}\sigma_{1}\sigma_{3}&\gamma_{23}\sigma_{2}\sigma_{3}&\sigma_{3}^{2}\end{array}\right)$
(4.11)

is constructed from the different correlators with444Notice that here for
clarity the index for the various $\sigma_{i}$ is related to the number of
derivatives of $\zeta_{\rm g}$, differently from the definition in the
previous section.

$\sigma_{i}^{2}=\langle x_{i}^{2}\rangle,\quad\gamma_{ij}=\frac{\langle
x_{i}x_{j}\rangle}{\langle x_{i}^{2}\rangle^{1/2}\;\langle
x_{j}^{2}\rangle^{1/2}}.$ (4.12)

Next, we need to convert all the relevant variables in terms of the Gaussian
ones $x_{i}$ ($i=0,\cdots,3)$. First we have

$C_{{\rm g}}=-\frac{4}{3}\,x_{1},$ (4.13)

and the derivatives of $C_{\zeta}$ can be written in terms of $x_{1}$ and
$x_{2}$ as

$\displaystyle C_{\zeta}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\;x_{1}F_{1}(x_{0}),$ $\displaystyle
rC_{\zeta}^{\prime}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\left(F_{1}(x_{0})(x_{1}+x_{2})+x_{1}^{2}F_{2}(x_{0})\right),$
$\displaystyle r^{2}C_{\zeta}^{\prime\prime}$ $\displaystyle=$
$\displaystyle-\frac{4}{3}\left[F_{1}(x_{0})(2x_{2}+x_{3})+2x_{1}^{2}F_{2}(x_{0})+3x_{1}x_{2}F_{2}(x_{0})+x_{1}^{3}F_{3}(x_{0})\right].$
(4.14)

The PBHs abundance for a given value of the curvature $q$ will read

$\beta(q)=\int_{D}\mathcal{K}\left(C-C_{c}(q)\right)^{\gamma}\;p(x_{0},C_{\zeta},x_{2},q)\;\delta\left(F_{1}(x_{0})(x_{1}+x_{2})+x_{1}^{2}F_{2}(x_{0})\right),$
(4.15)

where the domain of integration is

$D=\left\\{x_{2}\in\mathbb{R},C_{\zeta,c}(q)<C_{\zeta}<\frac{4}{3}\right\\},$
(4.16)

with

$C_{\zeta,c}(q)\simeq\frac{4}{3}\left(1-\sqrt{\frac{2-3C_{c}(q)}{2}}\right).$
(4.17)

We have reintroduced the scaling-law factor for critical collapse
$\mathcal{K}(C-C_{c}(q))^{\gamma}$ which accounts for the mass of the PBHs at
formation written in units of the horizon mass at the time of horizon re-
entry, with $\mathcal{K}\simeq 3.3$ for a log-normal power spectrum and
$\gamma\simeq 0.36$ [14, 15, 16] (see also Ref. [17]). By using the
conservation of probabilities we can finally write

$p\left[\zeta_{\rm
g},C_{\zeta},x_{2},q\right]=P\left[x_{0},x_{1},x_{2},x_{3}\right]|{\rm
Det}\,J|,$ (4.18)

where

${\rm
Det}\,J=\frac{3}{4}\left(\frac{4x_{1}+2F_{1}(x_{0})x_{1}^{2}}{1+F_{1}(x_{0})x_{1}}\right),$
(4.19)

and at the maximum

$x_{3}=\frac{-4q(1+\frac{1}{2}x_{1}F_{1}(x_{0}))x_{1}}{1+x_{1}F_{1}(x_{0})}-2x_{2}-2x_{1}^{2}\frac{F_{2}(x_{0})}{F_{1}(x_{0})}-3x_{1}x_{2}\frac{F_{2}(x_{0})}{F_{1}(x_{0})}-x_{1}^{3}\frac{F_{3}(x_{0})}{F_{1}(x_{0})}.$
(4.20)

We rewrite the Gaussian probability in the following form

$P(x_{0},x_{1},x_{2},x_{3})=\frac{1}{4\pi^{2}\sqrt{\det{\Sigma}}}\exp{\left(-\frac{(\sigma_{0}\sigma_{1}\sigma_{2}\sigma_{3})^{2}}{2\det{\Sigma}}\sum_{i,j=0}^{3}\frac{\kappa_{ij}x_{i}x_{j}}{\sigma_{i}\sigma_{j}}\right)},$
(4.21)

where the $\kappa_{ij}$’s will depend on all the $\gamma_{lm}$, and they can
be computed by performing the inverse of the matrix $\Sigma$, matching with
the definition. Performing the change of variables we get

$\displaystyle p(\zeta_{\rm g},C_{\zeta},x_{2},q)$ $\displaystyle=$
$\displaystyle\left|\frac{9C_{\rm g}\left(3C_{\rm g}F_{1}-8\right)}{8\
\left(3C_{\rm
g}F_{1}-4\right)}\right|\frac{1}{4\pi^{2}\sqrt{\det{\Sigma}}}\cdot$
$\displaystyle\cdot$
$\displaystyle\exp{\left(-\frac{(\sigma_{0}\sigma_{1}\sigma_{2}\sigma_{3})^{2}}{2\det{\Sigma}}\frac{1}{4096\
F_{1}^{4}}\left[A(\zeta_{\rm g},C_{\rm g})q^{2}+B(\zeta_{\rm g},C_{\rm
g})q+C(\zeta_{\rm g},C_{\rm g})\right]\right)},$

where we have defined the following functions of $\zeta_{\rm g}$ and $C_{\rm
g}$

$A(\zeta_{\rm g},C_{\rm g})=\frac{9216\kappa_{33}C_{\rm
g}^{2}F_{1}^{4}\left(8-3C_{\rm
g}F_{1}\right){}^{2}}{\sigma_{3}^{2}\left(4-3C_{\rm g}F_{1}\right){}^{2}},$
(4.23) $\displaystyle B(\zeta_{\rm g},C_{\rm g})$ $\displaystyle=$
$\displaystyle\frac{192C_{\rm g}F_{1}^{2}\left(3C_{\rm
g}F_{1}-8\right)}{\sigma_{0}\sigma_{1}\sigma_{2}\sigma_{3}^{2}\left(3C_{\rm
g}F_{1}-4\right)}\left[3\kappa_{33}\sigma_{0}\sigma_{1}\sigma_{2}C_{\rm
g}\left\\{9C_{\rm g}F_{1}\left(C_{\rm g}F_{3}+4F_{2}\right)-27C_{\rm
g}^{2}F_{2}^{2}-32F_{1}^{2}\right\\}\right.$ (4.24)
$\displaystyle\left.+4\sigma_{3}F_{1}\left\\{3\kappa_{23}\sigma_{0}\sigma_{1}C_{\rm
g}\left(4F_{1}-3C_{\rm
g}F_{2}\right)+4\sigma_{2}F_{1}\left(4\kappa_{30}\sigma_{1}\zeta_{\rm
g}-3\kappa_{13}\sigma_{0}C_{\rm g}\right)\right\\}\right],$ $\displaystyle
C(\zeta_{\rm g},C_{\rm g})$ $\displaystyle=$
$\displaystyle-\frac{\kappa_{33}}{\sigma_{3}^{2}}\left\\{4374C_{\rm
g}^{6}F_{1}F_{2}^{2}F_{3}-5832C_{\rm g}^{5}F_{1}^{2}F_{2}F_{3}+17496C_{\rm
g}^{5}F_{1}F_{2}^{3}-27216C_{\rm g}^{4}F_{1}^{2}F_{2}^{2}+\right.$
$\displaystyle\left.20736C_{\rm g}^{3}F_{1}^{3}F_{2}-729C_{\rm
g}^{6}F_{1}^{2}F_{3}^{2}+5184C_{\rm g}^{4}F_{1}^{3}F_{3}-9216C_{\rm
g}^{2}F_{1}^{4}-6561C_{\rm g}^{6}F_{2}^{4}\right\\}+$
$\displaystyle-\frac{1}{\sigma_{0}\sigma_{1}\sigma_{2}\sigma_{3}}24C_{\rm g}\
F_{1}\left(-9C_{\rm g}F_{1}\left(C_{\rm g}F_{3}+4F_{2}\right)+27C_{\rm
g}^{2}F_{2}{}^{2}+32F_{1}^{2}\right)$
$\displaystyle\left(3\kappa_{23}\sigma_{0}\sigma_{1}C_{\rm
g}\left(4F_{1}-3C_{\rm
g}F_{2}\right)+4\sigma_{2}F_{1}\left(4\kappa_{30}\sigma_{1}\zeta_{\rm
g}-3\kappa_{13}\sigma_{0}C_{\rm g}\right)\right)+$
$\displaystyle+16F_{1}^{2}\left(\frac{24C_{\rm g}F_{1}\left(3C_{\rm
g}F_{2}-4F_{1}\right)\left(3\kappa_{12}\sigma_{0}C_{\rm
g}-4\kappa_{20}\sigma_{1}\zeta_{\rm
g}\right)}{\sigma_{0}\sigma_{1}\sigma_{2}}+\right.$
$\displaystyle\left.+\frac{9\kappa_{22}C_{\rm g}^{2}\ \left(4F_{1}-3C_{\rm
g}F_{2}\right){}^{2}}{\sigma_{2}^{2}}+\frac{16F_{1}^{2}\left(-24\kappa_{10}\sigma_{1}\sigma_{0}C_{\rm
g}\zeta_{\rm g}+9\kappa_{11}\sigma_{0}^{2}C_{\rm
g}^{2}+16\kappa_{0}\sigma_{1}^{2}\zeta_{\rm
g}^{2}\right)}{\sigma_{0}^{2}\sigma_{1}^{2}}\right),$

and each function $F_{n}$ is intended to be $F_{n}(\zeta_{\rm g})$.

###  4.1 An illustrative example

We consider the following illustrative example which typically arises in
models in which the curvature perturbation is generated during a period of
ultra-slow-roll [10, 18, 19, 20]555For $\zeta_{\rm g}>\mu_{\star}$, Eq. (4.26)
does not capture the possibility of PBHs formed by bubbles of trapped vacuum
which requires a separate discussion [21, 22].

$\zeta({\bf x})=-\mu_{\star}\ln\left(1-\frac{\zeta_{\rm g}({\bf
x})}{\mu_{\star}}\right),$ (4.26)

with $\mu_{\star}$ a model-dependent parameter depending upon the transition
between the ultra-slow-roll phase and the subsequent slow-roll phase. In this
analysis we take $\mu_{\star}$ as a free parameter. We take the power spectrum
of the Gaussian component to be a log-normal power spectrum

${\cal P}_{\rm g}(k)=\frac{A}{\sqrt{2\pi}\Delta}{\rm
exp}\left[-\ln^{2}(k/k_{\star})/2\Delta^{2}\right].$ (4.27)

Our results are summarized in Fig. 4 where we have assumed $k_{\star}r_{m}=2$
for the plot666This is not strictly correct as the value of $r_{m}$ should
change for each profile, but it is nevertheless in the ballpark of the value
we have assumed [8].. The broadness of the power spectrum is controlled by the
parameter $\Delta$. We observe that by increasing the value of $\Delta$,
enlarging the power spectra, again the PBHs formation probability is dominated
by the broadest profiles. We have checked that for very peaked power spectrum,
as in the case for $\Delta=1/3$, the abundance is peaked again around the
average of $q$.

Figure 4: Mass fraction $\beta$ for the non-Gaussian scenario computed with
several values of $\Delta$, where we fix $\mu_{*}=5/2$, $k_{*}r_{m}=2$ and the
amplitude of the power spectrum $A=10^{-2}$.

##  5 Comparison with literature and impact on the physics of PBHs and Pulsar
Timing Arrays

In this section we compare the calculation presented above, accounting for the
curvature of the compaction function at its peak, with the prescription based
on threshold statistics on the compaction function, reported in Refs. [23,
24], where the only explicit dependence on $q$ is encoded in $C_{c}(q)$.
There, the formation probability is computed by integrating the joint
probability distribution function $P_{\mathrm{g}}$

$\beta=\int_{D}\mathcal{K}\left(C-C_{c}(q)\right)^{\gamma}\textrm{P}_{\rm
g}(C_{\rm g},\zeta_{\rm g}){\rm d}C_{\rm g}{\rm d}\zeta_{\rm g}\,,$ (5.1)

where the domain of integration is given by $\mathcal{D}=\left\\{C(C_{\rm
g},\zeta_{\rm g})>C_{\rm c}(q),~{}C_{\zeta}(C_{\rm g},\zeta_{\rm
g})<4/3\right\\}$. The Gaussian components are distributed as

$P_{\rm g}\left(C_{\rm g},\zeta_{\rm g}\right)=\frac{1}{2\pi\sigma_{\rm
a}\sigma_{\rm
c}\sqrt{1-\gamma_{*}^{2}}}\exp\left[-\frac{1}{2\left(1-\gamma_{*}^{2}\right)}\left(\frac{C_{\mathrm{g}}}{\sigma_{\rm
a}}-\frac{\gamma_{*}\zeta_{\mathrm{g}}}{\sigma_{\rm
c}}\right)^{2}\\!-\\!\frac{\zeta_{\mathrm{g}}^{2}}{2\sigma_{\rm
c}^{2}}\right],$ (5.2)

with correlators

$\displaystyle\langle C^{2}_{\rm g}\rangle$ $\displaystyle=\sigma_{\rm
a}^{2}=\frac{16}{81}\int_{0}^{\infty}\frac{{\rm
d}k}{k}\left(kr_{m}\right)^{4}W^{2}\left(k,r_{m}\right)T\left(k,r_{m}\right)P_{\zeta},$
(5.3a) $\displaystyle\langle C_{\rm g}\zeta_{\rm g}\rangle$
$\displaystyle=\sigma_{\rm
b}^{2}=\frac{4}{9}\\!\int_{0}^{\infty}\\!\\!\frac{{\rm
d}k}{k}\\!\left(kr_{m}\right)^{2}\\!W\\!\\!\left(k,r_{m}\right)\\!W_{s}\\!\left(k,r_{m}\right)T\left(k,r_{m}\right)\\!P_{\zeta},$
(5.3b) $\displaystyle\langle\zeta^{2}_{\rm g}\rangle$
$\displaystyle=\sigma_{\rm c}^{2}=\int_{0}^{\infty}\frac{{\rm
d}k}{k}W_{s}^{2}\left(k,r_{m}\right)T\left(k,r_{m}\right)P_{\zeta},$ (5.3c)

and $\gamma_{*}=\sigma_{\rm b}^{2}/\sigma_{\rm a}\sigma_{\rm c}.$ We have
defined $W\left(k,r_{m}\right)$ and $W_{s}\left(k,r_{m}\right)$ as the top-hat
window function and the spherical-shell window function [25]. To compare this
prescription with the one presented in this paper, we consider two cases:
$\beta_{0}$, in which we do not adopt any transfer function ($T=1$) since
everything is determined on superhorizon scales, and $\beta_{T}$, in which we
consider the radiation transfer function assuming a perfect radiation fluid,
as adopted in Ref. [23].

In Fig. 5, we show a comparison between the two prescriptions using the
typical non-Gaussian relation in the ultra-slow-roll scenario (see Eq. (4.26))
with a log-normal power spectrum (see Eq. (4.27)) with several benchmark
values for $\mu_{\star}$. We fix $\Delta=1$ in the plots, but we have found
analogous results also varying this parameter. As we can understand from Fig.
5, evaluating the quantities on superhorizon scales, i.e. the ratio
$\beta/\beta_{0}$, the old prescription marginally overestimates the abundance
of PBHs. This discrepancy arises because, unlike the prescription used in the
literature, where an average profile is employed, the effective threshold is
slightly larger than the averaged case, as evident from Eq. (3.23).
Nevertheless an equivalent amount of PBHs requires a slightly larger amplitude
of the curvature perturbation power spectrum.

The situation is different when we include the radiation transfer function,
i.e. the ratio $\beta/\beta_{T}$. The presence of the transfer function
decreases the values of the variances and, as a consequence, it reduces the
amount of PBHs.

This has important implications for the phenomenology related to PBHs respect
to the case discussed in this paper. Indeed in the standard formation scenario
PBH formation occurs as large curvature perturbations re-enter the Hubble
horizon after inflation and eventually collapse under the effect of gravity.
When such scalar perturbations cross the horizon they produce tensor modes as
a second-order effect, which appear to us today as a signal of stochastic
gravitational waves background (SGWB) (for a recent review, see Ref. [27]).
Recently in Ref.[26], where the old prescription was used, it was shown that
large negative non-Gaussianity are necessary in order to achieve high enough
amplitude, without overproducing PBHs, in order to relax the tension between
the PTA recent dataset (the most constrained dataset is the one released by
NANOGrav [28]) and the PBH explanation.

We demonstrate that even when correctly accounting for the impact of the
curvature of the compaction function and calculating all the relevant
quantities on superhorizon scales, thereby avoiding all concerns regarding
non-linearities in the radiation transfer function and the determination of
the true physical horizon, the tension between the PTA dataset and the PBH
hypothesis is even worse than what claimed in Ref. [26].

Figure 5: Ratio between mass fraction $\beta$ for the non-Gaussian case
between the prescriptions presented in this paper and the prescription
presented in Ref. [23]. We fix the shape parameter $q=0.5$ (as a consequence
also the threshold using Eq. (2.11)) and the shape of power spectrum
$\Delta=1$ while we vary the amplitude.

##  6  Conclusions and some further final considerations

In this paper we have shown that the abundance of PBHs is dominated by the
broadest profiles of the compaction function, even though they are not the
typical ones, unless the power spectrum of the curvature perturbation is very
peaked. The corresponding threshold is therefore always 2/5. We have also
discussed how this result makes the tension between overproducing PBHs and
fitting the recent PTA data on gravitational waves even worse than recent
analysis.

On more general grounds, given the dependence of the critical threshold on the
profile of the compaction function, the natural question is if it possible to
construct an observable whose critical threshold does not depend at all on the
profiles of the peaks. In Ref. [6] it has been proven numerically that the
volume average of the compaction function, calculated in a volume of sphere of
radius $R_{m}$

$\overline{C}(R_{m})=\frac{3}{R_{m}^{3}}\int_{0}^{R_{m}}{\rm d}x\,x^{2}\,C(x)$
(6.1)

has a critical threshold equal to 2/5 independently from the profile. In the
case of a broad compaction function, whose critical threshold is 2/5, and
since $\overline{C}(R_{m})\simeq C(R_{m})$, it is trivial that the volume
average has the same critical value 2/5. The case of a very spiky compaction
function corresponds to a flat universe with in it a sphere of radius $R_{m}$
and constant curvature $K(R)=C(R)/R^{2}$, that is $C(R)$ scales like $R^{2}$.
One then obtains

$\overline{C}(R_{m})=3\frac{C(R_{m})}{R_{m}^{5}}\int_{0}^{R_{m}}{\rm
d}x\,x^{4}=\frac{3}{5}C(R_{m})=\frac{3}{5}\cdot\frac{2}{3}=\frac{2}{5},$ (6.2)

where it is used that for very spiky compaction functions the critical value
is 2/3.

Assuming a universal threshold, one can then write the probability that the
volume average compaction function is larger than 2/5 even for the non-
Gaussian case as (we use here threshold statistics to make the point, one
could similarly use peak theory)777For non-Gaussian perturbations the
universal threshold remains 2/5 [29] for the realistic cases in which the non-
Gaussian parameter is positive [30]. Notice that one can construct easily
another observable whose threshold is independent from the profile. Indeed, as
we mentioned already, the compaction function is related to the local
curvature of the universe by the relation $C(R)=K(R)R^{2}$. Given a curvature
perturbation $\zeta(r)$, a compaction function $C(R)$ with maximum in $R_{m}$
and the corresponding curvature $K(R)$, one consider a new perturbation with
curvature  $\overline{K}=\Theta_{H}(R_{m}-R)\int_{0}^{R_{m}}{\rm
d}x\,x^{2}\,K(x),$ that is a spherical local closed universe with curvature
$\overline{K}$ with radius $R_{m}$ surrounded by a flat universe. This
corresponds to a new infinitely peaked compaction function equal to
$\Theta_{H}(R_{m}-R)R^{2}\overline{K}$ whose threshold will be always 2/3
[31], independently from the profile of the initial compaction function.

$\displaystyle P\left[\overline{C}(R_{m})>2/5\right]$ $\displaystyle=$
$\displaystyle\Big{<}\Theta_{H}\left[\overline{C}(R_{m})-2/5\right]\Big{>}$
(6.3) $\displaystyle=$
$\displaystyle\frac{1}{2\pi}\int\left[DC(r)\right]P\left[C(r)\right]\int_{2/5}^{\infty}{\rm
d}\alpha\int_{-\infty}^{\infty}{\rm
d}\phi\,e^{i\phi(\overline{C}(R_{m})-\alpha)}$

which can be written as

$P\left[\overline{C}(R_{m})>2/5\right]=\int_{2/5}^{\infty}{\rm
d}\alpha\int_{-\infty}^{\infty}{\rm d}\phi\,e^{-i\phi\alpha}\cdot Z[J],$ (6.4)

with

$\displaystyle Z[J]=\int\left[DC({\bf x})\right]P\left[C({\bf
x})\right]e^{i\int{\rm d}^{3}x\,J({\bf x})C({\bf x})},\quad J({\bf
x})=\,V^{-1}_{R_{m}}\,\phi\,\Theta_{H}(\overline{r}_{m}-r)\,,$ (6.5)

and the measure $\left[DC(r)\right]$ is such that

$\displaystyle\int\left[DC({\bf x})\right]P\left[C({\bf
x})\right]=\int\left[DC(r)\right]P\left[C(r)\right]=1.$ (6.6)

The correlators are determined by the expansion of the partition function
$Z[J]$ in terms of the source $J$, while the corresponding expansion of
$W[J]=\ln Z[J]$ generates the connected correlation functions. We will denote
the latter as

$\displaystyle\xi^{(n)}({\bf x}_{1},\cdots,{\bf x}_{n})=\frac{\delta}{\delta
J({\bf y}_{1})}\cdots\frac{\delta}{\delta J({\bf y}_{n})}\ln Z[J],$ (6.7)

and the connected cumulants of the volume average linear compaction function
as

$\displaystyle\langle\overline{C}^{n}(R_{m})\rangle$ $\displaystyle=$
$\displaystyle\frac{1}{V^{n}_{R_{m}}}\int{\rm d}^{3}x_{1}\cdots{\rm
d}^{3}x_{n}\prod_{i=1}^{n}\xi^{(n)}({\bf x}_{1},\cdots,{\bf
x}_{n})\,\Theta_{H}(R_{m}-x_{i})$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{n}\int\frac{{\rm d}^{3}k_{i}}{(2\pi)^{3}}P_{N}({\bf
k}_{1},\cdots,{\bf k}_{n})W(k_{1}R_{m})\cdots
W(k_{n}R_{m})\,\delta_{D}^{(n)}({\bf k}_{1}+\cdots+{\bf k}_{n}),$
$\displaystyle\langle C_{\zeta}({\bf k}_{1}),\cdots,C_{\zeta}({\bf
k}_{n})\rangle$ $\displaystyle=$ $\displaystyle P_{N}({\bf k}_{1},\cdots,{\bf
k}_{n})\delta_{D}^{(n)}({\bf k}_{1}+\cdots+{\bf k}_{n}).$ (6.8)

Then, we may write

$\displaystyle\ln Z[J]$ $\displaystyle=$
$\displaystyle\sum_{n=2}^{\infty}\frac{(-1)^{n}}{n!}\int{\rm d}^{3}{\bf
y}_{1}\cdots\int{\rm d}^{3}{\bf y}_{n}\,J_{i_{1}}({\bf y}_{1})\cdots
J_{i_{n}}({\bf y}_{n})\xi^{(n)}({\bf x}_{i_{1}},\cdots,{\bf x}_{i_{n}})$ (6.9)
$\displaystyle=$
$\displaystyle\sum_{n=2}^{\infty}\frac{(-1)^{n}}{n!}\phi^{n}\langle\overline{C}^{n}\rangle.$

Using the above expression for the connected partition function, we find that
the one-point statistics of Eq. (6.3) can be written as

$\displaystyle P\left[\overline{C}(R_{m})>2/5\right]=$
$\displaystyle(2\pi)^{-1/2}\int_{2/5}^{\infty}{\rm
d}a\,\exp\left\\{\sum_{n=3}^{\infty}\frac{(-1)^{n}}{n!}\langle\overline{C}^{n}\rangle\frac{\partial^{n}}{\partial
a^{n}}\right\\}\exp{\left(-\frac{a^{2}}{2\sigma_{\overline{C}}^{2}}\right)}$
(6.10) $\displaystyle=$ $\displaystyle(2\pi)^{-1/2}\int_{2/5}^{\infty}{\rm
d}a\,\left(1-\frac{1}{3!}\langle\overline{C}_{\zeta}^{3}\rangle\frac{{\rm
d}^{3}}{{\rm d}a^{3}}+\frac{1}{4!}\langle\overline{C}^{4}\rangle\frac{{\rm
d}^{4}}{{\rm
d}a^{4}}+\cdots\right)\exp{\left(-\frac{a^{2}}{2\sigma_{\overline{C}}^{2}}\right)}$
$\displaystyle=$ $\displaystyle h_{0}(2/5)+\frac{1}{\sqrt{2\pi}}\sum_{n\geq
3}\frac{1}{2^{\frac{n}{2}}n!}\frac{c_{n}}{\sigma_{\overline{C}}^{n-1}}e^{-4/50\sigma_{\overline{C}}^{2}}H_{n-1}\left(\frac{2/5}{\sqrt{2}\sigma_{\overline{C}}}\right),$

where

$\displaystyle h_{0}(2/5)=\frac{1}{2}{\rm
Erfc}\left(\frac{1}{\sqrt{2}}\frac{2/5}{\sigma_{\overline{C}}}\right),$ (6.11)

$\sigma_{\overline{C}}$ is the variance, $H_{n}$ are Hermite polynomials and
we have defined in Eq. (6.10) the parameters $c_{n}$ as

$\displaystyle
c_{n}=\sum_{\hat{p}[n]}\,\,\,\,\prod_{\begin{subarray}{c}p_{1}m_{1}+\cdots
p_{r}m_{r}=n\\\ p_{i}\geq 0,m_{i}\geq 3\end{subarray}}\frac{n!}{m_{1}!\cdots
m_{r}!\,p_{1}!\cdots
p_{r}!}\,\langle\overline{C}^{m_{1}}\rangle^{p_{1}}\cdots\langle\overline{C}^{m_{r}}\rangle^{p_{r}},$
(6.12)

where $\hat{p}[n]$ denotes the partitions of the integer $n$ into numbers
$m_{i}\geq 3$.

Given the statistics of the curvature perturbation, one can calculate the
abundance of PBHs using the volume average of the linear compaction function,
relying solely on superhorizon quantities. Generally, determining the
statistics of the curvature perturbation can be challenging and computing the
connected cumulants is highly non-trivial. We left this task for future
investigation.

Acknowledgements

We thank V. De Luca and G. Franciolini for useful comments on the draft. A.I.
and A.R. acknowledge support from the Swiss National Science Foundation
(project number CRSII5_213497). A.K. is supported by the PEVE-2020 NTUA
programme for basic research with project number 65228100. D. P. and A.R. are
supported by the Boninchi Foundation for the project “PBHs in the Era of GW
Astronomy”.

## References

  * [1] E. Bagui et al. [LISA Cosmology Working Group], [astro-ph.CO/2310.19857].
  * [2] G. Franciolini, V. Baibhav, V. De Luca, K. K. Y. Ng, K. W. K. Wong, E. Berti, P. Pani, A. Riotto and S. Vitale, Phys. Rev. D 105 (2022) no.8, 083526 [gr-qc/2105.03349].
  * [3] V. De Luca, A. Kehagias and A. Riotto, Phys. Rev. D 108 (2023) no.6, 063531 [astro-ph.CO/2307.13633].
  * [4] G. Franciolini, A. Ianniccari, A. Kehagias, D. Perrone and A. Riotto, [astro-ph.CO/2311.03239].
  * [5] T. Harada, C. M. Yoo and Y. Koga, Phys. Rev. D 108 (2023) no.4, 043515 [gr-qc/2304.13284].
  * [6] A. Escrivà, C. Germani and R. K. Sheth, Phys. Rev. D 101 (2020) no.4, 044022 [gr-qc/1907.13311].
  * [7] I. Musco, Phys. Rev. D 100 (2019) no.12, 123524 [gr-qc/1809.02127].
  * [8] I. Musco, V. De Luca, G. Franciolini and A. Riotto, Phys. Rev. D 103 (2021) no.6, 063538 [astro-ph.CO/2011.03014].
  * [9] Y. Hoffman and J. Shaham, Astrophys. J. 297 (1985), 16-22.
  * [10] Y. F. Cai, X. Chen, M. H. Namjoo, M. Sasaki, D. G. Wang and Z. Wang, JCAP 05 (2018), 012 [astro-ph.CO/1712.09998].
  * [11] M. Kawasaki, N. Kitajima and T. T. Yanagida, Phys. Rev. D 87, no.6, 063519 (2013) [hep-ph/1207.2550].
  * [12] M. Sasaki, J. Valiviita and D. Wands, Phys. Rev. D 74 (2006), 103003 [astro-ph/0607627].
  * [13] G. Ferrante, G. Franciolini, A. Iovino, Junior. and A. Urbano, JCAP 06 (2023), 057 [astro-ph.CO/2305.13382].
  * [14] M. W. Choptuik, Phys. Rev. Lett. 70 (1993), 9-12 [PhysRevLett.70.9].
  * [15] C. R. Evans, J. S. Coleman, Phys. Rev. Lett. 72 (1994), 1782-1785 [gr-qc/9402041].
  * [16] I. Musco, J. C. Miller Classical and Quantum Gravity, 30 (2013) no.14, 145009 [astro-ph.CO/1201.2379].
  * [17] M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, Classical and Quantum Gravity, 35 (2018) no.6, 063001 [astro-ph.CO/1801.05235].
  * [18] M. Biagetti, G. Franciolini, A. Kehagias and A. Riotto, JCAP 07 (2018), 032 [astro-ph.CO/1804.07124].
  * [19] M. Biagetti, V. De Luca, G. Franciolini, A. Kehagias and A. Riotto, Phys. Lett. B 820 (2021), 136602 [astro-ph.CO/2105.07810].
  * [20] E. Tomberg, Phys. Rev. D 108 (2023) no.4, 4 [astro-ph.CO/2304.10903].
  * [21] A. Escrivà, V. Atal and J. Garriga, JCAP 10 (2023), 035 [astro-ph.CO/2306.09990].
  * [22] K. Uehara, A. Escrivà, T. Harada, D. Saito and C. M. Yoo, [gr-qc/2401.06329].
  * [23] G. Ferrante, G. Franciolini, A. Iovino, Junior. and A. Urbano, Phys. Rev. D 107, no.4, 043520 (2023) [astro-ph.CO/2211.01728].
  * [24] A. D. Gow, H. Assadullahi, J. H. P. Jackson, K. Koyama, V. Vennin and D. Wands, EPL 142 (2023) no.4, 49001 [astro-ph.CO/2211.08348].
  * [25] S. Young, JCAP 05 (2022) no.05, 037 [astro-ph.CO/2201.13345].
  * [26] G. Franciolini, A. Iovino, Junior., V. Vaskonen and H. Veermae, Phys. Rev. Lett. 131 (2023) no.20, 201401 [astro-ph.CO/2306.17149].
  * [27] G. Domènech, Universe 7 (2021) no.11, 398 [gr-qc/2109.01398].
  * [28] G. Agazie et al. [NANOGrav], Astrophys. J. Lett. 951 (2023) no.1, L8 [astro-ph.HE./2306.16213]
  * [29] A. Escrivà, Y. Tada, S. Yokoyama and C. M. Yoo, JCAP 05, no.05, 012 (2022) [astro-ph.CO/2202.01028].
  * [30] H. Firouzjahi and A. Riotto, Phys. Rev. D 108, no.12, 123504 (2023) [astro-ph.CO/2309.10536].
  * [31] T. Harada, C. M. Yoo and K. Kohri, Phys. Rev. D 88, no.8, 084051 (2013) [erratum: Phys. Rev. D 89, no.2, 029903 (2014)] [astro-ph.CO/1309.4201].