Measurement incompatibility is one of the most striking examples of how quantum physics is different from classical physics. Two measurements are incompatible if they cannot arise via classical post-processing from a third one. A natural way to quantify incompatibility is in terms of noise robustness. In the present article, we review recent results on the maximal noise robustness of incompatible measurements, which have been obtained by the present authors using free spectrahedra, and rederive them using tensor norms. In this way, we make them accessible to a broader audience from quantum information theory and mathematical physics and contribute to the fruitful interactions between Banach space theory and quantum information theory. We also describe incompatibility witnesses using the tensor norm and matrix convex set duality, emphasizing the relation between the different notions of witnesses.

1.
N.
Bohr
, “
The quantum postulate and the recent development of atomic theory
,”
Nature
121
(
3050
),
580
590
(
1928
).
2.
W.
Heisenberg
, “
Über den anschaulichen inhalt der quantentheoretischen kinematik und mechanik
,”
Z. Phys.
43
(
3
),
172
198
(
1927
).
3.
O.
Gühne
,
E.
Haapasalo
,
T.
Kraft
,
J.-P.
Pellonpää
, and
R.
Uola
, “
Incompatible measurements in quantum information science
,” arXiv:2112.06784 (
2021
).
4.
T.
Heinosaari
,
T.
Miyadera
, and
M.
Ziman
, “
An invitation to quantum incompatibility
,”
J. Phys. A: Math. Theor.
49
(
12
),
123001
(
2016
).
5.
A.
Fine
, “
Hidden variables, joint probability, and the Bell inequalities
,”
Phys. Rev. Lett.
48
(
5
),
291
295
(
1982
).
6.
N.
Brunner
,
D.
Cavalcanti
,
S.
Pironio
,
V.
Scarani
, and
S.
Wehner
, “
Bell nonlocality
,”
Rev. Mod. Phys.
86
,
419
478
(
2014
).
7.
T.
Heinosaari
,
J.
Kiukas
, and
D.
Reitzner
, “
Noise robustness of the incompatibility of quantum measurements
,”
Phys. Rev. A
92
,
022115
(
2015
).
8.
P.
Busch
,
T.
Heinosaari
,
J.
Schultz
, and
N.
Stevens
, “
Comparing the degrees of incompatibility inherent in probabilistic physical theories
,”
Europhys. Lett.
103
(
1
),
10002
(
2013
).
9.
S.
Gudder
, “
Compatibility for probabilistic theories
,”
Mathematica Slovaca
66
(
2
),
449
458
(
2016
).
10.
A.
Bluhm
,
A.
Jenčová
, and
I.
Nechita
, “
Incompatibility in general probabilistic theories, generalized spectrahedra, and tensor norms
,” arXiv:2011.06497 (
2020
).
11.
A.
Bluhm
and
I.
Nechita
, “
Joint measurability of quantum effects and the matrix diamond
,”
J. Math. Phys.
59
(
11
),
112202
(
2018
).
12.
A.
Bluhm
and
I.
Nechita
, “
Compatibility of quantum measurements and inclusion constants for the matrix jewel
,”
SIAM J. Appl Algebra Geom.
4
(
2
),
255
296
(
2020
).
13.
M. A.
Jivulescu
,
C.
Lancien
, and
I.
Nechita
, “
Multipartite entanglement detection via projective tensor norms
,”
Ann. Henri Poincaré
(published online) (
2022
).
14.
A.
Jenčová
, “
Assemblages and steering in general probabilistic theories
,” arXiv:2202.09109 (
2022
).
15.
T.
Heinosaari
and
M.
Ziman
,
The Mathematical Language of Quantum Theory
(
Cambridge University Press
,
2011
).
16.
J.
Watrous
,
The Theory of Quantum Information
(
Cambridge University Press
,
2018
).
17.
P.
Busch
and
T.
Heinosaari
, “
Approximate joint measurements of qubit observables
,”
Quantum Inf. Comput.
8
(
8–9
),
797
818
(
2008
).
18.
T.
Brougham
and
E.
Andersson
, “
Estimating the expectation values of spin-1/2 observables with finite resources
,”
Phys. Rev. A
76
,
052313
(
2007
).
19.
P.
Busch
, “
Unsharp reality and joint measurements for spin observables
,”
Phys. Rev. D
33
(
8
),
2253
(
1986
).
20.
R.
Pal
and
S.
Ghosh
, “
Approximate joint measurement of qubit observables through an Arthur–Kelly model
,”
J. Phys. A: Math. Theor.
44
(
48
),
485303
(
2011
).
21.
R. A.
Ryan
,
Introduction to Tensor Products of Banach Spaces
(
Springer
,
2002
).
22.
K. R.
Davidson
,
A.
Dor-On
,
O. M.
Shalit
, and
B.
Solel
, “
Dilations inclusions of matrix convex sets, and completely positive maps
,”
Int. Math. Res. Not.
2017
(
13
),
4069
4130
.
23.
B.
Passer
,
O. M.
Shalit
, and
B.
Solel
, “
Minimal and maximal matrix convex sets
,”
J. Funct. Anal.
274
,
3197
3253
(
2018
).
24.
R.
Horodecki
,
P.
Horodecki
,
M.
Horodecki
, and
K.
Horodecki
, “
Quantum entanglement
,”
Rev. Mod. Phys.
81
(
2
),
865
(
2009
).
25.
B. M.
Terhal
, “
Bell inequalities and the separability criterion
,”
Phys. Lett. A
271
(
5–6
),
319
326
(
2000
).
26.
C.
Carmeli
,
T.
Heinosaari
, and
A.
Toigo
, “
Quantum incompatibility witnesses
,”
Phys. Rev. Lett.
122
(
13
),
130402
(
2019
).
27.
A.
Jenčová
, “
Incompatible measurements in a class of general probabilistic theories
,”
Phys. Rev. A
98
(
1
),
012133
(
2018
).
28.
Y.
Kuramochi
, “
Compact convex structure of measurements and its applications to simulability, incompatibility, and convex resource theory of continuous-outcome measurements
,” arXiv:2002.03504 (
2020
).
29.
A.
Bluhm
and
I.
Nechita
, “
Maximal violation of steering inequalities and the matrix cube
,”
Quantum
6
,
656
(
2022
).
30.
A.
Ben-Tal
and
A.
Nemirovski
, “
On tractable approximations of uncertain linear matrix inequalities affected by interval uncertainty
,”
SIAM J. Optim.
12
(
3
),
811
833
(
2002
).
31.
J. W.
Helton
,
I.
Klep
,
S.
McCullough
, and
M.
Schweighofer
, “
Dilations linear matrix inequalities, the matrix cube problem and beta distributions
,”
Mem. Am. Math. Soc.
257
(
1232
) (
2019
).
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