# Measurement incompatibility cannot be stochastically distilled

Huan-Yu Ku  Department of Physics, National Taiwan Normal University, Taipei
11677, Taiwan  Faculty of Physics, University of Vienna, Boltzmanngasse 5,
1090 Vienna, Austria  Institute for Quantum Optics and Quantum Information
(IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, 1090 Vienna, Austria
Chung-Yun Hsieh<EMAIL_ADDRESS>H.H. Wills Physics Laboratory,
University of Bristol, Tyndall Avenue, Bristol BS8 1TL, United Kingdom
Costantino Budroni  Department of Physics “E. Fermi” University of Pisa, Largo
B. Pontecorvo 3, 56127 Pisa, Italy

###### Abstract

We show that the incompatibility of a set of measurements cannot be increased
by subjecting them to a filter, namely, by combining them with a device that
post-selects the incoming states on a fixed outcome of a stochastic
transformation. This result holds for several measures of incompatibility,
such as those based on robustness and convex weight. Expanding these ideas to
Einstein-Podolsky-Rosen steering experiments, we are able to solve the problem
of the maximum steerability obtained with respect to the most general local
filters in a way that allows for an explicit calculation of the filter
operation. Moreover, our results generalize to nonphysical maps, i.e.,
positive but not completely positive linear maps.

Introduction.— A remarkable aspect of quantum mechanics is that certain
physical quantities, e.g., position and momentum of a particle, cannot be
simultaneously measured: They are said to be incompatible. Measurement
incompatibility is at the core of many quantum properties, from uncertainty
relations [1, 2], to Bell nonlocality [3, 4, 5, 6], Kochen-Specker
contextuality [7], Einstein-Podolsky-Rosen steering [8, 9, 10, 11], and so on.
It is also relevant in operational tasks such as cryptography [12], state
discrimination as well as measurement simulation [13, 14, 15, 16, 17], and
steering distillation [18].

Given its prominent role in quantum information theory, incompatibility has
been studied from the perspective of resource theories, with the goal of
understanding how it can be manipulated [19, 16, 17, 14, 15]. Different
quantifiers of incompatibility have been proposed, from robustness-type
measures [19, 10, 20, 21] to convex-weight-type measures [22, 23, 24]. A
central question is, then, which operations do not increase the
incompatibility of a set of measurements. Different operations have been
considered, from quantum pre-processing [19] to conditional classical pre- and
post-processing [14], and, finally, a combination of the two [15]. The latter
result, in particular, shows that incompatibility cannot be increased, on
average, by quantum instruments. This is the natural question both in
entanglement theory [25, 26] and steering [27]. In fact, by abandoning the
requirement that this holds on average, one finds that both entanglement [28,
29] and steerability [30, 18] can be distilled via local filters [31], namely,
they can be increased via local stochastic operations followed by a post-
selection on a fixed outcome. In the following, we show that, in contrast,
such operations are not able to increase the incompatibility of a set of
measurements. We first focus on the incompatibility robustness, but we show
how similar results apply to the incompatible weight and variants of the
robustness.

We extend this investigation to steering [32, 33], where our results solve the
problem of the maximum steerability obtainable under the most general local
filters. Previously, this problem was solved for the case of filters
consisting of a single Kraus operator [18]. We show that increasing the number
of Kraus operators does not increase the steerability, and consequently we are
able to compute the optimal filter explicitly from the results of [18].
Moreover, our results hold also for nonphysical filters, namely, linear maps
that are positive but not completely positive. Finally, we show that our
results hold for several measures of steerability present in the literature,
based on robustness and convex weights.

Preliminary notions.—The most general measurement in quantum mechanics is
represented by a positive-operator-valued measure (POVM) [34]
$\\{M_{a}\\}_{a}$; namely, a collection of operators satisfying $M_{a}\geq
0,\forall a$ and $\sum_{a}M_{a}=\openone$, where $a$ denotes the outcome of
the measurement. In general, we have multiple inputs $x$, so we have a
collection of POVMs $\vec{M}:=\\{M_{a|x}\\}_{a,x}$, which we call a
measurement assemblage (MA). If there exists a single POVM
$\\{G_{\lambda}\\}_{\lambda}$ that can fully characterize the set of POVMs by
classical post-processing, the MA is said to be jointly measurable (JM) i.e.,
$M_{a|x}=\sum_{\lambda}p(a|x,\lambda)G_{\lambda}$, where $p(a|x,\lambda)$ is a
conditional probability. If there is no such description, the measurement
assemblage is said to be incompatible. Incompatibility can be quantified in
different ways, for instance via the incompatibility robustness [10, 20]
defined as $\mathbb{IR}(\vec{M})=\min\\{t\geq 0\;|\;\exists\vec{N}~{}\text{MA
and}~{}\vec{D}\in\mathbb{JM}\ \text{s.t.}\ (M_{a|x}+tN_{a|x})/(1+t)=D_{a|x}\
\forall~{}a,x\\}$, where $\mathbb{JM}$ denotes the set of JM measurement
assemblages. Several other measures can be introduced by putting restrictions
on the MA $\vec{N}$; see, e.g., [21], or using different convex decompositions
[22].

Steering is a protocol in which one party, Alice, performs measurements
$\vec{A}=\\{A_{a|x}\\}_{a,x}$ on her half of a bipartite system in the initial
state $\rho_{\rm AB}$ and sends the measurement information, i.e., input $x$
and output $a$, to Bob. In this way, Bob obtains a collection of local states
labeled as $\vec{\sigma}:=\\{\sigma_{a|x}\\}_{a,x}$, where
$\sigma_{a|x}=\operatorname{tr}_{\rm A}[\rho_{\rm
AB}(A_{a|x}\otimes\openone)]$ is in an unnormalized form. We call
$\vec{\sigma}$ a state assemblage (SA). It satisfies $\sigma_{a|x}\geq
0,\forall a,x$ and $\sum_{a}\sigma_{a|x}=\rho_{\rm
B}=\operatorname{tr}_{A}[\rho_{\rm AB}],\forall x$. A SA is unsteerable if
there is a local-hidden-state (LHS) model, namely, if we can write
$\sigma_{a|x}=\sum_{\lambda}p(\lambda)p(a|x,\lambda)\rho_{\lambda},\forall
a,x$, and steerable otherwise. Steering can be quantified, for instance, by
the steering robustness [35] defined as
$\mathbb{SR}(\vec{\sigma})\coloneqq\min\\{t\geq
0\;|\;\exists\;\vec{\xi}\;\text{SA},\;\vec{\tau}\in\mathbb{LHS}~{}\text{s.t.}~{}(\sigma_{a|x}+t\xi_{a|x})/(1+t)=\tau_{a|x}\forall
a,x\\}$, where $\mathbb{LHS}$ is the set of SA admitting a LHS model.

The SA $\vec{\sigma}$ is unsteerable, whenever $\rho_{\rm AB}$ is separable or
$\vec{A}$ is JM, i.e., both entanglement and measurement incompatibility are
necessary for steering. This connection is indeed much deeper [8, 9] and
steering and incompatibility can be seen as equivalent problems [10]. A
central notion in this correspondence is that of steering-equivalent-
observables (SEOs) [10] which map a SA to a MA as

$B_{a|x}:=\rho_{\rm B}^{{-1/2}}\sigma_{a|x}\rho_{\rm B}^{{-1/2}},\text{ for
}\rho_{\rm B}\text{ full-rank,}$ (1)

or an analogous expression with an additional isometry in case $\rho_{\rm B}$
is not full rank; see [10] for more details. It is straightforward to see that
SEOs define equivalence classes among SAs, i.e., via the relation of having
the same SEO. It was recently shown [18] that such classes also define which
SAs can be transformed into each others via ${\rm LF_{1}}$ operations, i.e.,
local filters consisting of only one Kraus operator, and that the maximal
steerability within one class corresponds to the incompatibility of the SEO,
namely,

$\max_{\mathcal{E}_{\omega}\in{\rm
LF_{1}}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)=\max_{\rho_{\rm
B}}\mathbb{SR}\left(\rho_{\rm B}^{{1/2}}\vec{B}\rho_{\rm
B}^{{1/2}}\right)=\mathbb{IR}(\vec{B}),$ (2)

where $\vec{B}$ is the SEO of $\vec{\sigma}$, $\omega$ represents the outcome
on which one post-selects with probability
$p(\omega)=\operatorname{tr}[\mathcal{E}_{\omega}(\sum_{a}\sigma_{a|x})]$; see
App. A and [18]. In the following, when we talk about steerability
(incompatibility) we implicitly refer to the steerability (incompatibility)
robustness, unless explicitly stated otherwise. Several results generalize to
other steering (incompatibility) measures, as we discuss below.

Figure 1: (a) Channel applied before a measurement. The channel can be seen as
applied to the state (Schrödinger picture) or applied to the measurement
(Heisenberg picture). (b) Analogous setup, but with an instrument. The
instrument can still be seen equivalently as part of the state preparation or
the measurement procedure. However, the procedure now involves a post-
selection whose probability depends on the incoming state. The measurements
are part of a bipartite steering protocol, by which its incompatibility can be
operationally estimated.

Measurement incompatibility cannot be stochastically distilled.— The first
step to derive our result is to reason about what it means to distill
measurement incompatibility. A quantum channel $\mathcal{E}$, i.e., a
completely-positive (CP) trace-preserving linear map [34], that is applied to
a MA $\vec{A}$ corresponds to the circuit depicted in Fig. 1 (a). In
operational terms, we combine the original device implementing the MA
$\vec{A}$ with a quantum channel $\mathcal{E}$. This can be represented in the
operator form as the MA
$\mathcal{E}^{\dagger}(\vec{A}):=\\{\mathcal{E}^{\dagger}(A_{a|x})\\}_{a,x}$.
Its incompatibility $\mathbb{IR}(\mathcal{E}^{\dagger}(\vec{A}))$ can be
either computed directly or, more interestingly, indirectly through the
following argument. Given a MA $\vec{A}$ we define
$\widetilde{\mathbb{IR}}(\vec{A})$ as

$\widetilde{\mathbb{IR}}(\vec{A}):=\max_{\rho_{\rm AB}}\
\mathbb{SR}(\vec{\sigma}(\rho_{\rm AB})),$ (3)

with $\sigma_{a|x}(\rho_{\rm AB}):=\operatorname{tr}_{\rm A}[\rho_{\rm
AB}A_{a|x}\otimes\openone]$ and the maximization is taken over all bipartite
states $\rho_{\rm AB}$. The equivalence
$\widetilde{\mathbb{IR}}(\vec{A})=\mathbb{IR}(\vec{A})$ follows directly from
Th. 3 of [18], where it is proven that for each MA $\vec{A}$ one can find a
bipartite state such that the steerability of the obtained SA $\vec{\sigma}$
is arbitrary close to the incompatibility of $\vec{A}$; see Appendix A. In
other words, the incompatibility of $\vec{A}$ corresponds to the maximal
steerability obtained in all experiments involving $\vec{A}$ as measurements.

With this operational definition of incompatibility, we can compute
$\mathbb{IR}(\mathcal{E}^{\dagger}(\vec{A}))$. Consider the steering scenario
in Fig. 1 (a), where a quantum channel is applied before the measurement on
$\vec{A}$. This channel, interpreted above as a part of the measurement
procedure can be equivalently seen as part of the state preparation, i.e., the
Heisenberg vs Schrödinger picture. However, if one takes the optimal state
$\rho_{\rm AB}$ in Eq. (3), it is clear that any channel cannot increase the
steerability of the state. We therefore obtained that measurement
incompatibility cannot be increased by applying a quantum channel, i.e., a
deterministic operation, to a MA. This is, of course, a well known fact in the
resource theory of incompatibility [15].

The question, then, arises: What happens with filters? In operational terms, a
filter, corresponds to performing a measurement and then post-selecting on a
certain outcome. As before, this operation transforms the measurement
according to the mapping $\mathcal{E}_{\omega}^{\dagger}(\vec{A})$, where
$\omega$ is the specific outcome on which we post-select. This object,
however, is no longer a MA, since it does not respect the normalization
conditions, i.e.,
$\sum_{a}\mathcal{E}^{\dagger}_{\omega}(A_{a|x})\neq\openone$. Indeed, it is
impossible to define the filtered measurements as a MA that has a state-
independent form. Nevertheless, we can still make sense operationally of the
notion of filtering a measurement. As for the case of the channel, we perform
the filter between the state preparation and the measurement, but this time we
discard all experimental rounds where we do not observe the correct outcome
$\omega$. Again, the filter can be thought of as a part of the state
preparation, hence, given an already optimal initial state $\rho_{\rm AB}$,
the steerability of the associated SA cannot be improved. We thus obtained

######  Result 1.

Measurement incompatibility cannot be distilled via filter operations.

Notice that this result is independent of the specific task chosen, i.e.,
quantum steering, but depends only on the operational properties of the
filtered measurements. Indeed, the same result could be obtained by
considering instead of steering, state-discrimination tasks; see App. B for
more details. An analogous argument shows that both the incompatible weight
[22], which has an operational definition in terms of state-exclusion tasks
[23, 24], and the robustness of measurement [36], which is the minimal amount
of noise that needs to be added to a single measurement before it becomes
trivial and can be defined in terms of a state-discrimination task, cannot be
increased by filtering; see App. B for more details.

This also means that once $\mathcal{E}_{\omega}$ is a positive linear map, it
cannot distill incompatibility. Hence, at a mathematical level our no-go
result also rules out various non-physical linear maps such as positive linear
maps that are not CP or trace-increasing. Notice that in the latter case the
physical interpretation is more problematic since one cannot make sense of the
notion of post-selection and the associated renormalization of probability.
See App. B for more details. Finally, we remark that the case discussed does
not involve the joint use of multiple copies of the measurement device, as it
is the case in a recent discussion on simulability of a target measurement
from multiple copies of an imperfect version of it [37].

Filtering operations on quantum steering.—Here, we discuss the effect of
positive linear maps, which we later on interpret as filters, on the different
elements of a SA. The goal is to show that local filters described by a single
Kraus operator, denoted by ${\rm LF_{1}}$, are always enough to obtain the
maximum steerability over all filter operations, denoted by ${\rm LF}$. We
start with the following simple observation

######  Observation 1.

For a reduced state $\rho_{\rm B}$ such that $\mathbb{SR}(\rho_{\rm
B}^{{1/2}}\vec{B}\rho_{\rm B}^{{1/2}})=\mathbb{IR}(\vec{B})$, as in Eq. (2),
we can choose $\vec{\xi}$ and $\vec{\tau}$ appearing in the definition of
$\mathbb{SR}(\rho_{\rm B}^{{1/2}}\vec{B}\rho_{\rm B}^{{1/2}})$ such that their
reduced states coincide with $\rho_{\rm B}$, i.e.,
$\sum_{a}\xi_{a|x}=\sum_{a}\tau_{a|x}=\rho_{\rm B}$.

Proof (Sketch).— One can explicitly construct $\vec{\xi}$ and $\vec{\tau}$
from the optimal $\vec{N}$ and $\vec{D}$ appearing in the definition of
$\mathbb{IR}$ by multiplying them by $\rho_{\rm B}^{1/2}$. See App. C for
details. $\square$

This observation suggests that the definitions of $\mathbb{IR}$ and
$\mathbb{SR}$ are completely equivalent when we consider the maximal element
in a SEO class. Moreover, this observation implies that the consistent
steering robustness $\mathbb{SR}^{\rm c}(\vec{\sigma})$ [38] (a robustness
with the requirement that the noise term $\vec{\xi}$ has the same reduced
state as the original SA) quantifies the maximal steerability over the SEO
class of $\vec{\sigma}$, or equivalently, over all possible SA that can be
reached via local filters. Indeed, it has been shown that $\mathbb{SR}^{\rm
c}(\vec{\sigma})=\mathbb{IR}(\vec{B})$, see [39, 40].

From Obs. 1, another useful result follows

######  Result 2.

Given a MA $\vec{B}$ and a positive map $\mathcal{E}_{\omega}$

$\displaystyle\begin{split}\max_{\rho_{\rm{B}}}\mathbb{SR}\left(\frac{\rho_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{B})\rho_{\rm{B}}^{1/2}}{p(\omega)}\right)\leq\max_{\rho_{\rm{B}}}\mathbb{SR}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right),\end{split}$
(4)
$\displaystyle\begin{split}\max_{\rho_{\rm{B}}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2})}{p(\omega)}\right)\leq\max_{\rho_{\rm{B}}}\mathbb{SR}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right),\end{split}$
(5)

where $p(\omega)=\operatorname{tr}[\mathcal{E}_{\omega}(\rho_{\rm{B}})]$.

_Proof (Sketch).—_ Given the optimal decomposition for the robustness on the
r.h.s., i.e., either $(t,\vec{N},\vec{D})$, or $(t,\vec{\xi},\vec{\tau})$, one
can construct a valid decomposition for the l.h.s., which proves the
inequality. See App. D for details. $\square$

So far, this result remains at some abstract level. For instance, notice that
$\mathcal{E}_{\omega}$ is not necessarily a physical map, i.e., CP and trace-
non-increasing, but just a positive linear map. We can finally show that ${\rm
LF_{1}}$ are enough to obtain maximum steerability

######  Result 3.

The maximal steerability obtainable by ${\rm LF_{1}}$ filtering, is the same
as the one obtainable with general filters in ${\rm LF}$, namely,

$\max_{\mathcal{E}_{\omega}\in{\rm
LF_{1}}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)=\max_{\mathcal{E}_{\omega}\in{\rm
LF}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right).$
(6)

_Proof.—_ Fix a SA $\vec{\sigma}$, since ${\rm LF_{1}}\subset{\rm LF}$, we
have that $\max_{\mathcal{E}_{\omega}\in{\rm
LF_{1}}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)\leq\max_{\mathcal{E}_{\omega}\in{\rm
LF}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right).$
Conversely, take $\mathcal{E}_{\omega}^{\prime}\in{\rm LF}$ and take $\vec{B}$
the SEO of $\vec{\sigma}$. By Eqs. (2) and (5), we have

$\displaystyle\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}^{\prime}(\vec{\sigma})}{p^{\prime}(\omega)}\right)\leq\max_{\rho_{\rm{B}}}\mathbb{SR}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)=\max_{\mathcal{E}_{\omega}\in{\rm
LF_{1}}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)$
(7)

for all $\mathcal{E}_{\omega}^{\prime}$, which completes the proof of Eq. (6).
$\square$

Again, notice that even if we stated the results in terms of local filters,
i.e., $\\{\mathcal{E}_{\omega}\\}_{\omega}$ is a quantum instrument
($\mathcal{E}_{\omega}$ CP and $\sum_{\omega}\mathcal{E}_{\omega}$ trace-
preserving), Result 3 hold for any positive linear map $\mathcal{E}_{\omega}$.

This result completes the classification of steerability with respect to local
operations presented in Ref. [18]: the maximal steerability obtainable with
${\rm LF_{1}}$ filters is the same as the one obtainable with general ${\rm
LF}$ filters. Moreover, since optimal ${\rm LF_{1}}$ filters can be
efficiently computed as a semi-definite programming (SDP) [41, 42], this
completely solves the problem of local filtering for steering distillability.
Finally, notice that ${\rm LF}$ filters are able to change the SEO class of
the SA, contrary to what happens with two-way ${\rm LF_{1}}$ [18].

The second physical implication of Result 2 reads

######  Result 4.

Consider an experiment where Alice steers Bob’s state via a pure bipartite
state $\rho_{\rm AB}=|\psi\rangle\langle\psi|_{\rm AB}$. If Bob’s assemblage
$\vec{\sigma}$ is already optimal w.r.t. Bob’s local operations, i.e.,
$\mathbb{SR}(\vec{\sigma})=\max_{\mathcal{E}_{\omega}\in{\rm
LF}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)$,
then its steerability cannot be increased by performing any filter on Alice’s
side.

_Proof.—_ Alice performs a measurement $\vec{A}$ on her half of
$\rho_{\rm{AB}}$, generating $\vec{\sigma}$ on Bob’s side. Now Alice repeats
the same experiment with the filter $\\{\mathcal{E}_{\omega}\\}_{\omega}$ and
post-selection on an outcome $\omega$, generating the SA
$\vec{\sigma^{\prime}}$ for Bob. The state before her measurement is, then,
given by $\rho_{\rm{AB}}^{(\omega)}=(\mathcal{E}_{\omega}^{\rm A}\otimes{\rm
id}^{\rm B})(\rho_{\rm AB})/p(\omega)$, where $\rm{id}^{\rm B}$ is the
identity channel acting on ${\rm B}$ and
$p(\omega)=\operatorname{tr}[\mathcal{E}_{\omega}(\rho_{\rm{A}})]$. The
corresponding SA is given by

$\displaystyle\sigma^{\prime}_{a|x}=\tfrac{1}{p(\omega)}\operatorname{tr}_{\rm{A}}\left[A_{a|x}\otimes\openone\left(\mathcal{E}_{\omega}^{\rm{A}}\otimes{\rm
id}^{\rm{B}}\right)(\rho_{\rm AB})\right]$ $\displaystyle=\rho_{\rm
A}^{1/2}\left[\tfrac{\mathcal{E}_{\omega}^{\dagger}(A_{a|x})}{p(\omega)}\right]^{T}\rho_{\rm
A}^{1/2}=\left[\rho_{\rm
B}^{1/2}\tfrac{\mathcal{E}_{\omega}^{\dagger}(A_{a|x})}{p(\omega)}\rho_{\rm
B}^{1/2}\right]^{T},$ (8)

where we used that $\rho_{\rm AB}$ is a pure state, in the first equality to
write the assemblage 111This results follows directly from the Schmidt
decomposition of the pure state $\rho_{\rm
AB}=\sum_{ij}\lambda_{i}\lambda_{j}\left|ii\right\rangle\left\langle
jj\right|$; see [10] for more details. and the in the second to say that
$\rho_{\rm A}=\rho_{\rm B}$, and the transposition is taken with respect to
the basis where $\rho_{\rm B}$ is diagonal. An analogous calculation shows
that $\sigma_{a|x}=[\rho_{\rm B}^{1/2}A_{a|x}\rho_{\rm B}^{1/2}]^{T}$. We
identify Eq. (Measurement incompatibility cannot be stochastically distilled),
up to a global transposition of the assemblage which does not change its
steerability, with the l.h.s. of Eq. (4), while the r.h.s. is
$\mathbb{SR}(\vec{\sigma})$, by Eq. (2), Result 3, and the assumption that the
initial steerability of the assemblage is maximal over ${\rm LF}$ filters.
$\square$

This result states that for a bipartite pure state and whenever steerability
is already maximized by performing some local filter on Bob’s side, Alice
cannot further increase it by performing local filters. Again, this result can
be interpreted as the impossibility for Alice to distill the incompatibility
of the MA via local filters. Notice that this is no longer true if Alice and
Bob share a mixed state and indeed the steerability can be increased via local
filters on Alice’s side, as it is the case for the phenomenon of hidden
steerability [44, 45]. Indeed, trying to argue in terms of a purification of
the initial state does not work.

Extension to other incompatibility measures.— Here, we show that our results
are not limited to the (incompatibility/steering) robustness, but they apply
to several other measures. We start with the random and free-state robustness
defined similarly to the (general) robustness, but with additional
restrictions [38]. We recall the definitions of the general robustness
$\mathbb{IR}(\vec{M})=\min\\{t\geq 0\;|\;\exists\vec{N}~{}\text{MA
and}~{}\vec{D}\in\mathbb{JM}\ \text{s.t.}\ (M_{a|x}+tN_{a|x})/(1+t)=D_{a|x}\
\forall~{}a,x\\}$ and $\mathbb{SR}(\vec{\sigma})\coloneqq\min\\{t\geq
0\;|\;\exists\;\vec{\xi}\;\text{SA},\;\vec{\tau}\in\mathbb{LHS}~{}\text{s.t.}~{}(\sigma_{a|x}+t\xi_{a|x})/(1+t)=\tau_{a|x}\forall
a,x\\}$. For incompatibility (respectively, steering) random robustness
$\mathbb{IR}^{\rm R}$ (respectively, $\mathbb{SR}^{\rm R}$), we require
$N_{a|x}=\openone/|\mathcal{A}|$ (respectively,
$\xi_{a|x}=\rho_{\rm{B}}/|\mathcal{A}|$) for all $a,x$, with $|\mathcal{A}|$
being the cardinality of the set of outcomes. On the other hand, the
incompatibility (respectively, consistent steering) free-state robustness
$\mathbb{IR}^{\rm{JM}}$ (respectively, $\mathbb{SR}^{\rm{C-LHS}}$) is defined
by considering $\vec{N}\in\mathbb{JM}$ (respectively,
$\vec{\xi}\in\mathbb{LHS}$ and
$\sum_{a}\xi_{a|x}=\sum_{a}\sigma_{a|x}~{}\forall~{}x$). The result from the
previous section hold for these two types of robustness:

######  Result 5.

Eq. (2), Obs. 1, Res. 1, 2, 3, 4 still hold when $(\mathbb{SR},\mathbb{IR})$
is substituted with $(\mathbb{SR}^{\rm R},\mathbb{IR}^{\rm R})$ or
$(\mathbb{SR}^{\rm C-LHS},\mathbb{IR}^{\rm JM})$.

_Proof (Sketch).—_ The first statement can be proven by direct inspection of
the corresponding optimization problems; see App. E. Once this result is
obtained, all the others are obtained by performing exactly the same steps of
the original derivation for $(\mathbb{IR},\mathbb{SR})$; see App. E. $\square$

Finally, we extend these results to the incompatible weight [22] and steerable
weight [46], defined, respectively, as $\mathbb{IW}(\vec{B})=\min\\{t\geq
0\;|\;B_{a|x}=tN_{a|x}+(1-t)D_{a|x}~{}\forall~{}a,x,\text{ for }\vec{N}\text{
MA, and }\vec{D}\in\mathbb{JM}\\}$ and $\mathbb{SW}(\vec{\sigma})=\min\\{t\geq
0\;|\;\sigma_{a|x}=t\xi_{a|x}+(1-t)\tau_{a|x}~{}\forall~{}a,x,\text{ for
}\vec{\xi}\text{ SA, and }\vec{\tau}\in\mathbb{LHS}\\}$. More precisely, we
show that

######  Result 6.

For any MA $\vec{B}$,

$\mathbb{SW}(\rho_{\rm B}^{1/2}\vec{B}\rho_{\rm
B}^{1/2})\leq\mathbb{IW}(\vec{B})\forall\rho_{\rm B};\ \max_{\rho_{\rm
B}}\mathbb{SW}\left(\rho_{\rm B}^{1/2}\vec{B}\rho_{\rm
B}^{1/2}\right)=\mathbb{IW}(\vec{B}).$ (9)

Analogs of Obs. 1 and Res.1, 2, 3, 4 hold for $(\mathbb{SR},\mathbb{IR})$
substituted with $(\mathbb{SW},\mathbb{IW})$.

Proof (Sketch).— Given a MA $\vec{B}$, the inequality
$\mathbb{SW}\left(\rho_{\rm B}^{1/2}\vec{B}\rho_{\rm
B}^{1/2}\right)\leq\mathbb{IW}(\vec{B})~{}\forall~{}\rho_{\rm B}$ can be
obtained by applying the CP map $X\mapsto\rho_{\rm B}^{1/2}X\rho_{\rm
B}^{1/2}$, for any valid state $\rho_{\rm B}$, into the elements appearing in
the decomposition defining $\mathbb{IW}$. The optimal $\rho_{\rm B}$,
saturating $\mathbb{SW}(\rho_{\rm B}^{1/2}\vec{B}\rho_{\rm
B}^{1/2})\leq\mathbb{IW}(\vec{B})$, can be constructed from the feasible
solutions of the dual SDP of $\mathbb{IW}(\vec{B})$, similarly to the proof
for $(\mathbb{IR},\mathbb{SR})$ in Ref. [18]. Finally, this result
straightforwardly implies a new version of Obs. 1 and Res. 2, 3, 4 with
$\mathbb{SR}$ being substituted by $\mathbb{SW}$. A detailed proof is
presented in Appendix F. The proof of Res. 1 was discussed before, see App. B.
$\square$

Surprisingly, these different measures coincide when the maximization over
$\rho_{\rm B}$, i.e., over the SEO class, is considered. Results 5 and 6 both
suggest that the impossibility of distilling incompatibility via local filters
is general and does not depend on the specific measure of
incompatibility/steerability considered, and holds, at the mathematical level,
even for nonphysical maps, i.e., linear maps that are positive but not
necessarily CP or trance non-increasing. Moreover, they also extend the
classification of steerability, based on SEO classes and ${\rm LF_{1}}$
filters [18], to the case of different measures of steerability.

Conclusions and outlook.—We provided an operational definition of stochastic
distillation of measurement incompatibility based on quantum information
tasks, such as steering, state discrimination, and state exclusion, and we
proved that such a distillation of incompatibility is not possible, even with
stochastic, i.e., filter, operations. This definition overcomes the difficulty
in defining distillation just in terms of the measurement operators alone,
i.e., the MA, which is not possible in the probabilistic case due to the
dependence of the post-selection probability on the incoming state. This
result completes the previous investigations on the resource theory of
measurement incompatibility [19, 16, 17, 14, 15], showing that no physical
operation is able to stochastically increase the incompatibility of a MA, in
stark contrast with the case of entanglement and steering. Our findings thus
uncover a crucial difference between measurement incompatibility and steering
from the perspective of stochastic distillation [18]. This is rather
unexpected, since measurement incompatibility and steering have been viewed as
equivalent properties from various operational aspects for years [10]. An
important open question for future research is to understand the physics
behind this difference.

Moreover, the analysis of the effect of local filters on steering scenarios
completes the classification of steering assemblages with respect to the
maximum steerability obtained via the most general local filters, initiated in
[18]. In fact, we showed that maximum steerability is always obtainable via
${\rm LF_{1}}$ filters, namely, local filters described by a single Kraus
operator. This completely solves the problem of filtering of a SA, since the
problem of interconversion between assemblages via ${\rm LF_{1}}$ filters was
solved in [18] via the notion SEO classes, and it was shown that the explicit
form of the optimal filter can be efficiently obtained via SDP methods.

Many interesting questions remain open. For instance, in our study, we only
consider the two most well-known types of incompatibility measures. We
conjecture that our results still hold for any incompatibility measure e.g.,
trace-distance based measure [47, 48], and we leave this general type of no-go
statement as an open question. In addition, different types of incompatibility
have been introduced, e.g., [49, 50, 51, 52, 53, 54], which sometimes share
similarity with steering and measurement incompatibility [55, 56]. More
concretely, one can consider the problem of the stochastic distillation of the
incompatibility of collections of quantum instruments, also known as channel
assemblages (CA) [49], see also [50, 51]. A CA goes beyond a MA by considering
also the output state after the measurement. In contrast to the case of MA, it
seems that CA can be distilled by local filters. The simplest example can be
constructed considering a qubit-qubit CA with some nonzero incompatibility,
e.g., quantified by the general robustness. If we compose the associated
instrument with the channel $\Lambda^{\dagger}$, where $\Lambda$ is the qubit-
qutrit channel $\Lambda(\rho)=p(\rho\oplus
0_{\ket{2}})+(1-p)\left|2\right\rangle\left\langle 2\right|$, it is clear that
in the extreme case, $p=0$ the CA is compatible, since everything is mapped to
a fixed state, whereas for $p>0$ the incompatibility grows. To recover the
initial incompatibility, i.e., before the composition with $\Lambda$, it is
sufficient to filter the CA via the projector $\Pi_{0,1}$ on the initial qubit
subspace. This simple argument shows that the situation is again different for
CAs and requires a thorough investigation. We leave the characterization of
distillation of channel assemblages to a future investigation.

Acknowledgments.— The authors acknowledge fruitful discussions with Jessica
Bavaresco, Shin-Liang Chen, Yeong-Cherng Liang, Simon Milz, Miguel Navascues,
and Paul Skrzypczyk. This work is supported by the Ministry of Science and
Technology, Taiwan, (Grants No. MOST 111-2917-I-564-005), the Royal Society
through Enhanced Research Expenses (on grant NFQI), the Austrian Science Fund
(FWF) through Projects No. ZK 3 (Zukunftskolleg), and No. F7113 (BeyondC).

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##  Appendix A Local filters, maxmimum steerability, and SEO equivalence
classes

In this section, we recall for convenience some useful results from [18].
These statements, in fact, were presented in the proof of the main theorems of
Ref. [18], but not explicitly in their statements. The first result is the
following, which can be found in the proof of Th. 2 in [18]:

$\max_{\rho_{\rm B}}\mathbb{SR}\left(\rho_{\rm B}^{{1/2}}\vec{B}\rho_{\rm
B}^{{1/2}}\right)=\sup_{\text{{\hbox{\widetilde{\rho}_{\rm B}}}: full-
rank}}\mathbb{SR}\left(\widetilde{\rho}_{\rm
B}^{{1/2}}\vec{B}\widetilde{\rho}_{\rm
B}^{{1/2}}\right)=\mathbb{IR}(\vec{B}).$ (10)

It is proven by showing that from each feasible solution of one optimization
problem one can construct a feasible solution of the other optimization
problem, without changing the objective function. See Eqs. (31) and (33) in
[18] for details, and notice that ${\rm range}(\omega_{a|x})\subset{\rm
range(\eta)}$ for $\eta,\vec{\omega}$ in Eq. (31) of [18]. In the statement of
the theorem $\sup$ appeared instead of $\max$ since it is not guaranteed that
$\rho_{\rm B}$ is a full-rank state, so the assemblage $\rho_{\rm
B}^{{1/2}}\vec{B}\rho_{\rm B}^{{1/2}}$ may belong to a different SEO, than
$\vec{B}$. Notice that, in principle, we do not even know if a full-rank
solution is always achievable, since the constraint of being full-rank makes
the set of variables over which we optimize non-closed and, thus, non-compact.

The second result is that

$\sup_{\mathcal{E}_{\omega}\in{\rm
LF_{1}}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)=\max_{\mathcal{E}_{\omega}\in{\rm
LF_{1}}}\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)=\max_{\rho_{\rm
B}}\mathbb{SR}(\rho_{\rm B}^{{1/2}}\vec{B}\rho_{\rm B}^{{1/2}}).$ (11)

Notice that, a priori, it is not guaranteed that the supremum ($\sup$) can be
achieved in the maximization over $\mathcal{E}_{\omega}$. In our case,
however, due to the upper bound given by $\max_{\rho_{\rm
B}}\mathbb{SR}(\rho_{\rm B}^{{1/2}}\vec{B}\rho_{\rm B}^{{1/2}})$, we can
explicitly construct the Kraus operator associated with $\mathcal{E}_{\omega}$
from $\rho_{\rm B}$ and the state $\zeta_{\rm B}:=\sum_{a}\sigma_{a|x}$.

The third result is that

$\text{ for all }\vec{A}\ \exists\ \rho_{\rm AB}\text{ s.t.
}\mathbb{SR}(\vec{\sigma})=\mathbb{IR}(\vec{A})\text{ for
}\sigma_{a|x}=\operatorname{tr}_{\rm A}[A_{a|x}\otimes\openone\rho_{\rm AB}]$
(12)

Again, the proof is identical to the one presented in Th. 3 in [18], but uses
local filter associated with the optimal $\rho_{\rm B}$ appearing in the
expression in Eq. (10).

##  Appendix B No stochastic distillation of measurement incompatibility from
state discrimination and exclusion tasks

We show that Result 1 can be obtained by reasoning in terms of state
discrimination tasks instead of steering. We recall from [17, 16, 14, 15] that
the incompatibility of a MA $\vec{A}$ can be obtained as

$1+\mathbb{IR}(\vec{A})=\sup_{\mathcal{S}}\frac{p_{\rm succ}^{\rm
discr}(\vec{A},\mathcal{S})}{\max_{\vec{O}\in\mathbb{JM}}p_{\rm succ}^{\rm
discr}(\vec{O},\mathcal{S})},$ (13)

where $\mathcal{S}:=\\{p(x)p(a|x),\rho_{a|x}\\}$ is the state ensemble
involved in the state discrimination task with prior information [57], and
$p_{\rm succ}^{\rm discr}(\vec{A},\mathcal{S})$ is the success probability
associated with the MA $\vec{A}$ and the task $\mathcal{S}$. We refer the
reader to [17, 16, 14, 15] for formal details, and here we focus on
illustrating why stochastic distillation of incompatibility is forbidden in a
general sense. One can observe that, in this case, the filtering operation is
placed between the state preparation and the measurement. Again, we can
operationally interpret the filtering of the measurement as the filtering of
the incoming states. As a result of this post-selection operation, we obtain a
new valid state ensemble
$\mathcal{S}^{\prime}=\\{p(x)p(a|x),\mathcal{E}_{\omega}(\rho_{a|x})/p(\omega|a,x)\\}$,
where $p(\omega|a,x)=\operatorname{tr}[\mathcal{E}_{\omega}(\rho_{a|x})]$. The
effect of the filter, thus, is just to change the state ensemble. Substituting
this in Eq. (13) provides an optimization over a restricted subset of
ensembles, resulting in a lower or equal value for the r.h.s. of the equation.
We can conclude that the incompatibility of $\vec{A}$ cannot increase.

An analogous result can be obtained for the incompatible weight
$\mathbb{IW}(\vec{A})$ in state exclusion tasks [23, 24], namely,

$1-\mathbb{IW}(\vec{A})=\inf_{\mathcal{S}}\frac{p_{\rm succ}^{\rm
excl}(\vec{A},\mathcal{S})}{\max_{\vec{O}\in\mathbb{JM}}p_{\rm succ}^{\rm
excl}(\vec{O},\mathcal{S})},$ (14)

where $p_{\rm succ}^{\rm excl}(\vec{A},\mathcal{S})$ refers to the success
probability of such a task. The same argument as before shows that filter
operations can only transform the state ensemble $\mathcal{S}$ into a new one.
Hence, they cannot decrease the infimum on the r.h.s. of Eq. (14) and, thus,
cannot increase the incompatible weight.

Finally, we note that the same argument also applies to the robustness of
measurement (ROM) recently introduced in [36]. The ROM is defined as the
minimal amount of noise that can be added to a measurement before it becomes
completely uninformative, i.e.
$\mathbb{ROM}(M):=\min\\{t|(M_{a}+tN_{a})/(1+t)=q(a)\openone\\}$. As in the
previous cases, $\mathbb{ROM}(M)$ can be defined in terms of a discrimination
task [36] as

$1+\mathbb{ROM}(M)=\sup_{\mathcal{S}}\frac{p_{\rm succ}^{\rm
discr}(M,\mathcal{S})}{p_{\rm succ}^{\rm discr}(\mathcal{S})},$ (15)

where $\mathcal{S}:=\\{p_{x},\rho_{x}\\}$ is a state ensemble, and $p_{\rm
succ}^{\rm discr}(M,\mathcal{S})$ and $p_{\rm succ}^{\rm discr}(\mathcal{S})$
are the probability of guessing the correct label $x$, with and without
performing the measurement $M$, respectively. Again, one can show that a
filter $\mathcal{E}_{\omega}$ on the ensemble $\mathcal{S}$ only provides a
valid ensemble, so it is not able to increase $\mathbb{ROM}(M)$, which is
already maximized over all possible ensembles.

##  Appendix C Proof of Observation 1

_Proof.—_ It is helpful to recall the definition

$\mathbb{IR}(\vec{B})=\min\\{t\geq 0\;|\;(B_{a|x}+tN_{a|x})/(1+t)=D_{a|x}\
\forall~{}a,x,\text{ for }\vec{N}~{}\text{MA},~{}\vec{D}\in\mathbb{JM}\ \\}$
(16)

The central idea is that by using
$\mathbb{IR}(\vec{B})=\max_{\rho_{\rm{B}}}\mathbb{SR}(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2})$,
one can construct a decomposition appearing in the definition of
$\mathbb{SR}(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2})$ starting from one
in $\mathbb{IR}(\vec{B})$. Let $t^{*}$ be the minimal $t$ achieving Eq. (16);
namely, $t^{*}=\mathbb{IR}(\vec{B})$. Also, let $\tilde{\rho}_{\rm{B}}$ be the
optimal state for
$\max_{\rho_{\rm{B}}}\mathbb{SR}(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2})$.
Multiplying both sides of the $\mathbb{IR}(\vec{B})$ decomposition by
$\tilde{\rho}_{\rm{B}}^{1/2}$, we obtain

$\displaystyle\tilde{\rho}_{\rm{B}}^{1/2}\vec{B}\tilde{\rho}_{\rm{B}}^{1/2}$
$\displaystyle=(1+t^{*})\tilde{\rho}_{\rm{B}}^{1/2}\vec{D}\tilde{\rho}_{\rm{B}}^{1/2}-t^{*}\tilde{\rho}_{\rm{B}}^{1/2}\vec{N}\tilde{\rho}_{\rm{B}}^{1/2}=(1+t^{*})\vec{\tau}-t^{*}\vec{\xi},$
(17)

where
$\vec{\tau}=\tilde{\rho}_{\rm{B}}^{1/2}\vec{D}\tilde{\rho}_{\rm{B}}^{1/2}$ is
a SA in $\mathbb{LHS}$ since $\vec{D}\in\mathbb{JM}$, and
$\vec{\xi}=\tilde{\rho}_{\rm{B}}^{1/2}\vec{N}\tilde{\rho}_{\rm{B}}^{1/2}$ is a
generic SA. Note that $\vec{\tau}$ and $\vec{\xi}$ satisfy the conditions in
$\mathbb{SR}(\tilde{\rho}_{\rm{B}}^{1/2}\vec{B}\tilde{\rho}_{\rm{B}}^{1/2})$,
and we also have
$t^{*}=\mathbb{IR}(\vec{B})=\mathbb{SR}(\tilde{\rho}_{\rm{B}}^{1/2}\vec{B}\tilde{\rho}_{\rm{B}}^{1/2})$.
Consequently, $(t^{*},\vec{\tau},\vec{\xi})$ provide an optimal solution for
$\mathbb{SR}(\tilde{\rho}_{\rm{B}}^{1/2}\vec{B}\tilde{\rho}_{\rm{B}}^{1/2})$.
Finally, since $\vec{B},\vec{D},\vec{N}$ are MA,
$\tilde{\rho}_{\rm{B}}^{1/2}\vec{B}\tilde{\rho}_{\rm{B}}^{1/2}$, $\vec{\tau}$
and $\vec{\xi}$ all have the same marginal state $\tilde{\rho}_{\rm{B}}$.
$\square$

##  Appendix D Proof of Result 2

_Proof.—_ We recall that $\mathbb{SR}(\vec{\sigma})=\min\\{t\geq
0\;|\;(\vec{\sigma}+t\vec{\xi})/(1+t)=\vec{\tau}\in\mathbb{LHS},\text{ for
}\vec{\xi}\text{ SA }\\}$. Let $\tilde{\rho}_{\rm{B}}$ be an optimal state
achieving $\mathbb{IR}(\vec{B})=\max_{\rho_{\rm B}}\mathbb{SR}({\rho}_{\rm
B}^{1/2}\vec{B}{\rho}_{\rm B}^{1/2})$ and $t^{*}$ be the optimal parameter
achieving $t^{*}=\mathbb{SR}(\tilde{\rho}_{\rm
B}^{1/2}\vec{B}\tilde{\rho}_{\rm B}^{1/2})$. By Eq. (2), we know that the same
$t^{*}$ appear in the decomposition of $\vec{B}$ in Eq. (16) as

$\displaystyle\vec{B}=(1+t^{*})\vec{D}-t^{*}\vec{N},$ (18)

where $\vec{N}$ is a generic MA and $\vec{D}\in\mathbb{JM}$. We now apply the
filtering operation with an outcome $\omega$ and multiply for a generic state
$\hat{\rho}_{\rm{B}}$ with the normalization
$p(\omega)=\operatorname{tr}[\mathcal{E}_{\omega}(\hat{\rho}_{\rm{B}})]$. We
have that

$\displaystyle\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{B})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}$
$\displaystyle=(1+t^{*})\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{D})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}-t^{*}\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{N})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)},$
(19)

which is a valid decomposition for
$\mathbb{SR}\left(\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{B})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}\right)$.
To see this, first we note that, for every MA $\vec{M}$, we can write

$\displaystyle\operatorname{tr}\left[\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(M_{a|x})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}\right]=\frac{\operatorname{tr}[\mathcal{E}_{\omega}(\hat{\rho}_{\rm{B}})M_{a|x}]}{\operatorname{tr}[\mathcal{E}_{\omega}(\hat{\rho}_{\rm{B}})]},$
(20)

which is a valid conditional probability distribution. Since
$\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{M})\hat{\rho}_{\rm{B}}^{1/2}\geq
0$ for every $a,x$, we conlcude that
$\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{M})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}$
is a valid SA for every MA $\vec{M}$. Then all elements in Eq. (19) are valid
SAs, and
$\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{D})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}\in\mathbb{LHS}$
because of $\vec{D}\in\mathbb{JM}$.

Since the parameter $t^{*}$ may not be the minimum with respect to the
decomposition defining
$\mathbb{SR}\left(\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{B})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}\right)$,
we have

$\mathbb{SR}\left(\frac{\hat{\rho}_{\rm{B}}^{1/2}\mathcal{E}_{\omega}^{\dagger}(\vec{B})\hat{\rho}_{\rm{B}}^{1/2}}{p(\omega)}\right)\leq
t^{*}=\mathbb{IR}(\vec{B})=\mathbb{SR}\left(\tilde{\rho}_{\rm{B}}^{1/2}\vec{B}\tilde{\rho}_{\rm{B}}^{1/2}\right),$
(21)

which is true for every $\hat{\rho}_{\rm B}$. This concludes the proof of Eq.
(4).

The proof of Eq. (5) in the main text is analogous. For a generic state
$\hat{\rho}_{\rm{B}}$, we just apply $\mathcal{E}_{\omega}$ and the
normalization
$p(\omega)=\operatorname{tr}[\mathcal{E}_{\omega}(\hat{\rho}_{\rm{B}})]$ to
the decomposition $\vec{B}=(1+t^{*})\vec{D}-t^{*}\vec{N}$ appearing in the
definition of $\mathbb{IR}(\vec{B})$, where $\vec{D}\in\mathbb{JM}$ and
$\vec{N}$ is a generic MA. This means

$\frac{\mathcal{E}_{\omega}\left(\hat{\rho}_{\rm{B}}^{1/2}\vec{B}\hat{\rho}_{\rm{B}}^{1/2}\right)}{p(\omega)}=(1+t^{*})\frac{\mathcal{E}_{\omega}\left(\hat{\rho}_{\rm{B}}^{1/2}\vec{D}\hat{\rho}_{\rm{B}}^{1/2}\right)}{p(\omega)}-t^{*}\frac{\mathcal{E}_{\omega}\left(\hat{\rho}_{\rm{B}}^{1/2}\vec{N}\hat{\rho}_{\rm{B}}^{1/2}\right)}{p(\omega)},$
(22)

which is a valid decomposition for
$\mathbb{SR}\left(\frac{\mathcal{E}_{\omega}(\hat{\rho}_{\rm{B}}^{1/2}\vec{B}\hat{\rho}_{\rm{B}}^{1/2})}{p(\omega)}\right)$.
Since $\mathbb{IR}(\vec{B})=\mathbb{SR}(\tilde{\rho}_{\rm
B}^{1/2}\vec{B}\tilde{\rho}_{\rm B}^{1/2})=t^{*}$, the result follows.
$\square$

##  Appendix E Robustness-based measures and proof of Result 5

Here, we extend the result from [18], namely,
$\mathbb{IR}(\vec{B})=\max_{\rho_{\rm{B}}}\mathbb{SR}(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2})$,
to other measures of robustness, such as the random and free-state consistent
robustness, both in the incompatibility and steering scenarios.

Let us recall the definition of random incompatibility robustness [38]:

$\displaystyle\mathbb{IR}^{\rm R}(\vec{B})=\min\left\\{t\geq
0\;\middle|\;\frac{B_{a|x}+t\openone/|\mathcal{A}|}{1+t}=\sum_{\lambda}p(a|x,\lambda)G_{\lambda},G_{\lambda}\geq
0,\sum_{\lambda}G_{\lambda}=\openone\right\\},$ (23)

where $|\mathcal{A}|=n_{a}$ is the number of the outcomes $a$. On the other
hand, the random steering robustness (see the definition in Ref. [38]) of a SA
$\vec{\sigma}$ with reduced state $\sum_{a}\sigma_{a|x}=\rho_{\rm B}$ can be
expressed as

$\displaystyle\mathbb{SR}^{\rm R}(\vec{\sigma})=\min\left\\{t\geq
0\;\middle|\;\frac{\sigma_{a|x}+t\rho_{\rm
B}/|\mathcal{A}|}{1+t}=\sum_{\lambda}p(\lambda)p(a|x,\lambda)\rho_{\lambda},\rho_{\lambda}\geq
0,\operatorname{tr}[\rho_{\lambda}]=1\right\\}.$ (24)

Let us also recall the definition of free-state robustness measures. The JM
incompatibility robustness is given by (see, e.g., [38])

$\displaystyle\mathbb{IR}^{\rm{JM}}(\vec{B})=\min\left\\{t\geq
0\;\middle|\;\frac{B_{a|x}+t\sum_{\lambda}p^{\prime}(a|x,\lambda)G^{\prime}_{\lambda}}{1+t}=\sum_{\lambda}p(a|x,\lambda)G_{\lambda};\text{
with }G_{\lambda},G^{\prime}_{\lambda}\geq 0\
\forall\lambda,\sum_{\lambda}G_{\lambda}=\sum_{\lambda}G^{\prime}_{\lambda}=\openone\right\\}.$
(25)

Similarly, the consistent LHS steering robustness (see the definition in [38])
of a SA $\vec{\sigma}$ with reduced state $\sum_{a}\sigma_{a|x}=\rho_{\rm B}$
can be expressed as

$\displaystyle\mathbb{SR}^{\rm{C-LHS}}(\vec{\sigma})=\min\left\\{t\geq
0\;\middle|\;\frac{\sigma_{a|x}+t\sum_{\lambda}p^{\prime}(\lambda)p^{\prime}(a|x,\lambda)\rho^{\prime}_{\lambda}}{1+t}=\sum_{\lambda}p(\lambda)p(a|x,\lambda)\rho_{\lambda};\text{
$\rho_{\lambda},\rho^{\prime}_{\lambda}$: states s.t.
$\sum_{\lambda}p(\lambda)\rho_{\lambda}=\rho_{\rm B}$}\right\\}.$ (26)

###  E.1 Lemmas relevant to Result 5

Before proceeding to the proof of Result 5, we need some intermediate results.
More precisely, we need to prove that

######  Lemma 1.

The maximal consistent LHS steering robustness within an SEO class is the same
as the JM incompatibility robustness of the SEO; namely

$\mathbb{IR}^{\rm{JM}}(\vec{B})=\max_{\rho_{\rm{B}}}\mathbb{SR}^{\rm{C-LHS}}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right).$
(27)

Moreover, every full-rank state is an optimal $\rho_{\rm{B}}$.

_Proof.—_ Let $t^{*}=\mathbb{IR}^{\rm{JM}}(\vec{B})$ with the decomposition
$\vec{B}=(1+t^{*})\vec{D}-t^{*}\vec{D}^{\prime}$ appearing in the definition
of $\mathbb{IR}^{\rm JM}(\vec{B})$; that is, both $\vec{D},\vec{D}^{\prime}$
are in $\mathbb{JM}$. Then, for every $\rho_{\rm B}$, we have

$\displaystyle\rho_{\rm B}^{1/2}\vec{B}\rho_{\rm B}^{1/2}=(1+t^{*})\rho_{\rm
B}^{1/2}\vec{D}\rho_{\rm B}^{1/2}-t^{*}\rho_{\rm
B}^{1/2}\vec{D}^{\prime}\rho_{\rm B}^{1/2},$ (28)

which is a valid decomposition for
$\mathbb{SR}^{\rm{C-LHS}}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)$.
This shows that
$\max_{\rho_{\rm{B}}}\mathbb{SR}^{\rm{C-LHS}}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)\leq\mathbb{IR}^{\rm{JM}}(\vec{B})$.

To prove the tightness of this inequality, let us consider an arbitrary full-
rank state $\hat{\rho}_{\rm B}$. Let us write
$\hat{t}=\mathbb{SR}^{\rm{C-LHS}}\left(\hat{\rho}_{\rm
B}^{1/2}\vec{B}\hat{\rho}_{\rm B}^{1/2}\right)$ with the following
decomposition:

$\displaystyle\hat{\rho}_{\rm B}^{1/2}\vec{B}\hat{\rho}_{\rm
B}^{1/2}=\left(1+\hat{t}\right)\vec{\xi}-\hat{t}\vec{\tau},$ (29)

where, by the definition of $\mathbb{SR}^{\rm C-LHS}$, both
$\vec{\xi},\vec{\tau}$ are in $\mathbb{LHS}$, and both of them have the same
reduced state $\hat{\rho}_{\rm B}$ that is full-rank. By applying the map
$\hat{\rho}_{\rm B}^{-1/2}(\cdot)\hat{\rho}_{\rm B}^{-1/2}$, we can write

$\displaystyle\vec{B}=\left(1+\hat{t}\right)\vec{D}-\hat{t}\vec{N},$ (30)

where $\vec{D},\vec{N}$ are the SEOs of $\vec{\xi},\vec{\tau}$, meaning that
both of them are in $\mathbb{JM}$. This is therefore a valid decomposition for
$\mathbb{IR}^{\rm{JM}}(\vec{B})$. Consequently, we conclude that

$\displaystyle\mathbb{IR}^{\rm{JM}}(\vec{B})\leq\hat{t}=\mathbb{SR}^{\rm{C-LHS}}\left(\hat{\rho}_{\rm
B}^{1/2}\vec{B}\hat{\rho}_{\rm
B}^{1/2}\right)\leq\max_{\rho_{\rm{B}}}\mathbb{SR}^{\rm{C-LHS}}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right),$
(31)

which completes the proof.

$\square$

It is worth mentioning that our proof also shows that every full-rank state
can be used to construct optimal solution. This makes our proof also
constructive, since we know how to explicitly choose an optimal $\rho_{\rm
B}$. By using the same proof, a similar result also holds for the random
robustness:

######  Lemma 2.

The maximal random steering robustness within an SEO class is the same as the
random incompatibility robustness of the SEO; namely

$\mathbb{IR}^{\rm R}(\vec{B})=\max_{\rho_{\rm{B}}}\mathbb{SR}^{\rm
R}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right).$ (32)

Moreover, every full-rank state is an optimal $\rho_{\rm{B}}$.

_Proof.—_

One can use exactly the same proof of Lemma 1 to show this result. More
precisely, the argument goes through by replacing the free-state robustness by
random robustness. $\square$

###  E.2 Proof of Result 5

To prove Result 5, we need to show that all the statements we made for the
pair $(\mathbb{IR},\mathbb{SR})$ can be reformulated for the pairs
$(\mathbb{SR}^{\rm R},\mathbb{IR}^{\rm R})$ and $(\mathbb{SR}^{\rm
C-LHS},\mathbb{IR}^{\rm JM})$. The above lemmas show that Eq. (2) holds with
the substitution of $(\mathbb{IR},\mathbb{SR})$ with $(\mathbb{SR}^{\rm
R},\mathbb{IR}^{\rm R})$ and $(\mathbb{SR}^{\rm C-LHS},\mathbb{IR}^{\rm JM})$.
The same holds for the analogous of Obs. 1, by the definitions of
$\mathbb{SR}^{\rm C-LHS}$ and $\mathbb{SR}^{\rm R}$. Similarly, one can prove
the analogous of Res. 2, 3, 4, since the original proof is based on Eq. (2)
and Obs. 1. Regarding Res. 1, one can proceed by showing that also Eq. (3) can
be transformed into a version

$\widetilde{\mathbb{IR}}^{Y}(\vec{A}):=\max_{\rho_{\rm AB}}\
\mathbb{SR}^{X}(\vec{\sigma}(\rho_{\rm AB})),$ (33)

for the pairs $(X,Y)=({\rm C-LHS},{\rm JM})$ or $({\rm R},{\rm R})$. and
repeat the same argument that lead to Res. 1. $\square$

##  Appendix F Weight-based measure and proof of Result 6

In this section, we extend our Obs. 1 and Res. 2, 3, 4 from robustness-based
measures to the weight-based measures. We now recall their definition. The
incompatible weight [22] is defined as

$\displaystyle\mathbb{IW}(\vec{B})=\min\Big{\\{}$ $\displaystyle t\geq
0\;\Big{|}\;B_{a|x}=tN_{a|x}+(1-t)O_{a|x}~{}\forall a,x,\
\vec{N}\text{~{}~{}is any MA, }\vec{O}\in\mathbb{JM}\Big{\\}}.$ (34)

Similarly, the steerable weight [46] can be expressed as

$\displaystyle\mathbb{SW}(\vec{\sigma})=\min\Big{\\{}$ $\displaystyle t\geq
0\;\Big{|}\;\sigma_{a|x}=t\xi_{a|x}+(1-t)\tau_{a|x}~{}\forall a,x,\
\vec{\xi}\text{~{}~{}is any SA},\vec{\tau}\in\mathbb{LHS}\Big{\\}}.$ (35)

Before we show our statements in this section, we first derive the primal SDP
of the $\mathbb{IW}$ by reformulating the optimization problem as
$N_{a|x}=\frac{B_{a|x}-(1-t)O_{a|x}}{t}\geq 0$. If we define $(1-t)=s$, we
have $B_{a|x}-s\sum_{\lambda}D(a|x,\lambda)G_{\lambda}\geq 0$. Here, we use
the definition of the joint measureability
$O_{a|x}=\sum_{\lambda}D(a|x,\lambda)G_{\lambda}$ with $G_{\lambda}$ being a
POVM satisfying $\sum_{\lambda}G_{\lambda}=\openone$. Finally, by defining
$\tilde{G}_{\lambda}=sG_{\lambda}$, we arrive at the primal form of the SDP

Given $\displaystyle\vec{B}$ (36) Find $\displaystyle\max
s=1-\mathbb{IW}(\vec{B})$ s.t. $\displaystyle
B_{a|x}\geq\sum_{\lambda}D(a|x,\lambda)\tilde{G}_{\lambda},$
$\displaystyle\tilde{G}_{\lambda}\geq
0~{}~{}\text{and}~{}~{}\sum_{\lambda}\tilde{G}_{\lambda}=s\openone.$

Here, the last constraint is from
$\sum_{\lambda}\tilde{G}_{\lambda}=s\sum_{\lambda}G_{\lambda}=s\openone$. The
Lagrangian associated with the primal SDP is

$\displaystyle L$
$\displaystyle=s+\sum_{a,x}\operatorname{tr}\left[\omega_{a|x}\left(B_{a|x}-\sum_{\lambda}D(a|x,\lambda)\tilde{G}_{\lambda}\right)\right]+\operatorname{tr}\left[Z\left(s\openone-\sum_{\lambda}\tilde{G}_{\lambda}\right)\right]+\sum_{\lambda}\operatorname{tr}\left[Y_{\lambda}\tilde{G}_{\lambda}\right]$
(37)
$\displaystyle=s(1+\operatorname{tr}\left[Z\right])+\sum_{a,x}\operatorname{tr}\left[\omega_{a|x}B_{a|x}\right]+\sum_{\lambda}\operatorname{tr}\left[\tilde{G}_{\lambda}\left(-\sum_{a,x}D(a|x,\lambda)\omega_{a|x}-Z+Y_{\lambda}\right)\right].$

Here, $Z,\omega_{a|x}$, and $Y_{\lambda}$ are the Hermitian dual variables.
One can directly check that

$\displaystyle 1-\mathbb{IW}(\vec{B})=\max_{\begin{subarray}{c}s\geq 0,\\\
\\{\tilde{G}_{\lambda}\geq
0\\}\end{subarray}}\min_{\begin{subarray}{c}Z=Z^{\dagger},\\\
\\{\omega_{a|x}\geq 0\\},\\\ \\{Y_{\lambda}\geq
0\\}\end{subarray}}L\leq\min_{\begin{subarray}{c}Z=Z^{\dagger},\\\
\\{\omega_{a|x}\geq 0\\},\\\ \\{Y_{\lambda}\geq
0\\}\end{subarray}}\max_{\begin{subarray}{c}s\geq 0,\\\
\\{\tilde{G}_{\lambda}\geq 0\\}\end{subarray}}L,$ (38)

where the upper bound is the dual SDP. To write it down, note that the
Lagrangian is independent of the primal variable if
$\operatorname{tr}\left[Z\right]=-1$ and
$Y_{\lambda}=\sum_{a,x}D(a|x,\lambda)\omega_{a|x}+Z$. Together with the
criteria $\omega_{a|x}\geq 0,Y_{\lambda}\geq 0$, we obtain the dual SDP of the
$\mathbb{IW}$ as follows, where we define a new variable $X\coloneqq-Z$,

Given $\displaystyle\vec{B}$ (39) Find
$\displaystyle\min\sum_{a,x}\operatorname{tr}\left[\omega_{a|x}B_{a|x}\right]=1-\mathbb{IW}(\vec{B})$
s.t. $\displaystyle\sum_{a,x}D(a|x,\lambda)\omega_{a|x}\geq X,$
$\displaystyle\operatorname{tr}\left[X\right]=1,\omega_{a|x}\geq 0~{}\forall
a,x.$

We note that in general $X$ is not a quantum state because the first
constraint in Eq. (39) does not guarantee $X\geq 0$.

On the other hand, the dual SDP of the steerable weight $\mathbb{SW}$ is [46]

Given $\displaystyle\vec{\sigma}$ (40) Find
$\displaystyle\min\sum_{a,x}\operatorname{tr}\left[F_{a|x}\sigma_{a|x}\right]=1-\mathbb{SW}(\vec{\sigma})$
s.t. $\displaystyle\sum_{a,x}D(a|x,\lambda)F_{a|x}\geq\openone,$
$\displaystyle F_{a|x}\geq 0~{}~{}\forall~{}a,x.$

###  F.1 Lemmas relevant to Result 6

To show the main result of this subsection, we first show the following lemma:

######  Lemma 3.

Given a MA $\vec{B}$, the weight-based measures satisfy

$\mathbb{SW}\left(\rho_{\rm B}^{1/2}\vec{B}\rho_{\rm
B}^{1/2}\right)\leq\mathbb{IW}(\vec{B})\;\text{ for all }\rho_{\rm B}.$ (41)

_Proof.—_ We apply the positive map $\rho_{\rm B}^{1/2}(\cdot)\rho_{\rm
B}^{1/2}$, where $\rho_{\rm B}$ is an arbitrary quantum state, to the elements
of the decomposition appearing in Eq. (34). Then we define
$\vec{\xi}=\rho_{\rm B}^{1/2}\vec{N}\rho_{\rm B}^{1/2}$ and
$\vec{\tau}=\rho_{\rm B}^{1/2}\vec{O}\rho_{\rm B}^{1/2}$. The decomposition in
$\mathbb{IW}$, then, becomes a valid decomposition for $\mathbb{SW}(\rho_{\rm
B}^{1/2}\vec{B}\rho_{\rm B}^{1/2})$, since $\vec{O}\in\mathbb{JM}$ implies
that $\rho_{\rm B}^{1/2}\vec{O}\rho_{\rm B}^{1/2}\in\mathbb{LHS}$, which
proves the claim. $\square$

We now show our main result in this subsection:

######  Observation 2.

The maximal steerable weight in the SEO class is equivalent to the
incompatible weight of the SEO; i.e.,

$\max_{\rho_{\rm{B}}}\mathbb{SW}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)=\mathbb{IW}(\vec{B}),$
(42)

where the optimal $\rho_{\rm{B}}$ can be obtained by solving the dual SDP of
$\mathbb{IW}(\vec{B})$.

_Proof.—_ By Lemma 3, it is enough to prove that the any feasible solution of
the problem on the r.h.s. of Eq. (42) allows us to construct a feasible
solution of the problem on the l.h.s. with a lower objective function. In
other words we want to prove the opposite inequality with respect to Lemma 3.
Let us start with the feasible solutions, $X$ and $\vec{\omega}$, of
$\mathbb{IW}(\vec{B})$ defined in Eq. (39). Since the operator $X$ is
hermitian, it can always be decomposed as $X=X^{+}-X^{-}$ satisfying
$X^{\pm}\geq 0$ and $X^{+}X^{-}=X^{-}X^{+}=0$. Because of
$\operatorname{tr}(X)=1$, we have $\operatorname{tr}(X^{+})\geq 1$. We then
define the following variables:

$\displaystyle\eta_{\rm{B}}=\frac{X^{+}}{\operatorname{tr}(X^{+})},\quad\vec{\sigma}=\eta_{\rm{B}}^{1/2}\vec{B}\eta_{\rm{B}}^{1/2},\quad\vec{F}=\eta_{\rm{B}}^{-1/2}\vec{\omega}\eta_{\rm{B}}^{-1/2},$
(43)

where the inverse is defined after a projection on the range of
$\eta_{\rm{B}}$, which is also the range of $X^{+}$. Let $\Pi_{X^{+}}$ be the
projector onto $X^{+}$’s range. Then we have
$\eta_{\rm{B}}^{1/2}\eta_{\rm{B}}^{-1/2}=\eta_{\rm{B}}^{-1/2}\eta_{\rm{B}}^{1/2}=\Pi_{X^{+}}$.
The objective function of Eq. (39) can be re-written by

$\displaystyle\sum_{a,x}\operatorname{tr}\left(B_{a|x}\omega_{a|x}\right)$
$\displaystyle=\sum_{a,x}\operatorname{tr}\left[B_{a|x}\omega_{a|x}\left(\Pi_{X^{+}}+(\openone-\Pi_{X^{+}})\right)\right]\geq\sum_{a,x}\operatorname{tr}\left(B_{a|x}\omega_{a|x}\Pi_{X^{+}}\right)=\sum_{a,x}\operatorname{tr}\left[B_{a|x}\left(\Pi_{X^{+}}+(\openone-\Pi_{X^{+}})\right)\omega_{a|x}\Pi_{X^{+}}\right]$
$\displaystyle\geq\sum_{a,x}\operatorname{tr}\left(B_{a|x}\Pi_{X^{+}}\omega_{a|x}\Pi_{X^{+}}\right)=\sum_{a,x}\operatorname{tr}\left(\eta_{\rm{B}}^{1/2}{B}_{a|x}\eta_{\rm{B}}^{1/2}\eta_{\rm{B}}^{-1/2}\omega_{a|x}\eta_{\rm{B}}^{-1/2}\right)=\sum_{a,x}\operatorname{tr}\left(\sigma_{a|x}F_{a|x}\right),$
(44)

which is the objective function of
$1-\max_{\rho_{\rm{B}}}\mathbb{SW}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)$.
See Eq. (40) for comparison.

We now show that the constraints in Eq. (40) are satisfied. Applying the CP
map $\eta_{\rm{B}}^{-1/2}(\cdot)\eta_{\rm{B}}^{-1/2}$ on
$\sum_{a,x}D(a|x,\lambda)\omega_{a|x}\geq X$, which is the first constraint in
Eq. (39), we obtain

$\displaystyle\sum_{a,x}D(a|x,\lambda)F_{a|x}\geq\eta_{\rm{B}}^{-1/2}X\eta_{\rm{B}}^{-1/2}=\eta_{\rm{B}}^{-1/2}X_{+}\eta_{\rm{B}}^{-1/2}=\operatorname{tr}[X^{+}]\openone\geq\openone.$
(45)

Notice that, without loss of generality, we can restrict the space in Eq. (40)
to be the range of $X^{+}$, since by construction it includes the ranges of
both $\sigma_{a|x}$ and $F_{a|x}$ for all $a,x$. Consequently, $\vec{F}$ is a
feasible solution to Eq. (40), meaning that

$\displaystyle\sum_{a,x}\operatorname{tr}\left(\sigma_{a|x}F_{a|x}\right)\geq
1-\mathbb{SW}\left(\eta_{\rm{B}}^{1/2}\vec{B}\eta_{\rm{B}}^{1/2}\right)\geq
1-\max_{\rho_{\rm{B}}}\mathbb{SW}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right).$
(46)

Together with Eq. (F.1), we conclude that

$\displaystyle\sum_{a,x}\operatorname{tr}\left(B_{a|x}\omega_{a|x}\right)\geq
1-\max_{\rho_{\rm{B}}}\mathbb{SW}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)$
(47)

for every feasible solutions $X$ and $\vec{\omega}$ of
$\mathbb{IW}(\vec{B})$’s dual SDP in Eq. (39). In other words, by minimizing
over all feasible solutions, we obtain $1-\mathbb{IW}(\vec{B})\geq
1-\max_{\rho_{\rm{B}}}\mathbb{SW}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)$;
namely,

$\displaystyle\mathbb{IW}(\vec{B})\leq\max_{\rho_{\rm{B}}}\mathbb{SW}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right),$
(48)

which,using Lemma 3, concludes the proof. This also means that optimal
solutions to $\mathbb{IW}(\vec{B})$’s dual SDP in Eq. (39) can help us to
construct optimal state $\rho_{\rm B}$. $\square$

###  F.2 Proof of Result 6

The first part of the proof, i.e., Eq. (9), comes from the above lemmas. The
impossibility of distillability of $\mathbb{IW}$ has been proven in App. B. To
conclude, we need to show that Obs. 1 and Res. 2, 3, and 4 hold for
$(\mathbb{IR},\mathbb{SR})$ substituted with $(\mathbb{IW},\mathbb{SW})$. For
Obs. 1, it is enough to show that from any solution of $\mathbb{IW}(\vec{B})$
we can construct an assemblage
$\vec{\sigma}=\eta_{\rm{B}}^{1/2}\vec{B}\eta_{\rm{B}}^{1/2}$ as in Obs. 2
above. The corresponding solution for $\mathbb{SW}(\vec{\sigma})$ has the SA
$\vec{\xi}$ and $\vec{\tau}$ appearing in Eq. (35) with the same reduced state
as $\vec{\sigma}$. For Res. 2, one can proceed similarly to the proof in App.
D and construct a convex decomposition for the SA: $\rho_{\rm
B}^{\frac{1}{2}}\mathcal{E}_{\omega}^{\dagger}(\vec{B})\rho_{\rm
B}^{\frac{1}{2}}/p(\omega)$, necessary to compute its $\mathbb{SW}$, from the
convex decomposition of $\vec{B}$. One proceed similarly as in App. D also for
the SA: $\mathcal{E}_{\omega}(\rho_{\rm B}^{1/2}\vec{B}\rho_{\rm
B}^{1/2})/p(\omega)$. For Res. 3, it is enough to use Res. 2 together with
Obs. 2 and the fact that
$\max_{\rho_{\rm{B}}}\mathbb{SW}\left(\rho_{\rm{B}}^{1/2}\vec{B}\rho_{\rm{B}}^{1/2}\right)=\max_{\mathcal{E}\in{\rm
LF_{1}}}\mathbb{SW}\left(\frac{\mathcal{E}_{\omega}(\vec{\sigma})}{p(\omega)}\right)$,
since the ${\rm LF_{1}}$ filter operation can be directly computed from the
optimal $\rho_{\rm B}$ as in the case of $(\mathbb{SR},\mathbb{IR})$ (see also
Ref. [18]). From this, it follows directly also Res. 4 by repeating the same
construction as in the case of $(\mathbb{SR},\mathbb{IR})$. $\square$


# Bulk Superconductivity in the Dirac Semimetal TlSb

Yuxing Zhou  Department of Physics, Zhejiang University, Hangzhou $310027$,
China  Bin Li  New Energy Technology Engineering Laboratory of Jiangsu
Province and School of Science, Nanjing University of Posts and
Telecommunications, Nanjing $210023$, China  Zhefeng Lou  Huancheng Chen  Qin
Chen  Binjie Xu  Chunxiang Wu  Department of Physics, Zhejiang University,
Hangzhou $310027$, China  Jianhua Du  Department of Applied Physics, China
Jiliang University, Hangzhou $310018$, China  Jinhu Yang  Department of
Physics, Hangzhou Normal University, Hangzhou $310036$, China  Hangdong Wang
Department of Physics, Hangzhou Normal University, Hangzhou $310036$, China
Minghu Fang  Corresponding author<EMAIL_ADDRESS>Department of Physics,
Zhejiang University, Hangzhou $310027$, China  Collaborative Innovation Center
of Advanced Microstructures, Nanjing University, Nanjing $210093$, China

###### Abstract

A feasible strategy to realize the Majorana fermions is searching for a simple
compound with both bulk superconductivity and Dirac surface states. In this
paper, we performed calculations of electronic band structure, the Fermi
surface and surface states, as well as measured the resistivity,
magnetization, specific heat for TlSb compound with a CsCl-type structure. The
band structure calculations show that TlSb is a Dirac semimetal when spin-
orbit coupling is taken into account. Meanwhile, we first found that TlSb is a
type-$\rm{II}$ superconductor with _T_ c = 4.38 K, _H_ c1(0) = 148 Oe, _H_
c2(0) = 1.12 T and $\kappa_{GL}$ = 10.6, and confirmed it to be a moderately
coupled _s_ -wave superconductor. Although we can not determine which bands
near the Fermi level _E_ F to be responsible for superconductivity, its
coexistence with the topological surface states implies that TlSb compound may
be a simple material platform to realize the fault-tolerant quantum
computations.

###### pacs:

##  I i. INTRODUCTION

Topological superconductors host Majorana fermions described by a real wave
function, providing protection for quantum computations Beenakker and
Kouwenhoven (2016). So, realizing topological superconductivity (TSC) has
became one of the most interesting topics in the condensed matter physics in
the past decades. According to the discussion in Ref. Sato and Ando (2017),
there are intrinsic and artificial engineered topological superconductors. For
the intrinsic, the topological nontrivial gap function naturally shows up.
Sr2RuO4 Maeno  _et al._ (1994); Luke  _et al._ (1998); Kashiwaya  _et al._
(2011); Mackenzie and Maeno (2003) is the first proposed topological
superconductor although the existence of chiral _p_ -wave superconductivity
(SC) is still under debate. CuxBi2Se3 Hor  _et al._ (2010), as the first
material to show SC (_T_ c $\sim$ 4 K) upon doping charge carrier into a
topological insulator (TI), is a promising ground to look for two dimensional
(2D) TSC due to the topological surface states surviving in TI even when
carriers are doped. Many experiments, such as the conductance spectroscopy
Sasaki  _et al._ (2011), nuclear manetic resonance (NMR) measurements of the
Knight-shift Matano  _et al._ (2016), and specific heat in applied magnetic
fields Yonezawa  _et al._ (2017), have already given evidences for TSC seem
emerging in CuxBi2Se3. The nematic SC discovered in CuxBi2Se3 Matano  _et al._
(2016); Yonezawa  _et al._ (2017) was also observed in the similar
superconductors derived from Bi2Se3, such as in SrxBi2Se3 Liu  _et al._
(2015), NbxCu2Se3 Qiu  _et al._ (2015). Sn1-xInxTe Sasaki  _et al._ (2012) is
another superconductor upon doping charge carriers into a topological
crystalline insulator. In the cleanest sample (_x_ $\sim$ 0.04) with the
lowest _T_ c (1.2 K), a pronounced zero-bias conductance peak (ZBCP) similar
to that in CuxBi2Se3 has been observed by point contact spectroscopy Sasaki
_et al._ (2012). Another is the artificial engineered TSC in hybrid
structures. According to the idea proposed by Fu and Kane Fu and Kane (2008),
if _s_ -wave pairing is imposed on the topological surface states of a three
dimensional (3D) TI through superconducting proximity effect, the resulting
superconducting state should be a 2D _p_ -wave SC harboring a Majorana zero
mode in the vortex core. Experimentally, proximity-induced SC on the surface
of 3D TIs has been studied by many groupsWilliams  _et al._ (2012); Wang  _et
al._ (2012); Yang  _et al._ (2012); Wang  _et al._ (2013); Oostinga  _et al._
(2013); Finck  _et al._ (2014); Snelder  _et al._ (2014). The observation
Wiedenmann  _et al._ (2016) of 4$\pi$-period Josephson supercurrent in 3D HgTe
TI is encouraging, although, it is difficult to elucidate the topological
nature of the induced 2D SC.

However, there are controversies about TSC emerging and the observed ZBCP
being a Majorana zero-energy mode (MZM) in the doped topological material. To
realize TSC through a superconducting proximity effect in hybrid structures
has many engineering challenges. A feasible way is to realize TSC in a simple
compound, in which both the topological surface state and the bulk SC coexist,
thus Majorana fermions emerge at the edge of superconductor. Recently, the
observations of MZM in the core of vortex Wang  _et al._ (2018); Machida  _et
al._ (2019), at the end of the atomic defect line Chen  _et al._ (2020a), and
near the Bi islands Chen  _et al._ (2020b) and the interstitial Fe atoms Yin
_et al._ (2015) in the simplest Fe-based superconductor Fe1+yTe0.5Se0.5 (_T_ c
= 14 K) Fang  _et al._ (2008) motivate us to search for the similar system.
TlSb crystallizes in a cubic CsCl structure with space group _P_ _m_
$\bar{3}$_m_ (No. 221) (as shown in the inset of Fig. 1), from this structure
a large number of topological semimetal/metals (TMs) were designed Jin  _et
al._ (2019), ranging from triple nodal points, type-$\rm{I}$ nodal lines, and
critical type nodal lines to hybrid nodal lines. For example, CaTe is a
typical type-$\rm{I}$ nodal line and Dirac TM Du  _et al._ (2017); YIr is a
typical triple-nodal-point TM Jin  _et al._ (2019), YMg possesses multiple
types of band crossing Jin  _et al._ (2019). Therefore, we tried to grow TlSb
crystals for studying its topological natures and SC, unfortunately, only
polycrystalline TlSb samples were obtained.

In this paper, we performed calculations of the electronic band structure, the
Fermi surface and the surface states on (001) plane, as well as measured
resistivity, magnetization and specific heat for the polycrystalline TlSb
sample. The band structure calculations show TlSb is a Dirac semimetal with
4-fold degenerate nodes near $\Gamma$ and R points. It is also found that TlSb
is a type-$\rm{II}$ superconductor with the superconducting transition
temperature _T_ c = 4.38 K, the lower critical field _H_ c1(0) = 148 Oe, and
the upper critical field _H_ c2(0) = 1.12 T and the Ginzburg-Landau (GL)
parameter $\kappa_{GL}$ = 10.6. The obtained specific heat jump, $\Delta$_C_
el/$\gamma_{n}$_T_ c $\sim$ 1.42, indicates that TlSb is a conventional
phonon-mediated superconductor with _s_ -wave superconducting symmetry. These
results indicate that both _s_ -wave SC and surface states coexist in TlSb,
whether the Majorana fermions emerge or not on the edges is needed to confirm
in the future.

Figure 1: (a) Polycrystalline XRD pattern with its refinement profile at room
temperature of TlSb. The inset shows the schematic structure of TlSb, thallium
atoms are in gray while the antimony atoms are brown.

##  II ii. EXPERIMENTAL

Polycrystalline TlSb samples were synthesized by a peritectic reaction method.
The mixture of stoichiometric high purity Tl (99.99%) chunk and Sb (99.999%)
powder was placed in an alumina crucible, sealed in an evacuated quartz tube
and heated at 450 ∘C for 20 hrs, then decreases to 195 ∘C waiting for the
mixture melting completely. To avoid the decomposition of obtain TlSb phase at
191 ∘C Predel and Schwermann (1970), the quartz tube was quenched to room
temperature at 195 ∘C, however, the obtained TlSb samples always contain a
small amount of unreacted Sb impurities due to the precipitation of Sb before
the peritectic reaction. The obtained TlSb alloy is easily to cut for the
subsequent structure characterizations and property measurements.
Polycrystalline x-ray diffraction (XRD) was carried out on a PANalytical
diffractometer equiped with CuKα radiation. The TlSb XRD pattern is shown in
Fig. 1, in which the main peaks can be fitted by the CsCl-type structure with
space group _P_ _m_ $\bar{3}$_m_. The lattice parameters a = b = c = 3.86(5)
$\rm{\AA}$ were obtained by the Rietveld refinement by using general structure
analysis system (GSAS) Larson and Von Dreele (1994). A rectangular bar of the
sample was cut for the magnetization and resistivity mesurements, which were
performed on a magnetic property measurement system (Quantum Design, MPMS - 7
T) and a physical property measurement system (Quantum Design, PPMS - 9 T),
respectively. The band structure was calculated by using density function
theory (DFT) with the WIEN2k package Schwarz  _et al._ (2002). Generalized
gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE) Perdew  _et al._
(1996) was employed for the exchange correlation potential calculations. A
cutoff energy of 520 eV and a 13 $\times$ 15 $\times$ 15 _k_ -point mesh were
used to perform the bulk calculations. The Fermi surface (FS) was performed
with WannierTools Mostofi  _et al._ (2014) package which is based on the
maximally localized Wannier function tight-binding model Marzari and
Vanderbilt (1997); Souza  _et al._ (2001); Marzari  _et al._ (2012)
constructed by using the Wannier90 Wu  _et al._ (2018) package.

##  III iii. RESULTS AND DISCUSSIONS

We first discuss the electronic band structure without considering spin-orbit
coupling (SOC). As shown in Fig. 2(a), both conduction (red) and valence
(blue) band cross the Fermi level _E_ F. At the high symmetry $\Gamma$ and R
points, there are three bands crossing, with a threefold degenerate at 0.8 eV
and 2 eV below _E_ F, respectively. However, when SOC is taken into account
[see Fig. 2(b)], gaps open and leaves two twofold degenerate bands at both
high symmetry points. Since both time-reversal and inversion symmetries are
present, no spin-splitting occurs and then the twofold degenerated bands come
together to fourfold degenerate points, indicating TlSb is a Dirac semimetal.
We also calculated the density of state (DOS), as shown in the right panel of
Fig. 2(b), the DOS at _E_ F is mainly contributed by Sb orbits. To further
clarify the band structure of TlSb, we calculated its 3D bulk FS of the first
Brillouin zone (BZ) as shown in Fig. 2(d), exhibiting very complex 3D
characteristics. Figure 2(e) presents the FS on the _k_ z = 0 plane, which is
the cross section passing the $\Gamma$ point of the 3D FS. Figure 2(f)
displays the energy dispersion in the _k_ x \- _k_ y plane, in which the Dirac
dispersion is clearly seen, demonstrating further that TlSb is a Dirac
semimetal.

Figure 2: The electronic band structures of TlSb without (a) and with (b) SOC.
(c) 3D bulk Fermi surfaces and color-coded Fermi velocities (red is high
velocity). (d) Calculated Fermi surfaces cross section at the _k_ z = 0 plane.
(e) Calculated energy distribution at Dirac point in _k_ x \- _k_ y plane.
Figure 3: (a) Surface band structure for (001) plane along projected high
symmetry points. The surface spectra of (001) plane with (b) _E_ = _E_ F, (c)
_E_ = -1.5 eV and (d) _E_ = -4 eV.

Then we calculated the surface states on (001) plane by using a surface
Green’s function method Sancho  _et al._ (1985). As shown in Fig. 3(a), the
projected Dirac points are hidden in the continuous bulk states, the surface
states are shown as the red curves. The (001) surface energy contour is shown
in Fig. 3(b), (c) and (d) with _E_ = _E_ F, _E_ = -1.5 eV and _E_ = -4 eV,
respectively. The surface band at _E_ = -4 eV being deeply below _E_ F can be
ignored due to negligible contribution to the electronic properties of
material. Due to TlSb having inversion-symmetry, its topology can be described
by one strong topological index $\nu_{0}$ and three weak indices $\nu_{1}$,
$\nu_{2}$, $\nu_{3}$ Fu  _et al._ (2007). Thus we calculated the Wilson loops
on six time-reversal invariant planes using WannierTools Wu  _et al._ (2018),
and the results are shown in Fig. 4. According the the definition of Wilson
loops Yu  _et al._ (2011); Soluyanov and Vanderbilt (2011), the topological
indices are (1;000) indicating that TlSb is a strong topological material.

Figure 4: Wilson loops of six time-reversal invariant planes at (a) _k_ 1 =
0.0, (b) _k_ 1 = 0.5, (c) _k_ 2 = 0.0, (d) _k_ 2 = 0.5, (e) _k_ 3 = 0.0, (f)
_k_ 3 = 0.5, where _k_ 1, _k_ 2, _k_ 3 are in units of the reciprocal lattice
vectors. Figure 5: (a) The temperature dependence of resistivity, $\rho$(_T_),
of TlSb. The inset: enlarged $\rho$(_T_) near the superconducting transition.
(b) The temperature dependence of magnetic susceptibility, $\chi$(_T_),
measured at _H_ = 5 Oe.

Second, we focus on the SC emerging in the Dirac semimetal TlSb discovered
first by us. Figure 5(a) displays the temperature dependence of resistivity,
$\rho$(_T_), measured at zero field. With decreasing temperature, $\rho_{xx}$
decreases leisurely, exhibiting a poor metallic behavior, then drops to zero
at 4.32 K, a superconductivity transition occurring with the mid-temperature
_T_ ${}_{c}^{mid}$ = 4.38 K, $\Delta$_T_ c = 0.15 K. This superconducting
transition is also confirmed by the susceptibility measurement. Figure 5(b)
presents the temperature dependence of susceptibility, $\chi$(_T_), measured
at _H_ = 5 Oe with both zero-field cooling (ZFC) and field cooling (FC)
process. It is clear that a sharp diamagnetic transition emerges at 4.3 K, and
the complete diamagnetism (4$\pi\chi$ $\sim$ -1) below _T_ c indicates the
bulk superconductivity being from TlSb, since Sb element is non-
superconducting at ambient pressure.

Figure 6: (a) Field dependence of magnetization _M_(_H_) measured at 2 K. (b)
The low field magnetization of TlSb at different temperatures. The dashed line
indicates the initial linear magnetization curve. The inset shows the
temperature dependence of lower critical field, _H_ c1(_T_) determined by the
magnetization curve deviating from linear. The red line is the _H_ c1(_T_)
fitted by GL relation

Figure 6(a) shows the field dependence of magnetization, _M_(_H_), measured at
2 K for a TlSb sample, exhibiting a hysteresis, which indicates that TlSb is a
type-$\rm{II}$ superconductor. Then we measured the _M_(_H_) at various
temperatures below 4.5 K, as shown in Fig. 6(b). The lower critical field _H_
c1(_T_) can be estimated by the field, at which _M_(_H_) curve starts to
deviate from the linear relationship. The obtained _H_ c1(_T_) is shown in the
inset of Fig. 6(b), the lower critical field at zero temperature _H_ c1(0) =
148 Oe was obtained by the fitting using the GL relationship:

$\displaystyle\emph{H}_{c1}(\emph{T})=\emph{H}_{c1}(0)(1-(\frac{\emph{T}}{\emph{T}_{c}})^{2})$
(1)

Figure 7: The temperature dependence of upper critical field _H_ c2 determined
from resistivity measurements. The inset shows the low temperature $\rho$(_T_)
curves measured at various magnetic fields.

In order to get the upper critical field _H_ c2(0), we measured the
superconducting transition temperature (_T_ ${}_{c}^{mid}$) at various applied
magnetic fields. As shown in the inset of Fig. 7, the _T_ c decreases, and the
transition width $\Delta$_T_ c increases with increasing magnetic field. By
using the GL formula _H_ c2(_T_) = _H_ c2(0)(1-_t_ 2)/(1+_t_ 2), where _t_ is
the normalized temperature _t_ = _T_ /_T_ c, to fit the _H_ c2(_T_) data, the
zero temperature upper critical field _H_ c2(0) = 1.12 T was obtained, which
is much lower than the Pauli limit field _H_ ${}_{c2}^{P}$(0) = 1.86 _T_ c =
8.18 T. Then, the GL coherence length $\xi_{GL}$(0) = 15.3 nm was estimated by
using the formula _H_ c2(0) = $\Phi_{0}$/2$\pi$$\xi_{GL}^{2}$, where $\Phi$0
is the quantum flux (h/2e). The penetration depth $\lambda_{GL}$(0) = 162 nm
was estimated by using the formula _H_ c1(0) =
($\Phi_{0}$/4$\pi\lambda_{GL}^{2}$(0))$\ln$($\lambda_{GL}$(0)/$\xi_{GL}$(0)),
and the GL parameter $\kappa_{GL}$ = $\lambda_{GL}$(0)/$\xi_{GL}$(0) = 10.6.

Figure 8: The temperature dependence of specific heat of TlSb measured at 0 T
(black circles) and 3 T (red circles) plotted as _C_ p versus _T_ 2. The solid
blue line is a fit by the Debye model. The inset shows normalized electronic
specific heat _C_ el/$\gamma_{n}$_T_ versus _T_ /_T_ c at zero field.

We also measured the specific heat as a function of temperature, _C_ p(_T_),
for TlSb in the temperature range of 0.5 - 5 K at both zero field and 3 T,
respectively, as shown in Fig. 8. It is clear that the zero-field _C_ p(_T_),
compared with the _C_ p(_T_) measured at 3 T ($>$ _H_ c2, in this case bulk SC
is completely suppressed), exhibits a small and broad peak near _T_ c,
corresponding to the superconducting transition. No other anomaly was observed
except for the peak near _T_ c = 4.38 K, indicating that no Tl impurities (_T_
c = 2.39 K) emerge in our sample although a small amount of non-
superconducting Sb impurities was detected in the XRD. We fitted the low
temperature _C_ p(_T_) data measured at 3 T using the Debye model:

$\displaystyle\emph{C}_{p}/\emph{T}=\gamma_{n}+\beta_{3}\emph{T}^{2}+\beta_{5}\emph{T}^{4}$
(2)

where $\gamma_{n}$ is the Sommerfeld Coefficient, both the $\beta_{3}$_T_ 3
and $\beta_{5}$_T_ 5 are the phonon contributions to specific heat. The
parameters $\gamma_{n}$ = 5.56 mJ mol-1 K-2, $\beta_{3}$ = 1.39 mJ mol-1 K-4,
$\beta_{5}$ = 0.44 mJ mol-1 K-6 were obtained. The inset of Fig. 8 shows the
normalized $\frac{\Delta\emph{C}_{p}}{\gamma_{n}\emph{T}}$ =
$\frac{C_{p}(0\rm{T})-C_{\emph{p}}(3\rm{T})}{\gamma_{n}\emph{T}}$ as a
function of the normalized temperature _t_ = _T_ /_T_ c. The bulk
superconducting temperature _T_ c = 4.3 K was estimated by a entropy-balance
method, consistent with the results from the resistivity and susceptibility
measurements mentioned above. The normalized specific heat jump $\Delta$_C_
el/$\gamma_{n}$_T_ c = 1.42 was estimated, almost the same with the predicted
value (1.43) by the Bardeen-Cooper-Schrieffer (BCS) theory Bardeen  _et al._
(1957), indicating that TlSb is a _s_ -wave phonon-mediated superconductor.
The Debye temperature $\Theta_{D}$ = 141 K was estimated by using the formula
$\Theta_{D}$ = (12$\pi^{4}$_NR_ /5$\beta_{3}$)${}^{\frac{1}{3}}$, where _N_ =
2 is the number of atoms in an unit cell and the _R_ = 8.314 J mol-1 K-1 is
the molar gas constant. Using the obtained $\Theta_{D}$ and _T_ c values, we
calculated the electron-phonon coupling constant $\lambda_{ep}$ by using the
McMillan formula McMillan (1968):

$\displaystyle\lambda_{ep}=\frac{1.04+\mu^{*}\ln(\Theta_{D}/1.45\emph{T}_{c})}{(1-0.62\mu^{*})\ln(\Theta_{D}/1.45\emph{T}_{c})-1.04}$
(3)

where $\mu^{*}$ = 0.13 is a typical value of the Coulomb repulsion
pseudopotential for the intermetallic superconductors. The obtained
$\lambda_{ep}$ = 0.78 value is comparable to that of other superconductors
such as PbTaSe2 ($\lambda_{ep}$ = 0.74) Ali  _et al._ (2014) and Nb0.18Re0.82
($\lambda_{ep}$ = 0.73) Karki  _et al._ (2011), suggesting TlSb is a
moderately coupled superconductor. The obtained superconducting parameters are
summarised in Table I.

Table 1: Superconducting parameters of TlSb Parameters | unit | value
---|---|---
_T_ c | K | 4.3
_H_ c1(0) | Oe | 148
_H_ c2(0) | T | 1.12
_H_ ${}_{c2}^{P}$(0) | T | 8.18
$\xi_{GL}$ | nm | 15.3
$\lambda_{GL}$ | nm | 162
$\kappa_{GL}$ |  | 10.6
$\gamma_{n}$ | mJ mol-1 K-2 | 5.56
$\Delta$_C_ el/$\gamma_{n}$_T_ c |  | 1.42

##  IV iv. CONCLUSION

In summary, the calculations of the electronic band structure, the FS and the
surface states show that TlSb with a CsCl-type structure is a Dirac semimetal.
We measured the resistivity, magnetization, specific heat for the
polycrystalline TlSb sample. We first found that TlSb is a type-$\rm{II}$
superconductor with _T_ c = 4.38 K, _H_ c1(0) = 148 Oe, _H_ c2 = 1.12 T and
$\kappa_{GL}$ = 10.6. The specific heat results demonstrate it to be a
moderately coupled _s_ -wave superconductor. Although we can not determine
which bands near _E_ F to be responsible for SC, the coexistence of bulk SC
with _s_ -wave symmetry and the Dirac fermions on the surface in a single TlSb
compound provides an opportunity to realize Majorana zero energy mode.

##  V ACKNOWLEDGEMENTS

This research is supported by the National Key Program of China under Grant
No. 2016YFA0300402 and the National Natural Science Foundation of China
(Grants No. NSFC-12074335 and 11974095) the Fundamental Research Funds for the
Central Universities, an open program from the National Lab of Solid State
Microstructures of Nanjing University (Grant No. M32025).

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