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# Distributed quantum incompatibility

Lucas Tendick<EMAIL_ADDRESS>Hermann Kampermann  Dagmar Bruß  Institute
for Theoretical Physics, Heinrich Heine University Düsseldorf, D-40225
Düsseldorf, Germany

###### Abstract

Incompatible, i.e. non-jointly measurable quantum measurements are a necessary
resource for many information processing tasks. It is known that increasing
the number of distinct measurements usually enhances the incompatibility of a
measurement scheme. However, it is generally unclear how large this
enhancement is and on what it depends. Here, we show that the incompatibility
which is gained via additional measurements is upper and lower bounded by
certain functions of the incompatibility of subsets of the available
measurements. We prove the tightness of some of our bounds by providing
explicit examples based on mutually unbiased bases. Finally, we discuss the
consequences of our results for the nonlocality that can be gained by
enlarging the number of measurements in a Bell experiment.

The incompatibility of quantum measurements, i.e., the impossibility of
measuring specific observable quantities simultaneously, is one of quantum
physics’ most prominent and striking properties. First discussed by Heisenberg
Heisenberg (1927) and Robertson Robertson (1929), this counterintuitive
feature was initially thought of as a puzzling curiosity that represents a
drawback for potential applications. Nowadays, measurement incompatibility
Gühne  _et al._ (2021); Heinosaari  _et al._ (2016) is understood as a
fundamental property of nature that lies at the heart of many quantum
information processing tasks, such as quantum state discrimination Buscemi
_et al._ (2020); Carmeli  _et al._ (2019); Uola  _et al._ (2019); Oszmaniec
and Biswas (2019); Ducuara and Skrzypczyk (2020); Uola  _et al._ (2020a),
quantum cryptography Bennett and Brassard (2014); Pirandola  _et al._ (2020),
and quantum random access codes Carmeli  _et al._ (2020); Anwer  _et al._
(2020). Even more importantly, incompatible measurements are a crucial
requirement for quantum phenomena such as quantum contextuality Budroni  _et
al._ (2021), EPR-steering Uola  _et al._ (2020b); Cavalcanti and Skrzypczyk
(2016), and Bell nonlocality Brunner  _et al._ (2014).
Its fundamental importance necessitates gaining a deep understanding of
measurement incompatibility from a qualitative and quantitative perspective.
By its very definition, measurement incompatibility arises when at least
$m\geq 2$ measurements are considered that cannot be measured jointly by
performing a single measurement instead. Generally, adding more measurements
to a measurement scheme may allow for more incompatibility, hence increasing
advantages in certain applications.
However, it is unclear how much incompatibility can be gained from adding
further measurements to an existing measurement scheme and on what this
potential gain depends. Similarly, it is unclear how the incompatibility of
measurement pairs contributes towards the total incompatibility of the whole
set. Answering these questions is crucial to understanding specific protocols’
power over others, such as protocols involving different numbers of mutually
unbiased bases (MUB) in quantum key distribution Bennett and Brassard (2014);
Bruß (1998). While it is known Quintino  _et al._ (2019) that the different
incompatibility structures (e.g., genuine triplewise and pairwise
incompatibility) arising for $m\geq 3$ measurements set different limitations
on the violation of Bell inequalities and incompatibility structures beyond
two measurements have also been studied in Kunjwal  _et al._ (2014);
Heinosaari  _et al._ (2008); Liang  _et al._ (2011), so far, no systematical
way to quantify the gained advantage is known.
The systematical and quantitative analysis of incompatibility structures in
this work is inspired by the analysis of the distribution of multipartite
entanglement Coffman  _et al._ (2000) and coherence Radhakrishnan  _et al._
(2016), leading to the observation that these quantum resources behave
monogamously across subsets of systems. Despite the mathematical differences,
our work follows physically a similar path by studying the distribution of
quantum incompatibility across subsets of measurements. Namely, we show how an
assemblage’s incompatibility depends quantitatively on its subsets’
incompatibilities. More specifically, we show how the potential gain of adding
measurements to an existing measurement scheme is bounded by the
incompatibility of the parent positive operator valued measures that
approximate the respective subsets of measurements by a single measurement.
Our results reveal the _polygamous nature of measurement incompatibility_ in
the sense that an assemblage of more than two measurements can only be highly
incompatible if all its subsets and the respective parent POVMs of the closest
jointly measurable approximation of these subsets are highly non-jointly
measurable. Our considerations lead to a new notion of measurement
incompatibility that accounts only for a specific measurement’s
incompatibility contribution. We prove the relevance of our bounds on the
incompatibility that can maximally be gained by showing that they are tight
for particular measurement assemblages based on MUB. Finally, we show that our
results have direct consequences for steering and Bell nonlocality and discuss
future applications of our results and methods.
Preliminaries.—We describe a quantum measurement most generally by a POVM,
i.e., a set $\\{M_{a}\\}$ of operators $0\leq M_{a}\leq\mathds{1}$ such that
$\sum_{a}M_{a}=\mathds{1}$. Given a state $\rho$, the probability of obtaining
outcome $a$ is given by the Born rule $p(a)=\operatorname{Tr}[M_{a}\rho]$. A
_measurement assemblage_ is a collection of different POVMs with operators
$M_{a|x}$, where $x$ denotes the particular measurement. We write an
assemblage
$\mathcal{M}_{(1,2,\cdots,m)}=(\mathcal{M}_{1},\mathcal{M}_{2},\cdots,\mathcal{M}_{m})$
of $m$ measurements as an ordered list of POVMs, where $\mathcal{M}_{x}$
refers to the $x$-th measurement. For instance,
$\mathcal{M}_{(1,2,3)}=(\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{3})$
refers to an assemblage with three (different) measurements and
$\mathcal{M}_{(1,2,2)}=(\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{2})$
denotes an assemblage where the second and the third POVM are equal.
An assemblage $\mathcal{M}$ is called _jointly measurable_ if it can be
simulated by a single _parent POVM_ $\\{G_{\lambda}\\}$ and conditional
probabilities $p(a|x,\lambda)$ such that

$\displaystyle M_{a|x}=\sum_{\lambda}p(a|x,\lambda)G_{\lambda}\ \forall\ a,x,$
(1)

and it is called _incompatible_ otherwise. Here, we call $G(\mathcal{M})$ a
parent POVM of a jointly measurable assemblage $\mathcal{M}$. Various
functions can quantify measurement incompatibility Pusey (2015); Designolle
_et al._ (2019a); Cope and Uola (2022). The most suitable incompatibility
quantifier for our purposes is the recently introduced _diamond distance
quantifier_ Tendick  _et al._ (2023), given by

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})=\min\limits_{\mathcal{F}\in\mathrm{JM}}\sum_{x}p(x)\operatorname{D_{\diamond}}(\Lambda_{\mathcal{M}_{x}},\Lambda_{\mathcal{F}_{x}}),$
(2)

where $\mathrm{JM}$ denotes the set of jointly measurable assemblages,
$\Lambda_{\mathcal{M}_{x}}=\sum_{a}\mathrm{Tr}[M_{a|x}\rho]|a\rangle\langle
a|$ is the _measure-and-prepare channel_ associated to the measurement
$\mathcal{M}_{x},$ and
$\operatorname{D_{\diamond}}(\Lambda_{1},\Lambda_{2})=\max\limits_{\rho\in\mathcal{S}(\mathcal{H}\otimes\mathcal{H})}\dfrac{1}{2}\lVert((\Lambda_{1}-\Lambda_{2})\otimes\mathds{1}_{d})\rho\rVert_{1}$
is the _diamond distance_ Kitaev  _et al._ (2002) between two channels
$\Lambda_{1}$ and $\Lambda_{2}$, with the trace norm $\lVert
X\rVert_{1}=\operatorname{Tr}[\sqrt{X^{\dagger}X}]$. Furthermore,
$\mathcal{M}^{\mathbf{p}}=(\mathcal{M},\mathbf{p})$ denotes a _weighted
measurement assemblage_ , where $\mathbf{p}$ contains the probabilities $p(x)$
with which measurement $x$ is performed. Note that
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ is induced by the
general distance
$\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}})\coloneqq\sum_{x}p(x)\operatorname{D_{\diamond}}(\Lambda_{\mathcal{M}_{x}},\Lambda_{\mathcal{N}_{x}})$
between two assemblages $\mathcal{M}^{\mathbf{p}}$ and
$\mathcal{N}^{\mathbf{p}}$.
We denote by $\mathcal{M}^{\\#}_{(1,2,\cdots,m)}$ the closest jointly
measurable assemblage with respect to $\mathcal{M}_{(1,2,\dots,m)}$, i.e., the
_arg-min_ on the RHS in Eq. (2). While $\mathcal{M}^{\\#}_{(1,2,\cdots,m)}$
and its underlying parent POVM are generally not unique Heinosaari  _et al._
(2008); Guerini and Cunha (2018), all the results derived in this work hold
for any valid choice, as we do not assume uniqueness. If we only approximate a
subset of $n<m$ measurements of $\mathcal{M}_{(1,2,\dots,m)}$ by jointly
measurable ones, for instance the first $n$ settings, while keeping the
remaining measurements unchanged, we write
$\mathcal{M}^{\\#(1,2,\dots,n)}_{(1,2,\cdots,m)}$.
The diamond distance quantifier
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ Tendick  _et al._
(2023) is particularly well-suited for our purposes, as it is not only
monotonous under the application of quantum channels and classical simulations
but it also inherits all properties of a distance (in particular the triangle
inequality of
$\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}})$),
and it is written in terms of a convex combination of the individual
measurement’s distances.
Besides these technical requirements, the quantifier
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ admits the operational
interpretation of average single-shot distinguishability of the assemblage
$\mathcal{M}$ from its closest jointly measurable assemblage
$\mathcal{M}^{\\#}$. Furthermore, it can be used to upper bound the amount of
steerability and nonlocality that can be revealed by the measurements
$\mathcal{M}$ in Bell-type experiments Tendick  _et al._ (2023).
For pedagogical reasons, we focus in the main text on the scenario
$2\rightarrow 3$, i.e., we consider an assemblage of $m=2$ measurements that
is promoted to one with $m^{\prime}=3$ settings. Furthermore, we set $p(x)$ to
be uniformly distributed and simply use the symbol $\mathcal{M}$ for the
weighted assemblage in this case. We refer to the Supplemental Material (SM)
Sup  for all proofs, more background information, and generalizations to an
arbitrary number of measurements and general probability distributions.
Adding a third measurement $\mathcal{M}_{3}$ to the assemblage
$\mathcal{M}_{(1,2)}=(\mathcal{M}_{1},\mathcal{M}_{2})$ is mathematically
described by the concatenation of ordered lists, using the symbol
$\mathbin{+\mkern-10.0mu+}$, i.e., we write

$\displaystyle\mathcal{M}_{(1,2,3)}=\mathcal{M}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}_{3}=(\mathcal{M}_{1},\mathcal{M}_{2},\mathcal{M}_{3}).$
(3)

Using the concatenation of ordered lists, we formally define
$\mathcal{M}^{\\#(1,2)}_{(1,2,3)}$ such that

$\displaystyle\mathcal{M}^{\\#(1,2)}_{(1,2,3)}\coloneqq\mathcal{M}^{\\#}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}_{3}.$
(4)

Three measurements allow for incompatibility structures Kunjwal  _et al._
(2014); Heinosaari  _et al._ (2008); Liang  _et al._ (2011); Quintino  _et
al._ (2019) beyond Eq. (1). We define the sets $\mathrm{JM}^{(s,t)}$ with
$s\neq t\in\\{1,2,3\\}$ as those containing assemblages in which the
measurements $s$ and $t$ are jointly measurable. This allows us to define
_pairwise_ and _genuinely triplewise incompatible_ assemblages Quintino  _et
al._ (2019) as those that are _not_ contained in the intersection and the
convex hull of the sets $\mathrm{JM}^{(s,t)}$, respectively. See also Figure 1
for a graphical representation and more details.

Figure 1: Different structures of incompatibility for three measurements, see
also Ref. Quintino  _et al._ (2019). The sets $\mathrm{JM}^{(s,t)}$ contain
assemblages of measurements where the pairs $(s,t)$ are compatible. Their
intersection
$\mathrm{JM}^{\mathrm{pair}}\coloneqq\mathrm{JM}^{(1,2)}\cap\mathrm{JM}^{(1,3)}\cap\mathrm{JM}^{(2,3)}$
contains all pairwise compatible assemblages, with the set $\mathrm{JM}$ of
all jointly measurable assemblages as a proper subset. Assemblages not
contained in the convex hull
$\mathrm{JM}^{\mathrm{conv}}\coloneqq\mathrm{Conv}(\mathrm{JM}^{(1,2)},\mathrm{JM}^{(1,3)},\mathrm{JM}^{(2,3)})$
of the sets $\mathrm{JM}^{(s,t)}$, i.e., those that cannot be written as a
convex combination of assemblages from the sets $\mathrm{JM}^{(1,2)}$,
$\mathrm{JM}^{(1,3)}$, and $\mathrm{JM}^{(2,3)}$ are genuinely triplewise
incompatible. The incompatibility of $\mathcal{M}_{(1,2,3)}$ is given by the
distance to its closest jointly measurable approximation
$\mathcal{M}^{\\#}_{(1,2,3)}$. This distance can be upper bounded using the
triangle inequality via the assemblage $\mathcal{M}^{\\#(1,2)}_{(1,2,3)}$.

Incompatibility gain.—We investigate the _incompatibility gain_ obtained from
adding measurements to an already available assemblage. That is, for an
assemblage $\mathcal{M}_{(1,2,3)}$ defined via Eq. (3) we want to quantify the
gain

$\displaystyle\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}\coloneqq\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})-\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)}).$
(5)

Note that $\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}$ is the difference of
two quantities that can be computed via semidefinite programs Tendick  _et
al._ (2023), however, the purely numerical value of the gained incompatibility
does only provide limited physical insights by itself. While it seems
generally challenging to find an exact analytical expression for the
incompatibility gain, we will derive bounds on it in the following.
Our approach relies on a two-step protocol. First, we employ a _measurement
splitting_ , i.e., instead of considering the incompatibility of
$\mathcal{M}_{(1,2,3)}$, we consider the incompatibility
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})$. That is, each
measurement of $\mathcal{M}_{(1,2,3)}$ is now split up into two equivalent
ones, each occurring with a probability of $\tfrac{1}{6}$. Furthermore, it
holds
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})$
since the assemblages can be converted into each other by (reversible)
classical post-processing Sup  (Section II). The second step involves a
particular instance of the triangle inequality and uses specifically that
$\operatorname{I_{\diamond}}(\mathcal{M})$ is defined as convex combination
over the individual settings. More precisely, let

$\displaystyle\mathcal{N}=\mathcal{M}^{\\#}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{(1,3)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{(2,3)},$
(6)

be an assemblage that contains itself three assemblages (of two measurements
each) that are the closest jointly measurable approximations with respect to
the individual subsets of $\mathcal{M}_{(1,2,3)}$. We point out that
$\mathcal{N}$ itself can be incompatible in general. Using the triangle
inequality, it follows that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$
$\displaystyle=\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})$ (7)
$\displaystyle\leq\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)},\mathcal{N})+\operatorname{I_{\diamond}}(\mathcal{N}).$

Due to our choice of $\mathcal{N}$, the term
$\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)},\mathcal{N})$
evaluates to the average incompatibility of the subsets, as we can split the
sum over all six settings into three pairs, i.e. we obtain

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$
$\displaystyle\leq\dfrac{1}{3}\big{[}\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3)})$
(8)
$\displaystyle+\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3)})\big{]}+\operatorname{I_{\diamond}}(\mathcal{N}).$

That is, the incompatibility of $\mathcal{M}_{(1,2,3)}$ is upper bounded by
the average incompatibility of its two-measurement subsets plus the
incompatibility $\operatorname{I_{\diamond}}(\mathcal{N})$ that contains the
information about how incompatible the respective closest jointly measurable
POVMs are with each other. Notice that
$\operatorname{I_{\diamond}}(\mathcal{N})\leq\operatorname{I_{\diamond}}(\mathcal{G})$
holds, where

$\displaystyle\mathcal{G}=G(\mathcal{M}^{\\#}_{(1,2)})\mathbin{+\mkern-10.0mu+}G(\mathcal{M}^{\\#}_{(1,3)})\mathbin{+\mkern-10.0mu+}G(\mathcal{M}^{\\#}_{(2,3)})$
(9)

is the assemblage that contains the parent POVMs $G$ of the respective
subsets, as $\mathcal{N}$ is a classical post-processing of $\mathcal{G}$ Sup
(Section II). This shows that the incompatibility of $\mathcal{M}_{(1,2,3)}$
is limited on two different levels through its subsets. Moreover, it reveals a
type of _polygamous_ behavior of incompatibility. For high incompatibility of
$\mathcal{M}_{(1,2,3)}$ each of the subsets, as well as the underlying parent
POVMs of the respective jointly measurable approximations, have to be highly
incompatible. Coming back to the incompatibility gain, we are ready to present
our first main result.

######  Result 1.

Let $\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})\leavevmode\nobreak\
\geq\leavevmode\nobreak\
\max\\{\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3)}),\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3)})\\}$.
It follows that the incompatibility gain as defined in Eq. (5) is bounded such
that

$\displaystyle\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}\leq\operatorname{I_{\diamond}}(\mathcal{N})\leq\operatorname{I_{\diamond}}(\mathcal{G}).$
(10)

This means that the potential incompatibility gain is limited by the
incompatibility of the assemblage $\mathcal{N}$ in Eq. (6), i.e., the
concatenation of the respective closest jointly measurable approximations of
the subsets. Physically more intuitive, it is limited by the incompatibility
of the assemblage that contains the respective parent POVMs. The assumption
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})\geq\max\\{\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3)}),\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3)})\\}$
represents no loss of generality for all practical purposes, as one can simply
optimize over all possible two-measurement subsets.
We show in the SM Sup  that Result 1 can be generalized to

$\displaystyle\Delta\mathrm{I}_{(1,\cdots,m)\rightarrow(1,\cdots,m,m+1)}\leq\operatorname{I_{\diamond}}(\mathcal{N})\leq\operatorname{I_{\diamond}}(\mathcal{G}),$
(11)

by appropriately redefining $\mathcal{N}$ and $\mathcal{G}$.
We point out that Result 1 allows for the definition of a single _maximally
incompatible additional_ measurement, in the sense that it is the measurement
$\mathcal{M}_{3}$ that maximizes the incompatibility gain
$\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}$ for a given assemblage
$\mathcal{M}_{(1,2)}$. As an illustrative example, we consider the three
projective measurements $\\{\Pi_{a|x}\\}$ which represent the Pauli $X,Y,Z$
observables subjected to white noise, i.e., we analyze the incompatibility of
the assemblage
$\mathcal{M}^{\eta}_{(1,2,3)}=(\mathcal{M}^{\eta}_{1},\mathcal{M}^{\eta}_{2},\mathcal{M}^{\eta}_{3})$
defined via

$\displaystyle
M^{\eta}_{a|x}=\eta\Pi_{a|x}+(1-\eta)\operatorname{Tr}[\Pi_{a|x}]\dfrac{\mathds{1}}{2},$
(12)

where $(1-\eta)$ is the noise level. It holds in this particular case that
(see Figure 2):

$\displaystyle\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}(\eta)=\operatorname{I_{\diamond}}(\mathcal{N}(\eta)),$
(13)

which we prove analytically in the SM Sup  (Section VI). For the regime
$\tfrac{1}{\sqrt{2}}\leq\eta\leq 1$ we also show that
$\operatorname{I_{\diamond}}(\mathcal{N}(\eta))=\operatorname{I_{\diamond}}(\mathcal{M}^{1/\sqrt{2}}_{(1,2,3)})$,
which means that the gained incompatibility is exactly given by the
incompatibility of $\mathcal{M}^{\eta}_{(1,2,3)}$ at the noise threshold where
it becomes _pairwise compatible_.

Figure 2: Incompatibility gain for adding a third Pauli measurement. The
gained incompatibility is given by the red (dotted) line. In the regime where
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})\neq 0$, the gained
incompatibility remains constant. The red (dotted) curve and the blue curve
add up to the violet one.

Our methods can also be applied to obtain lower bounds. For instance, we show
Sup  (Section III) that $\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$
is bounded by the average subset incompatibility:

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})\geq\dfrac{1}{3}[\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3)})].$
(14)

In general,
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})<\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)}$
is possible, i.e., adding a measurement to an assemblage can actually decrease
the incompatibility, if we do _not_ optimize over the input distribution
$\mathbf{p}$. For instance, adding a measurement $\mathcal{M}_{3}$ that is
jointly measurable with $\mathcal{M}^{\\#}_{(1,2)}$, such as an identity
measurement, generally decreases the incompatibility.
Another way to see how the incompatibility of an assemblage
$\mathcal{M}_{(1,2,3)}$ can be upper bounded in terms of the incompatibility
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})$ plus the gained
incompatibility due to measurement $\mathcal{M}_{3}$ relies on directly
applying specific instances of the triangle inequality without splitting the
measurements.
A new notion of incompatibility.—Consider the general assemblage
$\mathcal{M}_{(1,2,3)}$ as defined in Eq. (3). Due to the triangle inequality,
see also Figure 1, it holds

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})\leq\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,3)},\mathcal{N}_{(1,2,3)})+\operatorname{I_{\diamond}}(\mathcal{N}_{(1,2,3)}),$
(15)

for any assemblage $\mathcal{N}_{(1,2,3)}$. By choosing
$\mathcal{N}_{(1,2,3)}=\mathcal{M}^{\\#(1,2)}_{(1,2,3)}\coloneqq\mathcal{M}^{\\#}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}_{3}$,
we obtain our second main result.

######  Result 2.

Let
$\mathcal{M}_{(1,2,3)}=\mathcal{M}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}_{3}$
be a concatenated measurement assemblage and $\mathcal{M}^{\\#}_{(1,2)}$ the
closest jointly measurable approximation of $\mathcal{M}_{(1,2)}$. It holds

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})\leq\dfrac{2}{3}\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2)}_{(1,2,3)}).$
(16)

This means that the incompatibility of $\mathcal{M}_{(1,2,3)}$ is upper
bounded by the incompatibility of the subset $\mathcal{M}_{(1,2)}$, weighted
with the probability $p=\tfrac{2}{3}$, plus the incompatibility of the added
measurement $\mathcal{M}_{3}$ with the closest jointly measurable
approximation $\mathcal{M}^{\\#}_{(1,2)}$ of $\mathcal{M}_{(1,2)}$. In Sup
(Section III) we also show that the incompatibility of $\mathcal{M}_{(1,2,3)}$
is lower bounded by

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})\geq\tfrac{2}{3}\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)}).$
(17)

The only incompatibility that contributes to
$\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2)}_{(1,2,3)})$ is the
incompatibility of $\mathcal{M}_{3}$ with the assemblage
$\mathcal{M}^{\\#}_{(1,2)}$, which itself is jointly measurable. Therefore,
this term in Eq. (16) can be understood as a new notion of incompatibility of
the assemblage $\mathcal{M}_{(1,2,3)}$, where all incompatibilities apart of
the contribution that comes from the presence of measurement $\mathcal{M}_{3}$
are omitted.
We show analytically in the SM Sup  (Section VI) that the bound in Eq. (16) is
tight for depolarized Pauli measurements (see Eq. (12)). Moreover, we show
analytically that a similar bound is tight for certain measurements based on
$d$-dimensional MUB in cases where the number of measurements $m$ is changed
such that $m=2\rightarrow m^{\prime}=d$, $m=2\rightarrow m^{\prime}=d+1$, and
$m=d\rightarrow m^{\prime}=d+1$. Namely, we prove and analyze the
generalization of Eq. (16):

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,\cdots,m)})\leq\dfrac{\lvert
C|}{m}\operatorname{I_{\diamond}}(\mathcal{M}_{C})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#C}_{(1,2,\cdots,m)}),$
(18)

for any assemblage $\mathcal{M}_{(1,2,\cdots,m)}$ and any subset $C$ of
measurements with cardinality $\lvert C|$.
Incompatibility decomposition.—Looking at the results in Figure 2 leads to the
question of whether there exists a more general decomposition of
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$ into different
incompatibility structures. Indeed, since
$\operatorname{I_{\diamond}}(\mathcal{M})$ is a _distance-based_
incompatibility quantifier, our final main result follows.

######  Result 3.

The incompatibility of any assemblage $\mathcal{M}$ of $m=3$ measurements is
upper bounded such that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M})$
$\displaystyle\leq\operatorname{I_{\diamond}}^{\mathrm{gen}}(\mathcal{M})+\operatorname{I_{\diamond}}^{\mathrm{pair}}(\mathcal{M})+\operatorname{I_{\diamond}}^{\mathrm{hol}}(\mathcal{M}),$
(19)

where $\operatorname{I_{\diamond}}^{\mathrm{gen}}(\mathcal{M})$ is the
_genuine triplewise incompatibility_ of $\mathcal{M}$, i.e., its minimal
distance to an assemblage
$\mathcal{M}^{\mathrm{conv}}\in\mathrm{JM}^{\mathrm{conv}}$. Furthermore, we
define
$\operatorname{I_{\diamond}}^{\mathrm{pair}}(\mathcal{M})\coloneqq\operatorname{D_{\diamond}}(\mathcal{M}^{\mathrm{conv}},\mathcal{M}^{\mathrm{pair}})$
to be the _pairwise incompatibility_ , where
$\mathcal{M}^{\mathrm{pair}}\in\mathrm{JM}^{\mathrm{pair}}$ is the closest
pairwise compatible assemblage with respect to $\mathcal{M}^{\mathrm{conv}}$.
We call the term
$\operatorname{I_{\diamond}}^{\mathrm{hol}}(\mathcal{M})\coloneqq\operatorname{I_{\diamond}}(\mathcal{M}^{\mathrm{pair}})$
the _hollow incompatibility_ , which implicitly depends on $\mathcal{M}$, see
also Figure 1 and Ref. Quintino  _et al._ (2019).

We emphasize that the bound in Eq. (19) relies crucially on the distance
properties of the quantifier $\operatorname{I_{\diamond}}(\mathcal{M})$ and
_cannot_ be adapted directly to robustness or weight quantifiers Pusey (2015);
Designolle  _et al._ (2019a). In the SM Sup  (Section VII) we show that the
decomposition in Eq. (19) is tight for the three Pauli measurements, and give
numerical indication that this is generally the case for measurements based on
MUB.
Implications for steering and Bell nonlocality.—Due to the mathematical
structure of our methods, they can directly be applied to quantum steering and
Bell nonlocality. Note that both of these phenomena occur in a scenario that
is similar to the one for measurement incompatibility. Namely, they depend on
the properties of a set of at least two measurements, while a single
measurement by itself does not contain any resource. This distinguishes the
above concepts from resource theories of single POVMs (see, e.g. Oszmaniec
_et al._ (2017); Skrzypczyk and Linden (2019); Oszmaniec and Biswas (2019))
where the resource gain can trivially be determined by considering averages of
single POVM resources Tendick  _et al._ (2023). We describe our results
regarding steering and nonlocality in more detail in the SM Sup  (Section IV).
The analysis of the gain in nonlocal correlations in Bell experiments is
particularly interesting as it seems fundamentally different from
incompatibility and steering. Consider a Bell experiment where Alice performs
$m_{A}=3$ and Bob $m_{B}=2$ measurements. Focusing on dichotomic measurements,
we observe the following intriguing effect: Alice cannot find three
measurements, such that the three Clauser-Horne-Shimony-Holt (CHSH)
inequalities Clauser  _et al._ (1969) $\mathrm{CHSH}_{(i,j)}\coloneqq\langle
A_{i}\otimes B_{1}\rangle+\langle A_{i}\otimes B_{2}\rangle+\langle
A_{j}\otimes B_{1}\rangle-\langle A_{j}\otimes B_{2}\rangle\leq 2$ with
$(i,j)\in\\{(1,2),(1,3),(2,3)\\}$ are simultaneously maximally violated. That
means,
$\mathrm{CHSH}_{(1,2,3)}\coloneqq\tfrac{1}{3}(\mathrm{CHSH}_{(1,2)}+\mathrm{CHSH}_{(1,3)}+\mathrm{CHSH}_{(2,3)})\leq\tfrac{4\sqrt{2}+2}{3}<2\sqrt{2}$
holds in quantum theory. This implies, that the average two-subset nonlocality
is lower than the maximal obtainable nonlocality with two measurements on
Alice’s side.
Conclusion and outlook.—In this work, we analyzed how much incompatibility can
maximally be gained by adding measurements to an existing measurement scheme.
We showed that this gain is upper bounded by the incompatibility of the
underlying parent POVMs that approximate subsets of measurements. We proved
the relevance of our bounds analytically by showing that they are tight for
specific measurements based on MUB. Moreover, we showed that our methods are
directly applicable to quantum steering and Bell nonlocality. For nonlocality
specifically, we discovered a promising path to understand better why using
more than two measurements may not provide any advantage for maximal nonlocal
correlations Araújo  _et al._ (2020); Brito  _et al._ (2018). Our results
reveal the polygamous nature of distributed quantum incompatibility, in stark
contrast to the monogamy of entanglement Coffman  _et al._ (2000) and
coherence Radhakrishnan  _et al._ (2016) across subsystems of multipartite
quantum states. While we focused in this text on $m=3$ measurements, all our
findings, in particular, Results 1-3 can be generalized to an arbitrary number
of measurements $m$ (see Sup , Section V).
Our work provides a foundation for several new directions of research. While
we focused on a particular distance-based quantifier here, the alternative
distance-based quantifiers proposed in Tendick  _et al._ (2023) do also
possess the necessary properties to be used in a similar way. It would be
interesting to see whether resource quantifiers such as the incompatibility
robustness Designolle  _et al._ (2019a) or weight Pusey (2015) can also be
used to analyze how the incompatibility of an assemblage depends on its
subsets. Our methods might also prove helpful to find better bounds on the
incompatibility of general assemblages and particularly maximally assemblages.
Finally, it would be interesting to analyze the performance gain of specific
cryptography Bruß (1998); Bennett and Brassard (2014) or estimation protocols
McNulty  _et al._ (2022) with different numbers of measurements.

###### Acknowledgements.

We thank Thomas Cope, Federico Grasselli, Martin Kliesch, Nikolai Miklin,
Martin Plávala, Isadora Veeren, Thomas Wagner, and Zhen-Peng Xu for helpful
discussions. This research was partially supported by the EU H2020 QuantERA
ERA-NET Cofund in Quantum Technologies project QuICHE, and by the Federal
Ministry of Education and Research (BMBF) within the funding program
“Forschung Agil - Innovative Verfahren für Quantenkommunikationsnetze“ in the
joint project QuKuK (grant number 16KIS1619).

AGF

    average gate fidelity
AMA

    associated measurement assemblage
BOG

    binned outcome generation
CGLMP

    Collins-Gisin-Linden-Massar-Popescu
CHSH

    Clauser-Horne-Shimony-Holt
CP

    completely positive
CPT

    completely positive and trace preserving
CPTP

    completely positive and trace preserving
CS

    compressed sensing
DFE

    direct fidelity estimation
DM

    dark matter
GST

    gate set tomography
GUE

    Gaussian unitary ensemble
HOG

    heavy outcome generation
JM

    jointly measurable
LHS

    local hidden-state model
LHV

    local hidden-variable model
LOCC

    local operations and classical communication
MBL

    many-body localization
ML

    machine learning
MLE

    maximum likelihood estimation
MPO

    matrix product operator
MPS

    matrix product state
MUB

    mutually unbiased bases
MW

    micro wave
NISQ

    noisy and intermediate scale quantum
POVM

    positive operator valued measure
PVM

    projector-valued measure
QAOA

    quantum approximate optimization algorithm
QML

    quantum machine learning
QMT

    measurement tomography
QPT

    quantum process tomography
QRT

    quantum resource theory
RDM

    reduced density matrix
SDP

    semidefinite program
SFE

    shadow fidelity estimation
SIC

    symmetric, informationally complete
SM

    Supplemental Material
SPAM

    state preparation and measurement
RB

    randomized benchmarking
rf

    radio frequency
TT

    tensor train
TV

    total variation
UI

    uninformative
VQA

    variational quantum algorithm
VQE

    variational quantum eigensolver
WMA

    weighted measurement assemblage
XEB

    cross-entropy benchmarking

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## Supplemental Material for "Distributed quantum incompatibility"

In this Supplemental Material, we give detailed background information on
measurement incompatibility, provide proofs for the results and statements in
the main text, and discuss how to apply our results to steering and
nonlocality. Furthermore, we show how to generalize our results to general
sets of $m$ measurements and weighted measurement assemblages
$\mathcal{M}^{\mathbf{p}}=(\mathcal{M},\mathbf{p})$ with general probability
distributions $\mathbf{p}$.

##  I Background information on incompatibility

Here, we give detailed background information on the important properties of
the diamond distance quantifier
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ defined in Eq. ($2$)
in the main text. To provide a relatively self contained overview in this
Supplemental Material, we also repeat the relevant definitions from the main
text. An assemblage
$\mathcal{M}_{(1,2,\cdots,m)}=(\mathcal{M}_{1},\mathcal{M}_{2},\cdots,\mathcal{M}_{m})$
of $m$ measurements with outcomes $a$ and settings $x$ is called _jointly
measurable_ if it can be simulated by a single _parent POVM_
$\\{G_{\lambda}\\}$ and conditional probabilities $p(a|x,\lambda)$ such that

$\displaystyle M_{a|x}=\sum_{\lambda}p(a|x,\lambda)G_{\lambda}\ \forall\ a,x,$
(20)

and it is called _incompatible_ otherwise. Note that the probabilities
$p(a|x,\lambda)$ can always be identified with deterministic response
functions $v(a|x,\lambda)$ since the randomness in $p(a|x,\lambda)$ can be
shifted to the parent POVM by appropriately redefining the $G_{\lambda}$. Let
us denote by $\mathrm{JM}$ the set of all jointly measurable assemblages. For
more than two measurements, there exist different sub-structures of
incompatibility. Focusing on the case of three measurements, we define the
sets $\mathrm{JM}^{(s,t)}$ with $s,t\in\\{1,2,3\\}$ such that $s\neq t$ as the
sets containing assemblages in which the measurement $s$ and $t$ are jointly
measurable. Their intersection
$\mathrm{JM}^{\mathrm{pair}}\coloneqq\mathrm{JM}^{(1,2)}\cap\mathrm{JM}^{(1,3)}\cap\mathrm{JM}^{(2,3)}$
contains all assemblages in which any pair of two measurements are compatible,
the so-called _pairwise_ compatible assemblages. On the other hand, the set
$\mathrm{JM}^{\mathrm{conv}}\coloneqq\mathrm{Conv}(\mathrm{JM}^{(1,2)},\mathrm{JM}^{(1,3)},\mathrm{JM}^{(2,3)})$
describes the convex hull of the sets $\mathrm{JM}^{(1,2)}$,
$\mathrm{JM}^{(1,3)}$, and $\mathrm{JM}^{(2,3)}$, i.e., it contains all
assemblage that can be written as a convex combination of assemblages where
one pair of measurements is compatible. More formally, it contains all
assemblages of the form

$\displaystyle\mathcal{M}_{(1,2,3)}=p_{(1,2)}\mathcal{J}^{(1,2)}_{(1,2,3)}+p_{(1,3)}\mathcal{J}^{(1,3)}_{(1,2,3)}+p_{(2,3)}\mathcal{J}^{(2,3)}_{(1,2,3)},$
(21)

where $\mathcal{J}^{(s,t)}_{(1,2,3)}\in\mathrm{JM}^{(s,t)}$ and the convex
combination is to be understood on the level of the individual POVM effects.
Finally an assemblage $\mathcal{M}_{(1,2,3)}\notin\mathrm{JM}^{\mathrm{conv}}$
is said to be _genuinely triplewise incompatible_. Note, these notions can
straightforwardly be generalized to more than three measurements. See also
Quintino  _et al._ (2019) and for a graphical representation Figure 1 in the
main text.
To quantify the incompatibility as a resource, we use the _diamond distance
quantifier_ Tendick  _et al._ (2023) given by

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})=\min\limits_{\mathcal{F}\in\mathrm{JM}}\sum_{x}p(x)\operatorname{D_{\diamond}}(\Lambda_{\mathcal{M}_{x}},\Lambda_{\mathcal{F}_{x}}),$
(22)

where
$\Lambda_{\mathcal{M}_{x}}=\sum_{a}\mathrm{Tr}[M_{a|x}\rho]|a\rangle\langle
a|$ is the _measure-and-prepare channel_ associated to the measurement
$\mathcal{M}_{x},$ and
$\operatorname{D_{\diamond}}(\Lambda_{1},\Lambda_{2})=\max\limits_{\rho\in\mathcal{S}(\mathcal{H}\otimes\mathcal{H})}\dfrac{1}{2}\lVert((\Lambda_{1}-\Lambda_{2})\otimes\mathds{1}_{d})\rho\rVert_{1}$
is the _diamond distance_ Kitaev  _et al._ (2002) between two channels
$\Lambda_{1}$, and $\Lambda_{2}$, with the trace norm$\lVert
X\rVert_{1}=\operatorname{Tr}[\sqrt{X^{\dagger}X}]$. Technically speaking,
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ quantifies the
incompatibility of a _weighted assemblage_
$\mathcal{M}^{\mathbf{p}}=(\mathcal{M},\mathbf{p})$ which contains the
information about the probabilities $p(x)$ with which the measurement $x$ is
performed. The distance between two assemblages $\mathcal{M}^{\mathbf{p}}$ and
$\mathcal{N}^{\mathbf{p}}$ that induces the quantifier
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ is given by

$\displaystyle\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}})\coloneqq\sum_{x}p(x)\operatorname{D_{\diamond}}(\Lambda_{\mathcal{M}_{x}},\Lambda_{\mathcal{N}_{x}}).$
(23)

Like in the main text, we will simply write
$\operatorname{I_{\diamond}}(\mathcal{M})$ to imply the case where
$p(x)=\tfrac{1}{m}\forall x$. We denote by
$\mathcal{M}^{\\#}_{(1,2,\cdots,m)}$ the closest jointly measurable assemblage
to $\mathcal{M}_{(1,2,\dots,m)}$, i.e., the _arg-min_ on the RHS in Eq. (22).
Therefore, $\mathcal{M}^{\\#}_{(1,2,\cdots,m)}$ can be seen as the closest
jointly measurable approximation of the assemblage
$\mathcal{M}_{(1,2,\dots,m)}$. If we only approximate a subset of $n<m$
measurements of $\mathcal{M}_{(1,2,\dots,m)}$ by jointly measurable
measurements, for instance the first $n$ settings, while keeping the remaining
measurements unchanged, we write
$\mathcal{M}^{\\#(1,2,\dots,n)}_{(1,2,\cdots,m)}$. Adding measurements
$\mathcal{M}^{\prime}_{(m+1,m+2,\dots,m+n)}=(\mathcal{M}^{\prime}_{m+1},\mathcal{M}^{\prime}_{m+2},\cdots,\mathcal{M}^{\prime}_{m+n})$
to the assemblage
$\mathcal{M}_{(1,2,\cdots,m)}=(\mathcal{M}_{1},\mathcal{M}_{2},\cdots,\mathcal{M}_{m})$
is mathematically described by the concatenation of ordered list, using the
symbol $\mathbin{+\mkern-10.0mu+}$, i.e., we write

$\displaystyle\mathcal{M}_{(1,2,\cdots,n+m)}=\mathcal{M}_{(1,2,\cdots,m)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\prime}_{(m+1,m+2,\cdots,m+n)}=(\mathcal{M}_{1},\mathcal{M}_{2},\cdots,\mathcal{M}_{m},\mathcal{M}^{\prime}_{m+1},\cdots,\mathcal{M}^{\prime}_{m+n}).$
(24)

Using the notion of concatenation of ordered lists, we formally define

$\displaystyle\mathcal{M}^{\\#(1,2,\dots,n)}_{(1,2,\cdots,m)}\coloneqq\mathcal{M}^{\\#}_{(1,2,\cdots,n)}\mathbin{+\mkern-10.0mu+}\mathcal{M}_{n+1}\mathbin{+\mkern-10.0mu+}\cdots\mathbin{+\mkern-10.0mu+}\mathcal{M}_{m}.$
(25)

The diamond distance quantifier in Eq. (22) is a _faithful_ resource
quantifier, i.e., it holds that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})=0\iff\mathcal{M}=\mathcal{M}^{\\#}\in\mathrm{JM}.$
(26)

For the above statement to be true, we assume that $p(x)\neq 0\ \forall x$,
which is no restriction, since measurements that are never performed can be
excluded from the assemblage before calculating the incompatibility.
Furthermore, $\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ is a
monotone under any unital quantum channel $\Lambda^{\dagger}$ (these are
exactly those channels that map POVMs to POVMs), i.e.,

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})\geq\operatorname{I_{\diamond}}(\Lambda^{\dagger}(\mathcal{M})^{\mathbf{p}}),$
(27)

which follows from the fact that the trace distance is contractive under the
application of completely positive and trace preserving (CPTP) maps. Note that
in the resource theory of incompatibility, all unital quantum channels
$\Lambda^{\dagger}$ are free. Indeed, it is straight forward to see that
$\\{\Lambda^{\dagger}(G_{\lambda})\\}$ is a parent POVM for the assemblage
$\Lambda^{\dagger}(\mathcal{M})$ whenever $\\{G_{\lambda}\\}$ is a parent POVM
for $\mathcal{M}$. That is, it holds

$\displaystyle\Lambda^{\dagger}(M_{a|x})=\sum_{\lambda}p(a|x,\lambda)\Lambda^{\dagger}(G_{\lambda}).$
(28)

Additionally, $\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ is non-
increasing under classical simulations $\mathcal{M}^{\prime}=\xi(\mathcal{M})$
with

$\displaystyle M^{\prime}_{b|y}=\sum_{x}p(x|y)\sum_{a}q(b|y,x,a)M_{a|x}\
\forall\ b,y,$ (29)

where $\mathcal{M}$ can be used to simulate Guerini  _et al._ (2017) the
assemblage $\mathcal{M}^{\prime}$ via the conditional probabilities $p(x|y)$
and $q(b|y,x,a)$ for all $y$, respectively for all $y,x,a$. Using the
classical simulations, one also obtains the possible probabilities $q(y)$ to
perform setting $y$ via $p(x)=\sum_{y}q(y)p(x|y)$. That means, it holds
Tendick  _et al._ (2023):

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})\geq\operatorname{I_{\diamond}}(\xi(\mathcal{M}^{\mathbf{p}})^{\mathbf{q}}),$
(30)

for all measurement simulations $\xi$. Eq. (30) follows ultimately from the
fact that $\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ is based on
a norm and that it is written as a convex combination over the settings.
Finally, since $\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ is
based on the diamond distance
$\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}})\coloneqq\sum_{x}p(x)\operatorname{D_{\diamond}}(\Lambda_{\mathcal{M}_{x}},\Lambda_{\mathcal{N}_{x}})$
between two weighted assemblages and the set $\mathrm{JM}$ of jointly
measurable assemblage is convex, it is a convex function. Even more the
distance
$\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}})$
fulfills the triangle inequality, i.e.,

$\displaystyle\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}})\leq\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{L}^{\mathbf{p}})+\operatorname{D_{\diamond}}(\mathcal{L}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}}),$
(31)

for any weighted measurement assemblages
$\mathcal{M}^{\mathbf{p}},\mathcal{L}^{\mathbf{p}},$ and
$\mathcal{N}^{\mathbf{p}}$. It therefore follows that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})\leq\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\\#,\mathbf{p}})\leq\operatorname{D_{\diamond}}(\mathcal{M}^{\mathbf{p}},\mathcal{N}^{\mathbf{p}})+\operatorname{I_{\diamond}}(\mathcal{N}^{\mathbf{p}}),$
(32)

for any assemblages $\mathcal{M}$ and $\mathcal{N}$, where
$\mathcal{N}^{\\#}\in\mathrm{JM}$ is the closest jointly measurable assemblage
with respect to $\mathcal{N}$. Note that the first inequality follows from the
fact that $\mathcal{N}^{\\#}$ is jointly measurable but not necessarily the
closest jointly measurable assemblage to $\mathcal{M}$, i.e.,
$\mathcal{N}^{\\#}\neq\mathcal{M}^{\\#}$.
To prove the tightness of our bounds in the main text, we rely on the SDP
formulation of $\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$, which
besides its numerical uses allows us, in some instances, to determine the
incompatibility of an assemblage analytically. In Tendick  _et al._ (2023) it
was shown that that $\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ is
equivalent to the optimal value of the SDP:

$\displaystyle\underline{\text{Primal problem (incompatibility):}}$ (33)
$\displaystyle\mathrm{given:}\ \mathcal{M}^{\mathbf{p}}$
$\displaystyle\underset{a_{x},Z_{x},G_{\lambda}}{\mathrm{minimize}}\sum_{x}p(x)a_{x}$
subject to: $\displaystyle a_{x}\mathds{1}-\mathrm{Tr}_{1}[Z_{x}]\geq 0\
\forall\ a,x,$ $\displaystyle Z_{x}\geq\sum_{a}|a\rangle\langle
a|\otimes(M_{a|x}-F_{a|x})^{T}\ \forall\ x,$ $\displaystyle
F_{a|x}=\sum_{\lambda}v(a|x,\lambda)G_{\lambda}\ \forall\ x,a,\
G_{\lambda}\geq 0\ \forall\ \lambda,\sum_{\lambda}G_{\lambda}=\mathds{1},$
$\displaystyle Z_{x}\geq 0,\ a_{x}\geq 0\ \forall\ x,$

where the $a_{x}$ are non-negative coefficients, the $Z_{x}$ are positive
semidefinite matrices and the $G_{\lambda}$ are the POVM effects of the parent
POVM. SDPs represent a special instance of convex optimization for which there
exist off-the-shelf software Grant and Boyd (2014, 2008); Toh  _et al._
(1999); ApS (2019) to efficiently solve them. Importantly, every SDP comes
with a _dual formulation_ that yields the same optimal value under some mild
assumptions (see e.g., Boyd and Vandenberghe (2004)). This is indeed the case
here Tendick  _et al._ (2023), i.e.,
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$ can also be understood
as the optimal value of the SDP:

$\displaystyle\underline{\text{Dual problem (incompatibility):}}$ (34)
$\displaystyle\mathrm{given}:\ \mathcal{M}^{\mathbf{p}}$
$\displaystyle\underset{C_{a|x},\rho_{x},L}{\mathrm{maximize}}\ \ \
\sum_{a,x}p(x)\mathrm{Tr}[M_{a|x}C_{a|x}]-\mathrm{Tr}[L]$ subject to:
$\displaystyle L\geq\sum_{a,x}p(x)v(a|x,\lambda)C_{a|x}\ \forall\ \lambda,$
$\displaystyle 0\leq C_{a|x}\leq\rho_{x}\ \forall\ a,x,\ \rho_{x}\geq
0,\mathrm{Tr}[\rho_{x}]=1\ \forall\ x,$

where the $C_{a|x}$, $\rho_{x}$, and $L$ are positive semidefinite matricies.
Since the primal problem in Eq. (33) corresponds to a minimization, every
feasible point (i.e., any set of variables that fulfills all constraints)
leads to an upper bound on
$\operatorname{I_{\diamond}}(\mathcal{M}^{\mathbf{p}})$. Similarly, every
feasible solution of the dual in Eq. (34) leads to a lower bound.

##  II Measurement splitting

In the main text, we argue that it is equivalent to consider the
incompatibility of the assemblage $\mathcal{M}_{(1,2,1,3,2,3)}$ instead of
$\mathcal{M}_{(1,2,3)}$, i.e., we use that
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})$
in order to derive the bound on the incompatibility gain in Eq. ($10$) in the
main text. Note that $\mathcal{M}_{(1,2,1,3,2,3)}$ is an assemblage in which
each of the measurements $\mathcal{M}_{1},\mathcal{M}_{2}$, and
$\mathcal{M}_{3}$ occurs twice with probability $\tfrac{1}{6}$ each. On the
other hand in $\mathcal{M}_{(1,2,3)}$ each of the measurements is used with a
probability of $\tfrac{1}{3}$. To show the equivalence
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})$,
we actually show that
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}^{\prime}_{(1,1,2,2,3,3)})$
and finally use that the set $\mathrm{JM}$ of jointly measurable measurements
is closed under relabeling. We first show that
$\mathcal{M}^{\prime}_{(1,1,2,2,3,3)}=\xi(\mathcal{M}_{(1,2,3)})$ for a
measurement simulation (see also Eq. (29)) of the form

$\displaystyle M^{\prime}_{b|y}=\sum_{x}p(x|y)\sum_{a}q(b|y,x,a)M_{a|x}\
\forall\ b,y,$ (35)

where we set $q(b|y,x,a)=\delta_{ba}$ for all $b,y,x,a$ with $\delta_{ba}$
being the Kronecker delta. Furthermore, we use mixing probabilities $p(x|y)$
such that $p(x=1|y=1)=p(x=1|y=2)=1$, $p(x=2|y=3)=p(x=2|y=4)=1$, and
$p(x=3|y=5)=p(x=3|y=6)=1$ with all other probabilities set to zero. This is
clearly a valid measurement simulation of
$\mathcal{M}^{\prime}_{(1,1,2,2,3,3)}$ using the measurements
$\mathcal{M}_{(1,2,3)}$. Finally, notice that due to
$p(x)=\sum_{y}q(y)p(x|y)$, it holds

$\displaystyle\dfrac{1}{3}=p(x=i)=q(y=2i-1)+q(y=2i),\ \text{for}\ i=1,2,3,$
(36)

which is clearly fulfilled for $q(y)=\tfrac{1}{6}\ \forall y$. The above
equation actually shows a more general statement, i.e., any probabilities
$p(y=1)+p(y=2)$ that sum to $\tfrac{1}{3}$ are allowed. This means, it is not
necessary to split a POVM into two equally likely POVMs, but one can introduce
an additional bias. This bias will not change the results in a qualitative
way, however, it can be used to fine-tune coefficients (multiplicative
prefactors) such as changing the weights of the average in Eq. $(8)$ in the
main text. The same holds for the other instances.
To show the other direction, i.e.,
$\mathcal{M}_{(1,2,3)}=\xi(\mathcal{M}^{\prime}_{(1,1,2,2,3,3)})$ we use again
$q(b|y,x,a)=\delta_{ba}$ for all $b,y,x,a$. For the mixing probabilities, we
set $p(x=1|y=1)=p(x=2|y=1)=\tfrac{1}{2}$,
$p(x=3|y=2)=p(x=4|y=2)=\tfrac{1}{2}$, and $p(x=5|y=2)=p(x=6|y=3)=\tfrac{1}{2}$
with all other probabilities set to zero. Again, it straightforward to check
that this a valid measurement simulation. From the equivalence

$\displaystyle p(x=1)=\dfrac{1}{6}=\sum_{y}q(y)p(1|y)=q(y=1)\dfrac{1}{2},$
(37)

it follows directly that $q(y=1)=\tfrac{1}{3}$ and similarly for the other
cases. Now, since
$\mathcal{M}^{\prime}_{(1,1,2,2,3,3)}=\xi(\mathcal{M}_{(1,2,3)})$ and
$\mathcal{M}_{(1,2,3)}=\xi(\mathcal{M}^{\prime}_{(1,1,2,2,3,3)})$, it holds
that
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}^{\prime}_{(1,1,2,2,3,3)})$.
Analogously follows the measurement splitting with more measurements.
Let us note here, that measurement simulations can also be used to show that
the incompatibility of the parent POVMs of different subsets of jointly
measurable assemblages is an upper bound on the incompatibility of these
assemblages. More formally, let $\mathcal{M}_{(1,2,3)}$ be an assemblage and
let
$\mathcal{N}=\mathcal{M}^{\\#}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{(1,3)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{(2,3)}$
be the assemblage that contains the closest jointly measurable assemblages for
the three subsets. Furthermore, let
$\mathcal{G}=G(\mathcal{M}^{\\#}_{(1,2)})\mathbin{+\mkern-10.0mu+}G(\mathcal{M}^{\\#}_{(1,3)})\mathbin{+\mkern-10.0mu+}G(\mathcal{M}^{\\#}_{(2,3)})$
be the assemblage that contains the parent POVMs of the respective subsets.
With the above methods (and by the definition of the parent POVM in Eq. (20))
it can be seen that there exists a measurement simulation $\xi$ such that
$\xi(\mathcal{G})=\mathcal{N}$, which directly implies that
$\operatorname{I_{\diamond}}(\mathcal{N})\leq\operatorname{I_{\diamond}}(\mathcal{G})$
holds.

##  III Lower bounds

Here, we prove the lower bounds stated in Eq. ($13$) and Eq. ($16$) in the
main text. Remember, we consider the case in which $p(x)=\tfrac{1}{3}$, i.e.,
the input probabilities are uniformly distributed. Let us start by showing
that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})\geq\dfrac{1}{3}[\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3)})],$
(38)

holds. We start by using that
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})$.
Now, the closest jointly measurable assemblage
$\mathcal{M}^{\\#}_{(1,2,1,3,2,3)}$ with respect to
$\mathcal{M}_{(1,2,1,3,2,3)}$ allows us to rewrite
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})$ such that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})=\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)},\mathcal{M}^{\\#}_{(1,2,1,3,2,3)}).$
(39)

Now, concerning the measurement pairs $(1,2),(1,3),$ and $(2,3)$ the subsets
of $\mathcal{M}^{\\#}_{(1,2,1,3,2,3)}$ are jointly measurable by definition
but not necessarily optimal for the respective subsets of
$\mathcal{M}_{(1,2,1,3,2,3)}$. Using that the distance
$\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)},\mathcal{M}^{\\#}_{(1,2,1,3,2,3)})$
is a convex combination over the individual settings, it follows that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,1,3,2,3)})\geq\dfrac{1}{3}[\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3)})].$
(40)

To show the second lower bound, i.e.,

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})\geq\tfrac{2}{3}\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)}),$
(41)

it is enough to notice that leaving out the contribution of the setting $x=3$
can only lead to lower values than
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$. Finally, we use again
that the remaining measurements (for the settings $x=1,2$) from the closest
jointly measurable assemblage $\mathcal{M}^{\\#}_{(1,2,3)}$ do not need to be
optimal.

##  IV Steering and nonlocality

Here, we show that our methods can directly be applied to quantum steering and
Bell nonlocality. We start by considering steering. Let
$\vec{\sigma}_{(1,2,\cdots,m)}=(\sigma_{1},\sigma_{2},\cdots,\sigma_{m})$ with
$\sigma_{x}=\\{\sigma_{a|x}\\}_{a}$ be the _steering assemblage_ that Alice
prepares for Bob by performing the measurements from a measurement assemblage
$\mathcal{M}_{(1,2,\cdots,m)}$ on a shared state $\rho$ such that
$\sigma_{a|x}=\mathrm{Tr}_{A}[(M_{a|x}\otimes\mathds{1})\rho]$. The
_consistent steering distance_ Ku  _et al._ (2018) given by

$\displaystyle\mathrm{S}(\vec{\sigma})=\min\limits_{\vec{\tau}\in\mathrm{CLHS}}\dfrac{1}{2}\sum\limits_{a,x}\dfrac{1}{m}\lVert\sigma_{a|x}-\tau_{a|x}\rVert_{1},$
(42)

can be used to quantify the steerability of any steering assemblage
$\mathrm{S}(\vec{\sigma})$. Here, $\vec{\tau}\in\mathrm{CLHS}$ denotes an
assemblage that admits a local hidden-state model (LHS) and fulfills the
consistency condition
$\sum\limits_{a}\tau_{a|x}=\sum\limits_{a}\sigma_{a|x}=\rho_{B}=\mathrm{Tr}_{A}[\rho]\
\forall x.$ A LHS for $\vec{\tau}$ is given by

$\displaystyle\tau_{a|x}=\sum_{\lambda}p(a|x,\lambda)\sigma_{\lambda},$ (43)

where the $\sigma_{\lambda}$ are sub-normalized states and the
$p(a|x,\lambda)$ resemble a classical post-processing, similarly to that in
Eq. (20) in the definition of jointly measurable assemblages. Note that we
directly used here that the choice of the settings is uniformly distributed,
i.e., $p(x)=\tfrac{1}{m}$. However, generally, we can use any distribution
with $p(x)\neq 0\ \forall x$, just like in the case for incompatibility. Note
further that our following arguments are independent, as it was also the case
for the incompatibility, of the number of outcomes $a$ in the steering
assemblage $\vec{\sigma}.$
Now, since $\mathrm{S}(\vec{\sigma})$ is based on a distance (the trace
distance) we can directly derive the steering analog to the incompatibility
bounds in the main text. In fact, our method relies only on the metric
properties of the respective quantifiers, the fact they are written as a
convex combination over the individual settings, and the general idea that a
measurement can be split in two separate copies of itself. We make the
following correspondence statements to our definitions for the incompatibility
case:

$\displaystyle\vec{\sigma}_{(1,2,\cdots,m)}\longleftrightarrow\mathcal{M}_{(1,2,\cdots,m)},$
(44a)
$\displaystyle\mathrm{S}(\vec{\sigma})\longleftrightarrow\operatorname{I_{\diamond}}(\mathcal{M}),$
(44b)
$\displaystyle\vec{\sigma}_{(1,2,\cdots,m)}^{\\#}\longleftrightarrow\mathcal{M}_{(1,2,\cdots,m)}^{\\#},$
(44c)
$\displaystyle\vec{\sigma}_{(1,2,\cdots,m)}^{\\#(1,2,\cdots,n)}\longleftrightarrow\mathcal{M}_{(1,2,\cdots,m)}^{\\#(1,2,\cdots,n)}.$
(44d)

That is, $\vec{\sigma}_{(1,2,\cdots,m)}^{\\#}$ is the closest assemblage in
the set $\mathrm{CLHS}$ to $\vec{\sigma}_{(1,2,\cdots,m)}$ with respect to the
distance

$\displaystyle\mathrm{D}_{A}(\vec{\sigma}_{(1,2,\cdots,m)},\vec{\sigma^{\prime}}_{(1,2,\cdots,m)})\coloneqq\sum_{a,x}\dfrac{1}{m}\lVert\sigma_{a|x}-\sigma^{\prime}_{a|x}\rVert_{1},$
(45)

which induces the steering distance in Eq. (42). Furthermore, it holds

$\displaystyle\vec{\sigma}_{(1,2,\cdots,m)}^{\\#(1,2,\cdots,n)}\coloneqq\vec{\sigma}_{(1,2,\cdots,n)}^{\\#}\mathbin{+\mkern-10.0mu+}\sigma_{n+1}\mathbin{+\mkern-10.0mu+}\cdots\mathbin{+\mkern-10.0mu+}\sigma_{m}.$
(46)

This implies, it holds that

$\displaystyle\mathrm{S}(\vec{\sigma}_{(1,2,3)})\leq\dfrac{1}{3}[\mathrm{S}(\vec{\sigma}_{(1,2)})+\mathrm{S}(\vec{\sigma}_{(1,3)})+\mathrm{S}(\vec{\sigma}_{(2,3)})]+\mathrm{S}(\vec{\tau}),$
(47)

where
$\vec{\tau}=\vec{\sigma}^{\\#}_{(1,2)}\mathbin{+\mkern-10.0mu+}\vec{\sigma}^{\\#}_{(1,3)}\mathbin{+\mkern-10.0mu+}\vec{\sigma}^{\\#}_{(2,3)}$
is a state assemblage (with $m=6$ settings) that contains itself three
assemblages (of two settings each) that are the closest consistent unsteerable
assemblages to the respective subsets. Note that $\vec{\tau}$ can be steerable
in general. Note further that it is crucial to use a _consistent_ steering
quantifier here, in order to avoid _signaling_ in the assemblage $\vec{\tau}$.
All the other bounds follow from here on directly. That is, it follows that

$\displaystyle\mathrm{S}(\vec{\sigma}_{(1,2,3)})$
$\displaystyle\geq\dfrac{1}{3}[\mathrm{S}(\vec{\sigma}_{(1,2)})+\mathrm{S}(\vec{\sigma}_{(1,3)})+\mathrm{S}(\vec{\sigma}_{(2,3)})],$
(48) $\displaystyle\mathrm{S}(\vec{\sigma}_{(1,2,3)})$
$\displaystyle\geq\dfrac{2}{3}\mathrm{S}(\vec{\sigma}_{(1,2)}).$

Moreover, using the assemblage $\vec{\sigma}^{\\#(1,2)}_{(1,2,3)}$ it holds
that

$\displaystyle\mathrm{S}(\vec{\sigma}_{(1,2,3)})\leq\dfrac{2}{3}\mathrm{S}(\vec{\sigma}_{(1,2)})+\mathrm{S}(\vec{\sigma}^{\\#(1,2)}_{(1,2,3)}).$
(49)

For nonlocality, very similar arguments can be made. However, we will see that
additional constraints arise that distinguish nonlocality from steering and
incompatibility. Let $\mathbf{q}=\\{q(ab|xy)\\}$ be a general probability
distribution between two distant parties Alice and Bob. We consider the case
where both, Alice and Bob, have two different measurement settings already
available and Alice upgrades her measurement scheme with an additional third
measurement. We denote the resulting distribution by $\mathbf{q}_{(1,2,3)}$.
The nonlocality of a general distribution $\mathbf{q}$ can be quantified via
the _consistent_ version of the classical trace distance quantifier introduced
in Brito  _et al._ (2018), which is given by

$\displaystyle\mathrm{N}(\mathbf{q})=\dfrac{1}{2}\min\limits_{\mathbf{t}\in\mathrm{CLHV}}\sum\limits_{a,b,x,y}\dfrac{1}{m_{A}m_{B}}\lvert
q(a,b|x,y)-t(a,b|x,y)\rvert.$ (50)

Here, we denote by $\mathrm{CLHV}$ the set of consistent local hidden-variable
models, i.e., the set of those local distributions $\mathbf{t}\in\mathrm{LHV}$
that fulfill $\sum_{a}t(a,b|x,y)=t(b|y)=q(b|y)=\sum_{a}q(a,b|x,y)\ \forall\
b,y,x$ and similarly $\sum_{b}t(a,b|x,y)=t(a|x)=q(a|x)=\sum_{b}q(a,b|x,y)\
\forall\ a,y,x$. The (Bell) locality condition is expressed in terms of the
LHV:

$\displaystyle
t(a,b|x,y)=\sum\limits_{\lambda}\pi(\lambda)p_{A}(a|x,\lambda)p_{B}(b|y,\lambda)\
\forall a,b,x,y,$ (51)

for the distribution $\mathbf{t}$. Finally, we denote by $m_{A}$ the number of
measurement settings of Alice and by $m_{B}$ those of Bob, which we set to
$m_{B}=2$ here. Once again, we restrict our discussion to the case where the
input probabilities $p(x,y)=p(x)p(y)=\tfrac{1}{m_{A}}\tfrac{1}{m_{B}}$ are
uniformly distributed.
Since $\mathrm{N}(\mathbf{q})$ relies on a distance that is written as a
convex combination over the individual settings, we can use the triangle
inequality together with the measurement splitting method. We make the
following correspondence statements to our definitions in the incompatibility
case:

$\displaystyle\mathbf{q}_{(1,2,\cdots,m_{A})}\longleftrightarrow\mathcal{M}_{(1,2,\cdots,m)},$
(52a)
$\displaystyle\mathrm{N}(\mathbf{q})\longleftrightarrow\operatorname{I_{\diamond}}(\mathcal{M}),$
(52b)
$\displaystyle\mathbf{q}^{\\#}_{(1,2,\cdots,m_{A})}\longleftrightarrow\mathcal{M}_{(1,2,\cdots,m)}^{\\#},$
(52c)
$\displaystyle\mathbf{q}_{(1,2,\cdots,m_{A})}^{\\#(1,2,\cdots,n_{A})}\longleftrightarrow\mathcal{M}_{(1,2,\cdots,m)}^{\\#(1,2,\cdots,n)}.$
(52d)

That is, $\mathbf{q}^{\\#}_{(1,2,\cdots,m_{A})}$ is the closest consistent and
local distribution to $\mathbf{q}_{(1,2,\cdots,m_{A})}$ with respect the the
classical trace distance ($\ell_{1}$ distance) that induces the nonlocality
distance in Eq. (50). Furthermore,
$\mathbf{q}_{(1,2,\cdots,m_{A})}^{\\#(1,2,\cdots,n_{A})}=\mathbf{q}_{(1,2,\cdots,n_{A})}^{\\#}\mathbin{+\mkern-10.0mu+}\mathbf{q}_{n_{A}+1}\mathbin{+\mkern-10.0mu+}\cdots\mathbin{+\mkern-10.0mu+}\mathbf{q}_{m_{A}}$,
where we treat the probability vector that describes a distribution
$\mathbf{q}_{(1,2,\cdots,m_{A})}$ as ordered list. We would like to emphasize
that the indices $(1,2,\cdots,m_{A})$ refer to the measurements of Alice, and
Bob’s number of measurements remains fixed here.
These correspondence relations imply that it is possible to obtain the bounds

$\displaystyle\dfrac{2}{3}\mathrm{N}(\mathbf{q}_{(1,2)})\leq\mathrm{N}(\mathbf{q}_{(1,2,3)})\leq\dfrac{2}{3}\mathrm{N}(\mathbf{q}_{(1,2)})+\mathrm{N}(\mathbf{q}^{\\#(1,2)}_{(1,2,3)}),$
(53)

Furthermore, we obtain the bounds

$\displaystyle\dfrac{1}{3}[\mathrm{N}(\mathbf{q}_{(1,2)})+\mathrm{N}(\mathbf{q}_{(1,3)})+\mathrm{N}(\mathbf{q}_{(2,3)})]\leq\mathrm{N}(\mathbf{q}_{(1,2,3)})\leq\dfrac{1}{3}[\mathrm{N}(\mathbf{q}_{(1,2)})+\mathrm{N}(\mathbf{q}_{(1,3)})+\mathrm{N}(\mathbf{q}_{(2,3)})]+\mathrm{N}(\mathbf{t}),$
(54)

where
$\mathbf{t}=\mathbf{q}^{\\#}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathbf{q}^{\\#}_{(1,3)}\mathbin{+\mkern-10.0mu+}\mathbf{q}^{\\#}_{(2,3)}$
is a distribution (with $m_{A}=6$ settings for Alice) which contains the
closest local distributions with respect to the corresponding two-measurement
subsets of Alice’s measurement settings.
Interestingly, the term
$\dfrac{1}{3}[\mathrm{N}(\mathbf{q}_{(1,2)})+\mathrm{N}(\mathbf{q}_{(1,3)})+\mathrm{N}(\mathbf{q}_{(2,3)})]$
behaves differently from its steering and incompatibility counterpart. Namely,
it is limited by the fact that $\mathrm{N}(\mathbf{q}_{(1,2)})$,
$\mathrm{N}(\mathbf{q}_{(1,3)})$, and $\mathrm{N}(\mathbf{q}_{(2,3)})$ cannot,
in general, be maximal simultaneously. That is, contrary to incompatibility or
steering, where all of the subset resources can be maximal at the same time.
The reason for this is that there are not enough degrees of freedom for Alice
to violate a given Bell inequality with different measurements, given that Bob
keeps his settings fixed (besides the state that is also fixed). To exemplify
this, we consider the scenario where both parties have two outcomes for each
setting. In that case, the nonlocality of $\mathrm{N}(\mathbf{q}_{(1,2)})$,
$\mathrm{N}(\mathbf{q}_{(1,3)})$, and $\mathrm{N}(\mathbf{q}_{(2,3)})$ is
directly linked to the amount of violation of the CHSH inequality Clauser  _et
al._ (1969), as it was shown in Brito  _et al._ (2018). However, the CHSH
inequality requires very specific combinations measurements to get maximal
violation. Indeed, consider the three corresponding versions of the CHSH
inequality:

$\displaystyle\mathrm{CHSH}_{(1,2)}\coloneqq\langle A_{1}\otimes
B_{1}\rangle+\langle A_{1}\otimes B_{2}\rangle+\langle A_{2}\otimes
B_{1}\rangle-\langle A_{2}\otimes B_{2}\rangle\leq 2,$ (55)
$\displaystyle\mathrm{CHSH}_{(1,3)}\coloneqq\langle A_{1}\otimes
B_{1}\rangle+\langle A_{1}\otimes B_{2}\rangle+\langle A_{3}\otimes
B_{1}\rangle-\langle A_{3}\otimes B_{2}\rangle\leq 2,$
$\displaystyle\mathrm{CHSH}_{(2,3)}\coloneqq\langle A_{2}\otimes
B_{1}\rangle+\langle A_{2}\otimes B_{2}\rangle+\langle A_{3}\otimes
B_{1}\rangle-\langle A_{3}\otimes B_{2}\rangle\leq 2,$

and their average

$\displaystyle\mathrm{CHSH}_{(1,2,3)}\coloneqq\dfrac{1}{3}(\mathrm{CHSH}_{(1,2)}+\mathrm{CHSH}_{(1,3)}+\mathrm{CHSH}_{(2,3)})\leq
2.$ (56)

The inequality $\mathrm{CHSH}_{(1,2,3)}\leq 2$ can also be rewritten as

$\displaystyle\dfrac{2[\langle A_{1}\otimes B_{1}\rangle+\langle A_{1}\otimes
B_{2}\rangle+\langle A_{3}\otimes B_{1}\rangle-\langle A_{3}\otimes
B_{2}\rangle]+2\langle A_{2}\otimes B_{1}\rangle}{3}\leq 2,$ (57)

which directly implies that the Tsirelson bound Cirel'son (1980), i.e., the
quantum bound of $\mathrm{CHSH}_{(1,2,3)}$ is given by
$Q=\tfrac{4\sqrt{2}+2}{3}<2\sqrt{2}$, i.e., quantum mechanics cannot reach the
value $2\sqrt{2}$ that would correspond to all three contributions of Alice to
be maximal simultaneously. The same is true for no-signaling theories, where
the bound is given by $\mathrm{NS}=\tfrac{10}{3}$. Further, one needs to
consider all possible combinations of different versions of the CHSH
inequalities in Eq. (55). That is, one needs to consider all $8$ symmetries of
the CHSH inequality, corresponding to the $8$ CHSH facets of the local
polytope. However, going through all the combinations shows that there is no
combination which allows for a higher combined CHSH value than
$\mathrm{CHSH}_{(1,2,3)}$ in Eq. (56).
We want to emphasize again that such additional restrictions are not prevalent
for the incompatibility and steering quantifiers analog of Eq. (54), which
shows a clear separation of nonlocality to the other resources. Since the term
$\dfrac{1}{3}[\mathrm{N}(\mathbf{q}_{(1,2)})+\mathrm{N}(\mathbf{q}_{(1,3)})+\mathrm{N}(\mathbf{q}_{(2,3)})]$
is also used in upper bounding the nonlocality
$\mathrm{N}(\mathbf{q}_{(1,2,3)})$, this could be a promising path to
understanding why additional settings do not seem to increase the resource of
nonlocality Araújo  _et al._ (2020); Brito  _et al._ (2018), in strict
contrast to the resources of incompatibility and steerability. We expect that
the same is true for more than two outcomes, however, more research in this
direction is necessary.

##  V Generalizations

In this section, we will generalize our framework from the main text in two
directions. First, we discuss the scenario for more measurements i.e, $m>3.$
Then, we will discuss the case in which the assemblage
$\mathcal{M}^{\mathbf{p}}=(\mathcal{M},\mathbf{p})$ is weighted by a general
probability distribution $\mathbf{p}$, instead of a uniform one.
Using the methods from the main text and from Section II, general bounds can
be derived. We demonstrate this in the following for the assemblage
$\mathcal{M}_{(1,2,3,4)}$ of $m=4$ uniformly distributed measurements. Further
generalizations follow straightforwardly then. Let
$\mathcal{M}^{\\#(1,2,3)}_{(1,2,3,4)}$ be the closest assemblage with respect
to the first three measurements of $\mathcal{M}_{(1,2,3,4)}$. Using the
triangle inequality we get

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3,4)})\leq\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,3,4)},\mathcal{M}^{\\#(1,2,3)}_{(1,2,3,4)})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2,3)}_{(1,2,3,4)})=\dfrac{3}{4}\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2,3)}_{(1,2,3,4)}),$
(58)

as a direct generalization of Eq. ($15$) in the main text.
In general, let $C_{0}=\\{1,2,\cdots,m\\}$ be the set of of all possible
measurements from an assemblage $\mathcal{M}_{(1,2,\cdots,m)}$. Furthermore,
let $C\in C_{0}$ be any non-empty subset of $C_{0}$ with cardinality $\lvert
C\rvert$. It follows that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,\cdots,m)})\leq\dfrac{\lvert
C|}{m}\operatorname{I_{\diamond}}(\mathcal{M}_{C})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#C}_{(1,2,\cdots,m)}),$
(59)

where $\lvert C|$ is the number of measurements contained in the subset $C\in
C_{0}$. Since Eq. (59) holds for any subset $C$, we can conclude that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,\cdots,m)})\leq\min_{C\in
C_{0}}\Big{[}\dfrac{\lvert
C|}{m}\operatorname{I_{\diamond}}(\mathcal{M}_{C})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#C}_{(1,2,\cdots,m)})\Big{]},$
(60)

which in particular includes the optimization over all $n$ measurement
subsets. Note the upper bound trivially results in an equality in the case
that $\lvert C\rvert=1$, i.e., for practical purposes, one might exclude these
cases from the minimization.
However, we can generalize our framework even more. Denote by $\\{C_{i}\\}$ a
set of disjoint subsets of $C_{0}$ such that $\cup_{i}C_{i}=C_{0}$. It can
directly be concluded that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,\cdots,m)})\leq\sum_{i}\dfrac{\lvert
C_{i}\rvert}{m}\operatorname{I_{\diamond}}(\mathcal{M}_{C_{i}})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#}_{C_{1}}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{C_{2}}\mathbin{+\mkern-10.0mu+}\dots\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{C_{n}}),$
(61)

where
$\mathcal{M}^{\\#}_{C_{1}}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{C_{2}}\mathbin{+\mkern-10.0mu+}\dots\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{C_{n}}$
is an assemblage that contains itself the closest jointly measurable
assemblages for the $n$ respective subsets $C_{i}$. Again, it is possible to
minimize Eq. (61) over a particular choice of different subsets and, in
particular, over all non-trivial sets of subsets.
Besides the generalization of our bounds based solely on particular instances
of the triangle inequality, we can also use the measurement splitting method
in a more general setup. For instance, by splitting each measurement from
$\mathcal{M}_{(1,2,3,4)}$ three times, we obtain the assemblage
$\mathcal{M}_{(1,2,3,1,2,4,1,3,4,2,3,4)}$. This lets us conclude that it holds

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3,4)})\leq\dfrac{1}{4}[\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,4)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3,4)})+\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3,4)})]+\operatorname{I_{\diamond}}(\mathcal{N}),$
(62)

with
$\mathcal{N}=\mathcal{M}^{\\#}_{(1,2,3)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{(1,2,4)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{(1,3,4)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#}_{(2,3,4)}$
as a direct generalization of Eq. ($8$) in the main text. This leads for the
generalization of the incompatibility gain in Eq. ($10$) in the main text to

$\displaystyle\Delta\mathrm{I}_{(1,2,3)\rightarrow(1,2,3,4)}\leq\operatorname{I_{\diamond}}(\mathcal{N})\leq\operatorname{I_{\diamond}}(\mathcal{G}),$
(63)

where we assume
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})\geq\max\\{\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,4)}),\operatorname{I_{\diamond}}(\mathcal{M}_{(1,3,4)}),\operatorname{I_{\diamond}}(\mathcal{M}_{(2,3,4)})\\}$
analogous to the condition stated in Result $1$ in the main text and
$\mathcal{G}$ is the assemblage containing the parent POVMs of the
corresponding subsets. Again, further generalizations of (63) for other
scenarios can be derived by applying our methods.
We show, in the following, that our results can be applied to any probability
distribution $\mathbf{p}$ with which an assemblage $\mathcal{M}$ is weighted.
Here, we focus on assemblages with $m=3$ measurements. Further generalizations
follow directly from the above discussion. Let
$\mathcal{M}^{\mathbf{p}}=(\mathcal{M},\mathbf{p})$ be a general weighted
measurement assemblage. Using the triangle inequality, it holds

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)}^{\mathbf{p}})\leq\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,3)}^{\mathbf{p}},\mathcal{N}_{(1,2,3)}^{\mathbf{p}})+\operatorname{I_{\diamond}}(\mathcal{N}_{(1,2,3)}^{\mathbf{p}}),$
(64)

for any assemblage $\mathcal{N}_{(1,2,3)}$. By setting
$\mathcal{N}=\mathcal{M}^{\\#(1,2)}_{(1,2,3)}\coloneqq\mathcal{M}^{\\#}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}_{3}$,
it follows that

$\displaystyle\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,3)}^{\mathbf{p}},\mathcal{N}_{(1,2,3)}^{\mathbf{p}})=[p(1)+p(2)]\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)}^{\mathbf{q}}),$
(65)

where $\mathbf{q}=(\tfrac{p(1)}{p(1)+p(2)},\tfrac{p(2)}{p(1)+p(2)})$ is the
probability distribution weighting the assemblage $\mathcal{M}_{(1,2)}$. It is
important to note here, that $\mathcal{M}^{\\#}_{(1,2)}$ refers specifically
to the closest assemblage to $\mathcal{M}_{(1,2)}$ with respect to the
distribution $\mathbf{q}$. Note further that the particular instance of a
uniform distribution can straightforwardly be recovered from here. This shows
that $\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)}^{\mathbf{p}})$ is
upper bounded by the incompatibility of its subset $\mathcal{M}_{(1,2)}$
weighted by the likelihood of choosing a measurement from that subset, plus
the incompatibility
$\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\mathbf{p}}_{(1,2,3)})$.
Similarly, if we want to use the measurement splitting method, we can chose
any initial distribution $\mathbf{p}$ and proceed as usual to obtain bounds.
As we noted in Section II the method is not limited to split a measurement
into two equally likely versions of itself. The only conditions that have to
be satisfied are the conditions in the second equality of Eq. (36).

##  VI Proofs regarding the incompatibility of mutually unbiased bases

In this section, we present the proofs related to statements in the main text
regarding the incompatibility of measurements based on MUB Durt  _et al._
(2010). Two orthonormal bases $\\{|v_{a}\rangle\\}_{0\leq a\leq d-1}$ and
$\\{|w_{b}\rangle\\}_{0\leq b\leq d-1}$ are said to be MUB if it holds that

$\displaystyle|\langle v_{a}|w_{b}\rangle|=\dfrac{1}{\sqrt{d}}\ \forall\ a,b.$
(66)

The set of projections onto the orthonormal bases $\\{|v_{a}\rangle\\}_{0\leq
a\leq d-1}$ form the measurement $\mathcal{M}=\\{M_{a}=\lvert
v_{a}\rangle\langle v_{a}\rvert\\}$. Now, an _MUB measurement assemblage_
Tendick  _et al._ (2023) is a set of measurements where the condition (66)
holds for any two projections from different bases. While it is generally
unknown how many MUB exist in a dimension $d$, it is known that for every
$d\geq 2$, there exist at least $m=p^{r}+1$ MUB, where $p^{r}$ is the smallest
prime power factor of $d$ Klappenecker and Rötteler (2004), and at most
$m=d+1$ MUB. In the case where $d$ is a prime-power there exist explicit
constructions of MUB Wootters and Fields (1989), which are known to be
operationally inequivalent Designolle  _et al._ (2019b); Tendick  _et al._
(2023). The possibly most simple construction of a complete set of MUB, i.e.,
$m=d+1$ bases, can be used whenever $d$ is a prime. In this case, we can use
the Heisenberg-Weyl operators

$\displaystyle\hat{X}=\sum_{k=0}^{d-1}\lvert k+1\rangle\langle k\rvert,\
\hat{Z}=\sum_{k=0}^{d-1}\omega^{k}\lvert k\rangle\langle k\rvert,$ (67)

for a specific construction. Here, $\\{\lvert k\rangle\\}_{0\leq k\leq d-1}$
is the computational basis and $\omega=\exp{\big{(}\dfrac{2\pi i}{d}\big{)}}$
is a root of unity. In prime dimensions $d$, the eigenbases of the $d+1$
operators
$\hat{X},\hat{Z},\hat{X}\hat{Z},\hat{X}\hat{Z}^{2},\cdots,\hat{X}\hat{Z}^{d-1}$
are mutually unbiased Bandyopadhyay  _et al._ (2002). Most notably, for $m=2$,
$m=d,$ and $m=d+1$ there exists an analytical expression for the
incompatibility of the MUB measurement assemblages obtained via this
construction Designolle  _et al._ (2019b); Tendick  _et al._ (2023). For
$d=2$, our MUB measurement assemblage reduces to the projective measurements
defined by the Pauli operators.

###  VI.1 Tightness proof for Eq. ($10$) in the main text

We start by proving that Eq. ($10$) in the main text is tight for a noisy MUB
measurement assemblage based on Pauli measurements, i.e., we show that

$\displaystyle\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}(\eta)\coloneqq\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})-\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})=\operatorname{I_{\diamond}}(\mathcal{N}(\eta)),$
(68)

with
$\mathcal{N}(\eta)=\mathcal{M}^{\\#\eta}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#\eta}_{(1,3)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#\eta}_{(2,3)}$
holds true for measurements of the form

$\displaystyle
M^{\eta}_{a|x}=\eta\Pi_{a|x}+(1-\eta)\operatorname{Tr}[\Pi_{a|x}]\dfrac{\mathds{1}}{2},$
(69)

where the $\Pi_{a|x}=M^{\eta=1}_{a|x}$ are projectors defined via the
eigenvectors of Pauli operators and $\eta$ defines the amount of noise in the
measurements.
We divide our proof into three different parameter regimes. Let $\eta^{*}_{2}$
and $\eta^{*}_{3}$ be the _white-noise robustness_ of $\mathcal{M}_{(1,2)}$,
respectively $\mathcal{M}_{(1,2,3)}$, i.e., the maximal $\eta$ where the noisy
assemblages are still jointly measurable. We consider the regimes $1):\
\eta\leq\eta^{*}_{3}\leq\eta^{*}_{2}$, $2):\
\eta^{*}_{3}<\eta\leq\eta^{*}_{2}$, and $3):\
\eta^{*}_{3}\leq\eta^{*}_{2}<\eta$ corresponding to the three regimes in
Figure $2$ in the main text.
Note that regime $1):\ \eta\leq\eta^{*}_{3}$ leads trivially to

$\displaystyle\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}(\eta)=\operatorname{I_{\diamond}}(\mathcal{N}(\eta))=0.$
(70)

For the second regime, i.e., $\eta^{*}_{3}<\eta\leq\eta^{*}_{2}$ it follows
directly that

$\displaystyle\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}(\eta)=\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{N}(\eta)).$
(71)

The first equality follows from the fact that
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})=0$ by definition. The
second equality follows from the fact that
$\mathcal{M}^{\\#\eta}_{(s,t)}=\mathcal{M}^{\eta}_{(s,t)}$ for any
$s,t\in\\{1,2,3\\}$ such that $s\neq t$, since the subset
$\mathcal{M}^{\eta}_{(s,t)}$ is jointly measurable by definition. Now, due to
the reverse direction of the measurement splitting method outlined in Section
II, it holds
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{N}(\eta))$.
That means, the only non-trivial case is regime $3):\
\eta^{*}_{3}\leq\eta^{*}_{2}<\eta$.
Our proof for this regime relies on solving the SDPs in Eq. (33) and Eq. (34)
analytically. Starting from the dual:

$\displaystyle\underline{\text{Dual problem (incompatibility):}}$ (72)
$\displaystyle\mathrm{given}:\ \mathcal{M}^{\eta},\mathbf{p}$
$\displaystyle\underset{C_{a|x},\rho_{x},L}{\mathrm{maximize}}\ \ \
\sum_{a,x}p(x)\mathrm{Tr}[M^{\eta}_{a|x}C_{a|x}]-\mathrm{Tr}[L]$ subject to:
$\displaystyle L\geq\sum_{a,x}p(x)v(a|x,\lambda)C_{a|x}\ \forall\ \lambda,$
$\displaystyle 0\leq C_{a|x}\leq\rho_{x}\ \forall\ a,x,\ \rho_{x}\geq
0,\mathrm{Tr}[\rho_{x}]=1\ \forall\ x,$

we choose the specific instance where $C_{a|x}=\tfrac{\Pi_{a|x}}{2}$,
$L=l\mathds{1}$, and $\rho_{x}=\sum_{a}C_{a|x}=\tfrac{\mathds{1}}{2}$ for some
appropriately chosen scalar-variable $l$. For a qubit assemblage $\mathcal{M}$
with POVM effects of the form

$\displaystyle
M^{\eta}_{a|x}=\eta\Pi_{a|x}+(1-\eta)\operatorname{Tr}[\Pi_{a|x}]\dfrac{\mathds{1}}{2}=\eta\Pi_{a|x}+(1-\eta)\dfrac{\mathds{1}}{2},$
(73)

this evaluates to the lower bound
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta})\geq\eta+\tfrac{(1-\eta)}{2}-\tfrac{T}{m}$,
where $T\coloneqq\lVert\sum_{a,x}v^{*}(a|x,\lambda)M_{a|x}\rVert_{\infty}$ and
$\\{v^{*}(a|x,\lambda)\\}_{a,x}$ is the deterministic strategy maximizing the
norm. Note that this bound results from choosing $l=\tfrac{T}{2m}$, which can
be shown to be always a valid choice Tendick  _et al._ (2023).
For (noise-free, i.e., $\eta=1$) MUB measurement assemblages it was proven in
Designolle  _et al._ (2019b) that whenever $m=2$, $m=d$, or $m=d+1$, it holds
that

$\displaystyle\eta^{*}_{m}=\dfrac{dT-m}{dm-m}.$ (74)

This lets us conclude (for the qubit case, i.e., d = 2) that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\eta})$
$\displaystyle\geq\eta+\dfrac{(1-\eta)}{2}-\dfrac{\eta^{*}_{m}+1}{2}$ (75)
$\displaystyle=\dfrac{1}{2}(\eta-\eta^{*}_{m}).$

For the upper bound of $\operatorname{I_{\diamond}}(\mathcal{M}^{\eta})$ we
invoke the primal SDP:

$\displaystyle\underline{\text{Primal problem (incompatibility):}}$ (76)
$\displaystyle\mathrm{given:}\ \mathcal{M}^{\eta},\mathbf{p}$
$\displaystyle\underset{Z_{x},G_{\lambda}}{\mathrm{minimize}}\sum_{x}p(x)\lVert\mathrm{Tr}_{1}[Z_{x}]\rVert_{\infty}$
subject to: $\displaystyle Z_{x}\geq\sum_{a}|a\rangle\langle
a|\otimes(M^{\eta}_{a|x}-F_{a|x})^{T}\ \forall\ x,$ $\displaystyle
F_{a|x}=\sum_{\lambda}v(a|x,\lambda)G_{\lambda}\ \forall\ x,a,\
G_{\lambda}\geq 0\ \forall\ \lambda,\sum_{\lambda}G_{\lambda}=\mathds{1},$
$\displaystyle Z_{x}\geq 0,\ \forall\ x,$

where we have explicitly replaced the constraints involving the variables
$a_{x}$ in the SDP in Eq. (33) by using the spectral norm (largest singular
value). By choosing
$F_{a|x}=\eta^{*}_{m}\Pi_{a|x}+(1-\eta^{*}_{m})\tfrac{\mathds{1}}{2}$ and
$Z_{x}=\tfrac{1}{2}(\eta-\eta^{*}_{m})\sum_{a}\lvert a\rangle\langle
a\rvert\otimes\Pi_{a|x}^{T}$ for $\eta\geq\eta^{*}_{m}$ all constraints can
directly be verified to hold. Therefore, we obtain the upper bound

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\eta})\leq\dfrac{1}{2}(\eta-\eta^{*}_{m}).$
(77)

That implies
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta})=\dfrac{1}{2}(\eta-\eta^{*}_{m})$
for any assemblage involving $m=2$ or $m=3$ noisy MUB measurement assemblages
with $\eta\geq\eta^{*}_{m}$ in $d=2$. Therefore, the incompatibility gain
$\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}(\eta)\coloneqq\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})-\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})$
evaluates to

$\displaystyle\Delta\mathrm{I}_{(1,2)\rightarrow(1,2,3)}(\eta)$
$\displaystyle=\dfrac{1}{2}[(\eta-\eta)+(\eta^{*}_{2}-\eta^{*}_{3})].$ (78)
$\displaystyle=\dfrac{1}{2}[(\eta^{*}_{2}-\eta^{*}_{3})].$

Note that the gain is constant in this regime, as it is also evident from
Figure $2$ in the main text. Now, to finish the proof, we have to show that
$\operatorname{I_{\diamond}}(\mathcal{N}(\eta))$ has the same incompatibility.
However, this follows almost directly, since
$\mathcal{N}(\eta)=\mathcal{M}^{\\#\eta}_{(1,2)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#\eta}_{(1,3)}\mathbin{+\mkern-10.0mu+}\mathcal{M}^{\\#\eta}_{(2,3)}$
contains the closest jointly measurable assemblages with respect to the
subsets. As it is known from Designolle  _et al._ (2019b) and Tendick  _et
al._ (2023) (and we confirmed it with the above calculation) all of these
subsets are again just noisy versions of MUB measurement assemblages, with the
same noise contained in every subset. From the reverse direction of the
measurement splitting method, it follows that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{N}(\eta))=\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})$
(79)

for $\eta=\eta^{*}_{2}=\tfrac{1}{\sqrt{2}}$. Therefore, it follows that
$\operatorname{I_{\diamond}}(\mathcal{N}(\eta))=\dfrac{1}{2}[(\eta^{*}_{2}-\eta^{*}_{3})]$
for $\eta\geq\eta^{*}_{2}$ which concludes the proof.

###  VI.2 Tightness proof for Eq. ($15$) in the main text

Here, we show that Eq. ($15$) in the main text is tight for the case of noisy
Pauli measurements (see Eq. (69)). That is, we show that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})=\dfrac{2}{3}\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,3)}),$
(80)

holds for the assemblage $\mathcal{M}^{\eta}_{(1,2,3)}$ that contains noisy
Pauli measurements. To give a better overview, we also plot the respective
incompatibility contributions of
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})$ in Figure 3.

Figure 3: Incompatibility bound from Eq. ($15$) in the main text for
measurements corresponding to the three Pauli measurements. The different
contributions (depicted by the dashed red and the blue line) result together
in the incompatibility
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})$. The two
discontinuities of
$\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,3)})$ (dashed
red line) indicate the points where $\mathcal{M}^{\eta}_{(1,2,3)}$ becomes
compatible, respectively pairwise compatible.

The proof reduces to show the equality for the case $\eta>\eta^{*}_{2}$, as
the other cases follow trivially from the discussions made in Section VI.1. We
already evaluated the values of
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})$ and
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})$, i.e., we only have
to show that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,3)})=\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,3)})-\dfrac{2}{3}\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})$
$\displaystyle=\dfrac{1}{2}(\eta-\eta^{*}_{3})-\dfrac{1}{3}(\eta-\eta^{*}_{2})$
(81)
$\displaystyle=\dfrac{1}{6}\eta+\dfrac{1}{3}\eta^{*}_{2}-\dfrac{1}{2}\eta^{*}_{3}.$

Since we already know that
$\dfrac{1}{6}\eta+\dfrac{1}{3}\eta^{*}_{2}-\dfrac{1}{2}\eta^{*}_{3}\leq\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,3)}),$
due to the general bound in Eq. (59), it is enough to show that
$\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,3)})\leq\dfrac{1}{6}\eta+\dfrac{1}{3}\eta^{*}_{2}-\dfrac{1}{2}\eta^{*}_{3}$
also holds true.
We rely again on the primal problem in Eq. (76) using the feasible point where
$F_{a|x}=\eta^{*}_{3}\Pi_{a|x}+(1-\eta^{*}_{3})\tfrac{\mathds{1}}{2}$ and
$Z_{x}=\tfrac{1}{2}(\eta^{*}_{2}-\eta^{*}_{3})\sum_{a}\lvert a\rangle\langle
a\rvert\otimes\Pi_{a|x}^{T}$ for $x=1,2$ and
$Z_{x}=\tfrac{1}{2}(\eta-\eta^{*}_{3})\sum_{a}\lvert a\rangle\langle
a\rvert\otimes\Pi_{a|x}^{T}$ for $x=3$. It can be checked again directly that
this point is indeed feasible. Moreover, we obtain a primal objective value of

$\displaystyle\sum_{x}\dfrac{1}{3}\lVert\mathrm{Tr}_{1}[Z_{x}]\rVert_{\infty}=2\dfrac{1}{3}\cdot\dfrac{1}{2}(\eta^{*}_{2}-\eta^{*}_{3})+\dfrac{1}{6}(\eta-\eta^{*}_{3})=\dfrac{1}{6}\eta+\dfrac{1}{3}\eta^{*}_{2}-\dfrac{1}{2}\eta^{*}_{3},$
(82)

which concludes the proof.

###  VI.3 Tightness proof for generalizations of Eq. ($15$) in the main text

Here, we show that in the scenarios $m=2\rightarrow m^{\prime}=d$,
$m=2\rightarrow m^{\prime}=d+1$, and $m=d\rightarrow m^{\prime}=d+1$ there
exists analog bounds to Eq. ($15$) in the main text that are tight for
$d$-dimensional noisy MUB measurement assemblages. As before, we only consider
the non-trivial case here and in the following, i.e., the noisy regime where
none of the incompatibilities vanish. Also, we refer to the noise-free
measurements, i.e., the projectors on the MUB by $\Pi_{a|x}=M_{a|x}^{\eta=1}$.
The corresponding bound (see Eq. (59)) for the instance $2\rightarrow d$ reads

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d)})\leq\dfrac{2}{d}\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,\cdots,d)}),$
(83)

where we know that
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})=(\tfrac{d-1}{d})(\eta-\eta^{*}_{2})$
by generalizing the previous qubit result. Indeed, carefully checking the
calculation for the $d=2$ case in the Section VI.1, reveals the general
(dimension dependant) prefactor for the incompatibility of two noisy MUB
measurements.
Furthermore, using essentially the same feasible points as before (simply
extended to the case of $m=d$ instead of $m=2$ measurements) we obtain that
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d)})=(\tfrac{d-1}{d})(\eta-\eta^{*}_{d})$.
With that, we know that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,\cdots,d)})\geq\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d)})-\dfrac{2}{d}\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})=\Big{(}\dfrac{d-1}{d}\Big{)}(\eta-\eta^{*}_{d})-\dfrac{2}{d}\Big{(}\dfrac{d-1}{d}\Big{)}(\eta-\eta^{*}_{2}),$
(84)

which means, it remains to show that
$\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,\cdots,d)})\leq(\tfrac{d-1}{d})(\eta-\eta^{*}_{d})-\tfrac{2}{d}(\tfrac{d-1}{d})(\eta-\eta^{*}_{2})$
also holds. This can directly be verified by using the feasible point
$F_{a|x}=\eta^{*}_{d}\Pi_{a|x}+(1-\eta^{*}_{d})\tfrac{\mathds{1}}{d}$ and
$Z_{x}=(\tfrac{d-1}{d})(\eta^{*}_{2}-\eta^{*}_{d})\sum_{a}\lvert
a\rangle\langle a\rvert\otimes\Pi_{a|x}^{T}$ for $x=1,2,$ and
$Z_{x}=(\tfrac{d-1}{d})(\eta-\eta^{*}_{d})\sum_{a}\lvert a\rangle\langle
a\rvert\otimes\Pi_{a|x}^{T}$ for $x=3,\cdots,d$. This concludes the proof.
The corresponding bound for the $2\rightarrow d+1$ scenario (see Eq. (59))
reads

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d+1)})\leq\dfrac{2}{d+1}\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,\cdots,d+1)}),$
(85)

with
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2)})=(\tfrac{d-1}{d})(\eta-\eta^{*}_{2})$.
Using the same feasible points as for the $m=2$ and $m=d$ case, it also
follows that
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d+1)})=(\tfrac{d-1}{d})(\eta-\eta^{*}_{d+1})$,
i.e, to prove tightness, we have to show that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2),\eta}_{(1,2,\cdots,d+1)})\leq\Big{(}\dfrac{d-1}{d}\Big{)}(\eta-\eta^{*}_{d+1})-\Big{(}\dfrac{2}{d+1}\Big{)}\Big{(}\dfrac{d-1}{d}\Big{)}(\eta-\eta^{*}_{2}),$
(86)

holds true. Using the same construction as before, this can be checked
directly. Namely, using the feasible point
$F_{a|x}=\eta^{*}_{d+1}\Pi_{a|x}+(1-\eta^{*}_{d+1})\tfrac{\mathds{1}}{d}$ and
$Z_{x}=(\tfrac{d-1}{d})(\eta^{*}_{2}-\eta^{*}_{d+1})\sum_{a}\lvert
a\rangle\langle a\rvert\otimes\Pi_{a|x}^{T}$ for $x=1,2,$ and
$Z_{x}=(\tfrac{d-1}{d})(\eta-\eta^{*}_{d+1})\sum_{a}\lvert a\rangle\langle
a\rvert\otimes\Pi_{a|x}^{T}$ for $x=3,\cdots,d+1$ it follows directly that Eq.
(86) is indeed true, which concludes the proof.
In the case $d\rightarrow d+1$, the corresponding bound reads

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d+1)})\leq\dfrac{d}{d+1}\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d)})+\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2,\cdots,d),\eta}_{(1,2,\cdots,d+1)}),$
(87)

with
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d+1)})=(\tfrac{d-1}{d})(\eta-\eta^{*}_{d+1})$
and
$\operatorname{I_{\diamond}}(\mathcal{M}^{\eta}_{(1,2,\cdots,d)})=(\tfrac{d-1}{d})(\eta-\eta^{*}_{d})$.
That means we have to check that

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}^{\\#(1,2,\cdots,d),\eta}_{(1,2,\cdots,d+1)})\leq\Big{(}\dfrac{d-1}{d}\Big{)}(\eta-\eta^{*}_{d+1})-\Big{(}\dfrac{d}{d+1}\Big{)}\Big{(}\dfrac{d-1}{d}\Big{)}(\eta-\eta^{*}_{d}),$
(88)

is true. Using
$F_{a|x}=\eta^{*}_{d+1}\Pi_{a|x}+(1-\eta^{*}_{d+1})\tfrac{\mathds{1}}{d}$ and
$Z_{x}=\tfrac{d-1}{d}(\eta^{*}_{d}-\eta^{*}_{d+1})\sum_{a}\lvert
a\rangle\langle a\rvert\otimes\Pi_{a|x}^{T}$ for $x=1,2,\cdots,d,$ and
$Z_{x}=\tfrac{d-1}{d}(\eta-\eta^{*}_{d+1})\sum_{a}\lvert a\rangle\langle
a\rvert\otimes\Pi_{a|x}^{T}$ for $x=d+1$ this can be verified, just as in the
above cases.

###  VI.4 Additional insights on Eq. ($18$) in the main text

In this subsection, we give additional insights to Eq. ($18$) from the main
text. That is, we analyse the incompatibility decomposition

$\displaystyle\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$
$\displaystyle\leq\operatorname{I_{\diamond}}^{\mathrm{gen}}(\mathcal{M}_{(1,2,3)})+\operatorname{I_{\diamond}}^{\mathrm{pair}}(\mathcal{M}_{(1,2,3)})+\operatorname{I_{\diamond}}^{\mathrm{hol}}(\mathcal{M}_{(1,2,3)}),$
(89)

for an arbitrary assemblage $\mathcal{M}_{(1,2,3)}$. Note that we defined here
$\operatorname{I_{\diamond}}^{\mathrm{gen}}(\mathcal{M})\coloneqq\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,3)},\mathcal{M}^{\mathrm{conv}})$
to be the _genuine triplewise incompatibility_ of $\mathcal{M}_{(1,2,3)}$,
i.e., its distance to the closest assemblage
$\mathcal{M}^{\mathrm{conv}}\in\mathrm{JM}^{\mathrm{conv}}\coloneqq\mathrm{Conv}(\mathrm{JM}^{(1,2)},\mathrm{JM}^{(1,3)},\mathrm{JM}^{(2,3)})$.
Furthermore,
$\operatorname{I_{\diamond}}^{\mathrm{pair}}(\mathcal{M})\coloneqq\operatorname{D_{\diamond}}(\mathcal{M}^{\mathrm{conv}},\mathcal{M}^{\mathrm{pair}}),$
is the _pairwise incompatibility_ , where
$\mathcal{M}^{\mathrm{pair}}\in\mathrm{JM}^{pair}\coloneqq\mathrm{JM}^{(1,2)}\cap\mathrm{JM}^{(1,3)}\cap\mathrm{JM}^{(2,3)}$
is the closest assemblage in which all measurements are pairwise-compatible
and
$\operatorname{I_{\diamond}}^{\mathrm{hol}}(\mathcal{M})\coloneqq\operatorname{I_{\diamond}}(\mathcal{M}^{\mathrm{pair}})$
is the _hollow incompatibility_ of $\mathcal{M}_{(1,2,3)}$. Note that the
pairwise and hollow incompatibility depend implicitly on
$\mathcal{M}_{(1,2,3)}$. See also Figure $1$ in the main text for the
different incompatibility structures. Indeed the incompatibilities defined
here, are nothing else but the distances to the next corresponding
compatibility structure in Figure $1$ in the main text.
We now show that the bound in Eq. (89) is tight for the three Pauli
measurements. For simplicity, we focus on the noise-free scenario in the
following. From the previous discussions, we know that
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})=\dfrac{1}{2}(1-\eta^{*}_{3})$.
For the contribution
$\operatorname{I_{\diamond}}^{\mathrm{gen}}(\mathcal{M}_{(1,2,3)})$ we can use
$\mathcal{M}^{\\#(1,2)}_{(1,2,3)}$ as (possibily sub-optimal) point in
$\mathrm{JM}^{\mathrm{conv}}$. Therefore, we obtain the bound

$\displaystyle\operatorname{I_{\diamond}}^{\mathrm{gen}}(\mathcal{M}_{(1,2,3)})\leq\operatorname{D_{\diamond}}(\mathcal{M}_{(1,2,3)},\mathcal{M}^{\\#(1,2)}_{(1,2,3)})=\tfrac{2}{3}\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2)})=\tfrac{1}{3}(1-\eta^{*}_{2}).$
(90)

For the contribution
$\operatorname{I_{\diamond}}^{\mathrm{pair}}(\mathcal{M}_{(1,2,3)})$ we use as
a guess for $\mathcal{M}^{\mathrm{pair}}$ the depolarized version of
$\mathcal{M}_{(1,2,3)}$ where it becomes pairwise compatible. That is,
$\mathcal{M}^{\mathrm{pair}}$ is of the form

$\displaystyle
M^{\mathrm{pair}}_{a|x}=\eta^{\mathrm{pair}}_{3}\Pi_{a|x}+(1-\eta^{\mathrm{pair}}_{3})\dfrac{\mathds{1}}{d}.$
(91)

Using the results from previous discussions, we therefore obtain

$\displaystyle\operatorname{I_{\diamond}}^{\mathrm{pair}}(\mathcal{M}_{(1,2,3)})\leq\dfrac{1}{3}(\eta^{*}_{2}-\eta^{\mathrm{pair}}_{3})+\dfrac{1}{6}(1-\eta^{\mathrm{pair}}_{3}),$
(92)

by bounding the distance
$\operatorname{D_{\diamond}}(\mathcal{M}^{\\#(1,2)}_{(1,2,3)},\mathcal{M}^{\mathrm{pair}})$
through the SDP for the diamond norm, i.e., we examine the SDP in Eq. (33)
with $\mathcal{F}=\mathcal{M}^{\mathrm{pair}}$.
Finally, for the contribution
$\operatorname{I_{\diamond}}^{\mathrm{hol}}(\mathcal{M}_{(1,2,3)})$ we use as
(possibly sub-optimal) candidate for the closest jointly measurable assemblage
simply the appropriately depolarized version of $\mathcal{M}_{(1,2,3)}$, i.e.,
we obtain the bound
$\operatorname{I_{\diamond}}^{\mathrm{hol}}(\mathcal{M}_{(1,2,3)})\leq\dfrac{1}{2}(\eta^{\mathrm{pair}}_{3}-\eta^{*}_{3})$.
Summing all these bounds up, we obtain that

$\displaystyle\operatorname{I_{\diamond}}^{\mathrm{gen}}(\mathcal{M}_{(1,2,3)})+\operatorname{I_{\diamond}}^{\mathrm{pair}}(\mathcal{M}_{(1,2,3)})+\operatorname{I_{\diamond}}^{\mathrm{hol}}(\mathcal{M}_{(1,2,3)})$
$\displaystyle\leq\dfrac{1}{3}(1-\eta^{*}_{2})+\dfrac{1}{3}(\eta^{*}_{2}-\eta^{\mathrm{pair}}_{3})+\dfrac{1}{6}(1-\eta^{\mathrm{pair}}_{3})+\dfrac{1}{2}(\eta^{\mathrm{pair}}_{3}-\eta^{*}_{3})$
(93) $\displaystyle=\dfrac{1}{2}(1-\eta^{*}_{3}),$

which equals the value for
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$ for the noise free Pauli
measurements, as calculated in section VI.1. Therefore Eq. (89) is tight. Note
that the proof crucially relies on knowing the incompatibility of
$\operatorname{I_{\diamond}}(\mathcal{M}_{(1,2,3)})$, i.e., without having an
analytical expression for this term in higher dimensions $d>2$, this way of
proving equality will not work generally. However, we can check numerically,
whether the bound in Eq. (89) is tight for higher dimensional MUB. Indeed, our
numerics suggest for up to $d=7$ that Eq. (89) is tight for MUB with a
deviation of the order $10^{-9}$.

  *[MUB]: mutually unbiased bases
  *[POVMs]: positive operator valued measure
  *[POVM]: positive operator valued measure
  *[ POVM]: positive operator valued measure
  *[SM]: Supplemental Material
  *[CHSH]: Clauser-Horne-Shimony-Holt
  *[CPTP]: completely positive and trace preserving
  *[SDP]: semidefinite program
  *[SDPs]: semidefinite program
  *[LHS]: local hidden-state model
  *[LHV]: local hidden-variable model