††thanks: These authors contributed equally to this work.††thanks: These
authors contributed equally to this work.

# Experimental investigation of quantum correlations in a two-qutrit spin
system

Yue Fu  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Wenquan Liu  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  CAS Center for Excellence in Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei 230026, China
Xiangyu Ye  CAS Key Laboratory of Microscale Magnetic Resonance and School of
Physical Sciences, University of Science and Technology of China, Hefei
230026, China  Ya Wang  CAS Key Laboratory of Microscale Magnetic Resonance
and School of Physical Sciences, University of Science and Technology of
China, Hefei 230026, China  CAS Center for Excellence in Quantum Information
and Quantum Physics, University of Science and Technology of China, Hefei
230026, China  Hefei National Laboratory, University of Science and Technology
of China, Hefei 230088, China  Chengjie Zhang  School of Physical Science and
Technology, Ningbo University, Ningbo 315211, China  State Key Laboratory of
Precision Spectroscopy, School of Physics and Electronic Science, East China
Normal University, Shanghai 200241, China  Chang-Kui Duan  CAS Key Laboratory
of Microscale Magnetic Resonance and School of Physical Sciences, University
of Science and Technology of China, Hefei 230026, China  CAS Center for
Excellence in Quantum Information and Quantum Physics, University of Science
and Technology of China, Hefei 230026, China  Xing Rong<EMAIL_ADDRESS>CAS
Key Laboratory of Microscale Magnetic Resonance and School of Physical
Sciences, University of Science and Technology of China, Hefei 230026, China
CAS Center for Excellence in Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei 230026, China  Hefei
National Laboratory, University of Science and Technology of China, Hefei
230088, China  Jiangfeng Du<EMAIL_ADDRESS>CAS Key Laboratory of Microscale
Magnetic Resonance and School of Physical Sciences, University of Science and
Technology of China, Hefei 230026, China  CAS Center for Excellence in Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei 230026, China  Hefei National Laboratory, University of Science
and Technology of China, Hefei 230088, China

###### Abstract

We report an experimental investigation of quantum correlations in a two-
qutrit spin system in a single nitrogen-vacancy center in diamond at room
temperatures. Quantum entanglement between two qutrits was observed at room
temperature and the existence of non-classical correlations beyond
entanglement in the qutrit case has been revealed. Our work demonstrates the
potential of the NV centers as the multi-qutrit system to execute quantum
information tasks and provides a powerful experimental platform for studying
fundamental physics of high-dimensional quantum systems in future.

Quantum correlation shed light on the most fundamental trait that
distinguishes a quantum correlated system from one fully ascribed to a joint
classical probability distribution and may reveal the origin of the quantum
enhancement in various quantum information tasks JPhys_2016_Adesso ;
review_2018_De ; book_2015_streltsov ; review_2018_Braun . Quantum
entanglement as a prolonged description of quantum correlation review_2009 ,
however, has been found cannot account for all quantum correlations, leading
to the introduction of quantum discord PRL_2001_Olivier ; JPA_2001_Henderson .
Quantum discord can describe non-classical correlations even in separable
states and may contribute to the quantum enhancement when these states are
applied to quantum information processing (QIP) PRL_1998_Knill ;
review_2012_Modi ; review_2018_Bera . In the last decades, quantum discord was
studied intensively but experimental investigations have been limited to qubit
systems PRA_2010_Soares ; PRL_2011_Auccaise ; PRA_2017_Singh ; PRL_2008_Lanyon
; NC_2010_Xu ; NC_2013_Xu ; PRL_2015_Bromley ; PRA_2016_Knoll ;
PRB_2011_Yurishchev ; PRB_2012_Rong ; PRB_2013_Rong ; NP_2013_Gessner ;
NC_2017_Abdelrahman ; PRA_2014_Wood . Recently, researches on systems with
$d$-level (${d>2}$), hereafter referred to as qudits, are emerging
AQT_2019_Cozzolino ; FP_2020_Wang ; review_2020_Erhard . Quantum information
processing with qudit may offer higher-dimensional Hilbert space, which
results in higher efficiency and flexibility in quantum computing
PRL_2019_Wang ; NP_2019_Reimer , larger channel capacity and better noise
tolerance in quantum communication PRL_2019_Luo ; PRL_2020_Hu  and relaxed
constraints in fundamental tests of the nature PRL_2010_Vertesi . Currently,
the experimental studies of quantum correlations in qudit systems have been
focused on quantum entanglement nature_2017_Kues ; Science_2018_Wang ;
NPJ_2020_Lu ; arxiv_2021_Ding ; LSA_2016_Ding  and quantum steering
PRL_2018_Zeng ; PRL_2019_Guo ; PRL_2021_Designolle . Quantum discord, a more
general description of the quantum correlation, remains untouched
experimentally in qutrit systems.

In this paper, we report an experimental investigation of quantum correlations
in a two-qutrit system. The nitrogen-vacancy (NV) center, which has a spin-1
electron spin with the complement of a spin-1 nuclear spin of the 14N
PhysRep_2013_Doherty , was utilized as a two-qutrit system to study the high
dimensional quantum correlations. The quantum correlations of two-qutrit
isotropic states (also known as qutrit Werner states QIP_2013_Ye ), which are
crucial in the research of quantum correlation PRA_2006_Baumgartner ;
review_2009 ; PRA_2012_Chitambar ; QIP_2013_Ye , were prepared and measured
here. The features of quantum discord and quantum entanglement with different
values of the state parameter are revealed. It is verified that there is a
threshold under which the quantum entanglement vanishes while the quantum
discord remains. Such a threshold differs from the one in the case of a two-
qubit system PRA_2006_Baumgartner ; PRA_2012_Chitambar ; QIP_2013_Ye ;
PRA_2021_Poxleitner .

Figure 1: The two-qutrit system constructed by the NV center. (a) Schematic
atomic structure of the NV center. (b) Ground state energy levels of the NV
center. The nine different states of the electron spin and the nuclear spin
constitute a two-qutrit system. The transitions between different electron
(nuclear) spin states can be steered by microwave (radio frequency) pulses
indicated by purple (red) arrows.

The total correlation of a bipartite system $\rho_{AB}$ quantified by the
quantum mutual information JPA_2001_Henderson ; PRL_2001_Olivier  is defined
as

$\displaystyle I(\rho_{AB})=S(\rho_{A})+S(\rho_{B})-S(\rho_{AB}),$ (1)

where $\rho_{A}$ ($\rho_{B}$) is the reduced density matrix of particle $A$
($B$) and $S(\rho)=-\rm{Tr}[\rho\rm{log_{2}}\rho]$ is the von Neumann entropy
of density matrix $\rho$. Both classical and quantum correlations are included
in $I(\rho_{AB})$. The classical correlation, which depends on the maximum
information gained by measuring one particle of the total system, is defined
as

$\displaystyle C(\rho_{AB})={\rm
max}_{B_{j}^{\dagger}B_{j}}[S(\rho_{A})-\sum_{j}q_{j}S(\rho_{A}^{j})],$ (2)

where $q_{j}={\rm Tr}[B_{j}\rho_{AB}B_{j}^{\dagger}]$ is the probability of
obtaining result $j$ when performing Positive Operator-Valued Measure (POVM)
$\\{B_{j}^{\dagger}B_{j}\\}$ on subsystem $B$, $\rho_{A}^{j}={\rm
Tr}_{B}[B_{j}\rho_{AB}B_{j}^{\dagger}]/q_{j}$ is the state of subsystem $A$
after obtaining outcome $j$. Therefore, the difference between the total
correlation and the classical correlation given by

$\displaystyle D(\rho_{AB})$ $\displaystyle=I(\rho_{AB})-C(\rho_{AB}),$ (3)

is quantum correlation, which is termed quantum discord.

We focus on a family of qutrit states referred to as isotropic states, which
have the form PRA_2006_Baumgartner

$\displaystyle\rho_{\rm
iso}=\frac{(1-p)}{9}\mathbb{I}_{3}\otimes\mathbb{I}_{3}+p|\psi\rangle\langle\psi|,$
(4)

where $|\psi\rangle=(|{+1,+1}\rangle+|0,0\rangle+|{-1,-1}\rangle)/\sqrt{3}$ is
the maximally entangled state, $p\in[0,1]$ and $\mathbb{I}_{n}$ denotes the
identity operator in $n$-dimensional Hilbert space. Physically, an isotropic
state can be regarded as a mixture of the maximally mixed state
$\mathbb{I}_{9}/9$ and the maximally entangled state
$|\psi\rangle\langle\psi|$ with parameter $p$ determining the components.
Theoretically, there is a threshold $p_{c}$ under which the states are
separable and otherwise entangled. The threshold varies with the dimension of
the isotropic states and $p_{c}=1/4$ for qutrits PRA_2006_Baumgartner ;
PRA_2012_Chitambar . However, the quantum discord remains non-vanishing for
all non-zero $p$. Thus, the isotropic states are a representative example of
quantum states with zero entanglement nevertheless exhibiting non-zero quantum
correlations.

Figure 2: The properties of the two-qutrit system. (a) Rabi oscillation
between different electron (left panel) or nuclear (right panel) spin states.
(b) Longitudinal relaxation time of the electron spin $T_{1,e}$ starting form
the state $|0,{+1}\rangle$. (c)-(d) Dephasing time of the electron spin and
the nuclear spin. All points with error bars are experimental data and the
curves are the fitting results.

A single negatively charged NV center in [100] face bulk diamond was utilized
to investigate the quantum discord and quantum entanglement of a two-qutrit
system. The diamond was isotopically purified ($[^{12}C=99.9\%]$) to enhance
the dephasing time of the electron spin. The NV center consists of a
substitutional nitrogen atom adjacent to a carbon vacancy (see Fig. 1a). When
an external magnetic field is applied along the NV symmetry axis, the
Hamiltonian of the NV center can be written as

$\displaystyle H_{\rm
NV}=2\pi(DS_{z}^{2}+\omega_{e}S_{z}+QI_{z}^{2}+\omega_{n}I_{z}+AS_{z}I_{z}),$
(5)

where $S_{z}\ (I_{z})$ is the spin operator of the electron (nuclear) spin,
$D=2.87$ GHz is the electronic zero-field splitting, $Q=-4.95$ MHz is the
nuclear quadrupolar interaction constant, and $A=-2.16$ MHz is the hyperfine
coupling constant. $\omega_{e}$ ($\omega_{n}$) corresponds to the Zeeman
frequency of the electron (nuclear) spin. This two-qutrit system contains nine
energy levels denoted as $|m_{S}\rangle_{e}|m_{I}\rangle_{n}$, with
$m_{S},m_{I}=0,\,\pm 1$ representing the states of the electron and nuclear
spins, respectively (see Fig. 1b). For simplicity,
$|m_{S}\rangle_{e}|m_{I}\rangle_{n}$ is hereafter labeled by
$|m_{S},m_{I}\rangle$. The magnetic field was set to 500 G and the NV center
was polarized into the state $|{0,+1}\rangle$ via applying a 532 nm laser
pulse PRL_2009_Jacques . As shown in Fig. 1b, microwave (MW) pulses labeled by
purple arrows were applied to manipulate the quantum states of the electron
spin of the NV center. For the controlling of the nuclear spins, radio
frequency (RF) pulses labeled by red arrows were utilized.

The Rabi frequency of the electron (nuclear) spin was set to $\Omega_{\rm
MW}=0.2$ MHz ($\Omega_{\rm RF}=25$ kHz) as depicted in Fig. 2a. The relaxation
times of both the electron spin and the nuclear spin were measured. For the
electron spin, the longitudinal relaxation time is $T_{1,e}=4\pm 1$ ms (Fig.
2b) and the dephasing time is $T_{2,e}^{*}=18\pm 2\ \mu$s (Fig. 2c). In Fig.
2d, we did not see evident decay of the oscillation amplitude during the
measurement, which indicated that the dephasing time of the nuclear spin
should be much longer than one millisecond. The overall decreasing background
probability is caused by the depolarization of the electron spin. The nuclear
longitudinal relaxation time $T_{1,n}$ is estimated to be $\sim 100$ times of
$T_{1,e}$ according to Ref. PRB_2013_Fischer, . $T_{1,n},T_{1,e}$ and
$T_{2,n}^{*}$ are much longer than $T_{2,e}^{*}$. Thus the off-diagonal
elements representing the coherence of the electron spin decay much faster
than any other elements of the density matrix. The dephasing of the electron
spin was the main effect we considered when preparing the isotropic states.

Figure 3: Experimental pulse sequence and reconstructed density matrix. (a)
Diagram of the pulse sequence, which includes polarization, state preparation
and tomography. In the procedure of state preparation, selective MW (purple)
and RF (red) pulses were performed to generate the isotropic state $\rho_{\rm
iso}(p)$. Frequencies of these pulses correspond to the transition frequencies
between energy levels shown in Fig. 1(b). The tomography comprises various
measurement sequences to readout different elements of the density matrix (see
Supplemental Material SM  for details). The grey box stands for MW and RF
pulses sequences for tomography, which is explained in detail in Supplemental
Material SM . (b) Experimental density matrix of the entangled state with
$p=0.94$. The bars show the experimental outcomes while wire grids represent
corresponding simulation results. Experimental results fit well with
simulations and the state fidelity is $96\%$.

Fig. 3a shows the diagram of the pulse sequence for studying the quantum
correlations of the isotropic states. It consists of three parts:
polarization, state preparation and tomography. The NV center was polarized
into the state $|0,+1\rangle$ via a green laser pulse. The preparation of the
isotropic states contains three steps: (I) applying four selective MW or RF
pulses followed by a free evolution time, $t_{wait}=90~{}\mu s$, to prepare
the NV center to a mixed state with the form $\rho_{\rm I}(p)=P_{+1,+1}^{\rm
I}(p)|{+1,+1}\rangle\langle{+1,+1}|+P_{0,0}^{\rm
I}(p)|{0,0}\rangle\langle{0,0}|+P_{-1,-1}^{\rm
I}(p)|{-1,-1}\rangle\langle{-1,-1}|$. (II) applying six selective MW pulses
and waiting the same waiting time in step (I) to manipulate the system to
mixed state $\rho_{\rm II}(p)=\sum_{i,j=0,\pm 1}P_{i,j}^{\rm
II}(p)|i,j\rangle\langle i,j|$. (III) applying selective MW and RF pulses to
generate the off-diagonal terms of the isotropic states such that $\rho_{\rm
III}(p)=\rho_{\rm iso}(p)$. These selective pulses correspond to transitions
between different energy levels displayed in Fig. 1b. The quantum state
tomography was performed after the state preparation and all the non-zero
elements of the isotropic states were measured. A maximum likelihood
estimation method was utilized to reconstruct the density matrix of the final
states PRA_2001_Daniel . In Fig. 3b, we show the result of the isotropic state
with $p=0.94$ as an example. The experimental results (colored bars) fit well
with the simulation results (wire grids, with the dephasing noise of the
electron spin considered). More details about the state preparation, the
measurement sequences and the state reconstruction can be found in
Supplemental Material SM .

Figure 4: Quantum correlations of the isotropic states. (a) and (b) displays
the experimental results of quantum entanglement and quantum discord,
respectively. Points are experimental data while solid lines show the
simulation results. The vertical error bars of the data points were calculated
by Monte Carlo simulation with the Gaussian statistics. The abscissas of the
data points and their error bars were obtained by comparing the experimental
results with simulations to find the most likely $p$ for each Monte Carlo run.
Some error bars are smaller than the size of dots so they are not visible. The
dashed line corresponds to $p=1/4$, which is the dividing line between
separable state and entangled state.

The results of the quantum entanglement and quantum discord measured in our
experiment are displayed in Fig. 4. The values of the quantum discord defined
by Eq. (3) were obtained from the experimentally reconstructed density
matrices using the approach introduced in Ref. PRA_2012_Rossignoli, . The
extremization in Eq. (2) should be taken over all possible complete sets of
projective measurements of subsystem $B$. We obtained the value of the quantum
discord with different measurement basis until it converged. For the isotropic
states, the measure of their entanglement can be given by the negativity
defined as $N(\rho_{\rm iso})=({||}\rho_{\rm iso}^{PT}{||}_{1}-1)/2$ where
$PT$ and ${||}\cdot{||}_{1}$ represent partial transposition and trace norm
calculation of the density matrix, respectively PRA_2007_Derkacz . Details
about the calculation of the quantum discord and the entanglement are given in
Supplemental Material SM . Theoretical results in Fig. 4 were obtained by
calculating the quantum entanglement and the quantum discord of the simulated
states. The experimental results agree well with the theoretical predictions.
In the region $0<p\leq 1/4$, the entanglement is zero while the non-zero
quantum discord still exists. This unambiguously means a type of quantum
correlation beyond quantum entanglement had been experimentally observed in a
two-qutrit system. When $p>1/4$, the state has non-zero quantum entanglement
and quantum discord that both monotonously increase with $p$. Our results show
that the qualitative behaviors of the quantum discord and the entanglement in
a two-qutrit system are similar to that of a two-qubit system. However, a
smaller threshold of the quantum entanglement is observed as predicted by
theories PRA_2012_Chitambar ; PRA_2021_Poxleitner .

$\mathit{Discussion}$. Quantum correlation is an essential issue in quantum
physics and recently high-dimensional quantum correlations have aroused
tremendous research interest. In this paper, we experimentally studied the
quantum correlations between two qutrits in a single NV center. Entanglement
in the two-qutrit system has been observed at room temperature, without
resorting to cryogenic conditions arxiv_2021_Ding . In particular, the non-
classical correlations beyond entanglement in the qutrit case are founded.
These results show that NV centers are a powerful platform for further
investigating the fundamental properties of high-dimensional quantum
correlations, such as the essence of quantum correlations PRL_1998_Knill ;
TCS_2004_Biham ; review_2018_De  and their relation with quantum superposition
and nonlocality review_2012_Modi ; review_2018_Bera . Experimentally studying
the dynamic behavior of high-dimensional quantum correlations and observing
whether there exists sudden death science_2007_Almeida  or sudden transition
PRL_2011_Auccaise  phenomena will also be interesting. Besides, qutrit systems
possess advantages in QIP review_2020_Erhard  and experimental investigations
of key procedures of QIP PRL_2021_Yurtalan ; PRX_2021 ; PRL_2021_Morvan  are
necessary. The NV center is a natural high-dimensional system and has long
coherence time even at room temperature. However, it has been utilized as
qubits in most studies, which indeed limited its potential. Our work promotes
the NV center as a high-dimensional system to execute quantum computation and
quantum sensing tasks AQT_2019_Cozzolino ; FP_2020_Wang ; PRL_2021_Designolle
. NV qutrit systems may play an important role in high-dimensional quantum
information processing since further coupling two NV centers as qutrits on top
of qubits science_2021 ; Nphys_2013 ; NC_2014 ; nature_2022  is foreseeable,
albeit challenging.

This work was supported by the Chinese Academy of Sciences (Grant No.
XDC07000000, No. GJJSTD20200001, No. QYZDY-SSW-SLH004, No. QYZDB-SSW-SLH005),
Innovation Program for Quantum Science and Technology (Grant No.
2021ZD0302200), the National Key R$\&$D Program of China (Grant No.
2018YFA0306600), the National Natural Science Foundation of China (Grant No.
81788101, No. 11734015), Anhui Initiative in Quantum Information Technologies
(Grant No. AHY050000), Hefei Comprehensive National Science Center, the
Fundamental Research Funds for the Central Universities, and the open funding
program from State Key Laboratory of Precision Spectroscopy (East China Normal
University). X. R thanks the Youth Innovation Promotion Association of Chinese
Academy of Sciences for the support.

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Supplementary Material

##  I Preparation, measurement and maximum likelihood estimation of the
isotropic state.

###  I.1 Preparation of the isotropic state.

Supplementary Figure S1: Rabi oscillation results. (a-f): Rabi oscillation
results of the electron spin corresponding to transition between energy levels
(a) $|{+1,+1}\rangle\leftrightarrow|{0,+1}\rangle$, (b)
$|{0,+1}\rangle\leftrightarrow|{-1,+1}\rangle$, (c)
$|{+1,0}\rangle\leftrightarrow|{0,0}\rangle$, (d)
$|{0,0}\rangle\leftrightarrow|{-1,0}\rangle$, (e)
$|{+1,-1}\rangle\leftrightarrow|{0,-1}\rangle$ and (f)
$|{0,-1}\rangle\leftrightarrow|{-1,-1}\rangle$, respectively. All the Rabi
frequencies were calibrated to ${\rm\Omega_{MW}}=0.2$ MHz. (g-h): Rabi
oscillation results of the nuclear spin corresponding to transition between
energy levels (g) $|{0,+1}\rangle\leftrightarrow|{0,0}\rangle$ and (h)
$|{0,0}\rangle\leftrightarrow|{0,-1}\rangle$. These Rabi frequencies were
calibrated to ${\rm\Omega_{RF}}=25$ kHz.

The two-qutrit isotropic state $\rho_{\rm iso}$ depending on the parameter $p$
is given by

$\rho_{\rm
iso}(p)=\frac{(1-p)}{9}\mathbb{I}_{3}\otimes\mathbb{I}_{3}+p|\psi\rangle\langle\psi|\
\ ,$ (S1)

where $p\in[0,1]$ and
$|\psi\rangle=(|{+1,+1}\rangle+|{0,0}\rangle+|{-1,-1}\rangle)/\sqrt{3}$ is the
maximally entangled state. In the two-qutrit system based on the NV center,
the energy levels $|+1\rangle_{e}$, $|0\rangle_{e}$ and $|-1\rangle_{e}$
($|+1\rangle_{n}$, $|0\rangle_{n}$ and $|-1\rangle_{n}$) correspond to the
state $|+1\rangle,|0\rangle$ and $|-1\rangle$ of the first (second) qutrit,
respectively. In this representation, the density matrix of $\rho_{\rm iso}$
takes the form

$\rho_{\rm
iso}(p)=\begin{pmatrix}\frac{1+2p}{9}&0&0&0&\frac{p}{3}&0&0&0&\frac{p}{3}\\\
0&\frac{1-p}{9}&0&0&0&0&0&0&0\\\ 0&0&\frac{1-p}{9}&0&0&0&0&0&0\\\
0&0&0&\frac{1-p}{9}&0&0&0&0&0\\\
\frac{p}{3}&0&0&0&\frac{1+2p}{9}&0&0&0&\frac{p}{3}\\\
0&0&0&0&0&\frac{1-p}{9}&0&0&0\\\ 0&0&0&0&0&0&\frac{1-p}{9}&0&0\\\
0&0&0&0&0&0&0&\frac{1-p}{9}&0\\\
\frac{p}{3}&0&0&0&\frac{p}{3}&0&0&0&\frac{1+2p}{9}\end{pmatrix}\ .$ (S2)

MW and RF pulses were applied to manipulate the state of the two-qutrit
system. The Rabi frequencies of the MW and RF pulses used in our experiment
were calibrated to ${\rm\Omega_{MW}}=0.2$ MHz and ${\rm\Omega_{RF}}=25$ kHz,
respectively. The experimental results of these Rabi oscillations are
displayed in Fig. S1.

The NV center was polarized into the state $|0,+1\rangle$ via laser pulses at
the beginning of the state preparation. Then the following three steps were
performed to prepare the $\rho_{\rm iso}$. Information of the pulses used in
the preparation process is displayed in Table. S1.

  * (i)

Two selective RF pulses with frequency $\omega_{n1}$ and $\omega_{n2}$
respectively were applied in sequence to manipulate the NV center into the
state
$|\psi_{1}\rangle=\cos\frac{\alpha_{1}}{2}|{0,+1}\rangle+\sin\frac{\alpha_{1}}{2}\cos\frac{\alpha_{2}}{2}|0,0\rangle+\sin\frac{\alpha_{1}}{2}\sin\frac{\alpha_{2}}{2}|{0,-1}\rangle$.
Then two selective MW pulses with frequency $\omega_{e2}$ and $\omega_{e5}$
respectively were performed to prepare the NV center into the state
$|\psi_{2}\rangle=\cos\frac{\alpha_{1}}{2}|{+1,+1}\rangle+\sin\frac{\alpha_{1}}{2}\cos\frac{\alpha_{2}}{2}|0,0\rangle+\sin\frac{\alpha_{1}}{2}\sin\frac{\alpha_{2}}{2}|{-1,-1}\rangle$.
The values of $\alpha_{j}$ ($j=1,2$) are displayed in the Table. S1. After a
waiting time of $t_{wait}=90~{}\mu$s, the coherence of the electron spin
dissipated, resulting in the NV center in the mixed state with the form

$\rho_{\rm I}(p)=\begin{pmatrix}\frac{1-p}{3}&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\
0&0&0&0&\frac{1+2p}{3}&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&\frac{1-p}{3}\end{pmatrix}\ .$ (S3)

  * (ii)

Six selective MW pulses with pulse phase being X were applied in sequence as
shown in Fig.3(a) of main text. The rotation angles of these pulses
$\theta_{j}$ ($j=1,2,3,4,5,6$) are shown in the Table. S1. Then after the same
waiting time $t_{wait}=90\ \mu s$, $\rho_{\rm I}(p)$ evolved to a mixed state
with the form

$\rho_{\rm II}(p)=\begin{pmatrix}\frac{1-p}{9}&0&0&0&0&0&0&0&0\\\
0&\frac{1-p}{9}&0&0&0&0&0&0&0\\\ 0&0&\frac{1-p}{9}&0&0&0&0&0&0\\\
0&0&0&\frac{1-p}{9}&0&0&0&0&0\\\ 0&0&0&0&\frac{1+8p}{9}&0&0&0&0\\\
0&0&0&0&0&\frac{1-p}{9}&0&0&0\\\ 0&0&0&0&0&0&\frac{1-p}{9}&0&0\\\
0&0&0&0&0&0&0&\frac{1-p}{9}&0\\\ 0&0&0&0&0&0&0&0&\frac{1-p}{9}\end{pmatrix}\
.$ (S4)

  * (iii)

Two selective RF pulses and two selective MW pulses were applied in sequence
to generate the off-diagonal terms of the isotropic state. The rotation angles
of these pulses are shown in the Table. S1. Then the state of the NV center
was $\rho_{\rm III}(p)=\rho_{\rm iso}(p)$. So the isotropic state was
generated in the two-qutrit system constructed by the NV center.

Supplementary Table S1:  Detailed information about the parameters of the MW and RF pulses used in the process of the isotropic state preparation. step | pulse frequency |  transition states |  pulse phase and rotation angle
---|---|---|---
$\rm I$ | $\omega_{n,1}$ |  $|0,+1\rangle\leftrightarrow|0,0\rangle$ |  X , $\alpha_{1}=2\arcsin\sqrt{\frac{2+p}{3}}$
| $\omega_{n,2}$ |  $|0,0\rangle\leftrightarrow|0,-1\rangle$ |  X , $\alpha_{2}=2\arcsin\sqrt{\frac{1-p}{2+p}}$
| $\omega_{e,2}$ |  $|0,+1\rangle\leftrightarrow|+1,+1\rangle$ |  X , $\pi$
| $\omega_{e,5}$ |  $|0,-1\rangle\leftrightarrow|-1,-1\rangle$ |  X , $\pi$
$\rm II$ | $\omega_{e,2}$ |  $|0,+1\rangle\leftrightarrow|+1,+1\rangle$ |  X , $\theta_{1}=2\arcsin\sqrt{\frac{2}{3}}$
| $\omega_{e,1}$ |  $|0,+1\rangle\leftrightarrow|-1,+1\rangle$ |  X , $\theta_{2}=\pi/2$
| $\omega_{e,4}$ |  $|0,0\rangle\leftrightarrow|+1,0\rangle$ |  X , $\theta_{3}=2\arcsin\sqrt{\frac{1-p}{3+6p}}$
| $\omega_{e,3}$ |  $|0,0\rangle\leftrightarrow|-1,0\rangle$ |  X , $\theta_{4}=2\arcsin\sqrt{\frac{1-p}{2+7p}}$
| $\omega_{e,5}$ |  $|0,-1\rangle\leftrightarrow|-1,-1\rangle$ |  X , $\theta_{5}=2\arcsin\sqrt{\frac{2}{3}}$
| $\omega_{e,6}$ |  $|0,-1\rangle\leftrightarrow|+1,-1\rangle$ |  X , $\theta_{6}=\pi/2$
$\rm III$ | $\omega_{n,1}$ |  $|0,+1\rangle\leftrightarrow|0,0\rangle$ |  Y , $\phi_{1}=2\arcsin\sqrt{\frac{1}{3}}$
| $\omega_{n,2}$ |  $|0,0\rangle\leftrightarrow|0,-1\rangle$ |  Y , $\phi_{2}=\pi/2$
| $\omega_{e,2}$ |  $|0,+1\rangle\leftrightarrow|+1,+1\rangle$ |  Y , $\pi$
| $\omega_{e,5}$ |  $|0,-1\rangle\leftrightarrow|-1,-1\rangle$ |  Y , $\pi$

###  I.2 Measurement of the isotropic state.

The isotropic state was measured via a series of measurement sequences. The
photoluminescence (PL) rate of the NV center depends on the state of both the
electron spin and the nuclear spin after being pumped by the laser pulse
PRL_V.Jacques . This property was utilized to readout the state of the two-
qutrit system. For simplicity, eigenstates $|+1,+1\rangle$, $|+1,0\rangle$,
$|+1,-1\rangle$, $|0,+1\rangle$, $|0,0\rangle$, $|0,-1\rangle$,
$|-1,+1\rangle$, $|-1,0\rangle$ and $|-1,-1\rangle$ are relabeled as
$|1\rangle$, $|2\rangle$, $|3\rangle$, $|4\rangle$, $|5\rangle$, $|6\rangle$,
$|7\rangle$, $|8\rangle$ and $|9\rangle$, respectively in the following.

We measured the PL rates of different eigenstates via pulse sequences given in
Fig. S2. Firstly, after performing a laser pulse, the NV center was prepared
into the state $\rho_{i}=\lambda|1\rangle\langle 1|+p_{e}|4\rangle\langle
4|+\lambda|7\rangle\langle 7|$ with $p_{e}$ denoting the polarization of the
electron spin and $\lambda=(1-p_{e})/2$. Polarization of the nuclear spin was
high and can be regarded as 1 here. When all the pulses sequences shown in
Fig. S2 are taken into account, we can obtain the following equations

$\begin{bmatrix}\lambda&0&0&p_{e}&0&0&\lambda&0&0\\\
\lambda&0&0&0&0&p_{e}&\lambda&0&0\\\ p_{e}&0&0&\lambda&0&0&\lambda&0&0\\\
\lambda&0&p_{e}&0&0&0&\lambda&0&0\\\ \lambda&0&0&\lambda&0&0&p_{e}&0&0\\\
\lambda&0&0&0&0&0&\lambda&0&p_{e}\\\ \lambda&0&0&0&p_{e}&0&\lambda&0&0\\\
\lambda&p_{e}&0&0&0&0&\lambda&0&0\\\ \lambda&0&0&0&0&0&\lambda&p_{e}&0\\\
\lambda&0&0&p_{e}&0&0&0&\lambda&0\\\
\end{bmatrix}\begin{bmatrix}L_{|1\rangle}\\\ L_{|2\rangle}\\\ L_{|3\rangle}\\\
L_{|4\rangle}\\\ L_{|5\rangle}\\\ L_{|6\rangle}\\\ L_{|7\rangle}\\\
L_{|8\rangle}\\\ L_{|9\rangle}\end{bmatrix}=\begin{bmatrix}N_{1}\\\ N_{2}\\\
N_{3}\\\ N_{4}\\\ N_{5}\\\ N_{6}\\\ N_{7}\\\ N_{8}\\\ N_{9}\\\
N_{10}\end{bmatrix}\ ,$ (S5)

where $L_{|m\rangle}$ denotes the PL rate of state $|m\rangle$ ($m\in[1,9]$),
$N_{j}$ is the PL rate of the state obtained via applying the pulse sequence
in Fig. S2 to initialized state $\rho_{i}$. The polarization $p_{e}$ and the
PL rates $L_{|m\rangle}$ can be obtained by solving equation (S5).

Supplementary Figure S2: Normalization pulse sequences. $\rho_{i}$ denotes the
initialized state after a optical excitation. ${\rm\Pi}_{m,n}$ denotes the
selective $\pi$ pulse applied between state $|m\rangle$ and state $|n\rangle$.
$N_{j}$ ($j\in[1,10]$) indicates the detected PL rate after applying these
pulse sequence.

Suppose that the density matrix of the isotropic state we prepared takes the
form

$\rho_{f}=\begin{pmatrix}\rho_{11}&\rho_{12}&\rho_{13}&\rho_{14}&\rho_{15}&\rho_{16}&\rho_{17}&\rho_{18}&\rho_{19}\\\
\rho_{12}^{*}&\rho_{22}&\rho_{23}&\rho_{24}&\rho_{25}&\rho_{26}&\rho_{27}&\rho_{28}&\rho_{29}\\\
\rho_{13}^{*}&\rho_{23}^{*}&\rho_{33}&\rho_{34}&\rho_{35}&\rho_{36}&\rho_{37}&\rho_{38}&\rho_{39}\\\
\rho_{14}^{*}&\rho_{24}^{*}&\rho_{34}^{*}&\rho_{44}&\rho_{45}&\rho_{46}&\rho_{47}&\rho_{48}&\rho_{49}\\\
\rho_{15}^{*}&\rho_{25}^{*}&\rho_{35}^{*}&\rho_{45}^{*}&\rho_{55}&\rho_{56}&\rho_{57}&\rho_{58}&\rho_{59}\\\
\rho_{16}^{*}&\rho_{26}^{*}&\rho_{36}^{*}&\rho_{46}^{*}&\rho_{56}^{*}&\rho_{66}&\rho_{67}&\rho_{68}&\rho_{69}\\\
\rho_{17}^{*}&\rho_{27}^{*}&\rho_{37}^{*}&\rho_{47}^{*}&\rho_{57}^{*}&\rho_{67}^{*}&\rho_{77}&\rho_{78}&\rho_{79}\\\
\rho_{18}^{*}&\rho_{28}^{*}&\rho_{38}^{*}&\rho_{48}^{*}&\rho_{58}^{*}&\rho_{68}^{*}&\rho_{78}^{*}&\rho_{88}&\rho_{89}\\\
\rho_{19}^{*}&\rho_{29}^{*}&\rho_{39}^{*}&\rho_{49}^{*}&\rho_{59}^{*}&\rho_{69}^{*}&\rho_{79}^{*}&\rho_{89}^{*}&\rho_{99}\end{pmatrix}\
,$ (S6)

and some off-diagonal elements of the density matrix were redefined as

$\left\\{\begin{aligned} \rho_{15}&=\mu_{1}+i\nu_{1}\\\
\rho_{59}&=\mu_{2}+i\nu_{2}\\\ \rho_{19}&=\mu_{3}+i\nu_{3}\\\
\end{aligned}\right.\ .$ (S7)

From the measurement sequences displayed in Fig. S3, we can obtain the
following equations

$\displaystyle\setcounter{MaxMatrixCols}{15}\begin{bmatrix}L_{1}&L_{2}&L_{3}&L_{4}&L_{5}&L_{6}&L_{7}&L_{8}&L_{9}&0&0&0&0&0&0\\\
L_{1}&L_{2}&L_{3}&L_{5}&L_{6}&L_{4}&L_{7}&L_{8}&L_{9}&0&0&0&0&0&0\\\
L_{4}&L_{2}&L_{3}&L_{1}&L_{5}&L_{6}&L_{7}&L_{8}&L_{9}&0&0&0&0&0&0\\\
L_{1}&L_{2}&L_{4}&L_{5}&L_{6}&L_{3}&L_{7}&L_{8}&L_{9}&0&0&0&0&0&0\\\
L_{1}&L_{2}&L_{3}&L_{7}&L_{5}&L_{6}&L_{4}&L_{8}&L_{9}&0&0&0&0&0&0\\\
L_{1}&L_{2}&L_{3}&L_{5}&L_{6}&L_{9}&L_{7}&L_{8}&L_{4}&0&0&0&0&0&0\\\
L_{1}&L_{2}&L_{3}&L_{5}&L_{4}&L_{6}&L_{7}&L_{8}&L_{9}&0&0&0&0&0&0\\\
L_{1}&L_{4}&L_{3}&L_{5}&L_{2}&L_{6}&L_{7}&L_{8}&L_{9}&0&0&0&0&0&0\\\
L_{1}&L_{2}&L_{3}&L_{5}&L_{8}&L_{6}&L_{7}&L_{4}&L_{9}&0&0&0&0&0&0\\\
L_{11}&L_{2}&L_{3}&L_{1}&L_{11}&L_{6}&L_{7}&L_{8}&L_{9}&L_{13}&0&0&0&0&0\\\
L_{11}&L_{2}&L_{3}&L_{1}&L_{11}&L_{6}&L_{7}&L_{8}&L_{9}&0&L_{13}&0&0&0&0\\\
L_{1}&L_{2}&L_{3}&L_{5}&L_{11}&L_{9}&L_{7}&L_{8}&L_{12}&0&0&L_{14}&0&0&0\\\
L_{1}&L_{2}&L_{3}&L_{5}&L_{11}&L_{9}&L_{7}&L_{8}&L_{12}&0&0&0&L_{14}&0&0\\\
L_{11}&L_{2}&L_{3}&L_{1}&L_{6}&L_{9}&L_{7}&L_{8}&L_{11}&0&0&0&0&L_{13}&0\\\
L_{11}&L_{2}&L_{3}&L_{1}&L_{6}&L_{9}&L_{7}&L_{8}&L_{11}&0&0&0&0&0&L_{13}\end{bmatrix}\cdot\begin{bmatrix}\rho_{11}\\\
\rho_{22}\\\ \rho_{33}\\\ \rho_{44}\\\ \rho_{55}\\\ \rho_{66}\\\ \rho_{77}\\\
\rho_{88}\\\ \rho_{99}\\\ \mu_{1}\\\ \nu_{1}\\\ \mu_{2}\\\ \nu_{2}\\\
\mu_{3}\\\ \nu_{3}\end{bmatrix}=\begin{bmatrix}E_{1}\\\ E_{2}\\\ E_{3}\\\
E_{4}\\\ E_{5}\\\ E_{6}\\\ E_{7}\\\ E_{8}\\\ E_{9}\\\ E_{10}\\\ E_{11}\\\
E_{12}\\\ E_{13}\\\ E_{14}\\\ E_{15}\end{bmatrix}\ ,$ (S8)

where

$\left\\{\begin{aligned} L_{11}&=(L_{4}+L_{5})/2\\\ L_{12}&=(L_{4}+L_{6})/2\\\
L_{13}&=L_{5}-L_{4}\\\ L_{14}&=L_{4}-L_{6}\end{aligned}\right.\ ,$ (S9)

and $E_{j}$ is the PL rate of the state obtained via applying the pulse
sequence in Fig. S3 to state $\rho_{f}$. The theoretical non-zero elements of
the isotropic state can be obtained by solving equation (S8). We also measured
a few matrix elements that were supposed to be zero theoretically and the
experimental data show that these elements were very close to zero.

Supplementary Figure S3: Measurement pulse sequences of the isotropic state.
$\rho_{f}$ denotes the density matrix of the prepared isotropic state.
${\rm\Pi}_{m,n}^{\rm X(Y)}$ (${\rm h}{\rm\Pi}_{m,n}^{\rm X(Y)}$) denotes the
selective $\pi$ ($\pi/2$) pulse applied between state $|m\rangle$ and state
$|n\rangle$ along X (Y) axis. $E_{j}$ ($j\in[1,15]$) indicates the detected PL
rate after applying these pulse sequence.

###  I.3 Maximum likelihood estimation of the isotropic state.

The obtained density matrix can violate basic physical properties, so we
employed a maximum likelihood estimation (MLE) method to acquire the most
possible physical state PRA_2001_Daniel . The likelihood function we adopted
reads:

$\displaystyle\mathcal{L}=\sum_{ij}\frac{(\sigma_{ij}-\rho_{ij})^{2}}{2\sigma_{ij}},$
(S10)

where $\rho_{ij}$ ($\sigma_{ij}$) is the element of the experimental
(estimated) density matrix. The matrix $\sigma$ is constructed by

$\displaystyle\sigma$ $\displaystyle=\frac{T^{\dagger}(k)T(k)}{{\rm
tr}(T^{\dagger}(k)T(k))},$ (S11)

with

$\displaystyle T(k)$
$\displaystyle=\left(\begin{array}[]{ccccccccc}k_{1}&0&0&0&0&0&0&0&0\\\
0&k_{2}&0&0&0&0&0&0&0\\\ 0&0&k_{3}&0&0&0&0&0&0\\\ 0&0&0&k_{4}&0&0&0&0&0\\\
k_{10}&0&0&0&k_{5}&0&0&0&0\\\ 0&0&0&0&0&k_{6}&0&0&0\\\
0&0&0&0&0&0&k_{7}&0&0\\\ 0&0&0&0&0&0&0&k_{8}&0\\\
k_{11}&0&0&0&k_{12}&0&0&0&k_{9}\end{array}\right),$ (S21)

where $k_{i}$ ($i\in\\{1,2,3,...,12\\}$) is independent real parameter. The
final physical state can be reconstructed by varying parameter $k_{i}$ to
minimize the value of the Eq. (S10).

The photon counts of the measurement sequences fluctuate due to the shot
noise, which leads to uncertainties of the elements of the experimental
reconstructed density matrix. We generated 100 different experimental density
matrices ($\rho^{M}$, $M\in\\{1,2,3,\dots,100\\}$) by Monte Carlo Simulations
with Gaussian statistics. Maximum likelihood estimation was applied for each
$\rho^{M}$ to obtain the corresponding $\sigma^{M}$. The average estimated
density matrix $\overline{\sigma^{M}}$ was calculated as the final state
(Fig.3b in the main text as an instance). We calculated the fidelity
$F(\sigma^{M},\rho_{\rm simu}(p))$ to find $p^{M}$ that maximizes
$F(\sigma^{M},\rho_{\rm simu}(p))$. Then the horizontal coordinates of the
data in Fig.4 of the main text equal the average value of $p^{M}$ and the
horizontal error bars are the standard derivation of $p^{M}$.

##  II Calculation of the quantum discord and the quantum entanglement

The QD is defined by Eq.(3) in the main text. It is challenging to calculate
the QD of a $d$-level system because the optimization needs to be executed in
the parameter space with dimension $d(d-1)$ review_2018_Bera . We used the
method introduced in Ref. PRA_2012_Rossignoli,  to calculate the QD of our
two-qutrit system, where the measurement operators were parameterized by six
parameters $\alpha$, $\beta$, $\gamma$, $\psi$, $\theta$ and $\phi$.
Parameters $\alpha,\beta\in[0,\pi]$ and $\gamma\in(-\pi/2,\pi/2]$ were used to
construct a group of orthonormal bases of a qutrit system given by

$\left\\{\begin{aligned}
|{+1}_{r}\rangle&=\cos\beta(e^{-i\phi_{0}}\cos\alpha|{+1}\rangle+e^{i\phi_{0}}\sin\alpha|{-1}\rangle)-\sin\beta
e^{-i\gamma}|0\rangle\\\
|0_{r}\rangle&=\sin\beta(e^{-i\phi_{0}}\cos\alpha|{+1}\rangle+e^{i\phi_{0}}\sin\alpha|{-1}\rangle)+\cos\beta
e^{-i\gamma}|0\rangle\\\
|{-1}_{r}\rangle&=-e^{-i\phi_{0}}\sin\alpha|{+1}\rangle+e^{i\phi_{0}}\cos\alpha|{-1}\rangle\end{aligned}\right.\
,$

where $\tan\phi_{0}=\tan\gamma\tan(\pi/4-\alpha)$. Then parameters
$\psi,\theta,\phi\in[0,2\pi]$ were used to define a general rotation of a
qutrit system with the form

$R(\psi,\theta,\phi)=e^{-i\psi S_{z}}e^{-i\theta S_{y}}e^{-i\phi S_{z}},$
(S22)

where $S_{x}$, $S_{y}$ and $S_{z}$ are the spin operators of a spin-1 system.
A set of orthogonal measurement bases of a qutrit system can be constructed by
applying rotation $R(\psi,\theta,\phi)$ to the basis given in equation (III).

The QD was obtained by traversing all the projective measurements with the
basis defined above. We discretized the parameter space and used MATLAB to
execute the calculations. With the increment of the discretization of the
parameters, the calculation converges and then the QD was obtained.

The quantum entanglement was characterized by the negativity PRA_2007_Derkacz
of the density matrix defined as

$N(\rho_{\rm iso})=\frac{{||}\rho_{\rm iso}^{PT}{||}_{1}-1}{2},$ (S23)

where $\rho_{\rm iso}^{PT}$ is the partial transposition of state $\rho_{\rm
iso}$ and $||\cdot||_{1}$ denotes the trace norm. We calculated the quantum
discord and the quantum entanglement for isotropic states with different
values of the parameter $p$ and the results are shown in Fig.4 of the main
text.

##  III  Alignment of the magnetic field to the NV symmetry axis and
polarization of the nuclear spin.

At around 500 G, the PL rate of the NV center drops rapidly when the direction
of the magnetic field deviates from the NV symmetry axis PRL_V.Jacques . This
property was utilized to realize refined alignment of the magnetic field to
the NV symmetry axis. We adjusted the position of the magnet and meanwhile
recorded the PL rate of the NV center to find the best position of the magnet.
The results are shown in Fig.S4.

Supplementary Figure S4: Alignment of the magnetic field. (a) and (b) shows
the PL rate of the NV center versus the position of the X-axis and Y-axis
stages of the magnet respectively. Gauss fitting of the data gives the
position of the magnet that the direction of the magnetic field most aligns
with the NV symmetry axis.

Then, following the method introduced in the supplementary material of
Nature_2012_Sar , we measured the polarization of the nuclear spin via pulse
sequences shown in Fig S5. The pulse sequences in Fig S5 lead to the following
equations

$\left\\{\begin{aligned} N_{11}&=\lambda\cdot p_{n}\cdot
l_{1}+\lambda\cdot(1-p_{n})\cdot l_{2}+p_{e}\cdot p_{n}\cdot
l_{4}+p_{e}\cdot(1-p_{n})\cdot l_{5}+\lambda\cdot p_{n}\cdot
l_{7}+\lambda\cdot(1-p_{n})\cdot l_{8}\\\ N_{12}&=\lambda\cdot p_{n}\cdot
l_{1}+\lambda\cdot(1-p_{n})\cdot l_{2}+p_{e}\cdot p_{n}\cdot
l_{4}+\lambda\cdot(1-p_{n})\cdot l_{5}+\lambda\cdot p_{n}\cdot
l_{7}+p_{e}\cdot(1-p_{n})\cdot l_{8}\\\ N_{13}&=\lambda\cdot p_{n}\cdot
l_{1}+\lambda\cdot(1-p_{n})\cdot l_{2}+\lambda\cdot(1-p_{n})\cdot
l_{4}+p_{e}\cdot p_{n}\cdot l_{5}+p_{e}\cdot(1-p_{n})\cdot l_{7}+\lambda\cdot
p_{n}\cdot l_{8}\\\ N_{14}&=\lambda\cdot p_{n}\cdot
l_{1}+\lambda\cdot(1-p_{n})\cdot l_{2}+\lambda\cdot(1-p_{n})\cdot
l_{4}+\lambda\cdot p_{n}\cdot l_{5}+p_{e}\cdot(1-p_{n})\cdot l_{7}+p_{e}\cdot
p_{n}\cdot l_{8}\end{aligned}\right.\ .$

Solving the equations above, we can obtain

$p_{n}=\frac{N_{13}-N_{14}}{N_{11}-N_{12}+N_{13}-N_{14}}.$ (S24)

Our experimental result shows that $p_{n}=0.981(5)$ for three million loops of
these pulse sequences. The high polarization of the nuclear spin also
indicates that the misalignment of the magnetic field to the NV symmetry axis
should be very small since the nuclear polarization also decreases rapidly
when the direction of the magnetic field deviates from the NV symmetry axis.

Supplementary Figure S5: Measurement of the nuclear spin polarization.
$\rho_{i}$ denotes the initialized state after an optical excitation.
${\rm\Pi}_{m,n}$ denotes the selective $\pi$ pulse applied between state
$|m\rangle$ and state $|n\rangle$. $N_{j}$ ($j\in[11,14]$) indicates the
detected PL rate after applying these pulse sequence.

When calculating the experimental density matrix of the isotropic state
$\rho_{\rm iso}$ in section $\rm{I}$, we assumed that $p_{n}=1$. Utilizing
$p_{n}=0.981$, a modified density matrix can be obtained as $\rho_{\rm
iso}^{\prime}(p)$. The difference between $\rho_{\rm iso}$ and $\rho_{\rm
iso}^{\prime}(p)$ is very subtle. For example, when $p=0.5$, the fidelity
between $\rho_{\rm iso}$ and $\rho_{\rm iso}^{\prime}(p)$ is 0.9996. Thus the
assumption $p_{n}=1$ is reasonable.

## References

  * (1) V. Jacques et al, Dynamic Polarization of Single Nuclear Spins by Optical Pumping of Nitrogen-Vacancy Color Centers in Diamond at Room Temperature, _Phys. Rev. Lett._ 102, 057403 (2009).
  * (2) D. F. James, and P. G. Kwiat, W. J. Munro, and A. G. White, Measurement of qubits, _Phys. Rev. A_ 64, 052312 (2001).
  * (3) A. Bera, T. Das, D. Sadhukhan, S. S. Roy, A. Sen(De), and U. Sen, Quantum discord and its allies: a review of recent progress, _Rep. Prog. Phys._ 81, 024001 (2018).
  * (4) R. Rossignoli, J. M. Matera, and N. Canosa, Measurements, quantum discord, and parity in spin-1 systems, _Phys. Rev. A_ 86, 022104 (2012).
  * (5) Ł. Derkacz and L. Jakóbczyk, Entanglement versus entropy for a class of mixed two-qutrit states, _Phys. Rev. A_ 76, 042304 (2007).
  * (6) T.V. Sar et al, Decoherence-protected quantum gates for a hybrid solid-state spin register, _Nature_ 484, 82-86 (2012).


#  3d steady Gradient Ricci Solitons with linear curvature decay

Yuxing Deng  and  Xiaohua $\text{Zhu}^{*}$ Yuxing Deng
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875,
China
<EMAIL_ADDRESS>Xiaohua Zhu
School of Mathematical Sciences and BICMR, Peking University, Beijing, 100871,
China
<EMAIL_ADDRESS>
###### Abstract.

In this note, we prove that a 3-dimensional steady Ricci soliton is
rotationally symmetric if its scalar curvature $R(x)$ satisfies

$\frac{C_{0}^{-1}}{\rho(x)}\leq R(x)\leq\frac{C_{0}}{\rho(x)}$

for some constant $C_{0}>0$, where $\rho(x)$ denotes the distance from a fixed
point $x_{0}$. Our result doesn’t assume that the soliton is
$\kappa$-noncollapsed.

###### Key words and phrases:

Ricci flow, Ricci soliton, $\kappa$-solution

###### 2000 Mathematics Subject Classification:

Primary: 53C25; Secondary: 53C55, 58J05

* Partially supported by the NSFC Grants 11271022 and 11331001

##

##  1\. Introduction

In his celebrated paper [13], Perelman conjectured that all 3-dimensional
$\kappa$-noncollapsed steady (gradient) Ricci solitons must be rotationally
symmetric. The conjecture is solved by Brendle in 2012 [1]. For a general
dimension $n\geq 3$, under an extra condition that the soliton is
asymptotically cylindrical, Brendle also proves that any $\kappa$-noncollapsed
steady Ricci soliton with positive sectional curvature must be rotationally
symmetric [2]. In general, it is still open whether an $n$-dimensional
$\kappa$-noncollapsed steady Ricci soliton with positive curvature operator is
rotationally symmetric for $n\geq 4$. For $\kappa$-noncollapsed steady Kähler-
Ricci solitons with nonnegative bisectional curvature, the authors have
recently proved that they must be flat [10], [11].

Recall from [2],

######  Definition 1.1.

An $n$-dimensional steady Ricci soliton $(M,g,f)$ is called asymptotically
cylindrical if the following holds:

(i) Scalar curvature $R(x)$ of $g$ satisfies

$\frac{C_{0}^{-1}}{\rho(x)}\leq
R(x)\leq\frac{C_{0}}{\rho(x)},~{}\forall~{}\rho(x)\geq r_{0},$

where $C_{0}>0$ is a constant and $\rho(x)$ denotes the distance of $x$ from a
fixed point $x_{0}$.

(ii) Let $p_{m}$ be an arbitrary sequence of marked points going to infinity.
Consider rescaled metrics $g_{m}(t)=r_{m}^{-1}\phi^{*}_{r_{m}t}g,$ where
$r_{m}R(p_{m})=\frac{n-1}{2}+o(1)$ and $\phi_{t}$ is a one-parameter subgroup
generated by $X=-\nabla f$. As $m\to\infty,$ flows $(M,g_{m}(t),p_{m})$
converge in the Cheeger-Gromov sense to a family of shrinking cylinders
$(\mathbb{R}\times\mathbb{S}^{n-1}(1),\widetilde{g}(t)),t\in(0,1).$ The metric
$\widetilde{g}(t)$ is given by

(1.1)
$\displaystyle\widetilde{g}(t)=dr^{2}+(n-2)(2-2t)g_{\mathbb{S}^{n-1}(1)},$

where $\mathbb{S}^{n-1}(1)$ is the unit sphere of euclidean space.

In this note, we discuss 3-dimensional steady (gradient) Ricci solitons
without assuming the $\kappa$-noncollapsed conditon.111It is proved by Chen
that any 3-dimensional ancient solution has nonnegative sectional curvature
[7]. We prove

######  Theorem 1.2.

Let $(M,g,f)$ be a 3-dimensional steady Ricci soliton. Then, it is
rotationally symmetric if the scalar curvature $R(x)$ of $(M,g,f)$ satisfies

(1.2) $\displaystyle\frac{C_{0}^{-1}}{\rho(x)}\leq
R(x)\leq\frac{C_{0}}{\rho(x)},$

for some constant $C>0$, where $\rho(x)$ denotes the distance from a fixed
point $x_{0}$.

Under the condition (1.2), we need to check the property (ii) in Definition
1.1 to prove Theorem 1.2. Actually, we show that for any sequence
$p_{i}\rightarrow\infty$, there exists a subsequence
$p_{i_{k}}\rightarrow\infty$ such that

$\displaystyle(M,g_{p_{i_{k}}}(t),p_{i_{k}})\rightarrow(\mathbb{R}\times\mathbb{S}^{2},\widetilde{g}(t),p_{\infty}),~{}for~{}t\in(-\infty,1),$

where $g_{p_{i_{k}}}(t)=R(p_{i_{k}})g(R^{-1}(p_{i_{k}})t)$ and
$(\mathbb{R}\times\mathbb{S}^{2},\widetilde{g}(t))$ is a shrinking cylinders
flow, i.e.

$\widetilde{g}(t)=dr^{2}+(2-2t)g_{\mathbb{S}^{2}}.$

As in [10], we study the geometry of neighborhood $M_{p,k}=\\{x\in
M|~{}f(p)-\frac{k}{\sqrt{R(p)}}\leq f(x)\leq f(p)+\frac{k}{\sqrt{R(p)}}\\}$
around level set $\Sigma_{r}=\\{f(x)=f(p)=r\\}$ for any $p\in M$. We are able
to give a uniform injective radius estimate for $(M,R_{p_{i}}g)$ at each
sequence of $p_{i}$. Then we can still get a limit flow for rescaled flows
$(M,g_{p_{i}}(t))$, which will split off a line. By using a classification
result of Daskalopoulos-Hamilton-Sesum for ancient flows on a compact surface
[8], we finish the proof of Theorem 1.2.

We remark that the curvature condition in Theorem 1.2 cannot be removed, since
there does exist a 3-dimensional non-flat steady Ricci soliton with
exponential curvature decay. For example,
$(\mathbb{R}^{2}\times\mathbb{S}^{1},g_{cigar}+ds^{2})$, where
$(\mathbb{R}^{2},g_{cigar})$ is a cigar soliton. Also, Theorem 1.2 is not true
for dimension $n\geq 4$ by Cao’s examples of steady Kähler-Ricci solitons with
positive sectional curvature [3].

At last, we remark that it is still open wether there exists a 3-dimensional
collapsed steady Ricci soliton with positive curvature. Hamilton has
conjectured that there should exist a family of collapsed 3-dimensional
complete gradient steady Ricci solitons with positive curvature and
$S^{1}$-symmetry (cf. [5]). Our result shows that the curvature of Hamilton’s
examples could not have a linear decay.

Acknowledgements. The work was done partially when the second author was
visiting at the Mathematical Sciences Research Institute at Berkeley during
the spring 2016 semester. The author would like to thank her hospitality and
the financial supports, National Sciences Foundation under Grant No.
DMS-1440140, and Simons Foundation.

##  2\. Positivity of Ricci curvature

$(M,g,f)$ is called a gradient steady Ricci soliton if a Riemannian metric $g$
on $M$ satisfies

(2.1) $\displaystyle{\rm Ric}(g)=\nabla^{2}f,$

for some smooth function $f$. We first show the positivity of Ricci curvature
of $(M,g,f)$ under (1.2) of Theorem 1.2.

######  Lemma 2.1.

Under (1.2), $(M,g,f)$ has positive sectional curvature.

###### Proof.

We need to show that $(M,g)$ has positive Ricci curvature. On the contrary,
$(M,g)$ locally splits off a flat piece of line by Shi’s splitting theorem
[14]. Then, the universal cover $(\widetilde{M},\widetilde{g})$ of $(M,g)$ is
isometric to a product Riemannian manifold of a real line and the cigar
soliton. Namely,
$(\widetilde{M},\widetilde{g})=(\mathbb{R}^{2}\times\mathbb{R},g_{cigar}+ds^{2})$.
Let $\pi:\widetilde{M}\to M$ be a universal covering. We fix $x_{0}\in M$ and
$\widetilde{x}_{0}\in\widetilde{M}$ such that $\pi(\widetilde{x}_{0})=x_{0}$.
For any $x\in M$ and $\widetilde{x}\in\widetilde{M}$ such that
$\pi(\widetilde{x})=x$, one sees

(2.2)
$\displaystyle\rho(x,x_{0})\leq\widetilde{\rho}(\widetilde{x},\widetilde{x}_{0}),$

where $\rho$ and $\widetilde{\rho}$ are the distance functions w.r.t $g$ and
$\widetilde{g}$ respectively. Let $\\{\widetilde{x}_{i}\\}_{i\geq 1}$ be a
sequence of points so that
$\widetilde{x}_{i}=(p_{i},0)\in\mathbb{R}^{2}\times\mathbb{R}$ tend to
infinity. Then, one may check that

(2.3)
$\displaystyle\widetilde{R}(\widetilde{x}_{i})\rho(\widetilde{x}_{i},\widetilde{x}_{0})\to
0,~{}{\rm as}~{}i\to\infty.$

Since $R(x_{i})=\widetilde{R}(\widetilde{x}_{i})\to 0$, where
$x_{i}=\pi(\widetilde{x}_{i})$, we see $d(x_{i},x_{0})\to\infty$ by (1.2).
Again by (1.2) and (2.2), we get

(2.4) $\displaystyle C_{1}\leq
R(x_{i})d(x_{i},x_{0})\leq\widetilde{R}(\widetilde{x}_{i})d(\widetilde{x}_{i},\widetilde{x}_{0}).$

This is a contradiction to (2.3). Hence, the lemma is proved. ∎

######  Corollary 2.2.

$(M,g,f)$ in Theorem 1.2 has a unique equilibrium point $o$, i.e., $\nabla
f(o)=0$. As a consequence, $\Sigma_{r}=\\{f(x)=r\\}$ is diffeomorphic to
$\mathbb{S}^{2}$, for any $r>f(o)$.

###### Proof.

Note that

(2.5) $\displaystyle|\nabla f|^{2}+R=A.$

By taking covariant derivatives on both sides of (2.5), it follows

(2.6) $\displaystyle 2{\rm Ric}(\nabla f,\nabla f)=-\langle\nabla R,\nabla
f\rangle.$

On the the hand, by (1.2), there exists a point $o$ such that

$\sup_{M}R(x)=R(o)=R_{{\rm max}}.$

In particular, $\nabla R(o)=0.$ Thus

${\rm Ric}(\nabla f,\nabla f)=0.$

By Lemma 2.1, $\nabla f(o)=0$. The uniqueness also follows from the positivity
of Ricci curvature.

By the Morse theorem, $\Sigma_{r}=\\{f(x)=r>f(o)\\}$ is diffeomorphic to
$\mathbb{S}^{2}$ (cf. [10], Lemma 2.1).

∎

##  3\. Geometry of $M_{p,k}$

For any $p\in M$ and number $k>0$, we set

$M_{p,k}=\\{x\in M|~{}f(p)-\frac{k}{\sqrt{R(p)}}\leq f(x)\leq
f(p)+\frac{k}{\sqrt{R(p)}}\\}.$

Let $g_{p}=R(p)g$ be a rescaled metric and denote $B(p,r;g_{p})$ a
$r$-geodesic ball centered at $p$ with respect to $g_{p}$. Then by Corollary
2.2, we have (cf. [10], Lemma 3.3)

######  Lemma 3.1.

Under (1.2), for any $p\in M$ and number $k>0$ with
$f(p)-\frac{k}{\sqrt{R(p)}}>f(o)$, it holds

(3.1) $\displaystyle B(p,\frac{k}{\sqrt{R_{max}}};g_{p})\subset M_{p,k}.$

By Lemma 3.1, we prove

######  Lemma 3.2.

Under (1.2), there exists a constant $C$ such that

(3.2) $\displaystyle\frac{|\Delta R|(p)}{R^{2}(p)}\leq
C,~{}\forall~{}p\in~{}M.$

###### Proof.

Fix any $p\in M$ with $f(p)\geq r_{0}>>1$. Then

(3.3) $\displaystyle|f(x)-f(p)|\leq\frac{1}{\sqrt{R(p)}},~{}\forall~{}x\in
M_{p,1}.$

It is known by [4],

(3.4) $\displaystyle c_{1}\rho(x)\leq f(x)\leq
c_{2}\rho(x),~{}\forall~{}\rho(x)\geq r_{0}.$

Thus by (1.2), (3.3) and (3.4), we get

$\displaystyle c_{2}\rho(x)\geq f(p)-\frac{1}{\sqrt{R(p)}}\geq
c_{1}\rho(p)-\sqrt{C_{0}\rho(p)}.$

It follows

(3.5) $\displaystyle\frac{R(x)}{R(p)}\leq
C_{0}^{2}\frac{\rho(p)}{\rho(x)}\leq\frac{2c_{2}C_{0}^{2}}{c_{1}},~{}\forall
x\in M_{p,1}.$

On the other hand, by (3.1), we have

$\displaystyle B(p,\frac{1}{\sqrt{R_{\max}}};g_{p})\subseteq M_{p,1}.$

Hence

(3.6) $\displaystyle R(x)\leq C^{\prime}R(p),~{}\forall~{}x\in
B(p,\frac{1}{\sqrt{R_{\max}}};g_{p}).$

Let $\phi_{t}$ be generated by $-\nabla f$. Then $g(t)=\phi_{t}^{\ast}g$
satisfies the Ricci flow,

(3.7) $\displaystyle\frac{\partial g(t)}{\partial t}$ $\displaystyle=-2{\rm
Ric}(g(t)).$

Also rescaled flow $g_{p}(t)=R(p)g(R^{-1}(p)t)$ satisfies (3.7). Since the
Ricci curvature is positive,

$\displaystyle B(p,\frac{1}{\sqrt{R_{\max}}};g_{p}(-t))\subseteq
B(p,\frac{1}{\sqrt{R_{\max}}};g_{p}(0)),~{}t\in~{}[-1,0].$

Combining with (3.6), we get

(3.8) $\displaystyle R_{g_{p}(t)}(x)\leq C^{\prime},~{}\forall~{}x\in
B(p,\frac{1}{\sqrt{R_{\max}}};g_{p}(0)),~{}t\in[-1,0].$

Thus, by Shi’s higher order estimates, we obtain

$\displaystyle|\Delta_{g_{p}(t)}R_{g_{p}(t)}|(x)\leq
C_{1}^{\prime},~{}\forall~{}x\in
B(p,\frac{1}{2\sqrt{R_{\max}}};g_{p}(-1)),~{}t\in[-\frac{1}{2},0].$

It follows

$\displaystyle|\Delta R|(x)\leq C_{1}^{\prime}R^{2}(p),~{}\forall~{}x\in
B(p,\frac{1}{2\sqrt{R_{\max}}};g_{p}(-1)).$

In particular, we have

$\displaystyle|\Delta R|(p)\leq C_{1}^{\prime}R^{2}(p),~{}{\rm
as}~{}\rho(p)\geq r_{0}.$

The lemma is proved. ∎

######  Remark 3.3.

Under (1.2), by the same argument as in the proof of Lemma 3.2, for each
$k\in\mathbb{N}$, there exists a constant $C(k)$ such that

(3.9) $\displaystyle\frac{|\nabla^{k}R|(p)}{R^{\frac{k+2}{2}}(p)}\leq
C(k),~{}\forall~{}p\in~{}M.$

Next, we want to show that $M_{p,k}$ is bounded by a finite ball
$B(p,Ck;g_{p})$, where $C$ is a uniform constant. We need to use the Gauss
formula,

$\displaystyle R(X,Y,Z,W)=\overline{R}(X,Y,Z,W)+\langle
B(X,Z),B(Y,W)\rangle-\langle B(X,W),B(Y,Z)\rangle,$

where $X,Y,Z,W\in T\Sigma_{r}$ and $B(X,Y)=(\nabla_{X}Y)^{\bot}$. Note that

$\displaystyle B(X,Y)=$ $\displaystyle\langle\nabla_{X}Y,\nabla
f\rangle\cdot\frac{\nabla f}{|\nabla f|^{2}}$ $\displaystyle=$
$\displaystyle[\nabla_{X}\langle Y,\nabla f\rangle-\langle Y,\nabla_{X}\nabla
f\rangle]\cdot\frac{\nabla f}{|\nabla f|^{2}}$ $\displaystyle=$
$\displaystyle-{\rm Ric}(X,Y)\cdot\frac{\nabla f}{|\nabla f|^{2}}.$

We choose a normal basis $\\{e_{1},e_{2}\\}$ on $(\Sigma_{r},\bar{g})$ with
the induced metric $\bar{g}$. Then $\\{e_{1},e_{2},\frac{\nabla f}{|\nabla
f|}\\}$ spans a normal basis of $(M,g)$. Thus

$\displaystyle R_{11}=$ $\displaystyle\overline{R}_{11}+R(\frac{\nabla
f}{|\nabla f|},e_{1},e_{1},\frac{\nabla f}{|\nabla
f|})-\frac{R_{11}R_{22}-R_{12}R_{21}}{|\nabla f|^{2}},$ $\displaystyle
R_{22}=$ $\displaystyle\overline{R}_{22}+R(\frac{\nabla f}{|\nabla
f|},e_{2},e_{2},\frac{\nabla f}{|\nabla
f|})-\frac{R_{11}R_{22}-R_{12}R_{21}}{|\nabla f|^{2}}.$

Since $(\Sigma_{r},\bar{g})$ is a surface,
$K=\overline{R}_{11}=\overline{R}_{22}$. Hence, we get

######  Lemma 3.4.

The Gauss curvature of $(\Sigma_{r},\bar{g})$ is given by

(3.10) $\displaystyle K=\frac{R}{2}-\frac{{\rm Ric}(\nabla f,\nabla
f)}{|\nabla f|^{2}}+\frac{R_{11}R_{22}-R_{12}R_{21}}{|\nabla f|^{2}}.$

######  Lemma 3.5.

Under (1.2), there exists a uniform $B>0$ such that the following is true: for
any $k\in\mathbb{N}$, there exists $\bar{r}_{0}=\bar{r}_{0}(k)$ such that

(3.11) $\displaystyle M_{p,k}\subset
B(p,2\pi\sqrt{B}+\frac{2k}{\sqrt{R_{\max}}};g_{p}),~{}\forall~{}\rho(p)\geq\bar{r}_{0}.$

###### Proof.

By (1.2) and (3.4), we have

(3.12) $\displaystyle\frac{R_{\max}}{2}\leq|\nabla f|^{2}(x)\leq
R_{\max},~{}\forall~{}x\in M_{p,k},$

as long as $\rho(p)\geq r_{0}>>1.$ Then by Lemma 3.4 and Lemma 3.2, we get

$\displaystyle|K-\frac{R}{2}|=$ $\displaystyle|-\frac{{\rm Ric}(\nabla
f,\nabla f)}{|\nabla f|^{2}}+\frac{R_{11}R_{22}-R_{12}R_{21}}{|\nabla
f|^{2}}|$ $\displaystyle\leq$ $\displaystyle\frac{|\langle\nabla R,\nabla
f\rangle|}{2|\nabla f|^{2}}+\frac{R^{2}}{|\nabla f|^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{|\Delta R+2|{\rm Ric}|^{2}|}{2|\nabla
f|^{2}}+\frac{R^{2}}{|\nabla f|^{2}}$ $\displaystyle\leq$
$\displaystyle\frac{(C+4)R^{2}}{R_{\max}}.$

It follows

(3.13) $\displaystyle\frac{R(x)}{4}\leq
K(x)\leq\frac{3R(x)}{4},~{}\forall~{}x\in M_{p,k},~{}\rho(p)\geq r_{0}.$

On the other hand, by (1.2), (3.3) and (3.4), we see

$\displaystyle c_{2}^{-1}\Big{(}c_{1}\rho(p)-k\sqrt{\rho(p)C_{0}}\Big{)}$
$\displaystyle\leq\rho(x)$ (3.14) $\displaystyle\leq
c_{1}^{-1}(c_{2}\rho(p)+k\sqrt{\rho(p)C_{0}}),~{}\forall~{}x\in M_{p,k},$

as long as $\rho(p)\geq r_{0}.$ Then similar to (3.5), there exists a
$\bar{r}_{0}\geq r_{0}$ such that

(3.15) $\displaystyle
R(x)\geq\frac{c_{1}}{2c_{2}C_{0}^{2}}R(p),~{}\forall~{}x\in M_{p,k}.$

Thus by (3.13), we get

$\displaystyle\overline{R}_{ij}\geq
B^{-1}R(p)\overline{g}_{ij},~{}\forall~{}x\in\Sigma_{f(p)},~{}\rho(p)\geq\bar{r}_{0},$

where $B>0$ is a uniform constant. By the Myer’s theorem, the diameter of
$\Sigma_{f(p)}$ is bounded by

$\displaystyle{\rm diam}(\Sigma_{f(p)},g)\leq{\rm
diam}(\Sigma_{f(p)},\overline{g}_{f(p)})\leq 2\pi\sqrt{\frac{B}{R(p)}}.$

As a consequence,

(3.16) $\displaystyle\Sigma_{f(p)}\subset B(p,2\pi\sqrt{B};R(p)g).$

For any $q\in M_{p,k}$, there exists $q^{\prime}\in\Sigma_{f(p)}$ such that
$\phi_{s}(q)=q^{\prime}$ for some $s\in\mathbb{R}$. Then by (3.16) and (3.12),
we have

$\displaystyle d(q,p)\leq$ $\displaystyle d(q^{\prime},p)+d(q,q^{\prime})$
$\displaystyle\leq$ $\displaystyle{\rm
diam}(\Sigma_{f(p)},g)+\mathcal{L}(\phi_{\tau}|_{[0,s]})$ $\displaystyle\leq$
$\displaystyle
2\pi\sqrt{\frac{B}{R(p)}}+|\int_{0}^{s}|\frac{d\phi_{\tau}(q)}{d\tau}|d\tau|$
$\displaystyle=$ $\displaystyle 2\pi\sqrt{\frac{B}{R(p)}}+\int_{0}^{s}|\nabla
f(\phi_{\tau}(q))|d\tau$ $\displaystyle\leq$ $\displaystyle
2\pi\sqrt{\frac{B}{R(p)}}+\int_{0}^{s}|\nabla
f(\phi_{\tau}(q))|^{2}\cdot\frac{2}{\sqrt{R_{\max}}}d\tau$ $\displaystyle=$
$\displaystyle
2\pi\sqrt{\frac{B}{R(p)}}+|\int_{0}^{s}\frac{d(f(\phi_{\tau}(q)))}{d\tau}\cdot\frac{2}{\sqrt{R_{\max}}}d\tau|$
$\displaystyle\leq$ $\displaystyle
2\pi\sqrt{\frac{B}{R(p)}}+|f(q)-f(p)|\cdot\frac{2}{\sqrt{R_{\max}}}$
$\displaystyle\leq$
$\displaystyle\Big{(}2\pi\sqrt{B}+\frac{2k}{\sqrt{R_{\max}}}\Big{)}\cdot\frac{1}{\sqrt{R(p)}}.$

Thus

$\displaystyle M_{p,k}\subset
B(p,2\pi\sqrt{B}+\frac{2k}{\sqrt{R_{\max}}};R(p)g).$

The lemma is proved.

∎

By Lemma 3.5, we get the following volume estimate of $B(p,s;g_{p})$.

######  Proposition 3.6.

Under (1.2) of Theorem 1.2, there exists $s_{0}$ and ${c}>0$ such that

(3.17) $\displaystyle{\rm Vol}B(p,s;g_{p})\geq cs^{3},~{}\forall~{}s\leq
s_{0}~{}{\rm and}~{}\rho(p)\geq r_{0}>>1.$

Moreover, the injective radius of $(M,g_{p})$ at $p$ has a uniform lower bound
$\delta>0$, i.e.,

(3.18) $\displaystyle{\rm inj}(p,g_{p})\geq\delta,~{}\forall~{}\rho(p)\geq
r_{0}.$

###### Proof.

By Lemma 3.5, we have

$\displaystyle M_{p,1}\subset
B(p,2\pi\sqrt{B}+\frac{2}{\sqrt{R_{\max}}};g_{p}).$

In the following, we give an estimate of ${\rm Vol}(\Sigma_{l},\bar{g})$ for
any $l$ with $f(p)-\frac{1}{\sqrt{R(p)}}\leq l\leq
f(p)+\frac{1}{\sqrt{R(p)}}$.

By (3.5) and (3.15), we see

$\displaystyle{C}_{1}^{-1}\leq\frac{R(x)}{R(p)}\leq{C}_{1},~{}\forall~{}\rho(p)\geq
r_{0}~{}{\rm and}~{}x\in M_{p,1}.$

By (3.13), it follows that the Gauss curvature $K_{l}$ of
$(\Sigma_{l},g_{p}|_{\Sigma_{l}})$ satisfies

$\displaystyle\frac{1}{4C_{1}}\leq K_{l}\leq\frac{3C_{1}}{4}.$

Thus

$\displaystyle{\rm Vol}(\Sigma_{l},\bar{g})=\frac{1}{R(p)}{\rm
Vol}(\Sigma_{l},g_{p}|_{\Sigma_{l}})\geq\frac{64\pi{C}_{1}}{R(p)}.$

By the Co-Area formula, we get

$\displaystyle{\rm Vol}(M_{p,1},g)=$
$\displaystyle\int_{f(p)-\frac{1}{\sqrt{R(p)}}}^{f(p)+\frac{1}{\sqrt{R(p)}}}\frac{{\rm
Vol}(\Sigma_{l},\bar{g})}{|\nabla f|}dl$ $\displaystyle\geq$ $\displaystyle
128\pi{C}_{1}R_{max}^{-\frac{1}{2}}R^{-\frac{3}{2}}(p).$

Hence

(3.19) $\displaystyle{\rm
Vol}(B(p,2\pi\sqrt{B}+\frac{2}{\sqrt{R_{\max}}};g_{p}))\geq{\rm
Vol}(M_{p,1},g_{p})\geq 128\pi{C}_{1}R_{max}^{-\frac{1}{2}}.$

By the volume comparison theorem, we derive from (3.19),

$\displaystyle\frac{{\rm Vol}(B(p,s;g_{p}))}{s^{3}}\geq$
$\displaystyle\frac{{\rm
Vol}(B(p,2\pi\sqrt{B}+\frac{2}{\sqrt{R_{\max}}};g_{p}))}{(2\pi\sqrt{B}+\frac{2}{\sqrt{R_{\max}}})^{3}}$
$\displaystyle\geq$
$\displaystyle\frac{128\pi{C}_{1}R_{max}^{-\frac{1}{2}}}{(2\pi\sqrt{B}+\frac{2}{\sqrt{R_{\max}}})^{3}},$

for any $s\leq 2\pi\sqrt{B}+\frac{2}{\sqrt{R_{\max}}}$. This proves (3.17). By
(3.17), we can apply a result of Cheeger-Gromov-Taylor for Riemannian
manifolds with bounded curvature to get the injective radius estimate (3.18)
immediately [6].

∎

##  4\. Proof of Theorem 1.2

First we prove the following convergence of rescaled flows.

######  Lemma 4.1.

Under (1.2), let $p_{i}\to\infty$. Then by taking a subsequence of $p_{i}$ if
necessary, we have

$\displaystyle(M,g_{p_{i}}(t),p_{i})\rightarrow(\mathbb{R}\times
N,\widetilde{g}(t);p_{\infty}),~{}for~{}t\in(-\infty,0],$

where $g_{p_{i}}(t)=R(p_{i})g(R^{-1}(p_{i})t)$, $\widetilde{g}(t)=ds\otimes
ds+g_{N}(t)$ and $(N,g_{N}(t))$ is an ancient solution of Ricci flow on $N$.

###### Proof.

For a fixed $\overline{r}$, as in (3.5), it is easy to see that there exists a
uniform $C_{1}$ independent of $\overline{r}$ such that

(4.1) $\displaystyle R(x)\leq C_{1}R(p_{i}),~{}\forall~{}x\in
M_{p_{i},\overline{r}\sqrt{R_{\max}}}$

as long as $i$ is large enough. By Lemma 3.1, it follows

$\displaystyle R_{g_{p_{i}}}(x)\leq C_{1},~{}\forall~{}x\in
B(p_{i},\overline{r};g_{p_{i}}),$

where $g_{p_{i}}=g_{p_{i}}(0)$. Since the scalar curvature is increasing along
the flow, we get

$\displaystyle|{\rm Rm}_{g_{p_{i}}(t)}(x)|_{g_{p_{i}}(t)}$ $\displaystyle\leq
3R_{g_{p_{i}}(t)}(x)$ $\displaystyle\leq 3R_{g_{p_{i}}}(x)\leq
3C_{1},~{}\forall~{}x\in B(p_{i},\overline{r};g_{p_{i}}),~{}t\in(-\infty,0].$

Thus together with the injective radius estimate in Proposition 3.6, we can
apply the Hamilton compactness theorem to see that $g_{p_{i}}(t)$ converges
subsequently to a limit flow $(\widetilde{M},\tilde{g}(t);p_{\infty})$ on
$t\in(-\infty,0]$ [12]. Moreover, the limit flow has uniformly bounded
curvature. It remains to prove the splitting property.

By Remark 3.3, we have

$|{\rm Ric}|(x)\leq CR(x),~{}\forall~{}x\in
B(p_{i},\overline{r};{g_{p_{i}}}).$

It follows from (4.1),

$\displaystyle|{\rm Ric}|(x)\leq CR(p_{i}),~{}\forall~{}x\in
B(p_{i},\overline{r};{g_{p_{i}}}).$

Let $X_{(i)}=R(p_{i})^{-\frac{1}{2}}\nabla f$. Then

$\displaystyle\sup_{B(p_{i},r_{0};{g_{p_{i}}})}|\nabla_{(g_{p_{i}})}X_{(i)}|_{g_{p_{i}}}$
$\displaystyle=\sup_{B(p_{i},r_{0};{g_{p_{i}}})}\frac{|{\rm
Ric}|}{\sqrt{R(p_{i})}}$ $\displaystyle\leq C\sqrt{R(p_{i})}\to 0.$

On the other hand, by Remark 3.3, we also have

$\sup_{B(p_{i},r_{0};{g_{p_{i}}})}|\nabla^{m}_{(g_{p_{i}})}X_{(i)}|_{g_{p_{i}}}\leq
C(n)\sup_{B(p_{i},r_{0};{g_{p_{i}}})}|\nabla^{m-1}_{(g_{p_{i}})}{\rm
Ric}({g_{p_{i}})}|_{g_{p_{i}}}\leq C_{1}.$

Thus $X_{(i)}$ converges subsequently to a parallel vector field
$X_{(\infty)}$ on $(\widetilde{M},\tilde{g}(0))$. Moreover,

$\displaystyle|X_{(i)}|_{g_{p_{i}}}(x)=|\nabla f|(p_{i})=\sqrt{R_{\rm
max}}+o(1)>0,~{}\forall~{}x\in B(p_{i},r_{0};{g_{i}}),$

as long as $f(p_{i})$ is large enough. This implies that $X_{(\infty)}$ is
non-trivial. Hence, $(\widetilde{M},\widetilde{g}(t))$ locally splits off a
piece of line along $X_{(\infty)}$. It remains to show that $X_{(\infty)}$
generates a line through $p_{\infty}$.

By Lemma 3.5,

$\displaystyle M_{p_{i},k}\subset
B(p_{i},2\pi\sqrt{B}+\frac{2k}{\sqrt{R_{\max}}};g_{p_{i}}(0)),~{}\forall~{}p_{i}\to\infty.$

Let $\gamma_{i,k}(s)$, $s\in(-D_{i,k},E_{i,k})$ be an integral curve generated
by $X_{(i)}$ through $p_{i}$, which restricted in $M_{p,k}$. Then
$\gamma_{i,k}(s)$ converges to a geodesic $\gamma_{\infty}(s)$ generated by
$X_{(\infty)}$ through $p_{\infty}$, which restricted in
$B(p_{\infty},2\pi\sqrt{B}+\frac{2k}{\sqrt{R_{\max}}};\widetilde{g}(0))$. If
let $L_{i,k}$ be lengths of $\gamma_{i,k}(s)$ and $L_{\infty,k}$ length of
$\gamma_{\infty}(s)$,

$\displaystyle L_{i,k}=$ $\displaystyle\int_{-D_{i,k}}^{E_{i,k}}|\nabla
f|_{g_{p_{i}}(0)}ds=\int_{f(p_{i})-\frac{k}{\sqrt{R(p_{i})}}}^{f(p_{i})+\frac{k}{\sqrt{R(p_{i})}}}\sqrt{R(p_{i})}\|\nabla
f\|_{g}$ $\displaystyle\geq
R_{max}^{-\frac{1}{2}}\int_{f(p_{i})-\frac{k}{\sqrt{R(p_{i})}}}^{f(p_{i})+\frac{k}{\sqrt{R(p_{i})}}}\sqrt{R(p_{i})}\|\nabla
f\|_{g}^{2}ds$
$\displaystyle=R_{max}^{-\frac{1}{2}}\int_{f(p_{i})-\frac{k}{\sqrt{R(p_{i})}}}^{f(p_{i})+\frac{k}{\sqrt{R(p_{i})}}}\sqrt{R(p_{i})}df=2R_{max}^{-\frac{1}{2}}k,$

and so,

$\displaystyle L_{\infty,k}\geq\frac{1}{2}L_{i,k}\geq
R_{max}^{-\frac{1}{2}}k.$

Thus $X_{(\infty)}$ generates a line $\gamma_{\infty}(s)$ through $p_{\infty}$
as $k\to\infty$. As a consequence, $(\widetilde{M},\widetilde{g}(0))$ splits
off a line and so does the flow $(\widetilde{M},\widetilde{g}(t);p_{\infty})$.
The lemma is proved.

∎

Next we estimate the curvature of $(N,g_{N}(t))$.

######  Lemma 4.2.

Under (1.2), there exists a constant $C$ independent of $t$ such that the
scalar curvature $R_{N}(t)$ of $(N,g_{N}(t))$ satisfies

(4.2) $\displaystyle\frac{R_{N}(x,t)}{R_{N}(y,t)}\leq C,~{}\forall~{}x,y\in
N,~{}t\in(-\infty,0].$

###### Proof.

Let $\widetilde{R}(x,t)$ be the scalar curvature of $(\mathbb{R}\times
N,\widetilde{g}(t))$. It suffices to prove the following is true:

(4.3) $\displaystyle\frac{\widetilde{R}(x,t)}{\widetilde{R}(y,t)}\leq
C,~{}\forall~{}x,y\in\mathbb{R}\times N,~{}t\in(\infty,0],$

for some constant $C$. For any $x,y\in\mathbb{R}\times N$, we choose
$\overline{r}>0$ such that $x,y\in
B(p_{\infty},\overline{r};\widetilde{g}(0))$. By the convergence of
$g_{p_{i}}(t)$, there are sequences $\\{x_{i}\\}$ and $\\{y_{i}\\}$ in
$B(p_{i},\overline{r};{g}_{p_{i}}(0))$ such that $x_{i}$ and $y_{i}$ converge
to $x$ and $y$ in the Cheeger-Gromov sense, respectively. By Lemma 3.1, we
have

$\displaystyle x_{i},y_{i}\subseteq
B(p_{i},\overline{r};{g}_{p_{i}}(0))\subseteq
M_{p_{i},\overline{r}\sqrt{R_{\max}}}.$

Thus

(4.4) $\displaystyle f(x_{i})=(1+o(1))f(p_{i})~{}{\rm
and}~{}f(y_{i})=(1+o(1))f(p_{i}),~{}{\rm as}~{}p_{i}\to\infty.$

On the other hand, for a fixed $t<0$,

$\displaystyle\frac{f(\phi_{R^{-1}(p_{i})t}(x_{i}))-f(x_{i})}{|R^{-1}(p_{i})t|}=\frac{\int_{R^{-1}(p_{i})t}^{0}|\nabla
f|^{2}ds}{|R^{-1}(p_{i})t|}\rightarrow R_{\max},~{}{\rm as}~{}p_{i}\to\infty$

and

$\displaystyle\frac{f(\phi_{R^{-1}(p_{i})t}(y_{i}))-f(y_{i})}{|R^{-1}(p_{i})t|}=\frac{\int_{R^{-1}(p_{i})t}^{0}|\nabla
f|^{2}ds}{|R^{-1}(p_{i})t|}\rightarrow R_{\max},~{}{\rm as}~{}p_{i}\to\infty.$

By (4.4) and the fact

$\displaystyle C_{1}\leq R(x)f(x)\leq C_{2},~{}\forall~{}f(x)>>1,$

we get

$\frac{f(\phi_{R^{-1}(p_{i})t}(x_{i}))}{f(\phi_{R^{-1}(p_{i})t}(y_{i}))}\to
1,~{}{\rm as}~{}p_{i}\to\infty.$

It follows

$\displaystyle\frac{R(\phi_{R^{-1}(p_{i})t}(x_{i}))}{R(\phi_{R^{-1}(p_{i})t}(y_{i}))}\leq\frac{C_{2}}{C_{1}}.$

Hence we obtain

$\displaystyle\frac{R_{N}(x,t)}{R_{N}(y,t)}$
$\displaystyle=\lim_{i\to\infty}\frac{R^{-1}(p_{i})R(x_{i},R^{-1}(p_{i})t))}{R^{-1}(p_{i})R(y_{i},R^{-1}(p_{i})t)}$
$\displaystyle=\lim_{i\to\infty}\frac{R^{-1}(p_{i})R(\phi_{R^{-1}(p_{i})t}(x_{i}))}{R^{-1}(p_{i})R(\phi_{R^{-1}(p_{i})t}(y_{i}))}\leq\frac{C_{2}}{C_{1}}.$

This proves (4.3).

∎

The proof of Theorem 1.2 is completed by the following lemma.

######  Lemma 4.3.

$(N,g_{N}(t))$ in Lemma 4.1 is a shrinking spheres flow. Namely,

$(N,g_{N}(t))=(\mathbb{S}^{2},(2-2t)g_{\mathbb{S}^{2}}).$

###### Proof.

By Lemma 4.2, the Gauss curvature of $(N,g_{N}(0))$ has a uniform positive
lower bound. Then $N$ is compact by Myer’s Theorem. On the other hand, by a
classification theorem of Daskalopoulos-Hamilton-Sesum [8], an ancient
solution on a compact surface $N$ is either a shrinking spheres flow or a
Rosenau solution. The Rosenau solution is obtained by compactifying
$(\mathbb{R}\times\mathbb{S}^{1}(2),h(x,\theta,t)=u(x,t)(dx^{2}+d\theta^{2}))$
by adding two points, where $u(x,t)=\frac{\sinh(-t)}{\cosh(x)+\cosh(t)}$ and
$t\in(-\infty,0)$. By a direct computation,

(4.5) $\displaystyle
R_{h(t)}=\frac{\cosh(t)\cosh(x)+1}{\sinh(-t)(\cosh(x)+\cosh(t))}.$

It is easy to check that $R_{h(t)}$ doesn’t satisfy (4.2) in Lemma 4.2 as
$t\to-\infty$. Hence, $(N,g_{N}(t))$ must be a shrinking spheres flow on
$\mathbb{S}^{2}$. Note that $\widetilde{R}(p_{\infty},0)=1$. Then it is easy
to see that $g_{\mathbb{S}^{2}}(t)=(2-2t)g_{\mathbb{S}^{2}}$.

∎

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