# All pure bipartite entangled states can be semi-self-tested with only one
measurement setting on each party

Lijinzhi Lin1 Zhenyu Chen1 Xiaodie Lin1 Zhaohui Wei2,3<EMAIL_ADDRESS>1Institute for Interdisciplinary Information Sciences, Tsinghua University,
Beijing 100084, China
2Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China
3Yanqi Lake Beijing Institute of Mathematical Sciences and Applications,
Beijing 101407, China

###### Abstract

It has been known that all bipartite pure quantum states can be self-tested,
i.e., any such state can be certified completely by initially measuring both
subsystems of this state by proper local quantum measurements and subsequently
verifying that the correlation between the measurement choices and the
outcomes satisfies a specific condition. In such a protocol, a key feature is
that the conclusion can still be reliable even if involved quantum
measurements are untrusted, where quantum nonlocality is crucial and plays a
central role, and this means that each party has to conduct at least two
different quantum measurements to produce a desirable correlation. Here, we
prove that when the underlying Hilbert space dimension is known beforehand, an
arbitrary $d\times d$ bipartite pure state can be certified completely (up to
local unitary transformations) by a certain correlation generated by a single
measurement setting on each party, where each measurement yields only $3d$
outcomes. Notably, our protocols do not involve any quantum nonlocality. We
believe that our result may provide us a remarkable convenience when
certifying bipartite pure quantum states in quantum labs.

##  I Introduction

In quantum physics and quantum information processing tasks, characterizing
unknown quantum states is a fundamental problem. For this, a possible approach
that one can take is quantum state tomography (QST), which aims to completely
reconstruct the density matrices of target quantum states by performing
tomographically complete quantum measurements on these states and then
analyzing the outcome statistics Chuang and Nielsen (1997); Poyatos et al.
(1997). However, QST suffers from two obvious drawbacks. First, it is
extremely costly and only feasible for small-scale quantum systems. Second, in
order to obtain reliable results one has to make sure that the involved
quantum measurements are conducted precisely, which is quite challenging due
to limited precision of quantum operations and the existence of quantum noise
and decoherence.

When characterizing unknown quantum states, it will be desirable to minimize
the reliance on the physical descriptions of quantum operations. Typical
schemes that operate on minimal physical assumptions are device-independent
quantum protocols that were first introduced in the area of quantum key
distribution Ekert (1991); Barrett et al. (2005); Acin et al. (2006), where
one can make various nontrivial and reliable characterizations for unknown
quantum states based only on the quantum nonlocality revealed by the outcome
statistics produced by measuring these quantum states locally. To illustrate,
consider a bipartite quantum state shared between Alice and Bob. Each party
possesses a set of measurement devices, and they pick up these devices
randomly to measure the subsystems they hold respectively. By repeating the
procedure many times and recording all the outcome statistics, one can find
out the correlation between the choices of measurement devices and the
measurement outcomes, which may exhibit quantum nonlocality. If the observed
quantum nonlocality is strong enough, it is possible to characterize the
target state safely even if the measurement devices cannot be trusted Bancal
and Bancal (2014); Moroder et al. (2013); Baccari et al. (2017); Zwerger et
al. (2019); Brunner et al. (2008); Sikora et al. (2016).

An extreme scenario of device-independent quantum protocol is quantum self-
testing. In Ref.Mayers and Yao (2004) (see also Ref.Popescu and Rohrlich
(1992)), Mayers and Yao found that when a quantum state violates the Clauser-
Horne-Shimony-Holt (CHSH) inequality maximally Clauser et al. (1969), both the
quantum state itself and the associated quantum measurements can be completely
determined, up to local isometries. The phenomenon of self-testing is very
surprising and valuable. On the one hand, it is well-known that a typical
$n$-qubit unknown quantum system needs a density matrix of an exponential size
to characterize completely, but in self-testing a single classical quantity on
Bell inequality violation can certify the underlying quantum state. On the
other hand, self-testing enables reliable benchmarking of the quality of
entangled quantum states, even if the involved quantum measurements are
untrustworthy. This is particularly useful in many quantum information
processing tasks Ekert (1991); Barrett et al. (2005); Acin et al. (2006).
After the phenomenon of quantum self-testing was discovered, various specific
bipartite and multipartite pure states were also shown to be self-testable
Yang and Navascués (2013); Bamps and Pironio (2015); Salavrakos et al. (2016);
Yang et al. (2014); Acin et al. (2002); McKague (2014); Wu et al. (2014); Pál
et al. (2014); Šupić et al. (2018); Baccari et al. (2020).

Recently, a seminal work in Ref.Coladangelo et al. (2017) proved that all
bipartite pure states can be self-tested. Furthermore, in another breakthrough
on quantum self-testing, by examining correlations generated by quantum
networks it has also been proved that all multipartite pure states can be
self-tested Šupić et al. (2023). These remarkable discoveries imply that the
possibility of self-testing is a general feature for pure quantum states. On
the other hand, it has been known that mixed quantum states cannot be self-
tested, as any correlation produced by a mixed state can always be generated
by another pure state Sikora et al. (2016).

Fundamentally, the crucial concept that enables quantum self-testing is
quantum nonlocality, which means that when self-testing a bipartite pure
quantum state, each party must involve at least two measurement settings
Sarkar et al. (2021). Indeed, if each party has only one measurement setting,
it is impossible to exhibit quantum nonlocality at all, and furthermore, the
resulting correlation can always be produced classically Zhang (2012), making
quantum self-testing impossible. In such a situation, a profound question
arises: Is quantum nonlocality a necessary condition for determining a quantum
state based only on the observed correlation? Surprisingly, as we shall
demonstrate, if the underlying Hilbert space dimension is known beforehand,
any bipartite pure state can be certified completely (up to local
transformations) based only on a certain correlation produced by measuring the
state with only one measurement setting on each party. We achieve this by
exploiting a close relationship between the mathematical structure of
correlations generated by local quantum measurements and that of positive
semidefinite factorizations, a concept originated from optimization theory
Fiorini et al. (2015); Gouveia et al. (2013); Jain et al. (2013).

We call such a phenomenon _semi-self-testing_. We stress that when semi-self-
testing a $d\times d$ bipartite quantum pure state, the only local quantum
measurement conducted by each party has only $3d$ outcomes, implying that they
are far from tomographically complete. We believe the discovery in the current
manuscript not only helps us to gain a better understanding of the essence of
quantum self-testing, but also provides us a potential new approach to
efficiently certify crucial quantum states in many quantum information
processing tasks.

##  II The setting

Suppose $\rho\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ is a $d\times d$
bipartite quantum state shared by Alice and Bob. Then both of them measure the
corresponding subsystem by a local quantum measurement, where the POVM
operators for Alice are $\\{A_{x}\\}$, and those for Bob are $\\{B_{y}\\}$.
Here $x,y\in[m]\equiv\\{1,2,...,m\\}$ and $\sum_{x}A_{x}=\sum_{y}B_{y}=I_{d}$,
where $I_{d}$ is the $d$-dimensional identity operator. We denote the
probability that Alice obtains the outcome $x$ and Bob obtains the outcome $y$
by $P_{xy}$, i.e.,

$P_{xy}=\mathrm{tr}(\rho(A_{x}\otimes B_{y})).$ (1)

We call the $m\times m$ matrix $P=[P_{xy}]$ a _classical correlation_
generated by measuring $\rho$ with the POVMs $\\{A_{x}\\}$ and $\\{B_{y}\\}$.

It turns out that there is a close relation between $\rho$ and the so-called
_positive semidefinite factorizations_ (PSD factorizations) of $P$ Jain et al.
(2013). Suppose two set of $r\times r$ PSD matrices $\\{C_{x}\\}$ and
$\\{D_{y}\\}$ satisfy that $P_{xy}=\mathrm{tr}(C_{x}D_{y})$ for any
$x,y\in[m]$, we say $\\{C_{x}\\}$ and $\\{D_{y}\\}$ form a PSD factorizations
for $P$, and the _PSD rank_ of $P$ is $r$ if the size of such $\\{C_{x}\\}$
and $\\{D_{y}\\}$ is minimal. Particularly, if $\\{C_{x}\\}$ and $\\{D_{y}\\}$
also satisfy that $\sum_{x=1}^{m}C_{x}=\sum_{y=1}^{m}D_{y}=\Lambda$, where
$\Lambda$ is a diagonal PSD matrix, we say $\\{C_{x}\\}$ and $\\{D_{y}\\}$
form a _canonical form of PSD factorization_ for $P$. The following conclusion
shows that every normal PSD factorization corresponds to a canonical form.

######  Proposition 1.

Suppose $\\{C_{x}\\}$ and $\\{D_{y}\\}$ form a PSD factorization for a
correlation $P$, where $S_{C}=\sum_{x}C_{x}$ and $S_{D}=\sum_{y}D_{y}$ are
full rank. Then this factorization corresponds to a normal form of PSD
factorization where the diagonal entries of $\Lambda$ are the eigenvalues of
$\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{1/2}$.

###### Proof.

Consider

$\displaystyle C^{\prime}_{x}=$
$\displaystyle\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{1/4}\left(S_{C}^{-1/2}C_{x}S_{C}^{-1/2}\right)\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{1/4},$
$\displaystyle D^{\prime}_{y}=$
$\displaystyle\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{-1/4}\left(S_{C}^{1/2}D_{y}S_{C}^{1/2}\right)\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{-1/4}.$

It can be verified that

$\displaystyle\mathrm{tr}(C^{\prime}_{x}D^{\prime}_{y})=$
$\displaystyle\mathrm{tr}(C_{x}D_{y}),$
$\displaystyle\sum_{x}C^{\prime}_{x}=\sum_{y}D^{\prime}_{y}=$
$\displaystyle\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{1/2}.$

Suppose a unitary matrix $U$ can diagonalize
$\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{1/2}$, then
$\\{UC^{\prime}_{x}U^{\dagger}\\}$ and $\\{UD^{\prime}_{y}U^{\dagger}\\}$ form
a normal form of PSD factorization for $P$, where the diagonal entries of
$\Lambda$ are the eigenvalues of
$\left(S_{C}^{1/2}S_{D}S_{C}^{1/2}\right)^{1/2}$. ∎

It has been known that if $\rho$ generates $P$, then there exists a pure state
$\ket{\psi}$ in
$\mathcal{H}_{A_{1}}\otimes\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{B_{1}}$
such that it is a purification for both $\rho$ and
$\sigma=\sum_{x,y=1}^{m}P_{xy}\cdot\ket{x}\bra{x}\otimes\ket{y}\bra{y}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
Jain et al. (2013). In addition, suppose the Schmidt rank of $\ket{\psi}$ with
respect to the partition $AA_{1}|BB_{1}$ is $r$, and the Schmidt coefficients
are $\sqrt{\lambda_{i}}$, $i\in[r]$, then $P$ has a canonical form of PSD
factorization, denoted $\\{C_{k}\\}$ and $\\{D_{l}\\}$, such that
$\Lambda=\sum_{k=1}^{m}C_{k}=\sum_{l=1}^{m}D_{l}=\text{diag}(\sqrt{\lambda_{1}},...,\sqrt{\lambda_{r}})$,
where $\text{diag}(a_{1},a_{2},...,a_{n})$ is a $d\times d$ diagonal matrix
with the diagonal entries $a_{i}$’s Chen et al. (2023).

##  III A Correlation that semi-self-tests the Maximally Entangled State

Let $\vec{e}\in\mathbb{R}^{d}$ be the $d$-dimensional column vector with all
entries equal to one. Consider the following $2d\times 2d$ classical
correlation specified in block matrix form:

$\displaystyle
P=\frac{1}{d(d+1)^{2}}\begin{pmatrix}d^{2}I_{d}&\vec{e}\vec{e}^{T}\\\
\vec{e}\vec{e}^{T}&I_{d}\\\ \end{pmatrix}.$ (2)

We now are ready to introduce the first main result of the current paper.

######  Theorem 2.

If Alice and Bob can generate $P$ by locally measuring a bipartite $d\times d$
quantum state $\rho$ with the POVMs $\\{A_{x}\\}$ and $\\{B_{y}\\}$, then
$\rho$ must be a $d\times d$ maximally entangled pure state.

To prove this conclusion, we need to examine all possible PSD factorizations
for $P$. Given a PSD factorization for $P$, we denote the matrices therein by
$\\{C_{1},C_{2},...,C_{2d}\\}$ and $\\{D_{1},D_{2},...,D_{2d}\\}$.

We first consider the lower right submatrix $\frac{1}{d(d+1)^{2}}I_{d}$ in
$P$. It can be verified that this implies that for $x,y\in[d]$, $C_{x}$ and
$D_{y}$ must be rank-1, i.e., $C_{d+x}=a_{x}\ket{\alpha_{x}}\bra{\alpha_{x}}$
and $D_{d+y}=b_{y}\ket{\beta_{y}}\bra{\beta_{y}}$ for certain pure states
$\ket{\alpha_{x}}$ and $\ket{\beta_{y}}$, and positive real number $a_{x}$ and
$b_{y}$. Furthermore, the vector set $\\{\ket{\alpha_{x}}\\}(x\in[d])$ is
linearly independent, and similar for $\\{\ket{\beta_{y}}\\}(y\in[d])$. To see
why this is the case, for each $x\in[d]$ we pick unit vectors
$a_{x}\in\mathrm{range}(C_{d+x})$ and $b_{{x}}\in\mathrm{range}(D_{d+{x}})$
such that $\langle a_{x},b_{x}\rangle\neq 0$; this is possible since
$\mathrm{tr}(C_{d+x}D_{d+x})>0$, where $\mathrm{range}(A)$ is the subspace
consisting of all linear combinations of the eigenvectors of $A$ with nonzero
eigenvalues. Suppose the vectors $b_{x}$ are linearly dependent; that is,
without loss of generality,

$\displaystyle b_{1}=\sum_{x=2}^{d}\alpha_{x}b_{x},$

where $\alpha_{x}$ are complex scalars. Then

$\displaystyle 0\neq\langle a_{1},b_{1}\rangle=\sum_{x=2}^{d}\alpha_{x}\langle
a_{1},b_{x}\rangle=0,$

which is impossible. This gives

$\displaystyle\mathrm{rank}(D_{d+2}+\cdots+D_{2d})\geq\mathrm{dim}(\mathrm{span}\\{b_{2},\cdots,b_{d}\\})=d-1,$

where $\mathrm{rank}(A)$ is the rank of the matrix $A$, $\mathrm{dim}(S)$ is
the dimension of a vector space $S$, and
$\mathrm{span}\\{b_{2},\cdots,b_{d}\\}$ is the vector space consisting of all
linear combinations of the vectors $b_{2},\cdots,b_{d}$. Since
$\mathrm{tr}(C_{d+1}(D_{d+2}+\cdots+D_{2d})=0$, we have
$\mathrm{rank}(C_{d+1})\leq 1$, hence $\mathrm{rank}(C_{d+1})=1$ due to
$C_{d+1}\neq 0$. By symmetry we have $\mathrm{rank}(C_{d+x})=1$ for all
$x\in[d]$, and by swapping the roles of $C_{d+x}$ and $D_{d+y}$ we get
$\mathrm{rank}(D_{d+y})=1$ for all $y\in[d]$.

Therefore, one can find an invertible matrix $H$ such that
$HC_{d+x}H^{\dagger}=\frac{1}{\sqrt{d}(d+1)}\ket{x-1}\bra{x-1}$, where
$x\in[d]$ and $\\{\ket{i}\\}(0\leq i\leq d-1)$ are the computational basis
states. For any $x,y\in[2d]$, we let $C_{x}^{\prime}=HC_{x}H^{\dagger}$ and
$D_{y}^{\prime}=(H^{\dagger})^{-1}D_{y}H^{-1}$, then $\\{C_{x}^{\prime}\\}$
and $\\{D_{y}^{\prime}\\}$ also form a PSD factorization for $P$. When
focusing on $y\in[d]$, by direct calculations we can see that
$D_{d+y}^{\prime}=\frac{1}{\sqrt{d}(d+1)}\ket{y-1}\bra{y-1}$.

Similarly, for any $x,y\in[d]$ the matrices $C_{x}^{\prime}$ and
$D_{y}^{\prime}$ must be of rank $1$ since they have to produce the submatrix
$\frac{d^{2}}{d(d+1)^{2}}I_{d}$ of $P$. Additionally, for any $x,y\in[d]$ the
diagonals of $C_{x}^{\prime}$ and $D_{y}^{\prime}$ are all
$\frac{1}{\sqrt{d}(d+1)}$ (from the two submatrices
$\frac{1}{d(d+1)^{2}}\vec{e}\vec{e}^{T}$ of $P$), hence all entries of
$C_{x}^{\prime}$ and $D_{y}^{\prime}$ have norms at most
$\frac{1}{\sqrt{d}(d+1)}$. For any $x\in[d]$, since
$\frac{d}{(d+1)^{2}}=\mathrm{tr}(C_{x}^{\prime}D_{x}^{\prime})\leq\sqrt{\mathrm{tr}(C_{x}^{\prime
2})\mathrm{tr}(D_{x}^{\prime 2})}=\frac{d}{(d+1)^{2}}$ with the equality
holding if and only if $C_{x}^{\prime}$ is proportional to $D_{x}^{\prime}$,
we have $C_{x}^{\prime}=D_{x}^{\prime}$. Thus the constraint
$\mathrm{tr}(C_{x}^{\prime}D_{y}^{\prime})=\frac{d}{(d+1)^{2}}\delta_{xy}$
implies that
$\sum_{x=1}^{d}C_{x}^{\prime}=\sum_{y=1}^{d}D_{y}^{\prime}=\frac{\sqrt{d}}{d+1}I_{d}$.

In fact, for any $x,y\in[d]$ if setting
$C_{x}^{\prime}=D_{x}^{\prime}=\frac{\sqrt{d}}{d+1}\ket{\gamma_{x}}\bra{\gamma_{x}}$
where $\ket{\gamma_{x}}$ is the $x$-th column of the $d\times d$ quantum
Fourier transform matrix, and then combining the above choices of
$\\{C_{d+x}^{\prime}\\}$ and $\\{D_{d+y}^{\prime}\\}$ together, one can verify
that $\\{C_{x}^{\prime}\\}$ and $\\{D_{y}^{\prime}\\}$ do form a valid PSD
factorization for $P$. Furthermore, we also have that
$\sum_{x=1}^{2d}C_{x}^{\prime}=\sum_{y=1}^{2d}D_{y}^{\prime}=\frac{1}{\sqrt{d}}I_{d}$,
implying that $\\{C_{x}^{\prime}\\}$ and $\\{D_{y}^{\prime}\\}$ form a
canonical form of PSD factorization for $P$. Interestingly, we also observe
the following key fact.

######  Lemma 3.

The canonical form of PSD factorization for $P$ is unique.

###### Proof.

Suppose $\\{C_{x}^{\prime\prime}\\}$ and $\\{D_{y}^{\prime\prime}\\}$ is
another canonical form of PSD factorization for $P$, and
$\sum_{x=1}^{2d}C_{x}^{\prime\prime}=\sum_{y=1}^{2d}D_{y}^{\prime\prime}=\Lambda^{\prime}$,
where $\Lambda^{\prime}$ is a diagonal PSD matrix. Then according to our
discussion above, all $C_{x}^{\prime\prime}$ and $D_{y}^{\prime\prime}$ must
be rank-1, and there exists an invertible matrix $M$ such that for each
$x\in[d]$, $MC_{d+x}^{\prime\prime}M^{\dagger}$ has the form of
$\frac{1}{\sqrt{d}(d+1)}\ket{x-1}\bra{x-1}$. Again based on our above
discussion, for $x,y\in[2d]$, $\\{MC_{x}^{\prime\prime}M^{\dagger}\\}$ and
$\\{(M^{\dagger})^{-1}D_{y}^{\prime\prime}(M)^{-1}\\}$ will be the canonical
form of PSD factorization $\\{C_{x}^{\prime}\\}$ and $\\{D_{y}^{\prime}\\}$
introduced above. That is,
$\sum_{x=1}^{2d}MC_{x}^{\prime\prime}M^{\dagger}=M\Lambda^{\prime}M^{\dagger}=\frac{1}{\sqrt{d}}I_{d}$,
and
$\sum_{x=1}^{2d}(M^{\dagger})^{-1}C_{x}^{\prime\prime}M^{-1}=(M^{\dagger})^{-1}\Lambda^{\prime}M^{-1}=\frac{1}{\sqrt{d}}I_{d}$.
This means that $MM^{\dagger}=I_{d}$, i.e.,
$\Lambda^{\prime}=M^{\dagger}\cdot\frac{1}{\sqrt{d}}I_{d}\cdot
M=\frac{1}{\sqrt{d}}I_{d}$. ∎

Note that $\rho$ can be a mixed state, and if this is the case, Bob can purify
$\rho$ to be a pure state $\ket{\phi}$ in
$\mathcal{H}_{A}\otimes\mathcal{H}_{B}\otimes\mathcal{H}_{R}$ by introducing
an ancillary system $R$. After that, Alice and Bob can generate $P$ by
measuring $\ket{\phi}$ with the POVMs $\\{A_{x}\\}$ and $\\{B_{y}\otimes
I_{R}\\}$, where $I_{R}$ is the identity operator on the system $R$. According
to Ref.Chen et al. (2023), we know that with respect to the partition $A|BR$,
the Schmidt coefficients of $\ket{\phi}$ is the diagonal entries of
$\frac{1}{\sqrt{d}}I_{d}$, i.e.,
$\\{\frac{1}{\sqrt{d}},\frac{1}{\sqrt{d}},...,\frac{1}{\sqrt{d}}\\}$. Since
the ancillary system $R$ can also be introduced by Alice, we obtain the
following conclusion.

######  Proposition 4.

$\mathrm{tr}_{A}(\rho)=\mathrm{tr}_{B}(\rho)=\frac{1}{d}I_{d}$.

Due to the above lemma, if we can next show that the state $\rho$ is pure,
then it must be a bipartite maximally entangled pure state. For this, we need
to use the following fact.

######  Proposition 5.

All measurement operators $A_{x}$ and $B_{y}$ have rank $1$, where
$x,y\in[2d]$.

###### Proof.

Let
$\tilde{B}_{y}=\mathrm{tr}_{B}((I_{A}\otimes\sqrt{B_{y}})\rho(I_{A}\otimes\sqrt{B_{y}}))$.
Then $P_{xy}=\mathrm{tr}(A_{x}\tilde{B}_{y})=\mathrm{tr}(\rho(A_{x}\otimes
B_{y}))$. Note that $\tilde{B}_{y}$ is a PSD matrix for any $y\in[2d]$, thus
we can apply the technique we have utilized before to prove that
$\mathrm{rank}(A_{x})=1$ for any $x\in[2d]$. By swapping the roles of $A_{x}$
and $B_{y}$ we get $\mathrm{rank}(B_{y})=1$ for all $y\in[2d]$. ∎

Then $A_{x}$ and $B_{y}$ can be further characterized as below.

######  Proposition 6.

For $i,j\in[d]$ with $i\neq j$, we have

$\displaystyle A_{i}A_{j}=B_{i}B_{j}=A_{d+i}A_{d+j}=B_{d+i}B_{d+j}=0.$

###### Proof.

We now show that $A_{1}A_{2}=0$, and the other cases can be proved similarly.
Recall that $P_{xy}=\mathrm{tr}(\rho(A_{x}\otimes B_{y}))$, then it holds that

$\sum_{y=1}^{2d}P_{1y}=\mathrm{tr}(\rho(A_{1}\otimes
I_{d}))=\mathrm{tr}(\mathrm{tr}_{B}(\rho)\cdot
A_{1})=\frac{\mathrm{tr}(A_{1})}{d}.$

Combined with $\sum_{y=1}^{2d}P_{1y}=\frac{1}{d+1}$, this means that
$\mathrm{tr}(A_{1})=\frac{d}{d+1}$.

Recall also that $A_{1}$ is rank-1, so without loss of generality we can
perform local unitaries on both $\rho$ and all POVM operators such that they
still generate $P$ and $A_{1}=B_{1}=\frac{d}{d+1}\ket{0}\bra{0}$. Let

$\displaystyle A_{2}=$
$\displaystyle\frac{d}{d+1}(\alpha\ket{0}+\beta\ket{\bar{1}})({\alpha^{*}}\bra{0}+{\beta^{*}}\bra{\bar{1}}),$

where $\ket{\bar{1}}$ is some vector orthogonal to $\ket{0}$ and
$|\alpha|^{2}+|\beta|^{2}=1$. Then

$\displaystyle\braket{00}{\rho}{00}=$
$\displaystyle\frac{(d+1)^{2}}{d^{2}}\mathrm{tr}(\rho(A_{1}\otimes B_{1}))$
$\displaystyle=$ $\displaystyle\frac{1}{d},$ $\displaystyle A_{2}\otimes
B_{1}=$
$\displaystyle\frac{d^{2}}{(d+1)^{2}}\left(|\alpha|^{2}\ket{00}\bra{00}+\alpha{\beta^{*}}\ket{00}\bra{\bar{1}0}+{\alpha^{*}}\beta\ket{\bar{1}0}\bra{00}+|\beta|^{2}\ket{\bar{1}0}\bra{\bar{1}0}\right).$

Since $\mathrm{tr}_{A}(\rho)=\frac{I_{d}}{d}$, we have

$\displaystyle\braket{00}{\rho}{00}+\braket{\bar{1}0}{\rho}{\bar{1}0}=$
$\displaystyle\mathrm{tr}(\rho((\ket{0}\bra{0}+\ket{\bar{1}}\bra{\bar{1}})\otimes(\ket{0}\bra{0})))$
$\displaystyle\leq$
$\displaystyle\mathrm{tr}(\rho(I_{d}\otimes\ket{0}\bra{0}))$ $\displaystyle=$
$\displaystyle\mathrm{tr}(\mathrm{tr}_{A}(\rho)\ket{0}\bra{0})$
$\displaystyle=$ $\displaystyle\mathrm{tr}(\frac{I_{d}}{d}\ket{0}\bra{0})$
$\displaystyle=$ $\displaystyle\frac{1}{d}.$

Therefore, by $\rho\geq 0$ and $\braket{00}{\rho}{00}=\frac{1}{d}$ we have
$\braket{\bar{1}0}{\rho}{\bar{1}0}=0$, which implies that
$\braket{\bar{1}0}{\rho}{00}=\braket{00}{\rho}{\bar{1}0}=0$. As a result, it
holds that

$\displaystyle 0=$ $\displaystyle\mathrm{tr}(\rho(A_{2}\otimes B_{1}))$
$\displaystyle=$
$\displaystyle\frac{d^{2}}{(d+1)^{2}}|\alpha|^{2}\cdot\frac{1}{d}$
$\displaystyle=$ $\displaystyle\frac{d}{(d+1)^{2}}|\alpha|^{2}.$

This implies that $\alpha=0$ and $A_{1}A_{2}=0$. ∎

Because of the above fact, we may perform two local unitaries on each parties
so that for $i\in[d]$ it holds that $A_{i}$ and $B_{i}$ are both proportional
to $\ket{i-1}\bra{i-1}$, where $\rho$ is also adjusted accordingly. Meanwhile,
using the same technique to calculate $\mathrm{tr}(A_{1})$, we can obtain that
$\mathrm{tr}(A_{i})=\frac{d}{d+1}$ for any $i\in[d]$. Combining these two
observations together, we can let $A_{i}=\frac{d}{d+1}\ket{i-1}\bra{i-1}$ and
$B_{i}=\frac{d}{d+1}\ket{i-1}\bra{i-1}$ for any $i\in[d]$.

Lastly, since $P_{ij}=\mathrm{tr}(\rho(A_{i}\otimes B_{j}))=0$ for any $i\neq
j\in[d]$, we have that
$\mathrm{range}(\rho)\subseteq\mathrm{span}\\{\ket{00},\ket{11},\cdots,\ket{(d-1)(d-1)}\\}$.
Similarly, it holds that $\braket{ii}{\rho}{ii}=\frac{1}{d}$, which also means
$|\braket{ii}{\rho}{jj}|\leq\frac{1}{d}$ since $\rho$ is a PSD matrix, here
$i,j\in\\{0,\cdots,d-1\\}$.

######  Proposition 7.

$|\braket{i}{A_{d+1}}{j}|=|\braket{i}{B_{d+1}}{j}|=\frac{1}{d(d+1)}$ for all
$i,j\in\\{0,\cdots,d-1\\}$.

###### Proof.

For any $i\in\\{0,\cdots,d-1\\}$ we have

$\displaystyle\frac{1}{d(d+1)^{2}}=$
$\displaystyle\mathrm{tr}(\rho({A_{i+1}}\otimes B_{d+1}))$ $\displaystyle=$
$\displaystyle\frac{d}{d+1}\mathrm{tr}(\rho(\ket{i}\bra{i}\otimes B_{d+1}))$
$\displaystyle=$
$\displaystyle\frac{d}{d+1}\sum_{jkj^{\prime}k^{\prime}}\braket{jk}{\rho}{j^{\prime}k^{\prime}}\bra{j^{\prime}k^{\prime}}(\ket{i}\bra{i}\otimes
B_{d+1})\ket{jk}$ $\displaystyle=$
$\displaystyle\frac{d}{d+1}\sum_{kk^{\prime}}\braket{ik}{\rho}{ik^{\prime}}\braket{k^{\prime}}{B_{d+1}}{k}$
$\displaystyle=$
$\displaystyle\frac{d}{d+1}\cdot\frac{1}{d}\braket{i}{B_{d+1}}{i}$
$\displaystyle=$ $\displaystyle\frac{1}{d+1}\cdot\braket{i}{B_{d+1}}{i},$

hence $\braket{i}{B_{d+1}}{i}=\frac{1}{d(d+1)}$. Since
$\mathrm{rank}(B_{d+1})=1$, the rest of the entries also have norms
$\frac{1}{d(d+1)}$. ∎

We now examine the $(d+1,d+1)$-th entry of $P$ using the Cauchy-Schwarz
inequality:

$\displaystyle\frac{1}{d(d+1)^{2}}=$
$\displaystyle\mathrm{tr}(\rho(A_{d+1}\otimes B_{d+1}))$ $\displaystyle=$
$\displaystyle\sum_{iji^{\prime}j^{\prime}}\braket{ij}{\rho}{i^{\prime}j^{\prime}}\braket{i^{\prime}j^{\prime}}{A_{d+1}\otimes
B_{d+1}}{ij}$ $\displaystyle=$
$\displaystyle\sum_{ii^{\prime}}\braket{ii}{\rho}{i^{\prime}i^{\prime}}\braket{i^{\prime}}{A_{d+1}}{i}\braket{i^{\prime}}{B_{d+1}}{i}$
$\displaystyle\leq$
$\displaystyle\left(\sum_{ii^{\prime}}|\braket{ii}{\rho}{i^{\prime}i^{\prime}}|^{2}\right)^{1/2}\left(\sum_{ii^{\prime}}|\braket{i^{\prime}}{A_{d+1}}{i}\braket{i^{\prime}}{B_{d+1}}{i}|^{2}\right)^{1/2}$
$\displaystyle\leq$ $\displaystyle\left(d^{2}\cdot
d^{-2}\right)^{1/2}\left(d^{2}\cdot(d(d+1))^{-4}\right)^{1/2}$
$\displaystyle=$ $\displaystyle\frac{1}{d(d+1)^{2}}.$

Therefore, there exists a scalar $\alpha$ such that

$\displaystyle\braket{ii}{\rho}{i^{\prime}i^{\prime}}=\alpha\braket{i^{\prime}}{A_{d+1}}{i}\braket{i^{\prime}}{B_{d+1}}{i}.$

This implies that $\mathrm{rank}(\rho)=1$, hence $\rho$ is a pure state. By
$\mathrm{tr}_{A}(\rho)=\mathrm{tr}_{B}(\rho)=\frac{I_{d}}{d}$, we conclude
that $\rho$ is the maximally entangled state.

##  IV A correlation that semi-self-tests a general bipartite pure entangled
state

We now generalize the techniques developed in Sec.III to show that any
$d\times d$ bipartite pure entangled state $\ket{\psi}$ can be semi-self-
tested. Without loss of generality, we assume that the Schmidt rank of
$\ket{\psi}$ is $d$.

We define $\vec{z}=(z_{1},z_{2},...,z_{d})\in\mathbb{R}^{d}$ to be a
$d$-dimensional column vector with all entries positive, and the values of its
entries $z_{i}$ will be fixed later. Consider the following classical
correlation of size $3d\times 3d$:

$\displaystyle Q=\begin{pmatrix}a\cdot P&\begin{pmatrix}\vec{e}\vec{z}^{T}\\\
\mathrm{diag}(\vec{z})\end{pmatrix}\\\
\begin{pmatrix}\vec{z}\vec{e}^{T}&\mathrm{diag}(\vec{z})\\\
\end{pmatrix}&\frac{d(d+1)^{2}}{a}\mathrm{diag}(\vec{z})^{2}\\\
\end{pmatrix},$ (3)

where $0<a<1$ is a positive real number that we will pin down later, and $P$
is the $2d\times 2d$ classical correlation defined in Eq.(2). As a classical
correlation, the sum of all entries of $Q$ must be 1, and this means that

$a+2(d+1){\|\vec{z}\|_{1}}+\frac{d(d+1)^{2}}{a}{\|\vec{z}\|_{2}^{2}}=1,$

where $\|\vec{z}\|_{1}=\sum_{i=1}^{d}z_{i}$ and
$\|\vec{z}\|_{2}^{2}=\sum_{i=1}^{d}z_{i}^{2}$.

Suppose $\\{C_{x}\\}$ and $\\{D_{y}\\}$ form a canonical form of PSD
factorization for $Q$, where $x,y\in[3d]$. Then using the same techniques as
in Sec.III, one can prove that we must have that
$\sum_{x=1}^{3d}C_{x}=\sum_{y=1}^{3d}D_{y}=\Lambda$, where

$\Lambda=\mathrm{diag}{(\gamma_{1},\gamma_{2},...,\gamma_{d})},$

and $\gamma_{i}=\sqrt{\frac{a}{d}}+\sqrt{\frac{d}{a}}(d+1)\cdot z_{i}$ for any
$i\in[d]$.

This can be explained as below. First, let us first focus on the submatrix
$a\cdot P$, then according to the discussions in Sec.III we know that
$\\{C_{x}\\}$ and $\\{D_{y}\\}$ must be rank-1 for any $x,y\in[2d]$. In
addition, for any $x,y\in[d]$, by a proper invertible matrix $S$ we can have
that $SC_{d+x}S^{\dagger}=\frac{\sqrt{a}}{\sqrt{d}(d+1)}\ket{x-1}\bra{x-1}$
and
${(S^{\dagger})^{-1}D_{d+y}S^{-1}}=\frac{\sqrt{a}}{\sqrt{d}(d+1)}\ket{y-1}\bra{y-1}$.
Similarly, $SC_{x}S^{\dagger}$ and $(S^{\dagger})^{-1}D_{y}S^{-1}$ can also be
characterized accordingly for any $x,y\in[d]$. For convenience, we still
denote the adjusted PSD factorizations for $Q$ by $\\{C_{x}^{\prime}\\}$ and
$\\{D_{y}^{\prime}\\}$, where $x,y\in[3d]$.

Next we focus on the two submatrices $\mathrm{diag}(\vec{z})$ of $Q$. Based on
this part we can know that when $x,y\in[d]$, it holds that
$C_{2d+x}^{\prime}=\frac{\sqrt{d}(d+1)}{\sqrt{a}}{\cdot
z_{x}}\ket{x-1}\bra{x-1}$, and
$D_{2d+y}^{\prime}=\frac{\sqrt{d}(d+1)}{\sqrt{a}}{\cdot
z_{y}}\ket{y-1}\bra{y-1}$. By direct calculations, one can verify that the
above choices of $\\{C_{x}^{\prime}\\}$ and $\\{D_{y}^{\prime}\\}$ are indeed
a valid PSD factorization for $Q$, and what is more, it holds that

$\sum_{x=1}^{3d}C_{x}^{\prime}=\sum_{y=1}^{3d}D_{y}^{\prime}=\mathrm{diag}{(\gamma_{1},\gamma_{2},...,\gamma_{d})},$
(4)

where $\gamma_{i}=\sqrt{\frac{a}{d}}+\sqrt{\frac{d}{a}}(d+1)\cdot z_{i}$ for
any $i\in[d]$.

Notice that for any PSD factorizations of $Q$, all the involved PSD matrices
must be rank-1, thus the technique utilized in Lemma 3 can also be applied
here, and as a result, $Q$ has a unique canonical form of PSD factorization,
which is given in Eq.(4).

We now suppose that Alice and Bob generate $Q$ by measuring a $d\times d$
bipartite quantum state $\rho_{Q}\in\mathcal{H}_{A}\otimes\mathcal{H}_{B}$
with two local POVM quantum measurements, and the operators for Alice are
$\\{A_{x}\\}$, and those for Bob are $\\{B_{y}\\}$, where $x,y\in[3d]$ and
$\sum_{x=1}^{3d}A_{x}=\sum_{y=1}^{3d}B_{y}=I_{d}$. Let
$A_{P}=\sum_{x=1}^{2d}A_{x}$ and $B_{P}=\sum_{y=1}^{2d}B_{y}$ be the sum of
measurement operators corresponding to the block $a\cdot P$ of $Q$. Since
$a\cdot P$ must be generated by a $d\times d$ state and $P$ contains submatrix
of PSD rank $d$, we have $\mathrm{rank}(A_{P})=\mathrm{rank}(B_{P})=d$; that
is, both $A_{P}$ and $B_{P}$ are invertible.

Then it can be seen that $\frac{1}{a}\cdot(A_{P}\otimes
B_{P})^{1/2}\rho_{Q}(A_{P}\otimes B_{P})^{1/2}$ generates $P$ when measured by
the POVMs with the measurement operators $\\{A_{P}^{-1/2}A_{x}A_{P}^{-1/2}\\}$
and $\\{B_{P}^{-1/2}B_{y}B_{P}^{-1/2}\\}$, where $x,y\in[2d]$. By the previous
section we have that $\frac{1}{a}\cdot(A_{P}\otimes
B_{P})^{1/2}\rho_{Q}(A_{P}\otimes B_{P})^{1/2}$ must be pure, hence the
bipartite state $\rho_{Q}$ is also pure. Furthermore, the Schmidt coefficients
of $\rho_{Q}$ is exactly the $\gamma_{i}$’s in Eq.(4).

Therefore, suppose one wants to semi-self-test a $d\times d$ pure quantum
state with the Schmidt decomposition

$\ket{\psi}=\sum_{k=1}^{d}\sqrt{\lambda_{i}}\ket{\alpha_{i}}\otimes\ket{\beta_{i}},$
(5)

where $\lambda_{1}\geq\lambda_{2}\geq...\geq\lambda_{d}>0$, and
$\\{\ket{\alpha_{i}}\\}$ and $\\{\ket{\beta_{i}}\\}$ are orthonormal states
for Alice and Bob respectively. Then what he/she needs to do is to set $a$ and
$\vec{z}$ properly such that the $\gamma_{i}$’s in Eq.(4) are exactly the same
with $\sqrt{\lambda_{i}}$. It can be verified that this can be achieved by
letting

$a=\frac{d}{4}\cdot\lambda_{d}$ (6)

and

${z_{i}}=\frac{1}{d+1}\cdot\frac{\sqrt{\lambda_{d}}}{2}\left(\sqrt{\lambda_{i}}-\frac{1}{2}\sqrt{\lambda_{d}}\right)$
(7)

for any $i\in[d]$.

##  V Semi-self-testing quantum states with inequalities

For a $d\times d$ classical correlation $A$, we denote its entries by $A_{ij}$
($i,j\in[d]$). By the Cauchy-Schwarz inequality, for any $3d\times 3d$
classical correlation $P$ it holds that

$\sum_{x,y=1}^{3d}\sqrt{P_{xy}}\cdot\sqrt{Q_{xy}}\leq 1$ (8)

and the equality holds if and only if $P=Q$, where $Q$ is the classical
correlation in Eq.(3), and we have utilized the relations
$\sum_{x,y}P_{xy}=\sum_{x,y}Q_{xy}=1$. Therefore, we have the following main
result.

######  Theorem 8.

If Alice and Bob generate a $3d\times 3d$ classical correlation $P$ by locally
measuring a $d\times d$ bipartite quantum state $\rho$ with the POVMs
$\\{A_{x}\\}$ and $\\{B_{y}\\}$ ($x,y\in[3d]$), and saturates the inequality
in Eq.(8), then $\rho$ must be a $d\times d$ pure state with the Schmidt
coefficients $\sqrt{\lambda_{i}}$ ($i\in[d]$) satisfying Eqs.(6) and (7).

Lastly, we would like to point out that some $d\times d$ pure states can be
semi-self-tested by classical correlations of smaller size, and such size can
be as small as $d+1$. Since any classical correlations of size $d\times d$ can
always be generated by a $d\times d$ separable state Zhang (2012), $d+1$ is
the smallest we can expect.

##  VI Discussion

In conclusion, we have proven that if the dimension of a bipartite quantum
system is known, any pure state of this system can be semi-self-tested by a
Bell experiment involving a single measurement setting on each party.
Importantly, our protocol does not rely on quantum nonlocality since an
entirely classical protocol is able to generate the same correlation with
strictly larger dimensions. Though our protocol needs the information on
underlying quantum dimension in constrast to normal quantum self-testing, our
results still suggest that quantum self-testing may be much more general than
what many people think, and could potentially manifest in various settings.

We stress that in order to semi-self-test a $d\times d$ bipartite pure quantum
state, each quantum measurement in our protocol generates only $3d$ outcomes,
which seems to be appealing for practical purposes. We believe that our
results imply a new potential approach to certify crucial quantum states for
many quantum information processing tasks in an efficient way.

Before applying our protocol in realistic quantum schemes, another important
issue we have to address is how robust to noise our protocol can be. That is,
if the correlation we observe in a quantum experiment is not exactly $P$ or
$Q$ we have discussed, but very close, can we guarantee that the underlying
quantum state has high fidelity with the ideal pure state that $P$ or $Q$
semi-self-tests? Moreover, if this is the case, can we give a useful lower
bound for this fidelity based on the observed correlation? Lastly, we also
expect that similar protocols can be designed to semi-self-test multipartite
pure quantum state. We leave these problems for future work.

###### Acknowledgements.

We thank Zhengfeng Ji, Zizhu Wang, Yu Guo, and Zhangqi Yin for helpful
comments on our manuscript. This work was supported by the National Key R&D
Program of China, Grants No. 2018YFA0306703, 2021YFE0113100, and the National
Natural Science Foundation of China, Grant No. 61832015, 62272259.

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# Polaronic Conductivity in Cr2Ge2Te6 Single Crystals

Yu Liu,1 Myung-Geun Han,1 Yongbin Lee,2 Michael O. Ogunbunmi,3 Qianheng Du,1,4
Christie Nelson,5 Zhixiang Hu,1,4 Eli Stavitski,5 David Graf,6 Klaus
Attenkofer,5 Svilen Bobev,3 Liqin Ke,2 Yimei Zhu,1 and C. Petrovic1,4
1Condensed Matter Physics and Materials Science Department, Brookhaven
National Laboratory, Upton, New York 11973, USA
2Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011, USA
3Department of Chemistry and Biochemistry, University of Delaware, Newark,
Delaware 19716, USA
4Materials Science and Chemical Engineering Department, Stony Brook
University, Stony Brook, New York 11790, USA
5National Synchrotron Light Source II, Brookhaven National Laboratory, Upton,
New York 11973, USA
6National High Magnetic Field Laboratory, Florida State University,
Tallahassee, Florida 32306-4005, USA

###### Abstract

Intrinsic, two-dimensional (2D) ferromagnetic semiconductors are an important
class of materials for spin-charge conversion applications. Cr2Ge2Te6 retains
long-range magnetic order in bilayer at cryogenic temperatures and shows
complex magnetic interactions with considerable magnetic anisotropy. Here, we
performed a series of structural, magnetic, X-ray scattering, electronic,
thermal transport and first-principles calculation studies which reveal that
localized electronic charge carriers in Cr2Ge2Te6 are dressed by surrounding
lattice and are involved in polaronic transport via hopping that is sensitive
on details of magnetocrystalline anisotropy. This opens possibility for
manipulation of charge transport in Cr2Ge2Te6 \- based devices by electron-
phonon- and spin-orbit coupling-based tailoring of polaron properties.

##  I INTRODUCTION

Ferromagnetic semiconductors are of great importance for the newly-developing
spintronics technology that allows for control of both charge and spin degrees
of freedom of charge carriers.Wolf  The mechanism of intrinsic long-range
magnetic order in monolayers of 2D materials such as CrI3 and bilayer
Cr2Ge2Te6 might be exploited for design of novel spin-related devices.Huang ;
Gong  They can be stacked into heterostructuresAlegria ; Yao ; Escolar ; Mogi
; Lix ; ParkBG  and their physical properties are highly tunable by carrier
doping,Tang ; Tang1 ; Peng ; Wang0  strain,Sivadas ; Chittari ; Zollner
vacancies,DY ; Hao ; Song0 ; Song ; Milosavljevic  pressureLin ; Sun ; Yu ; Ge
; Dong  and electric field.Xing ; Chen ; Wang ; Mak

Cr2Ge2Te6 exhibits a paramagnetic (PM) to ferromagnetic (FM) transition with
the Curie temperature ($T_{c}$) of 61 $\sim$ 68 K in bulk, an out-of-plane
easy axis and negligible coercivity.Gong ; Carteaux2  The $T_{c}$ is absent in
monolayer but persists in bilayer Cr2Ge2Te6 of a relatively high value of 30
K.Gong  Few-layer Cr2Ge2Te6 crystals hold an intrinsic FM order with high
tunability by gating, as required of a good functional quantum material in
spintronics.Wang  Cr2Ge2Te6 is proposed to be a two-dimensional (2D)
Heisenberg ferromagnet based on the spin wave theory,Gong  but was also found
to follow a 2D Ising-like or tricritical mean-field model,YL ; LinGT
indicating a complex magnetic mechanism.

Cr2Ge2Te6 has a semiconducting band gap of 0.2 $\sim$ 0.74 eV, which can be
tuned into metallic by applying pressure, organic ion intercalation, and
vacancies.Wang0 ; Song0 ; Yu  In a double-layer Cr2Ge2Te6 device subjected to
electrostatic gating metallic resistivity has been observed.Verzhbitskiy
Strong spin-phonon coupling has been verified in isostructural Cr2Si2Te6.Casto
Yet, the electronic transport mechanism is still unclear in conducting
Cr2Ge2Te6 single crystals.

In this letter we show that local bond distances obtained from X-ray
absorption spectroscopy exhibit little change when compared to published
average values, implying little difference in the basic structural units of
each atom. On the other hand, stacking of such units at longer atomic
distances results in stacking faults. We observe relatively low values of
thermal conductivity $\kappa$(T) below 300 K which is enhanced in magnetic
field pointing to strong spin-phonon coupling. Moreover, electrical and
thermal transport conduction reveal polaronic transport, commonly observed in
oxide materials.Palstra  Due to strong spin-orbit coupling which is the source
of magnetic anisotropy, polaronic transport mechanism could allow for tunable
spin current in future spintronic devices.Covaci ; Watanabe ; Ostwal

##  II RESULTS AND DISCUSSION

The obtained single crystals exhibits a 2D layered characteristic with the
cleave surface in the $\mathbf{ab}$-plane and the $\mathbf{c}$-axis is
perpendicular to this surface (Figure 1a).Experimental  The lattice constant
of about $c=20.57(2)$ Å can be extracted by the Bragg’s law. Cr2Ge2Te6
crystalizes in a layered structure with the $R\bar{3}h$ space group, in which
the layers stack along the (00l) direction (Figure 1b,c). There are different
types of chemical bonds in crystal structure, including strong intralayer Cr-
Te ionic bonds, Ge-Te covalent bonds, Ge-Ge dimers, and weak interlayer van
der Waals (vdW) forces.Ji  Each Cr is octahedrally surrounded by six Te, and
the edge-sharing CrTe6 octahedra form a honeycomb network. Dimers of Ge-Ge
occupy the void of each honeycomb, forming Ge2Te6 groups.

Figure 1d shows the atomic resolution high-angle annular dark-field (HAADF)
scanning transmission electron microscopy (STEM) image along the (00l)
direction. The Te atoms are hexagonally packed from the top view, while the Cr
ions and Ge-Ge dimers occupy Te-octahedral sites. The corresponding electron
diffraction pattern (inset in Figure 1d) confirms the space group. The
chemical composition gives elemental ratio Cr : Ge : Te = $20.09:20.05:59.86$
(Figure 1e).Experimental  We found the presence of stacking faults in cross-
section samples, as shown in Figure 1f-j. There are at least about 8 stacking
faults in 300 nm length scale along the $c$-axis (Figure 1g-j), which gives
8/(300 nm) = 0.027 nm-1 stacking fault density. This represents the upper
limit since most of stacking faults are found near the original crystal
surface.

The FM arises from the super-exchange between the nearest-neighbor (NN) Cr
linked by Te ligands through nearly-90∘ angle, however the exchange
interactions beyond the NN spins are also crucial in determining the magnetic
ground state.Sivadas  Pressure can compress the bond lengths of Cr-Cr and Cr-
Te and tilt the Cr-Te-Cr angle from 90∘, resulting in an enhancement of
antiferromagnetic (AFM) interaction.Ji  This can also induce spin
reorientation transition from an uniaxial to easy-plane anisotropy,Lin
indicating that the magnetism is strongly dependent on Cr local atomic
environment. To probe this local environment we performed X-ray absorption
near edge structure (XANES) and extended X-ray absorption fine structure
(EXAFS) experiments. Figure 2 shows the normalized Cr, Ge, and Te K-edge XANES
spectra and the Fourier transform magnitudes of EXAFS spectra of Cr2Ge2Te6,
respectively. The XANES spectra (Figure 2a-c) are close to that of Cr2Te3 with
Cr3+ and Te2+ states.Ofuchi  The local environment of indicated atoms is
revealed in the EXAFS spectra of Cr2Ge2Te6 measured at room temperature
(Figure 2d-f). In a single-scattering approximation, the EXAFS can be
described by:Prins

$\displaystyle\chi(k)=\sum_{i}\frac{N_{i}S_{0}^{2}}{kR_{i}^{2}}f_{i}(k,R_{i})e^{-\frac{2R_{i}}{\lambda}}e^{-2k^{2}\sigma_{i}^{2}}sin[2kR_{i}+\delta_{i}(k)],$

where $N_{i}$ is the number of neighbouring atoms at a distance $R_{i}$ from
the photoabsorbing atom. $S_{0}^{2}$ is the passive electrons reduction
factor, $f_{i}(k,R_{i})$ is the backscattering amplitude, $\lambda$ is the
photoelectron mean free path, $\delta_{i}$ is the phase shift, and
$\sigma_{i}^{2}$ is the correlated Debye-Waller factor measuring the mean
square relative displacement of the photoabsorber-backscatter pairs. The main
peaks have been corrected by standards of the NN Cr-Te [2.76(7) Å] and Ge-Te
[2.45(13) Å] in the Fourier transform magnitudes of EXAFS. We summarized the
local bond distances from different central atom ranging from 1.5 to 4.5 Å in
Experimental . There are six Te NN for the Cr atomic site with two different
distances [2.76(7) Å and 2.87(7) Å]; the next NN is one Cr [3.95(25) Å], and
then are six Ge with two different distances [4.11(25) Å and 4.17(25) Å]. Ge
atomic site has three NN Te [2.45(13) Å] and one Ge [2.52(18) Å], and also
three Te [3.84(18) Å].Carteaux2  Experimental bond distances agree very well
with each other when using different absorbing atom EXAFS spectra. The NN Te-
Te distances are 3.81(90) Å and 3.86(90) Å, and then there are three Te-Te
[4.03(6) Å]. The peaks above 4.2 Å are due to multiple scattering involving
different near neighbours and longer bond distances. Overall, local structure-
derived bond distances (Table I in Experimental ) are in agreement with bond
distances from neutron powder Rietveld refinement of the average crystal
structure.Carteaux2

Figure 3a exhibits the temperature-dependent zero-field specific heat
$C_{p}(T)$ of Cr2Ge2Te6. A $\lambda$-shape peak is observed at $T_{c}$ = 66 K,
corresponding well to the second-order PM-FM transition. The low temperature
part from 2 to 10 K can be well fitted by $C_{p}(T)=\gamma T+\beta
T^{3}+\delta T^{3/2}$ [inset in Fig. 3(a)], where the first term is the
Sommerfeld electronic part, the second term is low-temperature limit of Debye
phonon part, and the third term is low-temperature spin wave
contribution.Gopal  The derived $\gamma$, $\beta$, and $\delta$ are 9(5) mJ
mol-1 K-2, 2.52(4) mJ mol-1 K-4, and 11(3) mJ mol-1 K-5/2, respectively. The
Debye temperature $\Theta_{D}$ = 198(1) K can be calculated from $\beta$ using
$\Theta_{D}=(12\pi^{4}NR/5\beta)^{1/3}$, where $N=10$ is the number of atoms
per formula unit.

Figure 3b shows the temperature-dependent in-plane thermal conductivity
$\kappa(T)$ of Cr2Ge2Te6. In general, the $\kappa(T)$ consists of the
electronic part $\kappa_{e}$ and the phonon term $\kappa_{ph}$, i. e.,
$\kappa=\kappa_{e}+\kappa_{ph}$. The $\kappa_{e}$ part can be estimated from
the Wiedemann-Franz law $\kappa_{e}=L_{0}T/\rho$ with $L_{0}$ = 2.45 $\times$
10-8 W $\Omega$ K-2 and $\rho$ is the measured electrical resistivity. The
estimated $\kappa_{e}$ is less than 0.03 % of $\kappa(T)$ due to the large
electrical resistivity of Cr2Ge2Te6, indicating a predominantly phonon
contribution. At 300 K, the value of $\kappa(T)$ is about 5.34 W K-1 m-1,
larger than that of 3 W K-1 m-1 for Cr2Si2Te6 and, as expected due to the
absence of dense grain boundaries, also larger when compared to Cr2Ge2Te6
polycrystal.DY ; Casto  In the absence of magnetic field above 100 K, the
$\kappa(T)$ shows weak temperature dependence, different from phonon-dominated
$\kappa$ph($T$) calculated within Debye model (solid line in Figure 3b) that
takes $\sim$1/$T$ behavior at high temperatures.Experimental  The difference
is likely due to phonon scattering by magnetic fluctuations and strong spin-
lattice coupling producing phonon glass.Casto  With decreasing temperature, a
rapid increase occurs at $T_{c}$ and a typical phonon peak was observed at 25
K, however stacking faults have only minor influence on
$\kappa$($T$).Experimental  Strong positive field-dependent behavior of
$\kappa(T)$ in Figure 3b is similar to RuCl3 where it was described within the
model of phonon scattering off Kitaev-Heisenberg excitations that are either
fractionalized or incoherent type originating from strong magnetic
anharmonicity.Hentrich  Since Kitaev-type magnetic interactions have been
discussed in Cr2Ge2Te6 it is likely that $\kappa$($T$) in 9 T is connected
with the same mechanism.XuC1 ; XuC2

Figure 3c presents the temperature dependence of in-plane electrical
resistivity $\rho(T)$ for Cr2Ge2Te6, showing an obvious semiconducting
behavior with $\rho_{300K}=0.45$ $\Omega$ cm. For resistivity mechanism we
consider the thermally activated model $\rho(T)=\rho_{0}exp(E_{\rho}/k_{B}T)$,
the adiabatic small polaron hopping model $\rho(T)=ATexp(E_{\rho}/k_{B}T)$
where $k_{B}$ = 8.617 eV K-1 is the Boltzmann constant and $E_{\rho}$ is
activation energy, and the Mott variable-range hopping (VRH) model
$\rho(T)=\rho_{0}exp(T_{0}/T)^{1/4}$.Austin  Figure 3d shows the fitting
result of the adiabatic small polaron hopping model in two temperature ranges
380-150 K and 100-60 K. The extracted activation energy $E_{\rho}$ is about
118.9(3) meV for 380-150 K and 26.2(7) meV for 100-60 K, respectively. Whereas
the $\rho(T)$ curve can not be well fitted with the VRH model it can be
explained by the thermally activated model, i.e., plot of $ln\rho$ vs
$1/k_{B}T$ and $ln(\rho/T)$ vs $1/k_{B}T$ overlap with each other (Figure 3d).

Temperature-dependent thermopower $S(T)$ gives further insight into electronic
transport. The $S(T)$ exhibits positive values in the entire temperature range
of our measurement with $S_{300K}=582$ $\mu$V K-1 (Figure 3e), indicating
dominant hole-type carriers. The value of $S$(T) at 300 K is quite similar to
polycrystal, which argues for the absence of phonon drag effect and dominant
electronic diffusion mechanism of thermal conduction.DY  The $S(T)$ from 300
to 150 K can be fitted with the equation $S(T)=(k_{B}/e)(\alpha+E_{S}/k_{B}T)$
(Figure3f),Austin  where $E_{S}$ is activation energy and $\alpha$ is a
constant. The derived activation energy for thermopower is $E_{S}$ = 20(1)
meV. This is much smaller than that for high temperature resistivity
$E_{\rho}$ = 118.9(3) meV, and also smaller when compared to activation energy
associated with low temperature resistivity $E_{\rho}$ = 26.2(7) meV. The
large discrepancy between $E_{S}$ and $E_{\rho}$ typically reflects the
polaron transport mechanism of carriers.

According to the polaron model, the $E_{S}$ is the energy required to activate
the hopping of carriers, while the $E_{\rho}$ is the sum of the energy needed
for creation of carriers and activating the hopping of carriers.Austin
Therefore, within the polaron hopping model the activation energy $E_{S}$ is
smaller than $E_{\rho}$. Whereas the temperature-dependent thermopower $S(T)$
shows a change of slope anomaly at $T_{c}$ at 70 K, an additional change of
slope is observed at 150 K, coincident with the rapid rise of electrical
resistivity $\rho(T)$ on cooling through this temperature range (Figure 3c,e).
This points to connection between electrical transport and anisotropy of 2D
magnetic correlations above $T_{c}$, consistent with magnetic resonance
experiments in low magnetic fields.Zeisner  Close inspection of $S(T)$,
however, reveals an additional peak in (40 - 50) K range on cooling below
$T_{c}$ (Figure 3e). To investigate this further, we turn to low-field
magnetic susceptibility and X-ray scattering measurements.

Figure 4a,b shows the temperature dependence of dc magnetization $M(T)$ for
Cr2Ge2Te6 single crystal measured at a low field $H$ = 10 Oe applied in the
$\mathbf{ab}$-plane and along the $\mathbf{c}$-axis, respectively. A PM-FM
transition was observed at $T_{c}$ = 65 K. The $M(T)$ is nearly isotropic
above, while magnetic anisotropy is observed below $T_{c}$. The value of
$M(T)$ for $\mathbf{H\parallel c}$ is larger than that for $\mathbf{H\parallel
ab}$ below $T_{c}$, confirming the easy $\mathbf{c}$ axis. The splitting of
ZFC and FC curves at low temperature mostly originates from the anisotropic FM
domain effect, in line with previous reports.Zhang ; Selter ; YuLiu ; Liu
Figure 4c,d shows the temperature dependence of ZFC ac susceptibility measured
with oscillated ac field of 3.8 Oe and frequency of 499 Hz. A sharp peak is
observed at $T_{c}$ in the real part $m^{\prime}(T)$ for both directions. An
additional anomaly occurs around 40 K just like in the $S(T)$ (Figure 3e), as
seen a broad hump in both real part $m^{\prime}(T)$ and imaginary part
$m^{\prime\prime}(T)$ (Figure 4d) due to the interplay of magnetocrystalline
anisotropy and external field.Selter  The existence of anomaly below $T_{c}$
is confirmed by the temperature evolution of magnetic order parameter (Figure
4e) measured using X-ray scattering. Below $T_{c}$, enhanced X-ray scattering
was observed at the (006) Bragg peak at a photon energy of 6 keV in the
rotated, $\sigma$-to-$\pi$, scattering channel. The energy and polarization
dependencies of this X-ray scattering indicate its origin as resonant magnetic
scattering due to dipole transitions from the Cr 1$s$ to unoccupied 4$p$
states, which are spin-polarized by hybridization with Cr 3$d$ states. This
confirms influence of magnetocrystalline anisotropy on transport behavior not
only above but also below magnetic $T_{c}$.

Stacking faults are found in variety of 2D vdW materials such as Bi2Te3 or
RuCl3.MedlinD ; YamauchiI  They originate from dislocations, core regions with
large atomic displacement from the ideal position, and propagate as strain
field in the crystal lattice as covalently bonded crystal units stack with vdW
forces along the $c$-axis. Magnetic anisotropy in ordered state and local
electronic structure may be affected by stacking faultsYamauchiI ; LiuYBiSe
but also polaronic defects can be created by stacking faults as a result of
sub-coordinated metal atoms and electron transfer via structural sub-
units.BuenoR

Whereas surface Ge vacancies have been predicted to contribute to enhanced
electrical conductivity,Song  stacking faults observed in structurally related
material In2Si2Te3 were associated with Ge(Si) pairs atomic positions that
could modulate the bands near the Fermi level via Ge(Si)-Te electron transfer
that indirectly alters $p$-$d$ hybridization between Cr 3$d$ and Te $p$
orbitals.Song ; LefevreR  It was proposed that stacking faults lower local
structure symmetry at short range distances.LefevreR  Good agreement of bond
distances Figure 2d-fExperimental  with the average structure interatomic
distancesYL  imply that the stacking fault influence on the unit cell symmetry
is likely at the intermediate range distances, calling for detailed total
scattering analysis. Interestingly, stacking faults do not influence
significantly either $\kappa$($T$) mechanism nor magnetic stripe domains that
are the source of skyrmionic bubbles.Experimental ; YanX ; GiannoK ; HanMG
Stacking faults are visible in cross-sectional Lorentz TEM sample at 13 K
(Figure 4f). Note that dark and white lines parallel to the $c$-axis arising
due to 180∘ stripe domains do not show significant changes upon crossing
stacking faults (dark lines running perpendicular to the $c$-axis). This
indicates that the faults do not perturb magnetic state or topological
magnetic spin textures.HanMG  On the other hand, variation of crystal
distortion induces changes in electrical transport as confirmed in an
independently grown crystal.Experimental

To qualitatively understand the potential effects of structural distortion, we
calculate the band structures and conductivities in distorted (Figure 4h)and
undistorted structures.Carteaux2 ; Experimental  The overall band structure of
the distorted structure share a great similarity with that of undistorted
structure reported previously.LeeY ; MenichettiG . Figure 5a,b show scalar-
relativistic band structure and partial density of states (PDOS) of the
distorted structure calculated with a first-principles method. An indirect gap
of 0.38 eV is obtained and that is reduced by spin-orbit interaction down to
0.15 eV. The valence band maximum (VBM) is located at $\Gamma$-point in
minority spin state; it is slightly off from the $\Gamma$-point in the
majority spin channel. The conduction band minimums (CBM) are located between
$T$ and $H_{0}$ in the majority spin channel and between $H_{0}$ and $L$ in
the minority spin channel. Cr-$d$ states peak at -1 eV below Fermi level
($E_{F}$) in the majority spin channel and 2 eV above $E_{f}$ in the minority
spin channel, corresponding to a spin splitting of $\sim 3$ eV. There are flat
bands along the $\Gamma$–$T$ direction right below $E_{F}$, which may
significantly contribute to transport properties. Interestingly, the positions
of these flat bands are sensitive to structure distortion. Figure 5c,d compare
the band structures along $\Gamma$–$T$ direction calculated using the
distorted (crystal 1) and undisortedCarteaux2  structures. The distortion
slightly shifts the flat bands BD1 and BD2 above the more dispersive band BD3
along the $\Gamma$–$T$ path. Bands BD1 and BD2 mostly consist of in-plane
Te-$p_{x}$ and Te-$p_{y}$ states,Experimental  resulting in small exchange
splitting and negligible dispersion along the out-of-plane direction
($\Gamma$-$T$ path). In contrast, band BD3 consists of mainly Te-$p_{z}$ and
Cr-$d_{z^{2}}$ states, showing a much stronger dispersion along $k_{z}$ and
larger spin splitting. Considering the proximity of the flat bands near
$E_{F}$, their changes induced by the distortion can affect the transport
properties significantly, especially at higher temperatures and with hole
doping.

To explore the effects of bandstructure change on transport, we calculate the
conductivities for distorted (crystal 1) and undistortedCarteaux2  structure
using BoltzTraP.MadsenGK  Figure 5e shows the calculated $\sigma_{xx}$, the
in-plane component of electrical conductivity tensor, as a function of
temperature. The conductivities in both structures increase with temperature,
as expected for intrinsic semiconductors. The distorted structure has a higher
in-plane conductivity than the undistorted structure. The important
contribution of BD1 and BD2 on conductivity can be further illustrated by
calculating the predicted conductivities in the hole-doped case, as shown in
Figure 5f. The chemical potential $\mu_{2}$ is lowered to 0.05 eV below
$E_{f}$, as denoted in Figure 5c by green dash-doted line, to mimic the hole-
doping effects. As is expected, the conductivities of all cases increase at
lower temperatures. The $\sigma_{xx}$ of distorted structure, however, is
increased more than 100 times. Therefore, first-principle calculations
indicate that the local distortion with appropriate doping can profoundly
affect electronic and magnetic properties of Cr2Ge2Te6, leading to ever higher
conductivity.

##  III CONCLUSIONS

In summary, our results indicate strong influence of crystal structure
distortion and magnetocrystalline anisotropy on electrical and thermal
transport both above and below $T_{c}$ in 2D vdW magnetic Cr2Ge2Te6 crystals.
Electronic transport is dominated by polaronic effects commonly observed in
oxide materials, confirming strong electron-phonon coupling. Moreover,
interplay between spin-orbit electron-phonon coupling may tailor the spin
polarization since the polaron could retain only one of the spin-polarized
bands in its coherent spectrum.Covaci  This could be used in spin-orbitronic
and magneto-thermal devices that exploit current propagation via spin-orbit
torque mechanism.Watanabe ; Ostwal

##  IV Methods

Crystals were synthesized from the self-flux method Experimental . Crystals
were cut in suitable geometry for in-plane resistivity measurements.
Pulverized crystals were used XANES and EXAFS measaurement.
ReferenceExperimental  describes experimental methods in more details.

## Acknowledgements

Work at Brookhaven National Laboratory (BNL) is supported by the Office of
Basic Energy Sciences, Materials Sciences and Engineering Division, U.S.
Department of Energy under Contract No. DE-SC0012704. TEM sample preparation
was carried out at the Center for Functional Nanomaterials (BNL), which is
supported by the US Department of Energy, Office of Basic Energy Sciences
under Contract No. DE-AC02-98CH10886. This research used beamlines 4-ID and
8-ID of the National Synchrotron Light Source II, a U.S. Department of Energy
(DOE) Office of Science User Facility operated for the DOE Office of Science
by Brookhaven National Laboratory under Contract No. DE-SC0012704. The work
carried out at the University of Delaware was supported by the U.S. Department
of Energy, Office of Science, Basic Energy Sciences, under Award DE-SC0008885.
First principle calculations were supported by the U.S. Department of Energy,
Office of Science, Office of Basic Energy Sciences, Materials Sciences and
Engineering Division. Ames Laboratory is operated for the U.S. Department of
Energy by Iowa State University under Contract No. DE-AC02-07CH11358. A
portion of this work was performed at the National High Magnetic Field
Laboratory, which is supported by the NSF Cooperative Agreement No.
DMR-1644779 and the State of Florida.

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Figure 1: Crystal structure and nanostructure characterization of Cr2Ge2Te6.
a) 2$\theta$ X-ray diffraction scan along the (00l) direction. Inset shows
optical image of Cr2Ge2Te6 single crystals. $R\bar{3}h$ unit cell of Cr2Ge2Te6
shown from b) the side view and c) the top view. d) High-angle annular dark-
field (HAADF) scanning transmission electron microscopy (STEM) image obtained
along the $\mathbf{c}$-axis. Inset shows the electron diffraction pattern. e)
X-ray energy-dispersive spectrum of the selected area. HAADF STEM image f) of
cross-sectional TEM sample imaged along the $\mathbf{a}$-axis. Dark-field TEM
image g) obtained with the reflection of stacking A (orange arrowed peak in
h). Selected area electron diffractions (h and i) from the green-boxed area
(h) and blue-boxed area (i). In (i) the Stacking A only exists. Atomic
resolution HAADF STEM image j) from the green-boxed area. Two insets show the
enlarged area showing the two different stackings separated by a stacking
fault, indicated with a dotted line. Two inclined yellow and red lines show
the stacking sequence of the Te/Ge columns along the $\mathbf{c}$-axis.

Figure 2: Local atomic structure. Normalized a) Cr, b) Ge, and c) Te K-edge
XANES spectra with insets depicting local atomic structure. Fourier transform
magnitudes of the EXAFS oscillations (symbols) for d) Cr, e) Ge, and f) Te
K-edge with the phase shifts correction. The model fits are shown as solid
lines. Insets show the corresponding filtered EXAFS (symbols) with $k$-space
model fits (solid lines).

Figure 3: Thermodynamic, electronic and thermal transport properties.
Temperature dependence of a) specific heat $C_{p}(T)$, b) thermal conductivity
$\kappa$($T$) and c) electrical resistivity $\rho(T)$ of our Cr2Ge2Te6 single
crystals. Inset in a) shows the fitted (red line) low-temperature $C_{p}(T)$
from 2 to 10 K (see text). Green solid curve in b) shows the fitting of the
lattice thermal conductivity using the Debye model. The ln($\rho$) vs
$1/k_{B}T$ (left axis) and ln($\rho/T$) vs $1/k_{B}T$ (right axis) d) fitted
by adiabatic small polaron hopping model (see text). e) Temperature dependence
of $S(T)$thermopower. f) The $S(T)$ vs, $1000/T$ curve fitted by polaron
transport model (see text).

Figure 4: Magnetocrystalline anisotropy. Temperature dependence of zero field
cooling (ZFC) and field cooling (FC) dc magnetization $M(T)$ measured at H =
10 Oe for Cr2Ge2Te6 single crystal with a) $\mathbf{H\parallel ab}$ and b)
$\mathbf{H\parallel c}$, respectively. Temperature dependence of ac
susceptibility real part $m^{\prime}(T)$ and imaginary part
$m^{\prime\prime}(T)$ measured with oscillated ac field of 3.8 Oe and
frequency of 499 Hz applied c) in the $\mathbf{ab}$-plane and d) along the
$\mathbf{c}$-axis, respectively. e) The integrated intensities of longitudinal
scans through the (006) magnetic Bragg peak of Cr2Ge2Te6. f) Lorentz TEM image
of Cr2Ge2Te6 obtained perpendicular to the $\mathbf{c}$-axis. at 13 K, showing
180∘ stripe domains separated by the Bloch walls due to uniaxial anisotropy
(Ref. 61). Two stacking faults are visible as dark horizontal lines, which do
not affect the stripe domain configuration. g) Unit cell of distorted
Cr2Ge2Te6 showing thermal ellipsoids from single crystal refinement and Ge-Ge
bonds (red areas).

Figure 5: (a) Scalar-relativistic band structure of Cr2Ge2Te6 with lattice
distortion. Blue (red) color indicates the majority (minority) spin band. (b)
The corresponding partial density of states (PDOS) projected on Te-$p$,
Ge-$p$, and Cr-$d$ states. Scalar-relativistic band structures calculated in
(c) distorted and (d) undistorted structures. BD1, BD2, and BD3 are selected
bands to analysis band characters (See Ref. 40). Orange and green dash-dotted
lines denote two chemical potentials, $\mu$1 and $\mu$2, respectively, that
are employed for conductivity calculations. Electrical conductivity as a
function of temperature calculated without (e) and with (f) hole-doping in
Cr2Ge2Te6 distorted and undistorted structures. The chemical potential is set
to 0.05 eV below VBM to mimic the hole-doped case. The red and blue colors
denote the distorted and undistorted structures, respectively.