#  The ground-state potential and dipole moment of carbon monoxide:
contributions from electronic correlation, relativistic effects, QED,
adiabatic, and non-adiabatic corrections

D. P. Usov  Department of Physics, St. Petersburg State University, 7/9
Universitetskaya nab., 199034 St. Petersburg, Russia  Y. S. Kozhedub
<EMAIL_ADDRESS>Department of Physics, St. Petersburg State University, 7/9
Universitetskaya nab., 199034 St. Petersburg, Russia  V. V. Meshkov
Department of Chemistry, Lomonosov Moscow State University, 1/3 Leninskie
gory, 119991 Moscow, Russia  A. V. Stolyarov  Department of Chemistry,
Lomonosov Moscow State University, 1/3 Leninskie gory, 119991 Moscow, Russia
N. K. Dulaev  Department of Physics, St. Petersburg State University, 7/9
Universitetskaya nab., 199034 St. Petersburg, Russia  National Research Centre
“Kurchatov Institute” B.P. Konstantinov Petersburg Nuclear Physics Institute,
Gatchina, Leningrad district 188300, Russia  N. S. Mosyagin  National Research
Centre “Kurchatov Institute” B.P. Konstantinov Petersburg Nuclear Physics
Institute, Gatchina, Leningrad district 188300, Russia  A. M. Ryzhkov
Department of Physics, St. Petersburg State University, 7/9 Universitetskaya
nab., 199034 St. Petersburg, Russia  National Research Centre “Kurchatov
Institute” B.P. Konstantinov Petersburg Nuclear Physics Institute, Gatchina,
Leningrad district 188300, Russia  I. M. Savelyev  Department of Physics, St.
Petersburg State University, 7/9 Universitetskaya nab., 199034 St. Petersburg,
Russia  V. M. Shabaev  Department of Physics, St. Petersburg State University,
7/9 Universitetskaya nab., 199034 St. Petersburg, Russia  National Research
Centre “Kurchatov Institute” B.P. Konstantinov Petersburg Nuclear Physics
Institute, Gatchina, Leningrad district 188300, Russia  I. I. Tupitsyn
Department of Physics, St. Petersburg State University, 7/9 Universitetskaya
nab., 199034 St. Petersburg, Russia

###### Abstract

The ground X${}^{1}\Sigma^{+}$ state potential energy curve (PEC) and dipole
moment curve (DMC) of CO molecule have been revisited within the framework of
the relativistic coupled-cluster approach, which incorporates non-perturbative
single, double, and triple cluster amplitudes (CCSDT) in conjunction with a
finite-field methodology. The generalized relativistic pseudo-potential model
was used for the effective introducing the relativity in all-electron
correlation treatment and accounting the quantum-electrodynamics (QED)
corrections within the model-QED-operator approach. The diagonal Born-
Oppenheimer correction to PEC has been evaluated using the CCSD approach. The
sensitivity of resulting PEC and DMC to variations in basis set parameters and
regular intramolecular perturbations were considered as well. The present ab
initio results are in a reasonable agreement with their most accurate semi-
empirical counterparts.

##  1 Introduction

Carbon monoxide (CO) stands out as one of the most resilient diatomic
molecules in the Universe, owing to its robust ’triple’ chemical bond. Indeed,
the CO($\displaystyle J=2\to 1$) rotational emission is observed in Early
Universe at high red-shift $\displaystyle z\sim 5-7$ Riechers  _et al._
(2019). In fact, next to molecular hydrogen, CO ranks among the most prevalent
diatomic species Combes (1991). Spectral signatures of CO have been detected
in diverse celestial settings, including solar Lyons  _et al._ (2018) and
stellar Abia, C.  _et al._ (2012) atmospheres, and have even been observed on
celestial bodies such as Mars Olsen  _et al._ (2021) and Venus Vandaele  _et
al._ (2016). Contemporary studies, employing methods like cross-correlation
Giacobbe  _et al._ (2021), are identifying CO spectral lines in exoplanet’s
atmospheres. In addition to its cosmic presence, CO is a notable terrestrial
pollutant, particularly in the troposphere, stemming from both natural sources
like biomass combustion and anthropogenic origins, such as automobile
emissions Beale  _et al._ (2016). Consequently, an exhaustive understanding of
CO’s spectral characteristics is imperative for precise modeling and
monitoring of a broad spectrum of astrophysical, environmental, and
atmospheric phenomena (see, for instance, the HITRAN molecular spectroscopic
database Gordon  _et al._ (2022)).

Moreover, the carbon monoxide molecule serves as a crucial benchmark system
for high-accuracy studies of absorption line intensities Bielska  _et al._
(2022); Balashov  _et al._ (2023). Experimentally, CO offers distinct
advantages, given its ease of concentration determination and stability. The
diatomic nature of the molecule and its relatively large reduced mass ensures
well-isolated absorption rotational lines for most abundant isotopologues. Ab
initio electronic structure calculations of CO benefit from relatively small
number of electrons to be correlated explicitly Mosyagin  _et al._ (2021) and
the absence of light (H) atoms, diminishing mass-dependant non-adiabatic
effects that dominate in more light diatomic systems, such as molecular
hydrogen and hydrids.

Given its key role in numerous domains, there has been a wealth of
experimental and theoretical investigations dedicated to CO spectroscopy over
the past century, e.g., Refs. Chackerian Jr and Goorvitch (1982); Ermilov  _et
al._ (1990); Coxon and Hajigeorgiou (2004a) to mention a few. Recently, a
series of theoretical studies has been conducted to investigate the ground-
state potential and dipole moment of carbon monoxide Bielska  _et al._ (2022);
Peng-Fei  _et al._ (2013); Chen  _et al._ (2015); Chołuj and Bartkowiak
(2016); Konovalova  _et al._ (2018); Ushakov  _et al._ (2020); Lykhin  _et
al._ (2021); Balashov  _et al._ (2023); Meshkov  _et al._ (2022, 2023); Dulaev
(2023). However, the extensive dataset available may not be considered
exhaustive due to its specificity or lack of systematic coverage. In order to
provide a precise theoretical description of the potential energy levels and
various properties of CO at a modern level of accuracy, careful consideration
of correlation and relativistic effects is required. Despite the molecule
being relatively light, the contributions from relativistic and potentially
quantum electrodynamics (QED) corrections prove to be significant.
Furthermore, this molecule was proposed for studying the variation of the
fine-structure constant Konovalova  _et al._ (2018); Mosyagin  _et al._
(2021). It is also worth noting the high sensitivity of the intensity
distribution in the rovibrational spectrum to a permanent dipole moment curve
of the molecule Medvedev  _et al._ (2016).

The primary objective of this study is to present a comprehensive and
contemporary ab initio evaluation of ground state potential energy curve (PEC)
and dipole moment curve (DMC) for carbon monoxide with the utmost precision.
The investigation employs the exhausted approach rooted in contemporary
quantum theory to accurately address correlation, relativistic and QED effects
as well as adiabatic correction and a rotational part of non-adiabatic
correction within the ground-state of molecule. Specifically, we adopt the
relativistic coupled-cluster method encompassing nonperturbative single (S),
double (D), and, notably, entirely triple (T) cluster amplitudes (CCSDT).
These computations are conducted employing the all-electron Dirac-Coulomb (DC)
Hamiltonian as well as its two-component analogue (X2C) Sikkema  _et al._
(2009). The assessment of the Breit and QED corrections is conducted utilizing
the framework of generalized relativistic pseudo-potentials (GRPP) Titov and
Mosyagin (1999); Oleynichenko  _et al._ (2023). Incorporation of the QED
contribution is based on the paradigm of the model-QED-operator approach, as
exemplified in references Shabaev  _et al._ (2013, 2018).

A challenging aspect inherent to theoretical explorations within this domain
is the absence of a straightforward methodology for the robust evaluation of
uncertainties. Nonetheless, as it has been recently demonstrated by fine ab
initio calculation of rovibrational intensities in a very weak (0-7) overtone
Balashov  _et al._ (2023) that the estimation of accuracy remains essential to
guarantee reliable prediction of carbon monoxide properties, as well as
interpretation of experimental data. In this context, uncertainties derived
from pure theoretical analyses are typically given preeminence. A methodology
that exhibits transparency, both in terms of the approximations employed and
the considered effects, while also facilitating systematic enhancements,
constitutes a essential for a reliable theoretical uncertainty assessment.

The manuscript is organized as follows. Section 2 provides a concise
exposition of our methodological approaches, accompanied by a short overview
of their implementation details. In Section 3, an exhaustive description of
the calculation methodology is presented. This section further engages in a
comprehensive discussion of the obtained outcomes, together with a comparative
analysis with existing literature values.

##  2 Theoretical approaches and methods

###  2.1 Accounting for relativistic and quantum-electrodynamics corrections

The calculations are performed within the framework of the relativistic four-
component Dirac–Coulomb (DC) Hamiltonian, employing the computational software
package DIRAC DIR ; Saue  _et al._ (2020) and its relevant extensions. A
diatomic molecule is considered within the Born-Oppenheimer approximation,
where the nuclei are the sources of electrostatic potential, fixed at the
positions $\displaystyle{\mbox{\boldmath$\displaystyle R$}}_{1,2}$ with the
internuclear distance $\displaystyle R=|{\mbox{\boldmath$\displaystyle
R$}}_{2}-{\mbox{\boldmath$\displaystyle R$}}_{1}|$. The DC Hamiltonian of the
electronic system (in atomic units) is given by

$H_{\text{DC}}=\Lambda^{+}\left[\sum_{i}c{\mbox{\boldmath$\displaystyle\alpha$}}_{i}\cdot{\mbox{\boldmath$\displaystyle
p$}}_{i}+\beta_{i}c^{2}+V({\mbox{\boldmath$\displaystyle
r$}}_{i})+\sum_{i<j}\frac{1}{|{\mbox{\boldmath$\displaystyle
r$}}_{i}-{\mbox{\boldmath$\displaystyle r$}}_{j}|}\right]\Lambda^{+},$ (1)

where $\displaystyle\beta$ and $\displaystyle\alpha$ are the standard
$\displaystyle 4\times 4$ Dirac matrices,
$\displaystyle{\mbox{\boldmath$\displaystyle r$}}_{i}$ and
$\displaystyle{\mbox{\boldmath$\displaystyle p$}}_{i}$ are the position vector
and momentum of the $\displaystyle i$-th electron, $\displaystyle
V({\mbox{\boldmath$\displaystyle
r$}}_{i})=V^{(1)}_{\text{nucl}}(|{\mbox{\boldmath$\displaystyle
R$}}_{1}-{\mbox{\boldmath$\displaystyle
r$}}_{i}|)+V^{(2)}_{\text{nucl}}(|{\mbox{\boldmath$\displaystyle
R$}}_{2}-{\mbox{\boldmath$\displaystyle r$}}_{i}|)$ is the total nuclear
binding potential (the finite nuclear size effect is taken into account within
the Gaussian model of charge distribution), the summation goes over all
electrons of the system, and $\displaystyle\Lambda^{+}$ is the projector on
the positive-energy Dirac-Fock one-electron states.

There also exists the so-called eXact-2-Component (X2C) Hamiltonian, which
serves as an approximation to the DC Hamiltonian $\displaystyle H_{\text{DC}}$
(1), yet precisely replicates the positive-energy spectrum of the four-
component one-electron Hamiltonian Kutzelnigg and Liu (2005); Iliaš and Saue
(2007). Utilizing a 2-component framework, this Hamiltonian notably offers
enhanced numerical efficiency. Furthermore, within the computational framework
of the DIRAC package, the X2C Hamiltonian stands as the exclusive option that
facilitates the incorporation of the Gaunt interaction as spin-same and spin-
other orbit mean-field operators within the context of the Atomic Mean-Field
Integral (AMFI) approximation. A comprehensive exploration of this aspect is
available in Ref. Iliaš  _et al._ (2001) and the associated references. The
Gaunt term, denoted as $\displaystyle V^{\mathrm{G}}$

$V_{ij}^{\mathrm{G}}=-\frac{(\bm{\alpha}_{i}\cdot\bm{\alpha}_{j})}{r_{ij}},$
(2)

constitutes the foremost relativistic correction to the Coulomb interaction.
In this study, we employ the molecular mean-field X2C modification Sikkema
_et al._ (2009).

Another optimization used in the present work is based on calculations with
the generalized relativistic pseudopotentials (GRPP) Titov and Mosyagin
(1999); Oleynichenko  _et al._ (2023). This method has proven to be highly
effective as it allows for the description of interactions with internally
"core" electrons that are excluded from the calculation, accurately accounting
for relativistic corrections, Breit interaction, and other corrections without
performing a full four-component calculation. In this study, a nonlocal
formulation of the GRPP with a zero core was employed. In addition to
relativistic corrections and the Breit interaction, the QED correction was
included in the GRPP potential Meshkov  _et al._ (2023). Describing the QED
effects is based on the model-QED-operator approach Shabaev  _et al._ (2015,
2018). This single-electron operator is independently constructed for each
atom of the molecule as a sum of the vacuum polarization operator and the
self-energy operator. The vacuum polarization operator is described by a sum
of local Uehling and Wichmann-Kroll potentials, while the self-energy operator
is approximated by a sum of short-range quasi-local and nonlocal potentials.
The molecular QED operator is represented as a superposition of atomic
operators. This approximation is justified as the QED operator has a small
range of action and is concentrated in the internal "core" region of the
atoms, while we are interested in the range of internuclear distances
$\displaystyle R>0.6$ Å, which is far enough from the United Atom and, hence,
responsible for the formation of the so-called "chemical" bond in the
molecule. The calculations were performed using the DIRAC program and the
LIBGRPP library Oleynichenko  _et al._ (2023), which is necessary for
accounting for the nonlocal part of the potential.

###  2.2 Electronic correlation treatment

Utilizing the frameworks of DC, X2C, or GRPP Hamiltonians, the electronic
structure is addressed through the application of the single-reference
relativistic coupled cluster (CC) methodology. For the current investigation,
we implement the CC method as realized in the EXP-T program Oleynichenko  _et
al._ ; Oleynichenko  _et al._ (2020a). In our paper, we adopt the designation
CCSD for calculations incorporating the single (S) and double (D) cluster
amplitudes, while CCSDT signifies calculations involving both SD and
nonperturbative triple (T) amplitudes. Instances where the triple amplitudes
are perturbatively evaluated are identified as CCSD(T). The Dirac-Hartree-Fock
(DHF) calculations, alternatively known as relativistic Hartree-Fock
computations, alongside the subsequent integral transformation procedures, are
carried out utilizing the DIRAC package.

The CC calculations are performed across diverse configurations, involving
varying numbers of correlated electrons and virtual orbitals. For the
conclusive version of the computations, the standard basis sets from the cc-
pV$\displaystyle N$Z family ($\displaystyle N=3,4,5,6$) Dunning (1989);
Kendall  _et al._ (1992) are employed.

###  2.3 Extrapolation to the complete basis set

A comprehensive exploration of these calculations entails augmenting the
aforementioned basis sets through the sensible addition of further diffuse
(low exponent) basis functions, incorporated in an even-tempered manner. To
address the inherent limitations of basis-set completeness, an extrapolation
approach to attain the complete basis set (CBS) limit is employed. For the
outcomes stemming from the Dirac-Hartree-Fock methodology, the extrapolation
scheme detailed below is adopted:

$E_{\rm DHF}(N)=E_{\rm DHF}^{\rm CBS}+Ae^{-\beta N}.$ (3)

In the context of correlation corrections, the following formula is applied:

$E_{\rm corr}(N)=E_{\rm corr}^{\rm CBS}+\frac{A}{N^{3}}.$ (4)

The rationale underpinning of the utilization of this extrapolation technique
within molecular computations involving correlation-consistent basis sets is
expounded upon in the work referenced as Helgaker  _et al._ (1997). In the
pursuit of methodological robustness, Dyall’s relativistic basis sets Dyall
(2009); Gomes  _et al._ (2010); Dyall (2016); DYA , varying in quality and
employed in an uncontracted manner, are also incorporated within the realm of
test calculations.

###  2.4 The finite-field calculation of DMC

The dipole moment $\displaystyle d$ is computed employing the finite-field
(FF) methodology. This involves the expansion of the energy of the molecule
subjected to a weak and uniform electrostatic field
$\displaystyle{\mbox{\boldmath$\displaystyle F$}}=(F_{x},F_{y},F_{z})$ through
the Taylor series:

$E({\mbox{\boldmath$\displaystyle F$}})=E_{0}-\sum_{i}d_{i}F_{i}+\dots$ (5)

Subsequently, the components of $\displaystyle d$ are defined from this series
(5) as

$d_{i}=-\left.\frac{\partial E({\mbox{\boldmath$\displaystyle F$}})}{\partial
F_{i}}\right|_{{\mbox{\boldmath$\displaystyle F$}}=0},$ (6)

where $\displaystyle i=x,y,z$. The numerical determination of derivatives, as
given in equation (6), is achieved through the energies $\displaystyle
E({\mbox{\boldmath$\displaystyle F$}})$ obtained across multiple instances of
$\displaystyle F$. In principle, the finite-field approach stands as an exact
method. Nevertheless, a judicious selection of the field strength is
imperative to ensure precise numerical differentiation. Given the axial
symmetry characterizing the studied diatomic molecule, solely the components
of $\displaystyle d$, aligned with the molecular axis, exhibit nontrivial
values.

###  2.5 Evaluation of adiabatic correction to PEC

The adiabatic correction to the Born-Oppenheimer (BO) approximation
implemented in the quantum chemical treatment above can be easily computed
using perturbation theory (PT). The diagonal correction to the ground-state
PEC is obtained in 1-st order and takes the form

$U_{\rm ad}=\sum_{I={\rm C},{\rm O}}-\frac{\hbar^{2}}{2M_{I}}\langle\Psi^{\rm
el}_{X}|\Delta_{I}|\Psi^{\rm el}_{X}\rangle,$ (7)

where $\displaystyle M_{I}$ are the nuclear mass of C and O atoms,
$\displaystyle\Psi^{\rm el}_{X}$ is the electronic wave function of the ground
$\displaystyle X$-state depending on the internuclear distance $\displaystyle
R$ as a parameter, $\displaystyle\Delta_{I}$ is the Laplace operator.
Differentiations in Eq. (7) are over nuclear Cartesian coordinates while the
integration is over electronic coordinates.

The mass-dependent function $\displaystyle U_{\rm ad}(R)$, commonly referred
to as the "diagonal Born-Oppenheimer correction" (DBOC), introduces variations
in the effective interatomic PECs for different isotopologues of a molecule
Meshkov  _et al._ (2018); Coxon and Hajigeorgiou (2004a). The adiabatic
correction to the $\displaystyle X$-state PEC, specifically for the primary
isotopologue 12C16O, was assessed using analytic derivative techniques. This
computational approach was implemented in the CFOUR program package Stanton
_et al._ . The calculations were performed at the non-relativistic Hartree-
Fock (HF) and CCSD levels, employing the aug-cc-pCV$\displaystyle N$Z
($\displaystyle N$=3,4) basis sets. Notably, two lowest molecular orbitals
were kept frozen during the CCSD calculations.

###  2.6 Propagation of the intramolecular perturbations into the PEC and DMC

The energy-isolated ground state of CO molecule still undergoes the regular
intramolecular interactions with the higher-lying singlet and triplet states
Lefebvre-Brion and Field (2004). The impact of these very weak perturbations
on energy of the $\displaystyle X$-state can be estimated in the framework of
the $\displaystyle 2$-nd order PT:

$\displaystyle\displaystyle\delta
E_{X}\approx\sum_{j}\sum_{v_{j}}\frac{|V^{\rm
pert}_{v_{X}v_{j}}|^{2}}{E_{v_{X}}-E_{v_{j}}}$ (8)

as the sum over an infinite number of the bound and embedding in continuum
vibronic states. The tedious summation and integration over vibrational
$\displaystyle v_{j}$-states can be avoided due to the approximated
vibrational sum rule Stolyarov and Pupyshev (1994); Pazyuk  _et al._ (1994).
Then, the regular perturbation effect of the remote states manifold on the BO
PEC and DMC can be represented as

$\displaystyle\displaystyle\delta U_{X}\approx\sum_{j}\frac{|V^{\rm
pert}_{Xj}|^{2}}{\Delta U_{Xj}};\quad\delta d_{X}\approx 2\sum_{j}\frac{V^{\rm
pert}_{Xj}d_{Xj}}{\Delta U_{Xj}}$ (9)

where $\displaystyle\Delta U_{Xj}(R)=U^{\rm BO}_{X}-U^{\rm BO}_{j}$ is the
difference of the BO potentials, $\displaystyle d_{Xj}(R)$ is the spin-allowed
electronic transition dipole moment between the ground and excited singlet
states, whereas $\displaystyle V^{\rm pert}_{Xj}(R)$ is the relevant non-
adiabatic electronic matrix element assumed to be a multiplicative function of
$\displaystyle R$. The summation in Eq. (9) should be performed over all
excited states. However, under unique perturber approximation Lefebvre-Brion
and Field (2004) one can select a single electronic state giving the dominant
contribution to the sum.

In the case of homogeneous perturbations, which obey
$\displaystyle\Delta\Lambda=0$ selection rule, the non-adiabatic coupling
matrix elements $\displaystyle V^{\rm pert}_{Xj}$ are mainly determined by the
mass-invariant electrostatic interaction between the states of the same
$\displaystyle j\in~{}^{1}\Sigma^{+}$ symmetry. These matrix elements are
unambiguously related to the non-Hermitian radial coupling matrix elements
$\displaystyle B_{Xj}(R)=\langle\Psi^{\rm el}_{X}|\partial/\partial
R|\Psi^{\rm el}_{j}\rangle$ which can be evaluated using ab initio methods
Werner, HJ and Knowles, PJ and Knizia, G and Manby, FR and Schütz, M and
Celani, P and Korona, T and Lindh, R and Mitrushenkov, A and Rauhut, G and
others (2010). However, their transformation to the multiplicative Hermitian
function $\displaystyle V^{\rm pert}_{Xj}(R)$ is not straightforward.

In the case of heterogeneous perturbations fulfilling the selection rule
$\displaystyle\Delta\Lambda=\pm 1$, the function is

$\displaystyle\displaystyle V^{\rm pert}_{Xj}=-BL_{Xj}\sqrt{2J(J+1)};\quad
B\equiv\left(\frac{\hbar^{2}}{2\mu R^{2}}\right)$ (10)

where $\displaystyle\mu=\frac{M_{\rm C}M_{\rm O}}{M_{\rm C}+M_{\rm O}}$ is the
reduced molecular mass, $\displaystyle J$ is the rotational quantum number,
and $\displaystyle L_{Xj}(R)=\langle\Psi^{\rm el}_{X}|\hat{L}_{x}\pm
i\hat{L}_{y}|\Psi^{\rm el}_{j}\rangle/\sqrt{2}$ is the $\displaystyle
L$-uncoupling matrix element of electronic angular momentum operator
responsible for the so-called "Coriolis" electronic-rotational interaction.

Then, inserting of the operator (10) into the PT relations (9) leads to the
diagonal $\displaystyle J$-dependent correction to the BO PEC:

$\displaystyle\displaystyle\delta U_{X}\approx B\left[1+q\right]J(J+1);\quad
q=2B\left[\sum_{j}\frac{|L_{Xj}|^{2}}{\Delta U_{Xj}}\right]$ (11)

and BO DMC

$\displaystyle\displaystyle\delta
d_{X}\approx-2B\left[\sum_{j}\frac{L_{Xj}d_{Xj}}{\Delta
U_{Xj}}\right]\sqrt{2J(J+1)}$ (12)

where $\displaystyle d_{Xj}(R)=\langle\Psi^{\rm el}_{X}|\hat{d}_{x}\pm
i\hat{d}_{y}|\Psi^{\rm el}_{j}\rangle/\sqrt{2}$ is the electronic transition
dipole moment between the ground and excited $\displaystyle j\in~{}^{1}\Pi$
state.

The summations over the upper states in Eqs. (11) and (12) were restricted in
the present work by the lowest $\displaystyle j\in(1-3)^{1}\Pi$ terms. The
required electronic $\displaystyle L_{Xj}$ and $\displaystyle d_{Xj}$ matrix
elements, along with the relevant BO PECs, were obtained within the framework
of the internally contracted multi-reference configuration interaction (ic-MR-
CI) calculations implemented in the MOLPRO package Werner, HJ and Knowles, PJ
and Knizia, G and Manby, FR and Schütz, M and Celani, P and Korona, T and
Lindh, R and Mitrushenkov, A and Rauhut, G and others (2010). Both state-
averaged CASSCF and ic-MR-CI calculations have been accomplished with the aug-
cc-pCVQZ-DK basis set in the $\displaystyle 6/2/2/0$ active space while two
lowest orbitals were frozen.

##  3 Results and discussions

Evaluations of both BO PEC and DMC for the ground $\displaystyle
X^{1}\Sigma^{+}$ state of carbon monoxide have been performed in the range of
$\displaystyle R$ from $\displaystyle 0.6$ to $\displaystyle 1.6$ Å, which is
restricted to the convergence region of the single-reference CC methods used.
For DMC calculations within the finite-field scheme the optimal value of the
electric field strength $\displaystyle F=\pm 0.0001$ a.u. is found suitable
enough for accurate numerical differentiation.

To undertake a comprehensive exploration of correlation and relativistic
influences, our study encompasses various facets of the calculations and
associated approximations:

  1. 1.

Hamiltonian Comparisons: A comparative analysis is performed between the DC
and X2C as well as GRPP Hamiltonians to discern their respective impacts;

  2. 2.

Relativistic and QED Considerations: Both relativistic and QED contributions
are meticulously examined.

  3. 3.

Correlation Space Size: We scrutinize the dimensionality of the correlation
space, encompassing considerations, such as the total count of explicitly
correlated electrons and the energy threshold utilized for virtual orbitals
within the framework of the CC scheme.

  4. 4.

Basis Set Completeness: The effect of finite one-electron basis set
completeness, inclusive of diffuse orbital basis functions, is assessed.

  5. 5.

Higher-Order Excitations: A dedicated analysis of the influence arising from
triple excitations is conducted, aiming to gauge their relative contributions.

In such a way the results of calculations obtained within the DC Hamiltonian
and X2C approximation are almost identical. That is why using the last one is
fully justified for relativistic study of the molecule under consideration.

The PECs difference obtained for the X2C (with the Gaunt contribution) and
GRPP is presented on Fig. 1. The calculations are performed within the DHF and
CCSD(T) approximations. The difference is about $\displaystyle 2720$ cm-1, but
rather stable, within $\displaystyle 50$ cm-1 for the internulcear distance
range. We employ the GRPP approximation to investigate subtle effects such as
higher-order correlations or QED corrections, where extremely high accuracy is
not required.

Figure 1: The disparities in the PECs computed for the X2C Hamiltonian
(including the Gaunt contribution) and the GRPP Hamiltonian. These
calculations have been systematically conducted within both the DHF and
CCSD(T) approximations.

As mentioned earlier, the investigation makes use of the standard basis sets
from the cc-pV$\displaystyle N$Z family ($\displaystyle N=4,5,6$) Dunning
(1989); Kendall  _et al._ (1992) on both atomic nuclei. Additionally, we
employ Dyall’s basis sets DYA , which, after undergoing the extrapolation
procedure, yield analogous outcomes. If not stated otherwise, all the
$\displaystyle 14$ electrons are considered as the correlated ones, while the
virtual states with the energies larger than $\displaystyle 300$ a.u. (which
is more than enough) are excluded. The exploration of supplementary diffuse
functions, which holds particular significance, was also meticulously
undertaken, primarily due to their pronounced importance in the accurate
evaluation of the dipole moment. In order to gauge the impact, we introduced
an augmentation to the existing basis sets by inclusively appending additional
diffuse (low exponent) basis functions in a balanced manner. Basis sets of
this nature are designated by the prefix $\displaystyle n$-aug, signifying the
inclusion of $\displaystyle n$ supplementary diffuse basis functions within
each angular symmetry block. This augmentation strategy serves to
systematically assess the implications of such additional functions on the
computed outcomes.

A demonstrative illustration of the basis convergence analysis conducted under
the CCSD(T) approximation is depicted in Figure 2. The data set is procured
from the aug-cc-pV$\displaystyle N$Z basis set series, wherein the values are
rendered relative to the CBS outcome. The measure of convergence error bars,
amounting to approximately $\displaystyle 2000$ cm-1, is derived from the
disparity between the outcomes of the most extensive basis set and the CBS
values, specifically for the ground state energy. Given that the focus
pertains to the behavior of the PEC under scrutiny, while the absolute
magnitudes remain non-crucial, the uncertainty associated with the PEC is
gauged by the divergence in behavior between the PECs corresponding to the
most expansive basis set and the CBS values, approximating to around
$\displaystyle 300$ cm-1. As anticipated, the uncertainty pertinent to the
relative PEC manifests an improvement by nearly an order of magnitude in
comparison to that associated with the absolute values. It’s worth noting that
convergence is comparatively less favorable for smaller $\displaystyle R$,
possibly due to the inherent unoptimization of standard basis sets for cases
involving closely spaced atomic nuclei. For purposes of comparison, the figure
includes the CBS value derived from the Dyall basis set family, which exhibits
reasonable agreement with the aug-cc-pV$\displaystyle N$Z CBS result.

Figure 2: Basis set convergence for the ground state PEC. The results are
presented concerning the aug-cc-pV$\displaystyle N$Z basis set family relative
to the CBS values within the CCSD(T) approximation: $\displaystyle\Delta
E=E(\text{aug-cc-pV}N\text{Z})-E(\text{CBS})$. Additionally, for comparative
purposes, the CBS values obtained for the Dyall basis sets are also displayed.

Sensitivity analysis of the ground state PEC concerning the inclusion of
diffuse basis functions is exemplified in Figure 3. These investigations
encompass calculations conducted within both the DHF and CCSD(T)
approximations. The incremental addition of multiple augmentation functions
yields results that exhibit minimal deviation. It is worth highlighting that
the cc-pV$\displaystyle 6$Z basis set, which represents the most extensive
among those considered, demonstrates a commendable capability in
characterizing correlation effects on its own merit. Nonetheless, it is
crucial to underscore the continued relevance of diffuse functions,
particularly in enhancing the accuracy of the DHF contribution.

Figure 3: Contribution of diffuse orbital basis functions to the PEC of the
ground state. The results are presented within the cc-pV$\displaystyle N$Z
basis set family, considering both the DHF and CCSD(T) approximations. The
"Correlation" notation denotes the
$\displaystyle(E_{\text{CCSD(T)}}-E_{\text{DHF}})$ contribution.

In the context of this investigation, we have undertaken the first analysis of
the nonperturbative contributions stemming from the triple cluster amplitudes
in the case of carbon monoxide, to the best of our knowledge. The outcomes of
these calculations, relative to those within the framework of CCSD(T), are
shown in Figure 4. These data are obtained within the GRPP approximation,
considering explicit correlation effects among $\displaystyle 10$ electrons
and excluding the virtual states with the energies larger than $\displaystyle
30$ a.u. Our findings reveal that the residual T cluster amplitude
contribution $\displaystyle\Delta
E^{\text{T-(T)}}=E_{\text{CCSDT}}-E_{\text{CCSD(T)}}$ is nearly negligible at
small $\displaystyle R$ but steadily escalates, ultimately reaching a
magnitude of approximately $\displaystyle 250$ cm-1 at $\displaystyle R$=1.6
Å.

Figure 4:  Basis set convergence of the perturbative triple
$\displaystyle\Delta E^{\text{(T)}}=E_{\text{CCSD(T)}}-E_{\text{CCSD}}$ and
residual triple $\displaystyle\Delta
E^{\text{T-(T)}}=E_{\text{CCSDT}}-E_{\text{CCSD(T)}}$ cluster amplitudes
contributions to the PEC of the ground state. The $\displaystyle\Delta
E^{\text{(T)}}$ results are presented relative to the CBS values. These data
are obtained under the GRPP approximation and consider explicit correlation of
$\displaystyle 10$ electrons.

To gain a more comprehensive perspective on the relativistic corrections, we
have divided them into two distinct components: the contributions arising from
the DC Hamiltonian and the Gaunt inter-electronic term. The results of these
calculations, conducted within the framework of the CCSD(T) approximation, are
thoughtfully illustrated in Figure 5. It’s essential to note that these data
have been shifted to a region of zero reference point at the dissociation
limit, facilitating a clearer presentation of their relative behaviors and
trends.

Figure 5:  Relativistic corrections to the PEC of the ground state. The dashed
and dotted lines delineate the contributions arising from the DC Hamiltonian
and Gaunt corrections, respectively. The solid line represents the cumulative
effect. These data are acquired using the CCSD(T) approximation. The results
obtained in Ref. Konovalova  _et al._ (2018) are presented with a shift to
align them with the DC curve for the purpose of a clear comparison.

As anticipated, both contributions exhibit an increasing in magnitude at
smaller internuclear distances, and partly compensate each other. An
intriguing observation is the presence of extremal points within each of these
contributions. Specifically, the contribution from the DC Hamiltonian attains
its extreme value near the equilibrium internuclear distance, whereas the
Gaunt term reaches its extremum at more substantial internuclear separations.
The results obtained in Ref. Konovalova  _et al._ (2018) are also presented
here. It is worth noting a good agreement with our relativistic DC data, which
does not include the Gaunt term. To facilitate a clear comparison, the curve
has been shifted accordingly.

The dependency of the QED correction on the ground state of carbon monoxide
concerning internuclear distance is visually depicted in Figure 6. These
calculations have been meticulously conducted within both the DHF and CCSD
approximations. A discernible feature is the presence of a minimum within the
QED correction, positioned in the vicinity of the equilibrium internuclear
distance. This minimum manifests itself at approximately $\displaystyle 5$
cm-1 concerning its relative position with respect to the dissociation limit.
The incorporation of correlation effects engenders a slight (almost
negligible) shift of this minimum towards smaller values of $\displaystyle R$.
An approximate Lamb shift function, semi-empirically estimated Konovalova  _et
al._ (2018) by scaling the one-electron Darwin correction Pyykkö  _et al._
(2001), is included for comparison. Additionally, the QED correction, recently
obtained using the model-QED-operator approach, and configuration interaction
method based on Dirac-Sturm orbitals Dulaev (2023), is presented. While the
latter approach is more closely related to our method, it’s worth noting that
our results show a significantly better agreement with the findings from Ref.
Konovalova  _et al._ (2018).

Figure 6:  QED corrections to the PEC of the ground state. The calculations
are performed in the DHF and CCSD approximations. The data obtained by
Konovalova et al. Konovalova  _et al._ (2018) and Dulaev et al. Dulaev (2023)
are also presented.

Finally, in Figure 7, we present a comparative analysis of our computed
results with a semi-empirical potential Meshkov  _et al._ (2018). The PECs
correspond to various levels of considering correlation effects, including
DHF, CCSD, CCSD(T), CCSDT, while accounting for all other relevant
corrections. To facilitate a meaningful comparison, these PECs and the semi-
empirical potential are aligned to zero at their respective minimum points.
Within the vicinity of the minimum, a notable agreement between the approaches
is observed. However, as the internuclear distance increases, distinctions
from the CCSDT results become increasingly pronounced, reaching approximately
$\displaystyle 17000$ and $\displaystyle 6000$ cm-1 concerning deviations from
the DHF and CCSD outcomes, respectively.

Figure 7:  The PEC of the ground state obtained at various levels of
accounting for correlation effects: DHF, CCSD, CCSD(T), CCSDT, while including
all other relevant corrections. The semi-empirical potential curve is
extracted from Ref. Meshkov  _et al._ (2018). The minima of the potential
curves are normalized to zero.

Further insight into the deviations of the CCSD(T) and CCSDT curves from the
semi-empirical potential for the ground state is presented in Figure 8. To
ensure a meaningful comparison, these curves are synchronized at the
equilibrium points $\displaystyle R_{\rm e}\approx 1.128$ Å, with the
inclusion of relativistic and QED corrections. It is important to note that
the residual triple amplitude contributions primarily come into play at larger
$\displaystyle R$. The uncertainty associated with the PEC is predominantly
dictated by the basis set convergence in the CCSD(T) calculations and is
estimated to be approximately $\displaystyle 300$ cm-1. For purposes of
comparison, the figure includes the value recently derived by Meshkov et al.
within the averaged coupled pair functional (ACPF) approach Meshkov  _et al._
(2023).

Figure 8:  Deviation of the CCSD(T) and CCSDT PECs from the semi-empirical
potential Meshkov  _et al._ (2018) for the ground state. The results are
showcased for calculations conducted using the aug-cc-PV$\displaystyle 6$Z
basis set and CBS calculations. The relativistic and QED corrections are taken
into account. Additionally, for comparative purposes, the data obtained by
Mehskov et al. Ref. Meshkov  _et al._ (2023) obtained within the MR-ACPF
approach are also displayed.

Furthermore, our investigation was extended to an examination of the bond
distance’s dependence on various facets of the calculations and the associated
approximations. In pursuit of precision, additional evaluations in proximity
to the bond distance were conducted. The outcome of this scrutiny regarding
basis set convergence within the cc-pV$\displaystyle N$Z basis set series is
presented in Table 1. Notably, the accuracy of the CBS value is estimated to
be within the order of $\displaystyle 1$ mÅ. Furthermore, it is observed that
the influence of diffuse functions on the optimal geometry is minimal.
Interestingly, the residual triple and Gaunt contributions exhibit a
compensatory effect, collectively effecting a correction of less than
$\displaystyle 0.1$ mÅ to the final value.

Table 1: Basis set convergence for the equilibrium bond distance $\displaystyle R_{\rm e}$ (in Å) of the ground CO state. The results are presented for the cc-pV$\displaystyle N$Z basis set family within the CCSD, CCSD(T) and CCSDT approximations. Additionally, the impact of the Gaunt term correction noted as "+G" is also demonstrated. The semi-empirical $\displaystyle R_{\rm e}$=1.128217 (Å) value is adopted from Ref. Coxon and Hajigeorgiou (2004b). $\displaystyle N$ | CCSD | CCSD(T) | CCSD(T)+G | CCSDT+G
---|---|---|---|---
4 | 1.1225 | 1.1297 | 1.1299 | 1.1297
5 | 1.1213 | 1.1285 | 1.1287 | 1.1285
6 | 1.1208 | 1.1280 | 1.1282 | 1.1280
CBS | 1.1202 | 1.1274 | 1.1276 | 1.1274

The DBOC calculated for the ground state of the most abundant isotopologue
12C16O under the HF and CCSD approximations are compared on Figure 9 with the
relevant adiabatic correction derived empirically in Ref. Meshkov  _et al._
(2018). The original ab initio functions were vertically shifted to match
their pure empirical counterpart at the point of equilibrium distance
$\displaystyle R_{\rm e}$. The difference of the HF and CCSD curves clearly
observed at $\displaystyle R>1.2$ (Å) highlights the rapid increase of the
electronic correlation effect as $\displaystyle R$ increases.

Figure 9:  Comparison of the mass-dependant DBOC functions for the ground-
state of 12C16O molecule: the HF and CCSD methods versus empirical data from
Ref. Meshkov  _et al._ (2018).

Moreover, as can be seen on Figure 10a the 2-nd order PT correction to a
rotational part of the effective mass-dependant potential for the ground-state
of the 12C16O isotopologue, which was estimated using the ic-MR-CI
calculations, agrees very well with its empirical counterpart from Ref.
Meshkov  _et al._ (2018).

Figure 10:  (a) Comparison of the dimensionless mass-dependent $\displaystyle
q$-correction, evaluated according to Eq. (11), to rotational energy of the
12C16O ground-state with its empirical counterpart Meshkov  _et al._ (2018).
(b) The $\displaystyle J$-dependent contribution of the non-adiabatic effect
into the permanent dipole function for the CO ground-state.

A parallel analysis has been conducted for the DMC of the ground state of
carbon monoxide. The investigation centered on basis set convergence,
utilizing the aug-cc-pV$\displaystyle N$Z basis set series, is graphically
depicted in Figure 11.

Figure 11:  Basis set convergence for the DMC of the ground state. The results
are showcased concerning the aug-cc-pV$\displaystyle N$Z basis set family
relative to the CBS values within the CCSD(T) approximation:
$\displaystyle\Delta\mu=\mu(\text{aug-cc-pV}N\text{Z})-\mu(\text{CBS})$. For
comparative purposes, the CBS values obtained for the Dyall basis sets are
also presented.

Through an overarching assessment and a comparative evaluation with results
obtained using the Dyall basis sets, we estimate the associated uncertainty to
be within the range of $\displaystyle 3$ mD. Similar to the ground energy
scenario, it is noteworthy that the inclusion of just one diffuse orbital
basis function per symmetry proves to be adequate for attaining DMC accuracy
at the level of approximately $\displaystyle 1$ mD. The individual
contributions pertaining to these aspects are elucidated in Figure 12.

Figure 12:  Contribution of diffuse orbital basis functions to the ground-
state DMC. The results are illustrated for the cc-pV$\displaystyle 5$Z basis
set family within both DHF and CCSD(T) approximations.

Delving into the relativistic correction to the DMC, as delineated in Figure
13, we observe that this contribution amounts to a few mD, while the Gaunt
contribution registers an order of magnitude smaller. These calculations have
been conducted within the DHF, CCSD, and CCSD(T) approximations. The CCSD(T)
ones without the Gaunt contribution are in a reasonable agreement with the
Ref. Konovalova  _et al._ (2018).

Figure 13:  Relativistic correction to the ground-state DMC. The calculations
are executed within the DHF, CCSD and CCSD(T) approximations. Additionally,
for comparative purposes, the data obtained by Konovalova et al. Ref.
Konovalova  _et al._ (2018) are also displayed.

The rotational part of non-adiabatic correction to the DMC function, as
predicted by Eq. (12) for the 12C16O isotopologue, is illustrated in Figure
10b for different $\displaystyle J$-values. The absolute magnitude of this
mass-dependent correction is comparable to the relativistic correction to the
DMC (see Figure 13). The ab initio ic-MR-CI calculations revival that the
dominant perturbation is caused by the lowest excited $\displaystyle A^{1}\Pi$
state.

Finally, in Figure 14, we present a comparative analysis of our computed DMCs
with a semi-empirical curve Meshkov  _et al._ (2022).

Figure 14:  The ground-state DMC obtained at various levels of accounting for
correlation effects: DHF, CCSD, CCSD(T), CCSDT, while incorporating all other
relevant corrections. The semi-empirical curve is extracted from Ref. Meshkov
_et al._ (2022).

Within the realm of smaller internuclear distances, rather good agreement
among the different approaches is observed. However, as internuclear distances
increase, the influence of correlation effects becomes notably substantial. To
provide a detailed perspective, we have illustrated the deviations of the
CCSD, CCSD(T), and CCSDT DMCs from the semi-empirical curve in Figure 15.

Figure 15:  Deviation of the DMC obtained using the CCSD, CCSD(T), and CCSDT
methods from the semi-empirical curve Meshkov  _et al._ (2022) for the ground
state. The results of MR-ACPF and CCSD(T) calculations by Meshkov et al.
Meshkov  _et al._ (2022), MR-CISD(+Q)-FF by Balashov et al. Balashov  _et al._
(2023) and MR-ACPF by Chen et al. Chen  _et al._ (2015) are also presented.

The accuracy of the evaluated DMCs falls within the range of $\displaystyle 5$
mD and is primarily determined by basis set convergence. Nevertheless, it is
pertinent to note that the dipole moment contributions stemming from the
triple cluster amplitudes, both perturbative and residual, exhibit noteworthy
magnitudes of up to $\displaystyle 0.5$ and $\displaystyle 0.1$ debye,
respectively, particularly at larger internuclear distances. It is plausible
that the deviation of CCSDT results from the semi-empirical values is
attributed to the contributions from higher-order excitation. The results of
MR-ACPF calculations by Chen et al. Chen  _et al._ (2015), MR-ACPF and CCSD(T)
by Meshkov et al. Meshkov  _et al._ (2022) and multi-reference CI (MR-
CI(+Q)-FF) by Balashov et al. Balashov  _et al._ (2023), are also presented
for comparison.

##  4 Summary and Conclusions

The ground-state potential energy and dipole moment profiles of carbon
monoxide have been rigorously examined utilizing ab initio methods within the
framework of the relativistic coupled-cluster approach. This study
incorporates non-perturbative single, double, and triple cluster amplitudes,
complemented by a finite-field methodology. The nonperturbative triple-
amplitude corrections were evaluated for the first time. The calculations were
conducted using the implementation described in the references Oleynichenko
_et al._ (2020b, a, ). The generalized relativistic pseudo-potential model was
employed to effectively introduce relativistic effects into the all-electron
correlation treatment and to account for quantum-electrodynamics (QED)
corrections through the model-QED-operator approach.

We conducted a detailed investigation into the sensitivity of the results to
the parameters of the basis set and approximations employed. Our findings are
in satisfactory agreement with the most precise available semi-empirical data.

For further enhancement of calculation accuracy and the potential extension of
the range of considered internuclear distances, particularly towards larger
distances, it appears necessary to employ more computationally demanding
multiconfigurational variants of the coupled-cluster method (and/or
configuration interaction). These approaches would adequately account for the
significant increase in the multiconfigurational character of the electronic
wave function of the CO ground state at intermediate and, especially, large
internulcear distances.

### Acknowledgments

We are grateful to L.V. Skripnikov and A.V. Oleynichenko for helpful
discussions. The study was supported by the Russian Science Foundation
(interdisciplinary grant No. 22-62-00004).

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# Entropic uncertainty relations in Schwarzschild space-time

Tian-Yu Wang  School of Physics & Optoelectronic Engineering, Anhui
University, Hefei 230601, People’s Republic of China  Dong Wang
<EMAIL_ADDRESS>School of Physics & Optoelectronic Engineering, Anhui
University, Hefei 230601, People’s Republic of China

###### Abstract

The uncertainty principle is deemed as one of cornerstones in quantum
mechanics, and exploring its lower limit of uncertainty will be helpful to
understand the principle’s nature. In this study, we propose a generalized
entropic uncertainty relation for arbitrary multiple-observable in
multipartite system, and further derive a tighter lower bound by considering
Holevo quality and mutual information. Importantly, we detailedly discuss the
proposed uncertainty relations and quantum coherence in the context of
Schwarzschild space-time. It is interesting to find that Hawking radiation
will damage the coherence of the physically accessible region and increase the
uncertainty. Furthermore, we argue that the properties of the uncertainty in
Schwarzschild space-time can be explained from the systems’ purity and the
information redistribution of the different regions. Therefore, it is believed
that our findings provide the generalized entropic uncertainty relations in
multipartite systems, which may facilitate us deeper understanding of
quantumness and information paradox of the black holes.

##  I Introduction

The uncertainty relation is at the heart of quantum mechanics, which
demonstrates that one is unable to obtain the exact values of two non-
commutative observables at the same time, other than those in classical
counterpart. The concept of uncertainty relation was first proposed by
Heisenberg WH , and then Kennard EHK  strictly proved that the standard
deviation can be used to depict the uncertainty relation between momentum and
position:

$\begin{split}\Delta x\Delta p\geq\frac{\hbar}{2},\end{split}$ (1)

where $\Delta x$ ($\Delta p$) represents the standard deviation of position
(momentum). Based on this theory, Robertson HPR  generalizes it to the
relation for arbitrary two non-commutating observables.

Subsequently, some scholars have found that entropy is more suitable for
describing the uncertainty relation, because it can get rid of some
restriction from the standard-deviation-based uncertainty relation. Everett HE
and Hirschman IIH  originally brought information entropy into the uncertainty
principle in 1957. Later on, Deutsch DD  proposed the celebrated entropic
uncertainty relation (EUR) via Shannon entropy, i.e.,

$\begin{split}H(\hat{R})+H(\hat{Q})\geq\log_{2}\left(\frac{2}{1+\sqrt{c\left(\hat{R},\hat{Q}\right)}}\right)^{2},\end{split}$
(2)

where $H(\hat{X})=-\sum_{i}p_{i}\log_{2}p_{i}$ stands for Shannon entropy with
$p_{i}$ representing the probability of the $i$-th outcome.
$c\left(\hat{R},\hat{Q}\right)=\underset{i,j}{\max}\left|\left\langle{\psi_{i}^{\hat{R}}}\middle|{\psi_{j}^{\hat{Q}}}\right\rangle\right|^{2}$
is the maximum overlap between $\hat{R}$ and $\hat{Q}$ with
$\langle{\psi_{i}^{\hat{R}}}|$ ($|{\psi_{j}^{\hat{Q}}}\rangle$) represents the
eigenvector of observable $\hat{R}$ ($\hat{Q}$). Based on this, Kraus KK  and
Maassen and Uffink HM  proposed a more simplified EUR:

$\begin{split}H(\hat{R})+H(\hat{Q})\geq-\log_{2}c\left(\hat{R},\hat{Q}\right)=:q_{MU}.\end{split}$
(3)

Furthermore, some researchers have shown that the uncertainty can be reduced
when the measured subsystem is correlated with another JMR ; MB . To be
specific, the novel EURs with quantum memories was presented by Renes and
Boileau JMR , and Berta _et al._ MB :

$\begin{split}S(\hat{R}|B)+S(\hat{Q}|B)\geqslant q_{MU}+S(A|B),\end{split}$
(4)

and

$\begin{split}S(\hat{R}|B)+S(\hat{Q}|C)\geqslant q_{MU},\end{split}$ (5)

where $S(\hat{R}|B)=S(\rho_{\hat{R}B})-S(\rho_{B})$ is the conditional von
Neumann entropy of the system after $\hat{R}$ measurement of particle $A$, and
the state of the system after measurement can be expressed as
${{\rho}_{\hat{R}B}}=\sum_{i}{\left({{{\left|{\psi_{i}}\right\rangle}^{\hat{R}}}\left\langle{\psi_{i}}\right|\otimes{\mathds{I}_{B}}}\right){{\rho}_{AB}}\left({{{\left|{\psi_{i}}\right\rangle}^{\hat{R}}}\left\langle{\psi_{i}}\right|\otimes{\mathds{I}_{B}}}\right)}$.
Recently, various improvement for EURs has been put forward MAB ; SW ; AKP ;
MLH ; MNB ; TPP ; LM ; PJC ; LR1 ; SZ ; LR2 ; JZ ; SL ; YLX ; FA ; mf11 ; xbf
; xbf1 ; mf ; LLD ; QHZ ; TianYu . For examples, Pati _et al._ AKP  considered
the optimized EUR based on quantum discord, and Adabi _et al._ FA
strengthened the lower bound of EUR by considering Holevo quantities.
Furthermore, some promising experiments have been demonstrate the proposed
entropic uncertainty relations via all-optics and solid-state platforms Rp ;
Cfl ; ZY ; wcm11 ; zxc ; wml ; hyw11 ; zxc11 ; wcxy1 ; Yong .

Quantum systems exhibit many interesting traits within the framework of
relativity, and had attracted much attention. The EURs of curved space-time
have also been investigated by some authors JF ; JL ; DWW ; FMDW . Actually,
there are various curved space-time, including Schwarzschild space-time,
Reissner-Nordstr$\mathrm{\ddot{o}}$m space-time, Garfinkle-Horowitz-Strominger
space-time GWG ; GKM , etc. It is well known that, the Schwarzschild black
hole is regarded as one of basic and important black hole models. It assumes
that the black hole is a spherically symmetric celestial body without rotation
and charge. In order to better understand quantum properties of the black
hole, examining EUR and quantumness in Schwarzschild space-time is required.
To illustrate more clearly, we describe the EUR in the context of black holes
using the uncertainty game. Assuming that the system is prepared in the state
of $\rho_{ABC}$, and particles $A$, $B$, and $C$ are sent to Alice, Bob, and
Charlie, respectively. While particles $B$ and $C$ plunges into the black hole
and hovers closely to its event horizon, and particle $A$ is situated in a
flat space-time. After measuring her particles with $Q$ or $R$, Alice notifies
Bob and Charlie of her measurement choice. The aim is to estimate Alice’s
measurement results as precisely as they can. More generally, multi-observable
EURs in the many-body systems have been examined. For examples, some authors
LLD ; QHZ ; TianYu  have novelly derived EURs with $n$ measurements in
$n+1$-party systems. Herein, we derived a new and universal EUR with arbitrary
measurement, and specifically observed the properties of quantum measured
uncertainty and coherence in the background of Schwarzschild space-time.

The Letter is organized as follows: in Sec. II, we firstly present a new and
universal EUR for $m-$measurement within a $(n+1)-$party framework. In Sec.
III, the vacuum structure of the Schwarzschild black hole is reviewed briefly.
In Sec. IV, we investigate the intrinsic relationship among the entropic
uncertainty, quantum coherence and mutual information in Schwarzschild black
holes. Finally, we end up with a concise summary.

##  II ENTROPIC UNCERTAINTY RELATION FOR ARBITRARY MULTIPARTITE AND MULTI-
OBSERVABLE

In this section, we first derive a new and universal EUR for arbitrary
multiple measurements in multipartite systems, and then we present a stronger
EUR based on considering the Holevo quantity.

Theorem 1. By considering all possible measurements in arbitrary multipartite
system, we can derive a simply constructed entropic uncertainty relation in
the framework of multi-measurements:

$\begin{split}\sum_{j=1}^{n}\sum_{i=1}^{m}S(\hat{M}_{i}|B_{j})\geq&-\frac{n}{m-1}\sum_{i\neq
k,i=1}^{m}\log_{2}c_{ik}\\\ &+\frac{m}{2}\sum_{j=1}^{n}S(A|B_{j}),\end{split}$
(6)

where $m,n\in N^{+}$, $m$ represents the number of observable, while $n$
denotes the number of quantum memories.

Proof. Suppose there are $m-$measurement in $n-$party systems. Herein we can
list the conditional entropies for all possible measurements as below

$\begin{split}S(\hat{M}_{1}|B_{1})\ \ S(\hat{M}_{2}|B_{1})\ \
&S(\hat{M}_{3}|B_{1})\ ...\ S(\hat{M}_{m}|B_{1})\\\ S(\hat{M}_{1}|B_{2})\ \
S(\hat{M}_{2}|B_{2})\ \ &S(\hat{M}_{3}|B_{2})\ ...\ S(\hat{M}_{m}|B_{2})\\\
S(\hat{M}_{1}|B_{3})\ \ S(\hat{M}_{2}|B_{3})\ \ &S(\hat{M}_{3}|B_{3})\ ...\
S(\hat{M}_{m}|B_{3})\\\ &\vdots\\\ S(\hat{M}_{1}|B_{n})\ \
S(\hat{M}_{2}|B_{n})\ \ &S(\hat{M}_{3}|B_{n})\ ...\
S(\hat{M}_{m}|B_{n})\end{split}$ (7)

where, $\hat{M}_{i}\ (i=1,\cdots,m)$ is denoted as the observable to be
measured, and $B_{j}\ (j=1,\cdots,n)$ represents quantum memory.

Firstly, we consider all the possible measurement pairs
$(\hat{M}_{i},\hat{M}_{j})$ and particle $B_{1}$ as quantum memory in first
row in Eq. (7), and resort to the inequality in Eq. (4), we can obtain

$\begin{split}\sum_{i=1}^{m}S(\hat{M}_{i}|B_{1})\geq-\frac{1}{m-1}\sum_{i\neq
k,i=1}^{m}\log_{2}c_{ik}+\frac{m}{2}S(A|B_{1}).\end{split}$ (8)

In the same vein, we can also obtain the EUR outcomes of the other rows. By
adding up all the inequalities, we can achieve the construction of the desired
Eq. (6).

Theorem 2. By considering the mutual information and Holevo quantity and
combining the previous result (6), a new and tighter entropic uncertainty
relation within multiple measurements can be obtained as:

$\begin{split}\sum_{j=1}^{n}\sum_{i=1}^{m}S(\hat{M}_{i}|B_{j})\geq&-\frac{n}{m-1}\sum_{i\neq
k,i=1}^{m}\log_{2}c_{ik}\\\
&+\frac{m}{2}\sum_{j=1}^{n}S(A|B_{j})+\max\\{0,\Delta\\}\end{split}$ (9)

with
$\Delta=\frac{m}{2}\sum_{j=1}^{n}\mathcal{I}(A:B_{j})-\sum_{j=1}^{n}\sum_{i=1}^{m}\mathcal{H}(\hat{M}_{i}:B_{j}).$

Proof. In the absence of quantum storage $B$, the inequality in Eq. (6) can be
expressed as:

$\begin{split}\sum_{i=1}^{m}S(\hat{M}_{i})\geq-\frac{1}{m-1}\sum_{i\neq
k,i=1}^{m}\log_{2}c_{ik}+\frac{m}{2}S(A).\end{split}$ (10)

Linking the relation between conditional entropy and Holevo quantity
$S(\hat{M})=S(\hat{M}|B)+\mathcal{H}(\hat{M}:B)$ ($\mathcal{H}$ denotes the
Holevo quantity), the element $\sum_{i=1}^{m}S(\hat{M}_{i})$ of Eq. (10)
consequently can be written as:

$\begin{split}\sum_{i=1}^{m}S(\hat{M}_{i})=\frac{1}{n}\left[\sum_{j=1}^{n}\sum_{i=1}^{m}S(\hat{M}_{i}|B_{j})+\sum_{j=1}^{n}\sum_{i=1}^{m}\mathcal{H}(\hat{M}_{i}:B_{j})\right].\end{split}$
(11)

By substituting the above equation into the Eq. (10), we can obtain the
following inequality

$\begin{split}\sum_{j=1}^{n}\sum_{i=1}^{m}S(\hat{M}_{i}|B_{j})\geq-\frac{n}{m-1}\sum_{i\neq
k,i=1}^{m}\log_{2}c_{ik}\\\
+\frac{mn}{2}S(A)-\sum_{j=1}^{n}\sum_{i=1}^{m}\mathcal{H}(\hat{M}_{i}:B_{j}).\end{split}$
(12)

Because of the mutual information can written as the form of
$\mathcal{I}(A:B)=S(A)-S(A|B)$. Substituting $S(A)=\mathcal{I}(A:B)+S(A|B)$
into the above formula (12). As a result, our new entropic uncertainty
relation (i.e., Eq. (9)) can be obtained by comparing with Eq. (6) and taking
the optimization over the derived results. It is obvious that the lower bound
of Eq. (9) is tighter than that of Eq. (6), verifying that the latter EUR
outperforms the previous one.

##  III Vacuum structure of Dirac particles in the context of Schwarzschild
black holes;

We first go over the vacuum states of the Schwarzschild black hole. Generally,
one Schwarzschild black hole can be described by

$\begin{split}{ds^{2}=}&{\left(1-\frac{2M}{r}\right)^{-1}dr^{2}-\left(1-\frac{2M}{r}\right)dt^{2}}\\\
&{+r^{2}(d\theta^{2}+\sin^{2}\theta d\phi^{2}),}\end{split}$ (13)

where $M$ is the mass of the black hole. Additionally, the Hawking temperature
can be expressed as $T=\frac{1}{8\pi M}$ RKRB . For the sake of
simplification, we set the gravitational constant $G$, the reduced Planck
constant $\hbar$, the speed of light $c$, and the Boltzmann constant $k_{B}$
to unit $1$. In this case, Dirac’s equation can be written as:

$\begin{split}\left[\gamma^{a}e_{a}^{\mu}\left(\partial_{\mu}+\Gamma_{\mu}\right)\right]\psi=0,\end{split}$
(14)

where $\gamma^{a}$ stands for Dirac matrices and $\Gamma_{\mu}$ is the spin
connection coefficient. A set of solutions for positive frequency waves can be
obtained by solving the above equation:

$\displaystyle\psi_{k}^{I+}=\zeta e^{-i\omega\mu},$ (15)
$\displaystyle\psi_{k}^{II+}=\zeta e^{i\omega\mu}.$ (16)

They respectively represent the solutions with mode $k$ in the regions of
outside and inside the event horizon, and $\zeta$ is a four-component Dirac
spinor. $\mu=t-r_{*}$ stands for retarded time and $\nu=t+r_{*}$ stands for
advanced time, where $r_{*}=r+2M\ln[\frac{r-2M}{2M}]$ represents the tortoise
coordinate. By taking the above two positive-frequency wave solutions as a set
of completely orthogonal basis, and expanding the Dirac field, one can obtain

$\begin{split}\psi_{out}=\sum\limits_{\chi=I,II}\int
dk(\alpha_{k}^{\chi}\psi_{k}^{\chi+})(\beta_{k}^{\chi*}\psi_{k}^{\chi-}),\end{split}$
(17)

where $\alpha$ and $\beta$ are the fermion’s annihilation operator and
antifermion’s creation operator respectively. According to the relation
between Kruskal coordinates and black hole coordinates, a new set of
orthogonal basis can be obtained:

$\displaystyle{\phi_{k}^{I+}=e^{2\pi M\omega_{k}}\psi_{k}^{I+}+e^{-2\pi
M\omega_{k}}\psi_{-k}^{II-}}$ (18) $\displaystyle{\phi_{k}^{II+}=e^{-2\pi
M\omega_{k}}\psi_{-k}^{I-}+e^{2\pi M\omega_{k}}\psi_{k}^{II+}.}$ (19)

In this way, the new representation of the Dirac field in Kruskal space-time
is

$\begin{split}&\psi_{out}=\\\ &{\sum\limits_{\chi=I,II}\int
dk\frac{1}{\sqrt{2\cosh(4\pi
M\omega_{k})}}(a_{k}^{\chi}\phi_{k}^{\chi+}+b_{k}^{\chi*}\phi_{k}^{\chi-}),}\end{split}$
(20)

where $a_{k}^{\chi}$ and $b_{k}^{\chi*}$ are the fermion annihilation and
antifermion creation operators acting on the Kruskal vacuum respectively. By
Bogoliubov transform, based on Eqs. (17) and (20), we can get the annihilation
operator in the following form

$\begin{split}a_{k}^{I}=u\alpha_{k}^{I}-v\beta_{k}^{II*},\end{split}$ (21)

where, $u=(e^{-8\pi M\omega_{k}}+1)^{-\frac{1}{2}}$ and $v=(e^{8\pi
M\omega_{k}}+1)^{-\frac{1}{2}}$.

Figure 1: The Penrose diagram of the Schwarzschild space-time indicating Bob
and Anti-Bob trajectories. $i^{0}$ denotes the spatial infinities, $i^{+}$ and
$i^{-}$ represents timelike future and past infinity. $J^{+}$ and $J^{-}$
shows lightlike future and past infinity, respectively. $r=2M$ denotes the
event horizons and $r=0$ represents the singularity of the black hole.

The Penrose diagram of Schwarzschild space-time has been sketched as Fig. 1.
Since the physically accessible region $I$ and the physically inaccessible
region $II$ are causally disconnected, after some calculations for the
Bogoliubov transformation, the vacuum state and the only excited state of the
$k$-mode Kruskal particle can be expressed as

$\begin{split}\left|0\right\rangle_{k}^{+}=u\left|0\right\rangle_{I}^{+}\left|0\right\rangle_{II}^{-}+v\left|1\right\rangle_{I}^{+}\left|1\right\rangle_{II}^{-}\end{split}$
(22)

and

$\begin{split}\left|1\right\rangle_{k}^{+}=\left|1\right\rangle_{I}^{+}\left|0\right\rangle_{II}^{-}.\end{split}$
(23)

The superscripts “$+$” and “$-$” in the expressions of these two states stand
for particles and antiparticles, respectively, and the subscripts $I$ and $II$
stand for the outside and inside of the horizon.

Figure 2: Entropic uncertainty relations for two-measurement
($\hat{\sigma_{x}}$, $\hat{\sigma_{z}}$) under the W-like initial state
$\rho^{\rm W}_{AB_{I}C_{I}}$ in Schwarzschild black hole. The red circle
represents the entropic uncertainty, the green diamond represents
$\mathrm{Bound_{1}}$ in Eq. (24) and the blue triangle represents
$\mathrm{Bound_{2}}$ in Eq. (25). Graph (a) shows uncertainty and the lower
bounds vs the state’s parameter $x$ with the Hawking temperature $T=10$ and
state’s parameter $y=\frac{\pi}{4}$, and Graph (b) shows uncertainty and the
lower bounds versus $T$ with $x=\frac{\pi}{3}$ and $y=\frac{\pi}{4}$.

##  IV Entropic uncertainty relation, coherence and mutual information in
Schwarzschild black hole

In order to examine the connection between the entropic uncertainty relation,
quantum coherence, and mutual information in the context of Schwarzschild
black holes, a three-qubit system with state of $\rho_{ABC}$ is employed as
the initial state. To be explicit, particles $A$, $B$, and $C$ are located in
the asymptotically flat region outside the event horizon of the black hole.
Then, particles $B$ and $C$ fall toward the Schwarzschild black hole and hover
around its event horizon, while particle $A$ stays in the asymptotically flat
region. Consequently, the system’s state can be reexpressed as
$\rho_{A{B_{I}}{B_{II}}{C_{I}}{C_{II}}}$. We focus on obtaining the state
$\rho_{AB_{I}C_{I}}$ in the physically accessible region by tracing the
$B_{II}C_{II}$ in the quantum memory, since the two regions inside and outside
the event horizon are disconnected. In addition, the two-dimensional Pauli
operators are chosen as the incompatible measurements. We have
$q_{MU}=-\log_{2}c\left(\hat{R},\hat{Q}\right)=1$ with $\hat{R}$ and
$\hat{Q}\in\\{\hat{\sigma_{x}},\hat{\sigma_{y}},\hat{\sigma_{z}}\\}$, due to
that the Pauli operators are typical mutual unbiased bases.

With regard to two measurements, we can resort to $\hat{\sigma_{x}}$ and
$\hat{\sigma_{z}}$ as the measurement operators. As a result, our proposed
EURs can be given by

$\begin{split}{U_{1}}\geq 2+S(A|B)+S(A|C)=:\mathrm{Bound_{1}},\end{split}$
(24)

and

$\begin{split}{U_{1}}\geq
2+S(A|B)+S(A|C)+\max\\{0,\Delta_{1}\\}=:\mathrm{Bound_{2}}\end{split}$ (25)

where the uncertainty is denoted as
${U_{1}}:=S(\hat{\sigma_{x}}|B)+S(\hat{\sigma_{x}}|C)+S(\hat{\sigma_{z}}|B)+S(\hat{\sigma_{z}}|C)$
and
$\Delta_{1}=\mathcal{I}(A:B)+\mathcal{I}(A:C)-\mathcal{H}(\hat{\sigma_{x}}:B)-\mathcal{H}(\hat{\sigma_{x}}:C)-\mathcal{H}(\hat{\sigma_{z}}:B)-\mathcal{H}(\hat{\sigma_{z}}:C).$

In addition, $\hat{\sigma_{x}}$, $\hat{\sigma_{y}}$ and $\hat{\sigma_{z}}$ can
be employed as the measured observables in three-measurement scenario.
Thereby, the proposed uncertainty relations can be written as:

${U_{2}}\geq 3+\frac{3}{2}[S(A|B)+S(A|C)]=:\mathrm{Bound_{3}},$ (26)

and

$\begin{split}{U_{2}}\geq
3+\frac{3}{2}[S(A|B)+S(A|C)]+\max\\{0,\Delta_{2}\\}=:\mathrm{Bound_{4}}\end{split}$
(27)

where
${U_{2}}=S(\hat{\sigma_{x}}|B)+S(\hat{\sigma_{x}}|C)+S(\hat{\sigma_{y}}|B)+S(\hat{\sigma_{y}}|C)+S(\hat{\sigma_{z}}|B)+S(\hat{\sigma_{z}}|C)$
is the amount of the uncertainty, and
$\Delta_{2}=\frac{3}{2}[\mathcal{I}(A:B)+\mathcal{I}(A:C)]-\mathcal{H}(\hat{\sigma_{x}}:B)-\mathcal{H}(\hat{\sigma_{x}}:C)-\mathcal{H}(\hat{\sigma_{y}}:B)-\mathcal{H}(\hat{\sigma_{y}}:C)-\mathcal{H}(\hat{\sigma_{z}}:B)-\mathcal{H}(\hat{\sigma_{z}}:C)$.
Besides, we set the frequency $\omega_{k}=1$ in order to simplify the
calculation.

To verify our derived uncertainty relations (6) and (9), we proceed by taking
into account one three-qubit W-like state as $\rho_{ABC}$, which is a unique
kind of entangled state as:

$\begin{split}\rho^{\rm
W}_{ABC}={|\mathrm{W}\rangle}{\langle\mathrm{W}|}\end{split}$ (28)

with $|\mathrm{W}\rangle=\cos x|001\rangle+\sin x\cos y|010\rangle+\sin x\sin
y|100\rangle$.

Fig. 2 depicts the entropic uncertainty and the two kinds of bounds (${\rm
Bound}_{1}$ and ${\rm Bound}_{2}$) as functions of state’s coefficient $x$ and
Hawking temperature $T$, respectively. It is clear that our new form of
entropic uncertainty relation is applicable to two measurements within three-
qubit systems. Fig. 2(a) shows that the uncertainty and lower bound change
periodically with the increase of the state parameter $x$. The uncertainty
increases with the growth of $T$ as shown in Fig. 2(b). Meanwhile, the quantum
coherence is greatly broken and the uncertainty increases to maximum, which
will be further discussed later in this article. In addition,
$\mathrm{Bound_{2}}$ is obviously tighter than $\mathrm{Bound_{1}}$ shown in
Fig. 2. We can conclude: (1) our proposed inequalities (6) and (9) are hold;
(2) $\mathrm{Bound_{2}}\geqslant\mathrm{Bound_{1}}$ is satisfied all the time,
which implies the lower bound of the latter outperforms than that in the
former.

Figure 3: Entropic uncertainty relations of Werner state
$\rho^{\rm{Werner}}_{AB_{I}C_{I}}$ in the Schwarzschild black hole. The Graphs
show the case of three-measurement ($\hat{\sigma_{x}}$, $\hat{\sigma_{y}}$,
$\hat{\sigma_{z}}$). The red circle represents the entropic uncertainty, green
diamond represents $\mathrm{Bound_{3}}$ in Eq. (26) and blue triangle
represents $\mathrm{Bound_{4}}$ in Eq. (27). In Graph (a), uncertainty and the
lower bounds vs $z^{2}$ with $T=10$ and $p=0.5$; in Graph (b), uncertainty and
the lower bounds vs $T$ with $p=0.5$ and $z^{2}=0.5$; in graph (c),
uncertainty and the lower bounds vs the purity parameter $p$ with $z^{2}=0.5$
and $T=10$.

Figure 4: Entropic uncertainty $U$, coherence $C$ and purity $\cal P$ of
Werner state $\rho^{\rm Werner}_{AB_{I}C_{I}}$ in the Schwarzschild black
hole. Graphs (a), (b) and (c) show the uncertainty, coherence and purity
versus $T$ and $p$ with $z^{2}=0.5$; Graphs (d), (e) and (f) show the
uncertainty, coherence and purity as functions of $T$ and $z^{2}$ with
$p=0.5$, respectively. Figure 5:  Mutual information vs the Hawking
temperature $T$ for Werner state $\rho^{\rm Werner}_{AB_{I}B_{II}C_{I}C_{II}}$
as initial state in the Schwarzschild black hole with $p=0.5$ and $z^{2}=0.5$.
The red square, orange circle, yellow triangle, green inverted triangle, blue
diamond and purple pentagram represent the mutual information of $B_{I}C_{I}$,
$B_{II}C_{II}$, $AB_{I}$, $AB_{II}$, $AC_{I}$, and $AC_{II}$, respectively.

Figure 6: Mutual information as functions of the state’s parameters $p$ and
$z^{2}$ of Werner state $\rho^{\rm Werner}_{AB_{I}B_{II}C_{I}C_{II}}$ in the
Schwarzschild black hole. Graphs (a), (b) and (c) plot ${\cal I}_{B_{I}C_{I}}$
with $T=1,2,\infty$. Graphs (d), (e) and (f) plot ${\cal I}_{B_{II}C_{II}}$
with $T=1,2,\infty$, respectively.

Without loss of generality, we also consider a three-qubit Werner state, which
canonical is a type of mixed states:

$\begin{split}\rho^{\rm
Werner}_{ABC}=p{\left|\mathrm{GHZ}\right\rangle}{\left\langle\mathrm{GHZ}\right|}+\frac{(1-p)}{8}\mathbb{I}_{8\times
8},\end{split}$ (29)

with $|\mathrm{GHZ}\rangle=z|000\rangle+\sqrt{1-z^{2}}|111\rangle$, the purity
of the system $p\in[0,1]$. The state is the maximally mixed state for $p=0$,
while it becomes the Greenberger-Horne-Zeilinger-type (GHZ-like) state, a
typical pure state, for $p=1$. Explicitly, entropic uncertainty and two bounds
with respect to the state parameter $z^{2}$, $p$ and Hawking temperatur $T$
have been plotted in Fig. 3. As indicated in Fig. 3(a), the uncertainty
increases first and then decreases with the growing $z^{2}$. Fig. 3(b)
indicates that, near $T=0$, the uncertainty increases rapidly, which result is
similar to that in the W-like states. In Fig. 3(c), the uncertainty and lower
bounds decrease monotonically as the purity $p$ grows, and the uncertainty
will minimize as to $p=1$. What’s more, one can directly see that
$\mathrm{Bound_{4}}\geqslant\mathrm{Bound_{3}}$ in terms of the current
numerical analysis, which is essentially in agreement with the conclusion made
before.

To better interpret the dynamics of the uncertainty in the background of
Schwarzschild black hole, we will focus on examining coherence and purity of
the system of interest. Technically, $l_{1}$-norm is used to quantify quantum
coherence of a given system $\rho$ with the form of
$C_{l_{1}}{(\rho)}=\sum_{i\neq j}\left|\rho_{i,j}\right|$, and the purity of
the system is denoted as ${\cal P}=\mathrm{Tr}(\rho^{2})$. In this case, we
can work out $C_{l_{1}}{(\rho^{\rm
Werner}_{AB_{I}C_{I}})}=\frac{2pz\sqrt{(1-z^{2})}}{1+e^{-\frac{\omega_{k}}{T}}}$
and the analytical expression of ${\cal P}$ is provided as Appendix. As shown
in Figs. 4, 4, 4 and 4, the uncertainty and coherence with respect to the
Hawking temperature $T$ always shows the opposite trend, indicating that there
is an inverse correlation between uncertainty and quantum coherence regardless
of the initial state’s parameter $p$. In the physically accessible region, as
illustrated by Fig. 4 , the system’s purity is mainly determined by the
state’s parameter $p$; the purity declines as $T$ increases, which indicates
that an increase in Hawking temperature $T$ could disrupt the purity of the
system. The purity with the growing $T$ and $z^{2}$ has been plotted in Fig.
4. Following the figure, as the Hawking temperature $T$ increases, the
coherence $C$ of the system is gradually broken, resulting in a gradual
decrease in purity $\cal P$. However, it is interesting to note that when
$z^{2}$ is very small, which corresponds to that the initial state is the
direct product states $|111\rangle$, the purity of the system will increase.
This manifests that the existence of excited states will prevent the coherence
of the system from being destroyed due to the increase of $T$, so the purity
of the system will increase. Thereby, we can conclude: (1) an increasing
Hawking temperature will break the coherence of the system and increase the
entropic uncertainty; (2) the uncertainty and coherence exist an anti-
correlation characteristic with variation of Hawking temperature $T$; (3) when
the Hawking temperature $T$ is small, the purity of the state $\rho^{\rm
Werner}_{AB_{I}C_{I}}$ in the physically accessible region is dominated by the
purity of the original state $\rho^{\rm Werner}_{ABC}$, however the state
parameter $z^{2}$ will govern the system’s purity when the Hawking temperature
is large enough.

To clarify the dynamical mechanics of the uncertainty in Schwarzschild black
hole, we next turn to mutual information, which is able to show the
information distribution in the whole system. Fig. 5 has plotted the mutual
information related to the Hawking temperature $T$. It can be seen that the
physically accessible region mutual information ${\cal I}_{B_{I}C_{I}}$,
${\cal I}_{AB_{I}}$ and ${\cal I}_{AC_{I}}$ decrease as $T$ increases, while
the physically inaccessible region ones increase. This demonstrates the flow
of information from physically accessible region to physically inaccessible
region. Linking with the dynamics of uncertainty in Fig. 3(b), it is evident
that the mutual information and uncertainty exhibit fully opposite trends as
the Hawking temperature increases. Therefore, it is inferred that the Hawking
effect results in reduction of the correlation within the physically
accessible region, and consequently leads to inflation of the uncertainty.
Particularly, when $T\rightarrow\infty$, ${\cal I}_{B_{I}C_{I}}$ and ${\cal
I}_{B_{II}C_{II}}$ tend to be equal, which indicates that when the Hawking
temperature $T$ is large enough, the boundary between the physically
accessible region and the physically inaccessible region becomes blurred.
Mutual information $B_{I},C_{I}$ and $B_{II}C_{II}$ with respect to the
variation in $z^{2}$ and $p$ are plotted in Fig. 6. From the figure, the
mutual information generally increases with the increase of purity $p$. As $T$
increases, the mutual information between $B_{I}$ and $C_{I}$ decreases, the
mutual information between $B_{II}$ and $C_{II}$ increases, which indicates
the information redistribution of the Schwarzschild black hole.

##  V Conclusions

In summary, we have derived the generalized EUR for $m$ measurements with
$n+1$-party, and have further strengthened it via mutual information and
Holevo quantity. The proposed EURs have been verified by utilization of three-
qubit W-like and Werner states in the background of Schwarzschild space-time.
Besides, we observed that Hawking radiation destroys quantum coherence and
increases the uncertainty of the system. Furthermore, we revealed that the
uncertainty of the system is anti-correlation with the system’s purity. As the
Hawking temperature increases, the mutual information of systems outside the
black hole’s event horizon decreases, while ones within the event horizon
increases. This suggests that the Hawking effect leads to the redistribution
of information from a physically accessible region to a physically
inaccessible region, which can be used to explain the uncertainty dynamics of
the Schwarzschild black hole. Thus, it is believed that the current findings
provide insight into the generalized uncertainty relation, and are beneficial
to understand information paradox in the black holes.

###### Acknowledgements.

This study was supported by the National Natural Science Foundation of China
(Grant Nos. 12075001 and 61601002), and Anhui Provincial Key Research and
Development Plan (Grant No. 2022b13020004).

## APPENDIX

The specific form of purity is given by:

$\begin{split}{\cal
P}=&\frac{1}{64}\left[u^{8}(-1+p)^{2}+u^{8}(-1+p-8pz^{2})^{2}\right]\\\
&+\frac{1}{64}\left[-1+v^{2}(2+v^{2})(-1+p)-7p+8pz^{2}\right]^{2}\\\
&+\frac{1}{64}\left[(1+v^{2})^{2}(-1+p)-8v^{4}pz^{2}\right]^{2}\\\
&+\frac{1}{32}u^{4}\left\\{1-p+v^{2}[1+(-1+8z^{2})]\right\\}^{2}\\\
&+\frac{1}{32}u^{4}(1+v^{2})^{2}(-1+p)^{2}-2u^{4}p^{2}z^{2}(-1+z^{2})\end{split}$
(30)

with $u=[e^{-\omega_{k}/T}+1]^{-\frac{1}{2}}$ and
$v=[e^{\omega_{k}/T}+1]^{-\frac{1}{2}}$.

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