An operational approach to classifying measurement incompatibility

Abstract

Measurement incompatibility has proved to be an important resource for information processing tasks. In this work, we analyze various levels of incompatibility of measurement sets. We provide operational classification of measurement incompatibility with respect to two elementary classical operations, viz., coarse-graining of measurement outcomes and convex mixing of different measurements. We derive analytical criteria for determining when a set of projective measurements is fully incompatible with respect to coarse-graining or convex mixing. Robustness against white noise is investigated for mutually unbiased bases that can sustain full incompatibility. Furthermore, we propose operational witnesses for different levels of incompatibility subject to classical operations, using the input-output statistics of Bell-type experiments as well as experiments in the prepare-and-measure scenario.

Full text

An operational approach to classifying measurement incompatibility
Arun Kumar Das,1Saheli Mukherjee,1Debashis Saha,2Debarshi Das,3and A. S. Majumdar1
1S. N. Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India
2School of Physics, Indian Institute of Science Education and Research Thiruvananthapuram, Kerala 695551, India
3Department of Physics and Astronomy, University College London,
Gower Street, WC1E 6BT London, England, United Kingdom
Measurement incompatibility has proved to be an important resource for information processing
tasks. In this work, we analyze various levels of incompatibility of measurement sets. We provide
operational classification of measurement incompatibility with respect to two elementary classical
operations, viz., coarse-graining of measurement outcomes and convex mixing of different measure-
ments. We derive analytical criteria for determining when a set of projective measurements is fully
incompatible with respect to coarse-graining or convex mixing. Robustness against white noise is in-
vestigated for mutually unbiased bases that can sustain full incompatibility. Furthermore, we propose
operational witnesses for different levels of incompatibility subject to classical operations, using the
input-output statistics of Bell-type experiments as well as experiments in the prepare-and-measure
scenario.
I. INTRODUCTION
Measurement incompatibility is a concept relating ob-
servables which cannot be measured jointly with arbi-
trary accuracy [1]. It is a purely quantum effect of which
the most well-known example concerns the position and
momentum of a quantum particle. Being a fundamen-
tal concept of quantum theory, it takes a pivotal role in
explaining several quantum phenomena, such as, Bell-
nonlocality [2,3], EinsteinPodolskyRosen steering [4–
8], measurement uncertainty relations [9–12], quantum
contextuality [13–15], quantum violation of macroreal-
ism [16,17], and temporal and channel steering [18–20].
Measurement incompatibility finds applications in var-
ious quantum information processing tasks. Recently,
it has been shown that measurement incompatibility is
necessary for quantum advantage in any one-way com-
munication task [21].
The significance of measurement incompatibility in
various quantum effects calls for its in-depth charac-
terization. Towards this direction, a classification of
measurement incompatibility with respect to projection
onto subspaces has been recently performed [22]. In the
present work, our aim is to classify measurement in-
compatibility with respect to certain basic classical op-
erations. From an operational perspective, here we em-
ploy two such operations, namely coarse-graining of
measurement outcomes and convex mixing of measure-
ment settings. Coarse-graining of measurement out-
comes arises naturally [23] in several instances, such as,
for example, quite obviously in measurements on con-
tinuous variable system. Though the eigen spectra of
the observables are infinite-dimensional taking contin-
uous values, real-world experimental devices are lim-
ited by finite precision, leading to the measurement out-
comes taking a finite number of discrete values. This in-
accuracy in the recording of measurement outcomes is
manifested in the coarse-graining of measurement out-
comes, which is inevitable in practice. Coarse-graining
has also been employed to study the phenomenon of
quantum to classical transition. It is observed that quan-
tum phenomena may disappear due to imprecision of
measurement outcomes [16,24–26].
Device imperfection may also lead to the measure-
ment device performing probabilistically a set of mea-
surements, instead of always performing the desired
particular measurement. In such a case, a convex mix-
ing of the given set of measurements arises effectively.
As incompatibility of measurements is a quantum con-
cept, it is interesting to examine how this property be-
haves under such elementary classical operations. Our
study is motivated by the question as to how one may
compare the degree of incompatibility between two dif-
ferent sets of measurements. For instance, if the first
set remains incompatible for every possible non-trivial
coarse-graining of the measurement outcomes, but the
second set becomes compatible for a certain coarse-
graining, it follows that the first set of measurements ex-
hibits stronger incompatibility compared to the second
one. A similar argument holds for the case of convex-
mixing of measurements too.
As incompatible measurements are useful for per-
forming various information processing tasks, given a
measurement device claiming to produce incompatible
measurements, we must certify it before using it in an
information processing task. Verification of incompat-
iblity of the measurements is possible from the input-
output measurement statistics obtained from the de-
vice without knowing its internal functioning. Device-
independent protocols are conceptually most powerful,
relying only on the input-output statistics [27], with a
wide range of applications from quantum cryptogra-
phy [28,29] to communication complexity [30], self-
testing [31–34] and incompatibility witnesses [35]. How-
ever, they require shared entanglement, an expensive
resource, and prohibition of communication between
the involved parties. On the other hand, semi-device-
independent protocols are inspired by the standard
prepare-and-measure scenario, with an additional con-
arXiv:2401.01236v1 [quant-ph] 2 Jan 2024

2
straint on the dimension of the communicated quantum
states [36,37]. They have been applied in quantum key
distribution [38], quantum networks [39], randomness
certification [40], random access codes [41], and certifi-
cation of meassurements [42,43].
In the present work we investigate the issue of certifi-
cation of measurement incompatibility subjected to the
classical operations of coarse-graining and convex mix-
ing of measurements from both the device-independent
and the semi-device-independent perspective. An es-
sential ingredient in the performance of quantum in-
formation protocols is the presence of noise. Within
the context of our present study, noise is reflected in
the degrading of incompatibility properties of various
measurement sets. We examine the critical amount
of robustness of the noisy measurement above which
the measurements remain incompatible under the men-
tioned classical operations. We then study incompatibil-
ity witnesses in the device-independent and the semi-
device-independent scenarios.
Our paper is structured as follows. In Sec. (II) we
present the basic mathematical ingredients for classify-
ing measurement incompatibility under the two opera-
tions of coarse-graining of outcomes and convex-mixing
of measurements, respectively. In Sec. (III) we study the
robustness under noise for various levels of incompati-
bility subjected to the above classical operations. In Sec.
(IV) we define operational witnesses of incompatibility,
and furnish examples to study their performance in both
device-independent and the semi-device-independent
frameworks. Concluding remarks are presented in Sec.
(V).
II. CLASSIFYING MEASUREMENT INCOMPATIBILITY
UNDER ELEMENTARY CLASSICAL OPERATIONS
In the case of projective measurements, the ob-
servables whose corresponding operators commute are
jointly measurable. However, this is not true in the case
of general quantum measurements. The most general
quantum measurement is described by Positive Oper-
ator Valued Measures (POVM), which is a set of oper-
ators, {Oα}with 0 ≤Oα≤
1
, and ∑αOα=
1
. A
set of measurements defined by {Mzx|x}(here xcorre-
sponds to the measurement, zxdenotes the outcomes of
the measurement labeled by x) is compatible if there ex-
ists a parent POVM, Gλand classical post-processing for
each xgiven by {p(zx|x,λ)}such that
∀zx,x,Mzx|x=∑
λ
p(zx|x,λ)Gλ, (1)
where 0 ≤p(zx|x,λ)≤1, ∑zxp(zx|x,λ) = 1∀x,λ[1].
A. Coarse-graining of outcomes
Coarse-graining of the outcomes of a measurement is
a concept where some of the outcomes are treated on
the same footing, i.e., different outcomes are clubbed
together to form a single outcome. After the coarse-
graining of the outcomes, the effective number of out-
comes reduces. Coarse-graining may occur in prac-
tice due to measurement errors. Additionally, coarse-
graining is used in when one does not require a finer
description of the values of the measurement outcomes,
but rather a broad description suffices to understand the
nature of some physical property.
Let us introduce the notation [k]:={0, ··· ,k−1}for
any natural number k. Consider a doutcome measure-
ment {Mz}and an arbitrary coarse-graining that yields
adoutcome measurement {Mz}, where z∈[d]and z∈
[d]. The coarse-graining can be defined by conditional
probabilities {c(z|z)}such that
Mz=∑
z
c(z|z)Mz, (2)
and c(z|z)∈ {0, 1}. Trivial coarse-graining is the one
where the measurement becomes single outcome, that
is, c(z|z) = 1, ∀z. Two incompatible measurements may
not necessarily remain incompatible after certain coarse-
graining of their outcomes.
Definition 1 (Fully incompatible measurements w.r.t
coarse-graining).A set of measurements {Mzx|x}is fully
incompatible with respect to (w.r.t) coarse-graining if they
remain incompatible after all possible nontrivial coarse-
graining. That is, if the resultant set of measurements given
by
Mzx|x=∑
zx
cx(zx|zx)Mzx|x, (3)
after all possible sets of nontrivial coarse-graining
{cx(zx|zx)}pertaining to every setting x is incompati-
ble, then we call them fully incompatible.
Definition 2 (d-incompatible measurements w.r.t
coarse-graining).A set of measurements {Mzx|x}is
d-incompatible w.r.t coarse-graining if they remain incom-
patible after all possible nontrivial coarse-graining that give
rise to at least d outcome measurements. In other words, if
the resultant set of measurements
Mzx|x=∑
zx
cx(zx|zx)Mzx|x,zx∈[kx],kx⩾d,∀x(4)
after all possible coarse-graining resulting in at least d out-
come measurements is incompatible, then we call them d-
incompatible.
For example, the four outcome rank-one projec-
tive measurement pair, defined by the following (un-

3
normalized) vectors
M=n|0⟩,|1⟩,|2⟩,|3⟩o(5)
and
N=n|0+1⟩,|0−1⟩,|2+3⟩,|2−3⟩o(6)
is 3-incompatible, where the notation |0+1⟩=|0⟩+|1⟩,
and similarly for others.
Observation 1. A set of fully incompatible measurements
is equivalent to 2-incompatible measurements w.r.t coarse-
graining.
If a set of measurements is 2-incompatible, then it im-
plies that the set remains incompatible after all possible
coarse-graining of the outcomes such that the number of
outcomes of each measurement in the newly formed set
of measurements is greater than or equal to two. Also,
the lowest number of outcomes of measurement is two
for a nontrivial coarse-graining. Furthermore, if a set of
measurements is d-incompatible, then, by definition, it
is n-incompatible as well, where n>d, but the reverse
is not true. This proves Observation 1.
Observation 2. Consider two projective measurements, de-
fined by {Pi},and {Qj}, where i ∈[d]and j ∈[d′]. Let
{Mk}kbe the set of all proper subsets of [d], and {Nl}lbe
the set of all proper subsets of [d′]. Then these two measure-
ments are fully incompatible w.r.t coarse-graining if and only
if
"∑
i∈Mk
Pi,∑
j∈Nl
Qj#=0, ∀k,l. (7)
Proof. The result is a direct consequence of the fact that
for sharp measurement compatibility and commutativ-
ity are equivalent [44]. Suppose ∃k,l, such that the
lhs of (7) is zero. Consider the coarse-graining such
that the resulting measurements will be {∑i∈MkPi,
1
−
∑i∈MkPi}and {∑j∈NlQj,
1
−∑j∈NlQj}. The resultant
measurements will be compatible. The reverse direction
holds true from the definition of fully incompatible w.r.t
coarse-graining. ⊓⊔
Theorem 1. The following condition is necessary but not
sufficient for two rank-one projective measurements, defined
by {|ψi⟩} and {|ϕj⟩}, to be fully incompatible w.r.t coarse-
graining:
⟨ψi|ϕj⟩ =0, ∀i,j. (8)
The above condition is necessary and sufficient for two 3-
dimensional rank-one projective measurements.
Proof. First, note that for sharp measurement compati-
bility and commutativity are equivalent [44]. Suppose
∃i,j, such that ⟨ψi|ϕj⟩=0. Consider coarse-graining
of all other outcomes except iand jfor the two mea-
surements. Then, the resulting measurements will be
{|ψi⟩⟨ψi|,
1
−|ψi⟩⟨ψi|} and {|ϕj⟩⟨ϕj|,
1
−|ϕj⟩⟨ϕj|}, which
commute with each other (i.e., the resulting measure-
ments are compatible) since ⟨ψi|ϕj⟩=0. Thus, if
the measurements are fully incompatible, condition (8)
holds.
To show that (8) is not sufficient, consider the fol-
lowing two 4-dimensional rank-one projective measure-
ments (with the normalization factor 1/√2),
{|0⟩+|1⟩,|0⟩ − |1⟩,|2⟩+|3⟩,|2⟩− |3⟩}
{|0⟩+|2⟩,|0⟩ − |2⟩,|1⟩+|3⟩,|1⟩− |3⟩}. (9)
One can check that (8) holds for all i,j=0, 1, 2, 3. But
coarse-graining of outcomes 0, 1 and 2, 3 for both the
measurements leads to
{|0⟩⟨0|+|1⟩⟨1|,|2⟩⟨2|+|3⟩⟨3|},
{|0⟩⟨0|+|2⟩⟨2|,|1⟩⟨1|+|3⟩⟨3|}, (10)
which are compatible.
In 3-dimension, say, the measurements are M=
{|ψi⟩} and N={|ϕj⟩} with i,j∈ {1, 2, 3}.
Now any non-trivial coarse-graining yields binary-
outcome measurements of the form: {|ψi⟩⟨ψi|,
1
−
|ψi⟩⟨ψi|} and {|ϕj⟩⟨ϕj|,
1
− |ϕj⟩⟨ϕj|}. It is easy to see
that these two remain incompatible if and only if
[|ψi⟩⟨ψi|,|ϕj⟩⟨ϕj|]=0, which is equivalent to ⟨ψi|ϕj⟩ =
0, 1. In the case where ⟨ψi|ϕj⟩=1, there exists another
pair (i,j′)such that ⟨ψi|ϕj′⟩=0; thus, (8) implies fully
incompatible in dimension 3. ⊓⊔
B. Convex mixing of measurements
The concept of convex mixing of measurements may
be best understood by considering an example: suppose
we have a measurement device that produces three dif-
ferent measurements (M0,M1and M2)and has two in-
put switches. If the first switch is ON it performs mea-
surement M0with probability qand M1with probabil-
ity (1−q)and for the second switch, it always performs
M2. Thus, in principle, the device produces two mea-
surements: M2and a convex mixture of M0and M1.
Suppose M0={Mz|0},M1={Mz|1},M2={Mz|2}
such that they have same number of outcomes z∈
[d]. Now the measurement that is realized by a con-
vex combination of M0and M1with some weightage
qand 1 −q, where q∈[0, 1]is defined as
M(0,1,π)={qMz|0+ (1−q)Mπ(z)|1}, (11)
where π(z)is any permutation on outcome z. Here π(z)
is a bijective function from the set z∈[d]to itself. Even
if M2is incompatible with M0and M1separately, M2is
not necessarily incompatible with M(0,1,π)for all values

4
of q.
Definition 3. Three measurements M0,M1and M2are
fully incompatible w.r.t. convex mixtures if each of the pairs,
M0and M(1,2,π),M1and M(0,2,π),M2and M(0,1,π),are in-
compatible for all values of q and all possible permutations
π.
Consider three unbiased qubit measurements,
Mi:=1
2(
1
±
ni·
σ),i=0, 1, 2, (12)
and ||
ni|| ⩽1. These three measurements are fully
incompatible w.r.t convex mixtures if and only if (see
eq.(7) of [1]),
||
ni+q
nj±(1−q)
nk|| +||
ni−q
nj∓(1−q)
nk|| >2,
(13)
for all q, and for all (i,j,k)∈ {(0, 1, 2),(1, 2, 0),(2, 0, 1)}.
Observation 3. If three qubit measurements (12)are such
that 
niare in the same plane of the Bloch sphere, then they are
not fully incompatible w.r.t convex mixtures.
Proof. Note that permutation of the outcomes of Miin
(12) is nothing but taking −
niinstead of 
ni. If the three

niare in the same plane, then there exists at least one
triple (i,j,k)and permutations of outcomes such that

n0=1
c(q
n1+ (1−q)
n2)(14)
for some c∈(0, 1],q∈[0, 1]. In other words, there ex-
ists a triple (i,j,k)and permutation of outcomes such
that 
niis expressed as a linear combination of 
njand 
nk
with non-negative coefficients (here q/cand (1−q)/c),
such that sum of those two non-negative coefficients is
greater than or equal to 1. Substituting this into left hand
side of (13) taking +sign, we find
||(1+c)
ni|| +||(1−c)
ni|| ⩽|1+c|+|1−c|=2. (15)
This contradicts with (13). ⊓⊔
Observation 4. Suppose three qubit measurements (12)are
such that 
n0=ν0ˆ
x, 
n1=ν1ˆ
y, 
n2=ν2ˆ
z with 0≤
ν1,ν2,ν3≤1, that is, they are the noisy version of Pauli ob-
servables,
Mi:=1
2(
1
±νiσi)=νi
1
±σi
2+ (1−νi)
1
2, (16)
taking i =0, 1, 2 and (σ0,σ1,σ2) = (σx,σy,σz). Then they
are fully incompatible w.r.t convex mixture if and only if
min ν2
0+ν2
1ν2
2
ν2
1+ν2
2
,ν2
1+ν2
0ν2
2
ν2
0+ν2
2
,ν2
2+ν2
0ν2
1
ν2
0+ν2
1>1.
(17)
Proof. In terms of νi, the left hand side of (13) becomes
2qν2
i+q2ν2
j+ (1−q)2ν2
k. Note that the minimum of
q2ν2
j+ (1−q)2ν2
koccurs at ˜
q=ν2
k/(ν2
j+ν2
k). Since
0≤ν2
k/(ν2
j+ν2
k)≤1 for any νj,νk∈[0, 1], the above-
mentioned minimum can always be achieved with a
suitable choice of q. Hence, we have that
2qν2
i+q2ν2
j+ (1−q)2ν2
k≥2qν2
i+˜
q2ν2
j+ (1−˜
q)2ν2
k
=2v
u
u
tν2
i+ν2
jν2
k
ν2
j+ν2
k
(18)
Thus, the r.h.s of (13) is greater than 2 for all values of q
if and only if
ν2
i+ν2
jν2
k
ν2
j+ν2
k
>1. (19)
Taking all the three possible values of (i,j,k)we get the
condition (17). ⊓⊔
Clearly, the three Pauli observables are fully incom-
patible w.r.t. convex mixture, and moreover, if we take
an equal amount of noise ν0=ν1=ν2=ν, then (17)
implies ν>√2/3.
We can generalize this notion for a set of nmeasure-
ments.
Definition 4 (k-incompatible measurements w.r.t. con-
vex mixtures).A set of measurements is k-incompatible w.r.t
convex mixtures if after taking all possible convex mixtures
over all possible k partitions the resulting k number of mea-
surements from the set of n measurements (with relabelling of
outcomes) is incompatible.
Definition 5 (Fully incompatible measurements w.r.t.
convex mixtures).A set of measurements is fully incom-
patible w.r.t convex mixtures if it remains incompatible after
taking all possible convex mixtures over all possible partitions
of n measurements (with relabelling of outcomes).
Theorem 2. Fully incompatible measurements w.r.t convex
mixtures imply that every pair of measurements from that set
is incompatible. The reverse implication does not hold.
Consider a set of nmeasurements, M=
{M1,M2,M3, . . . , Mn}, in which there is a pair of
measurements Miand Mjwhich are compatible. Now
if we make two partitions where Miand Mjare in dif-
ferent partitions, the convex mixture of measurements
must be compatible. This is true for any compati-
ble pair Miand Mj. Thus, if the measurements are
fully incompatible w.r.t convex mixture, every pair of
measurements must necessarily be incompatible.
The reverse is not true. Consider the three noisy Pauli
measurements of Eq.(16). It can be shown by using
semi-definite programming that if 0.71 <ν≤0.81, the
measurements are pairwise incompatible but it is not
fully incompatible w.r.t convex mixture.

5
III. ROBUSTNESS UNDER NOISE FOR DIFFERENT
LEVELS OF INCOMPATIBILITY
In this section, we analyze the role of noise on quan-
tum measurements and study how the incompatibil-
ity properties depend on it. Due to the ubiquitous
nature of noise, it is pertinent to study the extent to
which noise could be tolerated by a set of measure-
ments while still retaining their incompatibility. We take
a noisy version of mutually unbiased bases measure-
ments, Mi|x≡ {ν|ϕx
i⟩⟨ϕx
i|+(1−ν)
d
1
d}, where {|ϕx
i⟩}
form mutually unbiased bases measurements in Cd.
Here νis the robustness parameter (or visibility parame-
ter) and (1−ν)is the noise parameter, 0 ≤ν≤1. When
the noise parameter is zero (i.e., robustness, ν=1),
the measurements are fully incompatible, and when the
noise parameter is one (i.e., the robustness, ν=0), the
measurements are trivial and compatible. Our aim is
to obtain the critical value of the robustness parameter
above which the measurements remain incompatible.
To check the compatibility, i.e., the existence of a par-
ent POVM, we use the method described in [1]. This can
be casted as a semi-definite programming (SDP) prob-
lem that takes a set of measurements {Mzx|x}and de-
terministic classical post-processings p(zx|x,λ)as input,
and checks whether the measurements are compatible
or not, subject to the constraints
∑
λ
p(zx|x,λ)Gλ=Mzx|x∀x,zx, (20)
∑
λ
Gλ=
1
, (21)
Gλ≥µ
1
, (22)
where µis the optimization parameter. This method
finds the maximum value of µfor each {p(zx|x,λ)}. If
this optimization returns a negative value of µ, then the
constraint of Eq.(22) can’t be fulfilled, which implies that
the measurements {Mzx|x,λ}are incompatible. Other-
wise, they are compatible.
A. Coarse-graining of outcomes
Dimension 3.Let {Pi}and {Qj},i,j∈[3]be two mea-
surements acting on C3, where
Pi=ν|i⟩⟨i|+ (1−ν)
1
3,
Qj=ν|ψj⟩⟨ψj|+ (1−ν)
1
3, (23)
with
|ψ0⟩=1
√3(|0⟩+|1⟩+|2⟩),
|ψ1⟩=1
√3(|0⟩+ω|1⟩+ω2|2⟩),
|ψ2⟩=1
√3(|0⟩+ω2|1⟩+ω|2⟩),
It is observed that for ν>0.683, the two measure-
ments are incompatible. For finding ”2-incompatible
w.r.t coarse-graining”, we consider all possible coarse-
graining of outcomes of the two measurements such
that after coarse-graining, the new set of measurements
have at least two outcomes each. We find that these
measurements are incompatible w.r.t all possible coarse-
graining for ν>0.711. Note that for 0.683 <ν≤0.711,
the incompatible measurements become compatible un-
der certain coarse-graining of outcomes (see TABLE I).
Dimension 4.For checking incompatibility w.rt
coarse-graining in C4, the same procedure is repeated
for two POVM measurements with 4 outcomes each.
The corresponding measurements are {P′
i},{Q′
j},i,j∈
[4].
P′
i=ν|i⟩⟨i|+ (1−ν)
1
4,
Q′
j=ν|ψ′
j⟩⟨ψ′
j|+ (1−ν)
1
4(24)
with
|ψ′
0⟩=1
2(|0⟩+|1⟩+|2⟩+|3⟩),
|ψ′
1⟩=1
2(|0⟩+|1⟩− |2⟩− |3⟩),
|ψ′
2⟩=1
2(|0⟩− |1⟩− |2⟩+|3⟩),
|ψ′
3⟩=1
2(|0⟩− |1⟩+|2⟩− |3⟩), (25)
for ν>0.666, these measurements are incompatible. We
find that for ν>0.675, the measurements are incompat-
ible under all possible non-trivial coarse-graining that
gives rise to at least 3 outcomes. The measurements are
fully incompatible for ν>0.72. The results are summa-
rized in TABLE I.
In dimension 3 In dimension 4
Incompatible 0.683 0.666
2-incompatible 0.711 0.720
3-incompatible 0.683 0.675
TABLE I. Critical values of robustness for MUBs w.r.t coarse-
graining in C3and C4.

6
B. Convex mixing of measurements
Dimension 3.For checking incompatibility w.r.t con-
vex mixtures (CM), a minimum of three measurements
are needed. The two measurements are {Pi}and {Qj},
as given in Eqs.(23), The third measurement is {Rk},k∈
[3]where
Rk=ν|ϕk⟩⟨ϕk|+ (1−ν)
1
3. (26)
with
|ϕ0⟩=ω|0⟩+|1⟩+|2⟩
√3,|ϕ1⟩=|0⟩+ω|1⟩+|2⟩
√3,
|ϕ2⟩=|0⟩+|1⟩+ω|2⟩
√3,
(27)
ωbeing the cube roots of unity.
The results are summarized in TABLE II.
Dimension 4.Consider the measurements {P′
i},{Q′
j}
as defined in Eqs.(24) and(25) and another measurement
{R′
k}where i,j,k∈[4].
{R′
k}=ν|ϕ′
k⟩⟨ϕ′
k|+ (1−ν)
1
4(28)
with
|ϕ′
0⟩=1
2(|0⟩− |1⟩−
i
|2⟩−
i
|3⟩),
|ϕ′
1⟩=1
2(|0⟩− |1⟩+
i
|2⟩+
i
|3⟩),
|ϕ′
2⟩=1
2(|0⟩+|1⟩+
i
|2⟩−
i
|3⟩),
|ϕ′
3⟩=1
2(|0⟩+|1⟩−
i
|2⟩+
i
|3⟩). (29)
The results are summarized in TABLE II.
In dimension 3 In dimension 4
Incompatible 0.537 0.692
2-incompatible 0.764 0.705
3-incompatible 0.537 0.692
TABLE II. Critical values of robustness for MUBs w.r.t convex-
mixing in C3and C4.
IV. OPERATIONAL WITNESSES OF INCOMPATIBILITY
A. Coarse-graining of outcomes
As we mentioned earlier, measurements that are fully
incompatible w.r.t coarse-graining show stronger in-
compatibility compared to the measurements that are
not fully incompatible w.r.t coarse-graining. To opera-
tionally certify whether a set of measurements are fully
incompatible w.r.t coarse-graining is important from the
perspective of determing the practical utility of such
a set for revealing phenomena such as Bell inequal-
ity, steering and contextuality [2–8,13–15], as well as
for checking the proficiency of such a set for informa-
tion processing tasks [21,45]. Operational certification
means that we need to infer the incompatibility of the
measurements from the input-output statistics of the
measurement device without knowing its internal func-
tioning. Below we study the two classes of operational
certification separately
Device-independent witness
Here one requires neither any prior knowledge of the
internal functioning of the measurement device, nor any
idea of the dimension of the system on which the mea-
surements act. In [3], it has been proven that any two
binary-outcome incompatible measurements violate the
Bell-CHSH inequality. We can use this result to witness
incompatibility for any two 3-outcome projective mea-
surements before and after coarse-graining, from the
input-output statistics.
Semi-Device-independent witness
Here, although we do not have any prior knowledge
of the internal functioning of the measurement device,
we do know the dimension of the system on which
the measurements act. Any two incompatible rank-
one projective measurements provide an advantage in
(2, d,d)−RAC [12]. If any two POVMs each with d
outcomes acting on Cdare jointly measurable, the av-
erage maximum success probability in (2, d,d)RAC is
bounded by [21]
P(2, d,d)≤1
2(1+d
d2) = PCB(2, d,d), (30)
where PCB (2, d,d)is the maximum average success
probability obtained in a (2, d,d)RAC using two com-
patible measurements. For full incompatibility of two
3-outcome rank-one projective measurements, we prove
the following:
Theorem 3. Two rank-one projective measurements, M =
n|ϕ0⟩,|ϕ1⟩,|ϕ2⟩oand N =n|ψ0⟩,|ψ1⟩,|ψ2⟩o, can be wit-
nessed to be fully incompatible w.r.t. coarse-graining via RAC
iff 0<|⟨ϕi|ψj⟩| <4
5,∀i,j=0, 1, 2.
Proof. Without loss of generality, any pair of three out-
come rank-one projective measurements can be written

7
up to unitary freedom as,
M=n|0⟩⟨0|,|1⟩⟨1|,|2⟩⟨2|o≡n|ϕi⟩⟨ϕi|o,
and
N=n|ψ1⟩⟨ψ1|,|ψ2⟩⟨ψ2|,|ψ3⟩⟨ψ3|o≡n|ψj⟩⟨ψj|o, (31)
where |ψj⟩=αj|0⟩+βj|1⟩+γj|2⟩.
Let us first coarse grain the second and third out-
comes. So, the new measurement pair becomes:
M(2,3)=n|0⟩⟨0|,|1⟩⟨1|+|2⟩⟨2|o, (32)
N(2,3)=n|ψ1⟩⟨ψ1|,|ψ2⟩⟨ψ2|+|ψ3⟩⟨ψ3|o.
The new compatibility bound in this case is,
PCB (2, 2, 3) = 1
2(1+3
22) = 7
8by Eq.(30),
and the bound is tight. M(2,3)and N(2,3)will give ad-
vantage in the (2,2,3) RAC if:
1
∑
x,y=0||M(2,3)(x) + N(2,3)(y)|| >7, which implies
||A|| +||B|| +||C|| +||D|| >7 (33)
with A=M(2,3)(0) + N(2,3)(0)
=|0⟩⟨0|+|ψ1⟩⟨ψ1|.
Now, |ψ1⟩can be written as:
|ψ1⟩=⟨0|ψ1⟩|0⟩+⟨u|ψ1⟩|u⟩with (34)
⟨0|u⟩=0 and |⟨0|ψ1⟩|2+|⟨u|ψ1⟩|2=1.
Acan be written as a (2×2)matrix in the basis
{|0⟩,|u⟩},
A= 1+|⟨0|ψ1⟩|2⟨0|ψ1⟩⟨ψ1|u⟩
⟨u|ψ1⟩⟨ψ1|0⟩ |⟨u|ψ1⟩|2!, (35)
So, the maximum eigenvalue of Ais 1 +|⟨0|ψ1⟩|. Thus,
||A|| =1+|⟨0|ψ1⟩|.
By Eq.(33) we have
B=M(2,3)(0) + N(2,3)(1)
=|0⟩⟨0|+|ψ2⟩⟨ψ2|+|ψ3⟩⟨ψ3|
=
1
+|0⟩⟨0|−|ψ1⟩⟨ψ1|. (36)
Similarly, Bcan be expressed in a block diagonal matrix
in the ortho-normal basis {|0⟩,|u⟩,|v⟩},
B= Γ0
0 1!where Γ= 2+|⟨0|ψ1⟩|2⟨0|ψ1⟩⟨ψ1|u⟩
⟨u|ψ1⟩⟨ψ1|0⟩1+|⟨u|ψ1⟩|2!,
(37)
and
||B|| =2+|⟨0|ψ1⟩|.
Also, from Eq.(33)
C=M(2,3)(1) + N(2,3)(0)
=|1⟩⟨1|+|2⟩⟨2|+|ψ1⟩⟨ψ1|
=
1
−|0⟩⟨0|+|ψ1⟩⟨ψ1|. (38)
Similarly, Ccan be expressed in a block diagonal matrix
in the basis {|0⟩,|u⟩,|v⟩},
C= Σ0
0 1!where Σ= |⟨0|ψ1⟩|2⟨0|ψ1⟩⟨ψ1|u⟩
⟨u|ψ1⟩⟨ψ1|0⟩1+|⟨u|ψ1⟩|2!,
(39)
and
||C|| =1+q1− |⟨0|ψ1⟩|2.
Similarly, from Eq.(33)
D=M(2,3)(1) + N(2,3)(1)
=|1⟩⟨1|+|2⟩⟨2|+|ψ2⟩⟨ψ2|+|ψ3⟩⟨ψ3|
=2
1
−|0⟩⟨0|−|ψ1⟩⟨ψ1|, (40)
and Dcan be expressed in a block diagonal matrix in the
basis {|0⟩,|u⟩,|v⟩},
D= Ξ0
0 2!where Ξ= 1−|⟨0|ψ1⟩|2−⟨0|ψ1⟩⟨ψ1|u⟩
−⟨u|ψ1⟩⟨ψ1|0⟩2−|⟨u|ψ1⟩|2!,
(41)
and ||D|| =2.
Substituting the values of ||A||,||B||,||C|| and ||D|| in
Eq.(33) and after simplification we get, |⟨0|ψ1⟩| <4
5.
Note that only the projectors or vectors that are not
coarse-grained, form the inner product. So, to be fully
incompatible w.r.t coarse-graining, the following con-
dition, |⟨ϕi|ψj⟩| <4
5∀i,jshould hold. Now. from
Theorem 1we know that for a pair of rank-one projec-
tive measurements in dimension 3, the necessary and
sufficient condition for fully-incompatible w.r.t coarse-
graining is ⟨ϕi|ψj⟩ =0∀i,j, this proves the Theorem 3.
⊓⊔

8
Device-independent witness of full-incompatibility of two
measurements w.r.t coarse-graining from a single experiment
We now study the following problem: given two
measurements of outcomes d1and d2respectively, how
can one witness the full-incompatibility w.r.t coarse-
graining in a device-independent way from a single
experiment? Here ”single experiment” implies the
Bell test which gives a Bell-inequality violation for the
two measurements of outcomes d1and d2respectively
as chosen as inputs of Alice and some measurement
choices for Bob and a shared entangled state. As we
know if two measurements are 2−incompatible w.r.t
coarse-graining then they are fully-incompatible w.r.t
coarse-graining. Also, we use the result of [3], that any
two binary outcome incompatible measurements violate
the Bell-CHSH inequality. Thus, from the original mea-
surement statistics we do all possible coarse-grainings
to make it two outcome measurements and check the
Bell-CHSH inequalities. If, all possible coarse-grained
two-outcome measurements violate the Bell-CHSH in-
equalities then the measurements are fully-incompatible
w.r.t coarse-graining.
To understand how this technique works, we consider
a well-known example of two three-outcome rank-one
projective measurements in C3which give maximum
violation of the CGLMP inequality [46], a well-studied
[47–49] Bell-type inequality. The projective measure-
ments of Alice are: Aa≡ {|ξ⟩A,a}, where
|ξ⟩A,a=1
√3
2
∑
j=0
exp
i
2π
3j(ξ+αa)|j⟩A, (42)
with a∈ {1, 2}corresponding to two different measure-
ment settings of Alice, and ξ∈ {0, 1, 2},α1=0, α2=1
2.
The projective measurements of Bob are: Bb≡ {|η⟩B,b},
where
|η⟩B,b=1
√3
2
∑
j=0
exp
i
2π
3j(−η+βb)|j⟩B, (43)
with, b∈ {1, 2}corresponding to two different measure-
ment settings of Bob, and η∈ {0, 1, 2},β1=1
4,β2=−1
4.
The incompatibility of the measurements of Eq.(42)
before coarse-graining is guaranteed by the CGLMP-
inequality violation. Now we investigate the incom-
patibility status of the measurements of Alice after all
possible coarse-graining from the measurement statis-
tics obtained in the CGLMP experiment. Alice’s mea-
surements now have two outcomes each and it is a
(2,2,2,3) scenario, i.e., two inputs for Alice and two in-
puts for Bob and each input of Alice has two outcomes
and each input of Bob has three outcomes. For this sce-
nario, Collins and Gisin have shown that there are a to-
tal of 72 CH-facet inequalities [50]. The CH-inequalities
[51] are of the form:
−1≤P(00|A1,B1) + P(00|A1,B2) + P(00|A2,B2)
−P(00|A2,B1)−P(0|A1)−P(0|B2)≤0. (44)
The other CH-inequalities are obtained by (1) inter-
changing A1with A2, (2) interchanging B1with B2, and
(3) interchanging both A1with A2and B1with B2. Con-
sider the scenario where we coarse-grain (0,1) outcomes
for both the inputs of Alice. Lets make the following re-
labeling (0, 1)≡0 and 2 ≡1 for the outcomes of A1and
A2and also consider the clubbing of (0, 1)≡0 outcomes
both for B1and B2. Under this relabelling Eq.(44) takes
the form:
−1≤P(0 0|A1,B1) + P(0 0|A1,B2) + P(0 0|A2,B2)
−P(0 0|A2,B1)−P(0|A1)−P(0|B2)≤0, (45)
where,
P(0 0|Ai,Bj) = P(00|Ai,Bj) + P(01|Ai,Bj)
+P(10|Ai,Bj) + P(11|Ai,Bj), (46)
P(0|Ai) = P(0|Ai) + P(1|Ai),
P(0|Bi) = P(0|Bi) + P(1|Bi),i,j∈ {1, 2}. (47)
Similarly, there are other 8 possible clubbings for
Bob’s measurement outcomes and for each clubbing,
we have facet inequalities similar to Eq.(45). One can
check that when there are the same coarse-graining of
outcomes for both measurement inputs of Alice, we get
a CH-inequality violation. Thus, under these coarse-
grainings, the two measurements of Alice remain in-
compatible. When there are different coarse-grainings,
for some cases we get CH-violation, but for other cases,
we do not get CH-violation. CH-violation under a par-
ticular coarse-graining signifies that the measurements
are incompatible. However, when there is no CH-
violation we can not conclude anything regarding the
incompatibility status. This is depicted in TABLE III.
B. Convex mixing of measurements
Device-independent witness
Here, one can simply use the result that any two
binary-outcome incompatible measurements violate the
Bell-CHSH inequality [3]. Considering the convex mix-
ture of measurements for any three binary-outcome
measurements, this result can be used to witness full in-
compatibility w.r.t. convex mixing of measurements.

9
A1A2Incompatible
(0,1) (0,1) yes
(0,1) (1,2) ?
(0,1) (0,2) yes
(1,2) (0,1) yes
(1,2) (1,2) yes
(1,2) (0,2) ?
(0,2) (0,1) ?
(0,2) (1,2) yes
(0,2) (0,2) yes
TABLE III. The incompatibility status for all possible coarse-
graining (CG) of outcomes for Alice’s measurements (A1and
A2) as inferred from the statistics of CGLMP experiment. Here
“yes” denotes that under the particular CG of A1and A2, the
measurement remains incompatible. On the other hand, “?”
denotes that we can not make any conclusion about their in-
compatibility.
Semi-device-independent witness
For three or more numbers of measurements we
can witness whether the measurements are fully-
incompatible w.r.t convex-mixture in a semi-device in-
dependent manner by constructing a suitable random
access codes task.
Theorem 4. Three noisy Pauli measurements of Eq.(16)are
witnessed to be fully incompatible w.r.t convex-mixing via
RAC iff: q2
3<ν≤1.
Proof. Let us denote the measurements of Eq.(16) as
M0=1
2(
1
+ν0σx),M1=1
2(
1
−ν0σx),N0=
1
2
1
+ν1σy,N1=1
2
1
−ν1σy,R0=1
2(
1
+ν2σz),
and R1=1
2(
1
−ν2σz), and consider the measurements
Qiwith i∈ {1, 2}, to be formed by the convex-mixing of
Mand Nas
Q1={p M0+ (1−p)N0,pM1+ (1−p)N1},
Q2={p M0+ (1−p)N1,p M1+ (1−p)N0}.
In order to witness the incompatibility status of (Qi,R),
one can construct a (2,2,2) RAC task. Now, when Bob
has to guess Alice’s 1st bit, he performs the measure-
ment Qiwhich can be realized by the convex mixing of
measurements Mand Nas follows: Bob tosses a bi-
ased coin which gives a ‘Head’ with probability pand a
‘Tail’ with probability (1−p). On obtaining ‘Head’, Bob
performs measurement M, and for ‘Tail’ he performs
measurement N. For the second bit, he performs mea-
surement R. The maximum average success probability
P(2, 2, 2)that can be obtained in the RAC task (consid-
ering ν0=ν1=ν2) is:
P(2, 2, 2) = 1
8
1
∑
j,k=0||Qi
j+Rk||
=1
4(2+q2ν2−2pν2+2p2ν2). (48)
It can be shown that for all possible convex-mixing the
maximum average success probability is the same as
Eq.(48). Now, to be fully incompatible w.r.t convex-
mixing P(2, 2, 2)>3
4, which implies ν2(p2−p+1)>
1
2(by Eq.(48)). Since the minimum value of (p2−p+1)
is 3
4, so for ν>q2
3the measurements are fully incom-
patible w.r.t convex-mixing.
In Observation 4, we have found that this kind of
three qubit measurements with noise become fully in-
compatible w.r.t convex mixture for ν>q2
3. So, we can
conclude that when these three measurements are fully
incompatible w.r.t convex mixing, quantum advantage
is obtained in a suitable Random Access Code task.
⊓⊔
Examples.— Consider the following three rank-one
projective measurements acting on C3:
X={|0x⟩⟨0x|,|1x⟩⟨1x|,|2x⟩⟨2x|},
Y={|0y⟩⟨0y|,|1y⟩⟨1y|,|2y⟩⟨2y|},
and Z={|0z⟩⟨0z|,|1z⟩⟨1z|,|2z⟩⟨2z|}, (49)
where,
|0x⟩=1
2|0⟩+1
√2|1⟩+1
2|2⟩,
|1x⟩=−1
√2|0⟩+1
√2|2⟩,
|2x⟩=1
2|0⟩− 1
√2|1⟩+1
2|2⟩,
|0y⟩=−
i
2|0⟩+1
√2|1⟩+
i
2|2⟩,
|1y⟩=
i
√2|0⟩+
i
√2|2⟩,
|2y⟩=−
i
2|0⟩− 1
√2|1⟩+
i
2|2⟩,
|0z⟩≡|0⟩,|1z⟩ ≡ |1⟩,|2z⟩ ≡ |2⟩. (50)
We take the following convex mixture of the mea-
surements Xand Y:A={p|0x⟩⟨0x|+ (1−
p)|0y⟩⟨0y|,p|1x⟩⟨1x|+ (1−p)|1y⟩⟨1y|,p|2x⟩⟨2x|+ (1−
p)|2y⟩⟨2y|} with 0 ≤p≤1. We now consider the
(2, 3, 3)RAC game involving the two measurements
Aand Z. The maximum average success probability

10
P(2, 3, 3)for this RAC task is given by [45],
P(2, 3, 3) = 1
18
2
∑
i,j=0||Ai+Zj||, (51)
which turns out to be greater than 2
3i.e. PCB (2, 3, 3)for
all p∈[0, 1]. Hence, the measurements Aand Zare
incompatible.
Next, consider the following measurement (tak-
ing an arbitrary convex mixture of Xand Y)
Ai,j,k,l,m,n={p|ix⟩⟨ix|+ (1−p)|jy⟩⟨jy|,p|kx⟩⟨kx|+ (1−
p)|ly⟩⟨ly|,p|mx⟩⟨mx|+ (1−p)|ny⟩⟨ny|} with 0 ≤p≤1,
i,j,k,l,m,n∈ {0, 1, 2},i=k,k=m,m=iand j=l,
l=n,n=j. Note that the aforementioned measure-
ment Ais nothing but A0,0,1,1,2,2 in the present notation.
Following a similar calculation, it can be shown that any
such Ai,j,k,l,m,nis incompatible with Zfor all p∈[0, 1].
Similarly, taking an arbitrary convex combination of
Xand Z(or, Yand Z), it can be shown that the new
measurement is incompatible with Y(or, X) for all p∈
[0, 1].
A generic description for the semi-device indepen-
dent witness of incompatibility.— Consider a mea-
surement assemblage of nmeasurements: M=
{M0,M1, . . . , Mn−1}. We make mpartitions S=
{Si}m−1
i=0,SiSi=M, and SiTSj=ϕ,∀i,jwith i=j.
Our purpose is to operationally witness the incompat-
ibility of mmeasurements produced from the convex
mixing of nmeasurements from each of the mpartitions.
We can construct a (m,d,d)RAC game where Alice has
an mdit input message, viz., x= (x0,x1, . . . , xm−1). De-
pending upon the message, she encodes it in a qudit
and sends it to Bob, who on the other hand, gets input
y∈ {0, 1, . . . , m−1}and accordingly, he has to predict
the value of the corresponding bit xy. He performs the
measurement which is obtained by the convex mixtures
of the measurements from the partition Syand declares
the outcome of the measurement. Now, if the success
probability P(m,d,d)is greater than PCB (m,d,d)[21] for
all possible convex mixtures, we can operationally de-
tect that the mpartitioned measurements are incompat-
ible under convex mixing.
V. CONCLUSIONS
Measurement incompatibility is the spectacle in quan-
tum theory that a set of measurements cannot be per-
formed jointly on arbitrary systems. It is one of the fun-
damental ingredients for non-classical correlations and
merits of quantum information science. Incompatibility
offers a complex structure with different layers as the
number of measurements and the dimension on which
the measurements act increases. Thus, understanding
the different levels of incompatibility with respect to el-
ementary classical operations is of paramount impor-
tance both from the foundational and practical perspec-
tives. In this work, we have considered the two most
general classical operations viz., coarse-graining of dif-
ferent outcomes of measurements, and convex-mixing
of different measurements.
Through our present analysis we have investigated
the different levels of incompatibility arising under the
above classical operations. Since environmental ef-
fects are ubiquitous in practical scenarios, here we have
investigated the tolerance thresholds for maintaining
measurement incompatibility against noise under clas-
sical operations. Furthermore, we have developed a
method to operationally witness different levels of mea-
surement incompatibility in the device-independent
framework involving Bell-type experiments, and also
in the semi-device-independent framework involving
prepare-and-measure experiments.
Several examples have been provided to illustrate the
efficacy of the operational witnesses proposed here. Our
results are particularly useful for the purpose of compar-
ing measurements in terms of their degree of incompat-
ibility. For example, measurements that remain fully in-
compatible with respect to coarse-graining of outcomes
(or convex-mixing of measurements) show stronger in-
compatibility compared to those that are not fully in-
compatible, thus facilitating legitimate choice of mea-
surements in an information-processing task where a
high degree of incompatibility is required.
Our present study motivates future work on several
open directions. The condition for full incompatibility
for two projective measurements with respect to coarse-
graining of outcomes can be extended for more num-
ber of projective measurements. Similarly, the criterion
for full incompatibility of projective measurements with
respect to convex mixing could be generalized for any
set of measurements. Further investigation would be
required to see if device-independent witnesses could
be formulated for more than three outcome measure-
ments with respect to coarse-graining and more than
three measurements with repect to convex mixing. It
will be interesting to look for examples where the full
incompatibility with respect to coarse-graining can be
inferred in a device-independent way from a single ex-
perimental statistics.
VI. ACKNOWLEDGEMENTS
AKD and ASM acknowledge support from Project
No. DST/ICPS/QuEST/2018/98 from the Department
of Science and Technology, Govt. of India. DD ac-
knowledges the Royal Society (United Kingdom) for the
support through the Newton International Fellowship
(NIF\R1\212007).

11
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