# Two concepts of noncontextuality
in quantum mechanics

Gábor Hofer-Szabó Research Center for the Humanities, Budapest, email:
<EMAIL_ADDRESS>
###### Abstract

There are two different and logically independent concepts of noncontextuality
in quantum mechanics. First, an ontological (hidden variable) model for
quantum mechanics is called noncontextual if every ontic (hidden) state
determines the probability of the outcomes of every measurement independently
of what other measurements are simultaneously performed. Second, an
ontological model is noncontextual if any two measurements which are
represented by the same self-adjoint operator or, equivalently, which have the
same probability distribution of outcomes in every quantum state also have the
same probability distribution of outcomes in every ontic state. I will call
the first concept simultaneous noncontextuality, the second measurement
noncontextuality. In the paper I will overview and critically analyze some of
the most significant accounts of contextuality in the literature and subsume
them under these two categories.

Keywords: contextuality, simultaneous measurability, Kochen-Specker theorem,
operational equivalence

##  1 Introduction

On Wayne Myrvold’s historical reconstruction,111I thank Wayne Myrvold for the
many valuable information on how the term “contextuality” appeared in the
foundations of quantum mechanics. the term “contextuality” has been introduced
into the foundations of quantum mechanics by Abner Shimony in his contribution
to the 1970 Lake Como conference (Shimony 1993, Ch. 7). Shimony first used the
somewhat contrived term "contextualistic" which has been shortened to
"contextual" in Stuart Freedman’s doctoral dissertation, Experimental Test of
Local Hidden-variable Theories (1972) and has been so used in Beltrametti and
Cassinelli’s book, The Logic of Quantum Mechanics (1981). In his interview for
the AIP oral history project222https://www.aip.org/history-programs/niels-
bohr-library/oral-histories/25643. Shimony ascribes the simplification to
Beltrametti and Cassinelli: “Why didn’t I think of that, instead of leaving it
to non-English speakers to simplify the words?” However, Shimony misremembers
since the term “contextual” already occurred in the Appendix of Clauser and
Shimony’s review article (1978, A.2.2). More interestingly, neither of the two
papers which are considered as the two seminal works on the subject issue,
namely Bell’s 1966 paper and Kochen and Specker’s 1976 paper uses explicitly
the term “contextuality.”

In this paper, however, I am not going to focus on the name of a concept but
rather on how the different authors interpreted this concept. The paper
consists of two parts. In the first part, I introduce the framework of
operational theories and ontological models and accommodate quantum mechanics
in this framework (Sect. 2). Then, I discern two concepts of noncontextuality:
one based on simultaneous measurability and another based on operational
equivalence (Sect. 3). Next, I show how the two concepts connects up to the
two different interpretations of the Kochen-Specker argument and how they are
related to the ontological, environmental and algebraic contextuality (Sect.
4). In the second half of the paper, I overview some of the most significant
accounts of contextuality in the literature and—after discounting
contextuality without interpretation (Sect. 5)—I group the accounts and
authors into two categories: simultaneous contextuality (Sect. 6) and
measurement contextuality (Sect. 7).

In the paper I will argue for the following claims: (i) Simultaneous
noncontextuality and measurement noncontextuality are two different and
logically independent concepts. (ii) To prove that quantum mechanics does not
admit a simultaneous or a measurement noncontextual ontological model requires
two different interpretations of the Kochen-Specker theorems. (iii) Formal and
the sheaf theoretic approaches are lacking a physical interpretation and hence
their contextuality concept is purely mathematical. (iv) Bell 1966 paper
excludes simultaneous noncontextual ontological models and Kochen and Specker
1967 paper excludes measurement noncontextual ontological models. (v)
Spekkens’ and the Bohmian account of noncontextuality is measurement
noncontextuality. (vi) Van Fraassen’s and Redhead’s ontological contextuality
is a special case of measurement contextuality; Shimony’s algebraic
contextuality is a special case of simultaneous contextuality; and Bohr’s
conceptual contextuality and Redhead’s and Shimony’s environmental
contextuality are uncommitted in the simultaneous-measurement noncontextuality
dichotomy.

##  2 Operational theories and ontological models

An operational theory is a theory which specifies the probability of the
outcomes of certain measurements performed on a physical system which was
previously prepared in certain states. Let $S$ be set of states or
preparations of the system, let $M$ be the set of measurements which can be
performed on the system, and let $X_{m}$ be the set of outcomes of the
measurement $m\in M$. Then, an operational theory is a set of conditional
probabilities of the outcomes for the various measurements in the various
preparations:

$\displaystyle\\{p(x_{m}|m\wedge s):\,\,x_{m}\in X_{m},\,\,m\in M,\,\,s\in
S\\}$ (1)

with the obvious normalizations. A preparation $s\in S$ is called an
eigenstate of the measurement $m\in M$ with eigenvalue $x_{ms}\in X_{m}$ if

$\displaystyle p(x_{ms}|m\wedge s)=1$ (2)

Two measurements $m$ and $m^{\prime}$ in $M$ are simultaneously measurable
(commeasurable), if they can be performed on the same system at the same time.
Simultaneous measurability is an empirical question. One can measure the width
and the length of a table at the same time but not whether a piece of wood is
combustible and whether it floats on water. Operationally one identifies
measurements by sets of (laboratory) instructions. Consequently, two
measurements $m$ and $m^{\prime}$ are simultaneously measurable if and only if
there is a measurement which can be identified by the conjunction of the sets
of instructions characterizing $m$ and $m^{\prime}$. We call this measurement
the simultaneous measurement of $m$ and $m^{\prime}$ and denote it by $m\wedge
m^{\prime}$ (which is again a measurement in $M$). If $m$ and $m^{\prime}$ are
not simultaneously measurable, we write $m\wedge m^{\prime}=\emptyset$. Note
that if measurements are identified by instructions and not by their outcome
statistics, the simultaneous measurement of $m$ and $m^{\prime}$ will not be
any measurement which is—after applying appropriate functions to the
outcomes—"operationally equivalent" (Spekkens, 2005) to $m$ and $m^{\prime}$.
If it exists, it will be $m\wedge m^{\prime}$ defined by the conjunction of
the instructions defining $m$ and $m^{\prime}$.

Denote the set of measurements in $M$ which are commeasurable with $m$ by
$M_{\sim m}\subseteq M$. We call an operational theory non-disturbing if no
conditional probability depends on whether the measurements are performed
alone or along with simultaneous measurements:

$\displaystyle p(x_{m}|m\wedge s)=p(x_{m}|m\wedge m^{\prime}\wedge
s)\quad\quad x_{m}\in X_{m},\,\,m\in M,\,\,m^{\prime}\in M_{\sim m},\,\,s\in
S$ (3)

Now, quantum mechanics, at least on the minimal interpretation, is a non-
disturbing operational theory in a linear algebraic representation. Thus, a
physical system is represented by a Hilbert space, each preparation $s\in S$
of the system by a density operator $\boldsymbol{\rho}$, each measurement
$m\in M$ by a self-adjoint operator $\textbf{O}_{m}$, and each outcome $x_{m}$
of $m$ by the eigenvalue $x_{m}$ and the associated spectral projection
$\textbf{P}_{m}^{x}$ of $\textbf{O}_{m}$. The representation is successful if
the Born rule holds:

$\displaystyle\mbox{Tr}(\boldsymbol{\rho}\,\textbf{P}_{m}^{x})=p(x_{m}|m\wedge
s)\quad\quad x_{m}\in X_{m},\,\,m\in M,\,\,s\in S$ (4)

where Tr is the trace function. Note, that the representation of measurements,
outcomes and preparations by self-adjoint operators, projections, and density
operators, respectively is not unique: all is required from the representation
is to satisfy (4).

Simultaneous measurements $m$ and $m^{\prime}$ get represented in quantum
mechanics by commuting operators $\textbf{O}_{m}$ and
$\textbf{O}_{m^{\prime}}$ such that333For the question as to whether
observables represented by noncommuting operators can be simultaneosly
measured, see e.g. (Park and Margenau, 1968).

$\displaystyle\mbox{Tr}(\boldsymbol{\rho}\,\textbf{P}_{m}^{x}\,\textbf{P}_{m^{\prime}}^{x^{\prime}})=p(x_{m}\wedge
x_{m^{\prime}}|m\wedge m^{\prime}\wedge s)\quad\quad x_{m}\in
X_{m},\,\,x_{m^{\prime}}\in X_{m^{\prime}},\,\,s\in S$ (5)

Consequently, $m$ and $m^{\prime}$ will be non-disturbing:

$\displaystyle p(x_{m}|m\wedge m^{\prime}\wedge
s)=\sum_{x_{m^{\prime}}}p(x_{m}\wedge x_{m^{\prime}}|m\wedge m^{\prime}\wedge
s)=\sum_{x_{m^{\prime}}}\mbox{Tr}(\boldsymbol{\rho}\,\textbf{P}_{m}^{x}\textbf{P}_{m^{\prime}}^{x^{\prime}})=$
$\displaystyle\mbox{Tr}(\boldsymbol{\rho}\,\textbf{P}_{m}^{x})=p(x_{m}|m\wedge
s)\quad\quad\quad x_{m}\in X_{m},\,\,s\in S$

That is the quantum mechanical representation of simultaneous measurements by
commuting operators implies that quantum mechanics is a non-disturbing (non-
signaling) operational theory.

Note that simultaneous measurements as used in this paper are not the same as
"joint measurements" standardly used in quantum mechanics. By the "joint
measurement" of $m$ and $m^{\prime}$ represented by the commuting operators
$\boldsymbol{O}_{m}$ and $\boldsymbol{O}_{m^{\prime}}$ in quantum mechanics
one usually refers to any measurement $m^{\prime\prime}$ which can be
represented by an operator $\boldsymbol{O}_{m^{\prime\prime}}$ such that there
are two Borel functions $f$ and $g$ for which
$\boldsymbol{O}_{m}=f(\boldsymbol{O}_{m^{\prime\prime}})$ and
$\boldsymbol{O}_{m^{\prime}}=g(\boldsymbol{O}_{m^{\prime\prime}})$. This joint
measurement $m^{\prime\prime}$, however, need not be the simultaneous
measurement $m\wedge m^{\prime}$ defined by the conjunction of the
instructions individually characterizing $m$ and $m^{\prime}$. It can well be
any other measurement procedure. On the other hand, the simultaneous
measurement $m\wedge m^{\prime}$ is a "joint measurement" in the sense that it
can be represented by the operator $\boldsymbol{O}_{m^{\prime\prime}}$
satisfying the above requirements.

The role of an ontological model (hidden variable model) is to account for the
conditional probabilities of an operational theory in terms of underlying
realistic entities of the measured system called ontic states (hidden
variables, elements of reality, beables). Let the set of ontic states be
$\Lambda$. An ontological model specifies a probability distribution over the
ontic states associated with each preparation:

$\displaystyle\\{p({\lambda}|s):\,\,{\lambda}\in\Lambda,\,\,s\in S\\}$ (6)

and a set of response functions that is a set of conditional probabilities
associated with every measurement and every ontic state:

$\displaystyle\\{p(x_{m}|m\wedge{\lambda}):\,\,x_{m}\in X_{m},\,\,m\in
M,\,\,{\lambda}\in\Lambda\\}$ (7)

again with the obvious normalizations.

Then, assuming no-conspiracy:

$\displaystyle p({\lambda}|s)=p({\lambda}|s\wedge m)\quad\quad\,\,m\in
M,\,\,s\in S$ (8)

(the independence of the probability distributions from the measurements
performed on the system) and ${\lambda}$-sufficiency:

$\displaystyle p(x_{m}|m\wedge{\lambda})=p(x_{m}|m\wedge{\lambda}\wedge
s)\quad\quad x_{m}\in X_{m},\,\,m\in M,\,\,{\lambda}\in\Lambda,\,\,s\in S$ (9)

(the independence of the response functions from the preparations in which the
ontic states are featuring) and using the theorem of total probability one can
recover the operational theory from the ontological model in terms of the
probability distributions and response functions:

$\displaystyle p(x_{m}|m\wedge s)$ $\displaystyle=$
$\displaystyle\sum_{{\lambda}\in\Lambda}p(x_{m}|m\wedge{\lambda}\wedge
s)\,p({\lambda}|s\wedge m)$ (10) $\displaystyle=$
$\displaystyle\sum_{{\lambda}\in\Lambda}p(x_{m}|m\wedge{\lambda})\,p({\lambda}|s)\quad\quad
x_{m}\in X_{m},\,\,m\in M,\,\,s\in S$

Similarly to (2), an ontic state ${\lambda}$ is called an eigenstate of the
measurement $m$ with eigenvalue $x_{m{\lambda}}\in X_{m}$ if

$\displaystyle p(x_{m{\lambda}}|m\wedge{\lambda})=1$ (11)

An ontological model is called outcome-deterministic (value-definite) if every
${\lambda}\in\Lambda$ is an eigenstate of every $m\in M$; otherwise it is
called outcome-indeterministic.

##  3 Two concepts of noncontextuality

There are two different and logically independent concepts of noncontextuality
in quantum mechanics. The first is motivated by the following question: Why is
an operational theory non-disturbing? Why simultaneous measurements do not
alter the outcome or the probability of an outcome of a measurement in the
various preparations. A possible answer is to say: because they don’t even
alter it in the ontic states. This idea is made precise in the concept of
noncontextuality:

An ontological model is noncontextual if every ontic state determines the
probability of the outcomes of every measurement independently of what other
measurements are simultaneously performed:

$\displaystyle p(x_{m}|m\wedge{\lambda})=p(x_{m}|m\wedge
m^{\prime}\wedge{\lambda})\quad\quad x_{m}\in X_{m},\,\,m\in
M,\,\,m^{\prime}\in M_{\sim m},\,\,s\in S$ (12)

Otherwise the model is contextual.

Thus, noncontextuality is a kind of inference to the best explanation for why
an operational theory is non-disturbing: if the ontological model for an
operational theory is noncontextual in the sense of (12), then—assuming no-
conspiracy (8) and ${\lambda}$-sufficiency (9)—one can show that the
operational theory is non-disturbing (3).

But what about disturbing operational theories?

If two measurements $m$ and $m^{\prime}$ are disturbing, then a reasonable
attitude444See Bridgman, 1958 and Hofer-Szabó, 2017, Sec. 7. is to say that
the measurements are not yet well-defined. In other words, one should cluster
the measurements in a more refined way: instead of taking two measurements:
$m$ and $m^{\prime}$, one should take three measurements: $m_{1}\equiv m\wedge
m^{\prime}$ (measuring both $m$ and $m^{\prime}$), $m_{2}\equiv
m\,\wedge\\!\sim\\!m^{\prime}$ (measuring $m$ but not $m^{\prime}$), and
$m_{3}\equiv m^{\prime}\,\wedge\\!\sim\\!m$ (measuring $m^{\prime}$ but not
$m$). By this move we can eliminate disturbance since the three new
measurements will be logically mutually exclusive. They cannot be
simultaneously measured and hence cannot disturb one another.

However, replacing the old measurements by the new ones alters the operational
theory. The new fine-grained measurements, being logically mutually exclusive,
cannot be simultaneously performed. Hence, noncontextuality cannot be defined
since $m_{i}\wedge m_{j}=\emptyset$ on the right hand side of (12). In short,
in a disturbing operational theory one faces the following dilemma: either we
leave the disturbing measurements as they are but then why to require
noncontextuality from the ontological model; or we replace the disturbing
measurements by mutually exclusive measurements but then noncontextuality will
hold vacuously. In both cases noncontextuality becomes meaningless.

Now, the term “context” in the above definition is coming from the everyday
discourse, where it refers to the circumstances in which a certain event or
measurement takes place. These circumstances are not constitutive in the
definition of the very event or measurement but can significantly influence
the occurrence of the event or the result of the measurement. The important
aspect of these circumstances is that they are simultaneously present with the
event or measurement. If I ask you whether you want an ice-cold coke, your
answer will greatly depend on whether you are on a sunny beach or on the
skating rink at the time of the question. A typical context in physics of a
measurement is another measurement which is performed simultaneously with the
first. In this sense noncontextuality refers to a kind of robustness of a
given system to respond in a definite way to a measurement when simultaneous
measurements are also performed on the system.

There is, however, another meaning of noncontextuality which is completely
independent of the above one. This noncontextuality relies not on simultaneous
measurability but on the association of measurements with observables. The
term “observable” is missing in the vocabulary of an operational theory since
operational theories directly connect measurements to the mathematical
representants. But observables are all the more important in standard physical
theories. Observables are measurable physical magnitudes which characterize a
given physical system. Thus, each observable is associated with a measurement
but this association is not necessarily one-to-one. One can measure the same
observable in different ways, for example distance by measuring rods and by
light signals or temperature by alcohol and gas thermometer. In this case the
two measurements have the same outcome or the same probability distribution of
outcomes in those preparations in which they both can be performed. In quantum
mechanics such measurements are represented by the same self-adjoint operator
via (4). This leads to the second concept of noncontextuality in quantum
mechanics which can be motivated by the following question: Why do different
measurements have the same probability distribution of outcomes in every
preparation? Again, a reasonable answer is to say that because they measure
the same observable which has a fixed value or a fixed probability
distribution of values in every ontic state and the faithful measurements
simply reveal this value or this distribution. Operationally, this means that
the “operationally equivalent” (Spekkens, 2005) measurements have the same set
of response functions. Thus, the second concept of noncontextuality reads as
follows:

An ontological model is noncontextual if any two measurements which have the
same probability distribution of outcomes in every preparation (and thus can
be represented by the same self-adjoint operator) also have the same
probability distribution of outcomes in every ontic state. Formally, let
$m,m^{\prime}\in M$ and suppose that

$\displaystyle p(x_{m}|m\wedge s)=p(x_{m^{\prime}}|m^{\prime}\wedge
s)\quad\quad x_{m}\in X_{m},\,\,x_{m^{\prime}}=h(x_{m})\in
X_{m^{\prime}},\,\,s\in S$ (13)

where $h:X_{m}\to X_{m^{\prime}}$ is a bijection between the outcomes of $m$
and $m^{\prime}$. Then

$\displaystyle
p(x_{m}|m\wedge{\lambda})=p(x_{m^{\prime}}|m^{\prime}\wedge{\lambda})\quad\quad
x_{m}\in X_{m},\,\,x_{m^{\prime}}=h(x_{m})\in
X_{m^{\prime}},\,\,{\lambda}\in\Lambda$ (14)

Otherwise the model is contextual.

This second noncontextuality is again a kind of inference to the best
explanation: (14)—together with no-conspiracy (8) and ${\lambda}$-sufficiency
(9)—implies (13). That is, if an ontological model is noncontextual in the
sense of (13)-(14), we obtain a neat explanation for why certain measurements
have the same probability distribution of outcomes.

Let us call the above first concept of noncontextuality (12) simultaneous
noncontextuality and the second concept (13)-(14) measurement noncontextuality
(following Spekkens, 2005).

Note that measurement noncontextuality and simultaneous noncontextuality are
two logically independent concepts. Measurement noncontextuality does not rely
on simultaneous measurability, while simultaneous contextuality does. If there
are no simultaneous measurements in an operational theory, then each
ontological model of the theory will be simultaneous noncontextual since (12)
fulfills vacuously. Still, the model can violate measurement noncontextuality
if there are measurements yielding the same probability distribution of
outcomes in every preparation, still differing in their response
functions.555For an example see (Hofer-Szabó, 2021a,b). Conversely, if premise
(13) is not satisfied in an operational theory, then measurement
noncontextuality fulfills vacuously. Still, the ontological model can be
simultaneous contextual—but only if the theory is disturbing. In a non-
disturbing operational theory for any two simultaneous measurements $m$ and
$m^{\prime}$ (13) fulfills for $m$ and $m\wedge m^{\prime}$. If the
ontological model is measurement noncontextual, then (14) also fulfills for
$m$ and $m\wedge m^{\prime}$. But this is just simultaneous noncontextuality
(12). In short, in a non-disturbing operational theory (like in quantum
theory) measurement noncontextuality implies simultaneous noncontextuality.

##  4 Noncontextuality in the Kochen-Specker arguments

In the previous section measurement noncontextuality and simultaneous
noncontextuality were defined at the general level of operational theories and
ontological models without any reference to quantum mechanics.666Modulo the
parenthetical note in the definition of measurement noncontextuality which can
be dropped. But quantum mechanics is an operational theory, hence both
definitions apply. Thus, one can ask: Is quantum mechanics simultaneous and
measurement contextual?

This question is commonly studied in the Kochen-Specker arguments.777See
(Bell, 1966/2004), (Kochen and Specker, 1967), (Redhead, 1989, Ch. 5) and
(Held, 2018). Kochen-Specker arguments have two parts: a Kochen-Specker
theorem and a physical interpretation. The Kochen-Specker theorems proceed in
the following steps: (i) one starts with a set of self-adjoint operators; (ii)
introduces value assignments on the operators sending each operator into one
of its eigenvalues; (iii) restricts these value assignments by the so-called
functional composition principle such that the values assigned to mutually
commuting operators are the eigenvalues of one of the common eigenstates of
these operators; and (iv) finally shows that there is no such value
assignment.

The Kochen-Specker arguments are not simply mathematical theorems. They intend
to prove that there is no noncontextual outcome-deterministic ontological
model underlying quantum mechanics. To this aim, however, the operators in the
Kochen-Specker theorems need to be physically interpreted. An interpretation
is an association of operators with measurements in an operational theory (1)
and with an underlying ontological model (6)-(7).

Now, there are two conceptually different interpretations of the Kochen-
Specker theorems. As we will shortly see, on the first interpretation the
Kochen-Specker theorems rule out simultaneous noncontextual ontological
models, on the second interpretation measurement noncontextual ontological
models. The difference between the interpretations in brief is the following:
In the first interpretation one directly associates each operator with a
measurement in a one-to-one way such that mutually commuting operators
represent simultaneous measurements (in the physical sense elucidated in
Section 2). In the second interpretation one does not associate each operator
with a different measurement but rather each (maximal) set of mutually
commuting operators with a measurement. Consequently, operators sitting in
more than one such set will be associated with more than one measurement. In
short, in this second interpretation the association of some operators with
measurements is of one-to-many type.

An example might help. Consider the Peres-Mermin-version of the Kochen-Specker
theorem. In this theorem one has 9 operators arranged in a 3 $\times$ 3 matrix
such that the operators are mutually commuting if and only if they are in the
same row or in the same column. Thus, one has 6 triples of mutually commuting
operators: the operators in the three rows and in the three columns. Now, the
two interpretations differ in the association of the operators with the
measurements. In the first interpretation, one associates each operator with a
measurement in a one-to-one way such that mutually commuting operators
represent simultaneous measurements. Thus, in this operational theory one has
altogether 9 different measurements. In the second interpretation, a triple of
mutually commuting operators is interpreted not as three different
measurements but as three different functions of one and the same measurement.
Thus, here one associates not each operator but each triple of mutually
commuting operators with a measurement. Thus, in this second operational
theory one has altogether 6 different measurements associated with the 6
different triples. The association of operators and measurements, however, is
not one-to-one: each operator is associated with two different measurements
via the two different commuting triples which it is an element of.

Now, the let’s see the two interpretations in a little more detail. In the
first interpretation, each ontic state ${\lambda}$ is associated with a value
assignment $v_{\lambda}$ such that the value assigned to an operator ${\bf
O}_{m}$ is an eigenvalue of ${\bf O}_{m}$ which conforms to the
eigenvalue888See eq. (11) above. $x_{m{\lambda}}$ in the ontic state
${\lambda}$ of the measurement $m$ (uniquely) associated with ${\bf O}_{m}$:

$\displaystyle v_{\lambda}({\bf O}_{m})=x_{m{\lambda}}\in X_{m}$ (15)

(15) renders the ontological model outcome-deterministic and also simultaneous
noncontextual since $p(x_{m{\lambda}}|m\wedge{\lambda})=1$ implies that
$p(x_{m{\lambda}}|m\wedge m^{\prime}\wedge{\lambda})=1$ for any measurement
$m^{\prime}\in M_{\sim m}$.

If ${\bf O}_{m},{\bf O}_{m^{\prime}},\dots$ are mutually commuting operators
in the Kochen-Specker theorem, then the associated measurements
$m,m^{\prime},\dots$ are simultaneously measurable, that is $m\wedge
m^{\prime}\wedge\dots\in M$. The functional composition principle restricts
the possible joint outcomes of this simultaneous measurement: only those joint
outcomes are allowed which are eigenvalues of one of the common eigenstates of
the operators ${\bf O}_{m},{\bf O}_{m^{\prime}},\dots$.

Now, since there is no value assignment on the Kochen-Specker operators
satisfying the functional composition principle, there will be no ontological
model satisfying (15). Thus, on the first interpretation, the Kochen-Specker
arguments rule out outcome-deterministic simultaneous noncontextual
ontological models.

The second interpretation is based on the following mathematical fact: for any
set of mutually commuting operators $\\{{\bf O}_{i}\\}$ there is a “fine-
grained” operator ${\bf O}^{\prime}$ and a set of Borel functions
$\\{f_{i}\\}$ such that

$\displaystyle{\bf O}_{i}=f_{i}({\bf O}^{\prime})$ (16)

(The prime indicates that ${\bf O}^{\prime}$ is not necessarily featuring in
the original Kochen-Specker theorem.) Now, the crux of the second
interpretation is that one does not directly associate a measurement with
every single operator in the set $\\{{\bf O}_{i}\\}$; rather one associates a
measurement $m^{\prime}$ only with the fine-grained operator ${\bf
O}^{\prime}$. Consequently, each operator in the set $\\{{\bf O}_{i}\\}$ will
represent a different function of the very same measurement $m^{\prime}$. For
example, ${\bf O}_{i}$ will represent $f_{i}(m^{\prime})$: “perform
$m^{\prime}$ and apply the function $f_{i}$ to the result.”

But then operators sitting in two mutually commuting sets will be “multiply
realized.” If ${\bf O}_{i}$ is both in $\\{{\bf O}_{i}\\}$ and $\\{{\bf
O}_{j}\\}$ where $\\{{\bf O}_{i}\\}$ and $\\{{\bf O}_{j}\\}$ are two different
sets of mutually commuting operators, then there will be two different fine-
grained operators ${\bf O}^{\prime}$ and ${\bf O}^{\prime\prime}$ such that
${\bf O}_{i}=f_{i}({\bf O}^{\prime})=g_{i}({\bf O}^{\prime\prime})$ and hence,
if ${\bf O}^{\prime}$ represents $m^{\prime}$ and ${\bf O}^{\prime\prime}$
represents $m^{\prime\prime}$, ${\bf O}_{i}$ will represent two different
measurements: $f_{i}(m^{\prime})$ and $g_{i}(m^{\prime\prime})$.

In this second interpretation, one then assigns values not directly to the
Kochen-Specker operators but to the fine-grained operator ${\bf
O}^{\prime},{\bf O}^{\prime\prime},\dots$ representing $m,m^{\prime},\dots$.
For the fine-grained operator ${\bf O}^{\prime}$ the assigned value in the
ontic state ${\lambda}$ is again the eigenstate $x_{m^{\prime}{\lambda}}$,
that is the unique outcome of the measurement $m^{\prime}$ in the ontic state
${\lambda}$:

$\displaystyle v_{\lambda}({\bf O}^{\prime})=x_{m^{\prime}{\lambda}}\in
X_{m^{\prime}}$ (17)

Now, (17) gives a well-defined value assignment on the original Kochen-Specker
operators only if each Kochen-Specker operators gets the same value
independently of which fine-grained operator it is derived from. In case of
${\bf O}_{i}$ this boils down to the requirement that

$\displaystyle
f_{i}(x_{m^{\prime}{\lambda}})=g_{i}(x_{m^{\prime\prime}{\lambda}})$ (18)

But this is exactly measurement noncontextuality (13)-(14): The two
measurements $f_{i}(m^{\prime})$ and $g_{i}(m^{\prime\prime})$ satisfy the
antecedent (13) since they are both represented by same operator ${\bf O}_{i}$
and they also satisfy the consequent (14) since from (18) it follows that:

$\displaystyle
p(f_{i}(x_{m^{\prime}{\lambda}})|f_{i}(m^{\prime})\wedge{\lambda})$
$\displaystyle=$ $\displaystyle
p(g_{i}(x_{m^{\prime\prime}{\lambda}})|g_{i}(m^{\prime\prime})\wedge{\lambda})\quad\quad{\lambda}\in\Lambda$
(19)

Now, since there is no value assignment satisfying the functional composition
principle, there will be no ontological model satisfying (16)-(19). Thus, on
this second interpretation, the Kochen-Specker arguments rule out outcome-
deterministic measurement noncontextual ontological models.

Some remarks on the two interpretations are in place here:

a. Note that simultaneous measurability pops up only in the first
interpretation. The measurements $\\{f_{i}(m^{\prime})\\}$ represented by
$\\{{\bf O}_{i}\\}$ in the second interpretation can be called “simultaneous
measurements” only metaphorically: $m^{\prime}$ is one single measurement to
the result of which one simply applies different functions. Conversely, in the
first interpretation there is no mention of measurement noncontextuality since
there are no two measurements associated with the same operator.

b. If a Kochen-Specker theorem has a first interpretation, it also has a
second interpretation: If each operator ${\bf O}_{i}$ represents a separate
measurement $m_{i}$ and commuting operators represent simultaneous
measurements, then one can associate with each set of mutually commuting
operators the simultaneous measurement $m^{\prime}=\wedge_{i}m_{i}$.
Consequently, ${\bf O}_{i}$ will be associated with all simultaneous
measurements which includes $m_{i}$.

c. The above interpretations are only the “pure” interpretations of the
Kochen-Specker theorems. To be sure, one can also provide a “mixed”
interpretation in which one associates measurements with Kochen-Specker
operators in certain commuting sets and with the fine-grained operators in
other commuting sets. For example, one can associate the three operators in
the first row of the Peres-Mermin square with (three different functions of)
one single measurement and uniquely associate the other six operators in the
second and third row with six different measurements. Kochen-Specker arguments
based on such mixed interpretations, however, neither rule out simultaneous
nor measurement noncontextual ontological models since either assumption can
be blamed for the no-go result.

The two interpretations of the Kochen-Specker theorems can also be expressed
in terms of observables $\\{{\cal O}_{i}\\}$ rather then measurements. The
first interpretation is simple: one just uniquely associates each operator
with an observable and a measurement. In the second interpretation, however,
one needs to choose: either one associates observables with operators or with
measurements in a one-to-one way. If ${\bf O}_{i}$ represents two measurements
$f_{i}(m^{\prime})$ and $g_{i}(m^{\prime\prime})$, then one can introduce
either one observable associated with the operator ${\bf O}_{i}$ or two
observables associated with the measurements $m^{\prime}$ and
$m^{\prime\prime}$.

In the first case, the operator ${\bf O}_{i}$ is uniquely associated with an
observable $\mathcal{O}_{i}$ which can be measured in two different ways:
either as $f_{i}(m^{\prime})$ or as $g_{i}(m^{\prime\prime})$. This is the
account of measurement noncontextuality given in Section 3. In the second
case, we have two observables $\mathcal{O}^{\prime}$ and
$\mathcal{O}^{\prime\prime}$ operationally associated with the measurements
$m^{\prime}$ and $m^{\prime\prime}$, respectively. But then ${\bf O}_{i}$ will
be associated with two observables: $f_{i}(\mathcal{O}^{\prime})$ and
$g_{i}(\mathcal{O}^{\prime\prime})$. I will refer to this one-to-many
association of operators and observables as general ontological contextuality.
In the literature,999See Subsection 7.2. ontological contextuality is used in
a narrower sense, namely when the association of operators and observables is
one-to-one for maximal (non-degenerate) operators but one-to-many for
nonmaximal (degenerate) operators. I will call this latter type of
contextuality special ontological contextuality. All in all, ontological
contextuality is just the sum of two facts: the existence of operationally
equivalent measurements plus the operational association of observables with
measurements.

The two cases lead to two different accounts of measurement contextuality in
terms of observables. Define the (measured) value of an observable
$\mathcal{O}$ in the ontic state ${\lambda}$ as the eigenvalue
$x_{m{\lambda}}$ of the measurement $m$ associated with the observable
$\mathcal{O}$ in the ontic state ${\lambda}$. Denote this value by
$[\mathcal{O}]_{\lambda}$. Now, in the first case one has only one observable
$\mathcal{O}_{i}$; its value, however, will depend on which measurement it is
measured with. In the second case one has two observables
$\mathcal{O}^{\prime}$ and $\mathcal{O}^{\prime\prime}$ with values
$x_{m^{\prime}{\lambda}}$ and $x_{m^{\prime\prime}{\lambda}}$, respectively.
Thus, in the first case the violation of measurement noncontextuality (18)
means that

$\displaystyle[\mathcal{O}_{i}]_{\lambda}(f_{i}(m^{\prime}))\neq[\mathcal{O}_{i}]_{\lambda}(g_{i}(m^{\prime\prime}))$
(20)

that is the value of $\mathcal{O}_{i}$ in a fixed ontic state depends on
whether the observable is measured via $f_{i}(m^{\prime})$ or
$g_{i}(m^{\prime\prime})$. This dependence is a special case of environmental
contextuality. Environmental contextuality is the dependence of the outcome of
a measurement or the value of an observable on the environment of the system.
Depending on how the term “environment” is characterized, environmental
contextuality can mean many different things. In the literature “environment”
typically means: (i) the measurement itself with which the observable is
measured, (ii) the macrostate of the measuring apparatus, (iii) the presence
or absence of other simultaneously performed measurements, etc. (20) is
environmental contextuality in the sense of (i).

In the second case, the meaning of measurement contextuality is much simpler:
due to ontological contextuality $\mathcal{O}^{\prime}$ and
$\mathcal{O}^{\prime\prime}$ are two different observables, therefore it is
not surprising that in a given ontic state their value differ:

$\displaystyle f_{i}([\mathcal{O}^{\prime}]_{\lambda})\neq
g_{i}([\mathcal{O}^{\prime\prime}]_{\lambda})$ (21)

To sum up, the two different associations of observables with operators and
measurements lead to two different accounts of measurement contextuality in
terms of observables, the one based on environmental contextuality in the
sense of (i), the other on ontological contextuality.

Finally, note that simultaneous contextuality is also a special case of
environmental contextuality but in the sense of (iii). Following Shimony
(2008) I will call environmental contextuality in the sense of (iii) algebraic
contextuality.101010See Subsection 7.2. Also note that simultaneous
contextuality is not related in any sense to ontological contextuality.

##  5 Contextuality without interpretation

In this second part of the paper I will overview some of the most significant
accounts of contextuality in the literature and cast them into two groups:
simultaneous contextuality (Sections 6) and measurement contextuality
(Sections 7). In categorizing the authors I will be careful not to rely too
much on how they use the term “simultaneous measurements” since in the
physicists’ parlance “simultaneous measurements” and “commuting operators” are
often used synonymously. Rather, I will identify the position of the authors
on the basis of their interpretation of the Kochen-Specker theorems. Namely,
if in interpreting a Kochen-Specker theorem they uniquely associate operators
with measurements (observables) and declare explicitly or think implicitly
that commuting operators represent simultaneous measurement, then they
presumably fall into the first group. If they associate some sets of mutually
commuting operators with one single measurement (observable) and consequently
they associate some operators with more than one measurement, then they
supposedly fall into the second group.

Let me make it clear, however, that the upcoming discussion is not meant as a
comprehensive historical analysis of contextuality in quantum theory. Rather
it is an attempt to “course-grain” the most prominent authors into two groups
around the two different concepts contextuality.

Before, however, I need to separate a third type of contextuality from the
above two which I will still call, for lack of a better word, contextuality:
contextuality without interpretation. Note that both simultaneous and
measurement noncontextuality were based on a physical interpretation of
quantum mechanics that is an association of the operators (operationally
defined) measurements. In the absence of such an association contextuality
remains uninterpreted, that is a pure mathematical concept. The upcoming two
approaches provide different examples to such a formal understanding of
contextuality.

###  5.1 Formal contextuality

In the mathematically oriented approach to the foundations of quantum
mechanics111111Especially in quantum information, see e.g. Klyachko et al.
2008; Bub and Stairs, 2009; Cabello, 2013. contextuality is often treated at
the purely formal level. In this approach contextuality is simply identified
with the construction of a Kochen-Specker theorem that is the impossibility to
provide a value assignment on a tricky set of self-adjoint operators
satisfying the functional composition principle.

Needless to say, that if the operators featuring in these theorems (along with
commutativity, value assignment, functional composition principle, etc.) are
not interpreted physically, then contextuality remains a purely mathematical
concept without any reference to the outer world. Sometimes this lack of
interpretation is concealed by an inadequate terminology which calls operators
“observables,” projections “events,” commutativity “simultaneous
measurability,” and a complete set of orthogonal projections a “context.”
These Janus-faced expressions might contribute to blur the boundaries between
mathematics and physics and convey the impression as if the formalism
automatically brought with it a physical interpretation. However, if the goal
is to prove that nature is contextual, then the physical interpretation of
these terms needs to be given explicitly. Note, however, that when I stick to
a physical interpretation I don’t expect a full-fledged interpretation of
quantum mechanics but simply a minimal one where operators are associated with
operationally defined measurements.

###  5.2 Sheaf theory

There is an elegant treatment of contextuality, the sheaf theoretic
approach.121212See Isham and Butterfield, 1998; Butterfield and Isham, 1999;
Abramsky et al. 2011, 2017. The basic ingredients of the approach are: the set
of measurements $M$ and the set of outcomes $X$; the set of contexts
$\\{C_{k}\\}=\big{\\{}\\!\\{m_{i_{k}}\\}\\!\big{\\}}$ that is the set of
maximal subsets of compatible measurements; and for each context $C_{k}$ a
probability distribution $p_{k}$ on the set of functions: $C_{k}\to X$. The
probability distributions are consistent in the sense that for any two
contexts $C_{k}$ and $C_{l}$, the distributions $p_{k}$ and $p_{l}$
marginalize to the same distribution on $C_{k}\wedge C_{l}$. These probability
distributions are said to satisfy the sheaf condition if there is a
probability distribution $d$ on $M$, called global section, which marginalizes
to the distribution $d_{k}$ on each contexts $C_{k}$. One of the theory’s goal
is to set up conditions on which a set of probability distributions on
contexts can have a global section.

Now, if measurements, outcomes, etc. in the sheaf theory represent physical
measurement, their outcomes, etc. and compatibility means simultaneous
measurability, then the above formalism boils down to an operational theory
(1) in a fixed preparation and the sheaf condition provides a necessary and
sufficient condition for the operational theory to have a simultaneous
noncontextual outcome-deterministic (or more generally a
factorizing131313Noncontextual non-factorizing ontological models are not
excluded by the sheaf-theoretic arguments (and neither by the Kochen-Specker
arguments). See for example Spekkens 2005, Sec. 8.B.) ontological model.

However, sheaf-theory, as a mathematical theory, leaves the compatibility
relation uninterpreted. Moreover, compatibility is intentionally devised to be
so general as to cover a wide range of mathematical and physical
interpretations: it can represent commutativity in quantum theory, consistency
in database theory, simultaneous measurement in operational theories, etc. As
Abramsky et al. put it:

> It should be noted that measurement covers provide a very general way of
> representing compatibility relationships. Of course, a physical
> interpretation in particular circumstances will give rise to specific
> structures of this kind. (Abramsky et al. 2011, p. 8)

In this respect, it is very instructive to see how Christopher Isham and
Jeremy Butterfield, the authors of one the programmatic papers on the sheaf
(topos) theoretic approach to quantum mechanics explicitly opt for a formal
treatment relying on operators rather than a physical approach relying on
physical observables:

> In a quantum theory, a physical quantity $A$ is represented by a self-
> adjoint operator $\bf{A}$ on the Hilbert space of the system, and the first
> thing one has to decide is whether to regard a valuation as a function of
> the physical quantities themselves, or on the operators that represent them.
> From a mathematical perspective, the latter strategy is preferable, and we
> shall therefore define a (global) valuation to be a real-valued function $V$
> on the set of all bounded, self-adjoint operators. (Isham and Butterfield,
> 1998, p. 2670)

And they go on to study value assignments to self-adjoint operators. This move
is very similar to the Kochen-Specker theorems where the point is to provide a
coloring no-go result on a set of operators. But as we saw before, neither
Kochen-Specker theorems nor sheaf theory per se can prove quantum
contextuality until the mathematical terms featuring in the arguments are not
interpreted physically.

But now let’s turn to the two contextuality concepts with physical
interpretation.

##  6 Simultaneous contextuality

###  6.1 Bell and Shimony

In his seminal paper reviewing the hidden variable program Bell (1966)
provided a no-go theorem against outcome-deterministic noncontextual
ontological models for quantum mechanics (without uttering the word
“contextual”). Bell’s no-go result is based on three one-dimensional
projections, the first two of which are orthogonal to the third but not to one
another (therefore the argument works only in at least three dimensions). Bell
assigns $0$ or $1$ values to these projections; applies the sum rule141414A
special case of the functional composition principle which requires that the
values assigned to a complete set of orthogonal projections should sum up to
$1$. to these projections; and derives a contradiction. In interpreting the
contradiction, Bell is wondering on how the sum rule, “an assumption in which
only commuting operators were explicitly mentioned,” can lead to a no-go
result making also use of noncommuting operators. He concludes:

> The danger in fact was not in the explicit but in the implicit assumptions.
> It was tacitly assumed that measurement of an observable must yield the same
> value independently of what other measurements may be made simultaneously.
> (Bell, 1966/2004, p. 9)

It becomes clear from Bell’s wording that he uses the term “simultaneous
measurements” as a synonym for “commuting operators.”151515At the beginning of
his paper Bell is more explicit on this: “Observables with commuting operators
can be measured simultaneously” (Bell, 1966/2004, p. 2). The observable
represented by the third projection can be simultaneously measured with the
observable represented by the first or second orthogonal projection but not
with both.

> These different possibilities require different experimental arrangements;
> there is no a priori reason to believe that the results for [the third
> projection] should be the same. (Bell, 1966/2004, p. 9)

That is measuring the third observable simultaneously with the first or with
the second observable (or just alone) requires two (three) physically
different measurements settings and one can avoid the contradiction by
assigning different values to the third observable depending on which
measurement it is simultaneously measured with.

In other words, Bell interprets his own no-go result along the lines of the
one operator-one measurement interpretation: he associates each of the three
projections with a different observable (and a corresponding measurement) such
that commuting operators represent simultaneous measurements. On this
interpretation, his no-go results rule out simultaneous noncontextual
(outcome-deterministic) ontological models for quantum theory. This reading of
Bell’s understanding of noncontextuality is all the more consistent with his
locality concept as the robustness of the system’ response to a measurement
against remote simultaneous measurements.

Following Bell, many authors working in the foundations of quantum mechanics
interpret noncontextuality as simultaneous noncontextuality. Here I mention
only Abner Shimony’s characterization of a noncontextual (outcome-
deterministic) hidden variable model:161616See also Shimony, 1984, p. 36.

> A noncontextual hidden variables model postulated that an isolated physical
> system is characterized by a complete state ${\lambda}$, which is the
> compendium of the real properties of the system at a definite time…When
> ${\lambda}$ is given, then the result of measuring any property $A$ of the
> system at the given time by an ideal measuring apparatus…is a function
> $A({\lambda})$. The outcome of the measurement is assumed to be independent
> of other properties $B$, $C$, …that may be measured simultaneously with $A$,
> and indeed such an independence may be tacitly assumed to be intrinsic to
> the ideal character of the measurement process. (Shimony, 2009, p. 289)

###  6.2 Bohr’s conceptual, Redhead’s environmental and Shimony’s algebraic
contextuality

Bell explicates simultaneous contextuality as follows:

> The result of an observation may reasonably depend not only on the state of
> the system (including hidden variables) but also on the complete disposition
> of the apparatus. (Bell, 1966/2004, p. 9)

Thus, Bell interprets simultaneous contextuality as a special case of
environmental contextuality, that is the impact of the measurement apparatus
on the system. This impact can be understood in two different ways: either as
a causal influence or a kind of conceptual or definitional relation. A
prominent representative of this latter type of influence is Niels Bohr to
whom Bell is referring at the end of his lastly quoted passage.171717Cf.
“Bell, by a judo-like maneuverer, cited Bohr in order to vindicate a family of
hidden variables theories in which the values of observables depend not only
upon the state of the system but also upon the context.” (Shimony, 1984, p.
41) Here Bohr draws attention to

> …the impossibility of any sharp distinction between the behavior of atomic
> objects and the interaction with the measuring instruments which serve to
> define the conditions under which the phenomena appear. (Bohr, 1949)

In another quote Bohr is more explicit on this point:

> …the procedure of measurement has an essential influence on the conditions
> on which the very definition of the physical quantities in question rests.
> (Bohr, 1935)

Most authors, however, opt for a causal reading of the influence of the
measuring apparatus on the system. Heywood and Redhead (1983) and Redhead
(1989), for example, write:

> [T]here is some nonquantum interaction between the system of interest and
> its surroundings which occurs before the act of measurement and alters the
> values of magnitudes of the system. (Heywood and Redhead, 1983 p. 488;
> Redhead, 1989 p. 140)181818The timing of the interaction is not important:
> “presumably there is no reason why they should only occur immediately prior
> to a measurement.” (Heywood and Redhead, 1983 p. 488; Redhead, 1989 p. 140)

Shimony (1984) characterizes environmental contextuality more generally:

> In practice, of course, the environment is very imprecisely known and
> characterized. Consequently there is motivation to let $C$ [the context] be
> a ’coarse-grained state’ of the environment, such as is specified by a
> macroscopic description of the apparatus.” (Shimony, 1984 p. 29)

At the end of Section 4 I discerned three different meanings of environmental
contextuality. Clearly, Redhead and Shimony (and maybe Bohr) employ the term
“environmental contextuality” in the sense of (ii) which is not related to
either simultaneous or measurement contextuality. I called environmental
contextuality in the sense of (iii) algebraic contextuality. This name goes
back to Shimony’s (2008)191919See also (Shimony, 1984, p. 29) and (Gudder
1970). definition of an algebraic context:

> There actually are two quite different versions of contextual hidden-
> variables theories, depending upon the character of the context: an
> “algebraic context” is one which specifies the quantities (or the operators
> representing them) which are measured jointly with the quantity (or
> operator) of primary interest, whereas an “environmental context” is a
> specification of the physical characteristics of the measuring apparatus
> whereby it simultaneously measures several distinct co-measurable
> quantities.

In this quote Shimony contrasts algebraic and environmental contextuality,
whereas I find it more proper to take the former to be a special case of
latter since a simultaneous measurement is also special environment. But be
that as it may, Shimony’s algebraic contextuality is a special case of
simultaneous contextuality (12).

##  7 Measurement contextuality

###  7.1 Kochen and Specker

In their pioneering paper, Kochen and Specker (1967) showed that there is no
value assignment on a set of 117 projections in a 3-dimensional Hilbert space
which respects the sum rule on every triplet of orthogonal projections. Each
triplet of projections are squared spin-1 operators associated with three
orthogonal spatial directions. What is the physical interpretation of these
projections?

The projections are commuting but Kochen and Specker are careful in
distinguishing the commutativity of the operators and the simultaneous
measurability (“commeasurability”) of the represented measurements. They
write:

> Of course, we have seen that these operators commute and it is a generally
> accepted assumption of quantum mechanics that commuting operators correspond
> to commeasurable observables. A rationale for this assumption …is that if
> ${\bf A}_{i},i\in I$ is a set of mutually pairwise commuting self-adjoint
> operators, then there exists a self-adjoint operator ${\bf B}$ and Borel
> functions $f_{i},i\in I$ such that ${\bf A}_{i}=f_{i}(\bf{B})$. However,
> this justification hinges on the existence of a physical observable which
> corresponds to the operator. (Kochen and Specker, 1967, 72)

What Kochen and Specker state here is that from the mathematical fact that for
a set of mutually commuting operators $\\{{\bf A}_{i}\\}$ one can find an
operator ${\bf B}$ such that ${\bf A}_{i}=f_{i}(\bf{B})$, it does not follow
that there also is a physical measurement realizing ${\bf B}$. Therefore, they
construct for every triplet of projections a self-adjoint operator, the spin-
Hamiltonian, and three functions. The spin-Hamiltonian represents a physical
observable: “the change in the energy of the lowest orbital state of
orthohelium resulting from the application of a small electric field with
rhombic symmetry” (Kochen and Specker, 1967, 73). They also associate a
measurement to this observable: the measurement of energy of the orthohelium.

What is important is that Kochen and Specker do not interpret the triplets of
spin operators as three different spin measurements on a spin-1 particle.
These measurements would not be simultaneous measurable. Instead they
associate with each triplet one single energy measurement performed on a
system of orthohelium perturbed by a rhombic electric field. When they speak
about “simultaneous measurement” of the three spin operators (p. 73), they
mean exactly this energy measurement. Therefore, I think, they use the term
“simultaneous measurement” only metaphorically,202020See Remark a in Section
4. namely performing one single measurement and, based on the outcome,
assigning values to the projections.

But if each triplet is associated with a different energy measurement, then
projection popping up in two different triplets will be associated with two
different measurements. Consequently, on the authors own interpretation the
Kochen-Specker theorem rules out measurement noncontextual (outcome-
deterministic) ontological models for quantum mechanics.

###  7.2 Van Fraassen’s and Redhead’s ontological contextuality

The concept (but not the name) of ontological contextuality goes back to Bas
van Fraassen. In his 1973 paper he writes:

> [M]aximal observables can be identified with the operators that represent
> them. All other observables are functions of maximal observables, and are
> identified jointly by a maximal observable and a function. (Van Fraassen,
> 1973, p. 107)

At the end of Section 4 I called this concept special ontological
contextuality where the non-unique association of operators and observables is
confined only to non-maximal operators. The general concept, which I called
general ontological contextuality, has been formulated in van Fraassen’s 1979
paper:

> [t]wo observables [$a$ and $b$] are statistically equivalent if they have
> the same probability distribution…In that case they are represented in
> physics by the same Hermitean operator…But that does not mean that $a=b$.
> (Van Fraassen, 1979, p. 158)

In other words, two observables can be represented by the same self-adjoint
operator without being the same. But then, one is not compelled to assign the
same value to the observables or the same response function to the associated
measurements in the Kochen-Specker argument. This a clear case of measurement
contextuality.

Michael Redhead picks up van Fraassen’s view in analyzing the Kochen-Specker
argument:

> [O]bservables corresponding to maximal operators are ontologically prior to
> those which correspond to nonmaximal operators. Knowing to which self-
> adjoint operator it corresponds is not sufficient to identify unambiguously
> a nonmaximal observable: we must know also to which maximal observable its
> values are related. It requires in this sense a context; and this splitting
> of nonmaximal observables, so that each nonmaximal operator now corresponds
> to many distinct observables, we shall refer to as Ontological
> Contextuality. Redhead (1989, p. 135)

Note, that since to block the Kochen-Specker arguments, it is enough to
confine ontological contextuality to nonmaximal operators, therefore Redhead
uses the special version of ontological contextuality.

###  7.3 Spekkens’ measurement contextuality

In the operational approach to quantum mechanics initiated by Robert Spekkens
(2005) noncontextuality is defined as follows:

> A noncontextual ontological model of an operational theory is one wherein if
> two experimental procedures are operationally equivalent, then they have
> equivalent representations in the ontological model. (Spekkens, 2005, p. 1)

If the “experimental procedure” is a measurement, then Spekkens’
noncontextuality boils down to measurement noncontextuality
(13)-(14).212121Spekkens applies the concept of noncontextuality also to
preparations and transformations. Many authors working in the operational
approach follow this measurement account of noncontextuality.222222See e.g.
Hermens, 2011; Liang et al., 2011; Leifer, 2014; Krishna et al., 2017. There
are also experiments devised to test noncontextuality in this
sense.232323Mazurek, 2016.

###  7.4 Bohmian mechanics

Bohmian mechanics is an outcome-deterministic measurement contextual
ontological model for quantum mechanics. The classical example for the
violation of measurement noncontextuality (13)-(14) is due to Albert (1992, p.
150).242424See also Hemmick and Shakur, 2012, Ch. 2.5 and 2.6; Dürr et al.,
2013, Ch. 3.8.3. Consider two scenarios of spin measurement performed on an
electron which differ only in the polarity of the Stern-Gerlach magnets. If
one identifies the spin outcomes associated with the upper path in the first
scenario with the spin outcomes associated with the lower path in the second
scenario, then the two spin measurements will have the same distribution of
outcomes in every quantum state and hence can be represented by the same spin
operator in quantum mechanics. Still, they provide opposite spin outcomes for
a fixed ontic state in Bohmian mechanics. Namely, in Bohmian mechanics the
ontic states are just the possible initial positions of the electron together
with the wave function of the electron. If the measurement situation has a
certain reflection symmetry with respect to a plane between the Stern-Gerlach
magnets, then the Bohmian equations of motion ensure that the electron cannot
cross the symmetry plane of the measurement. This means that electrons
starting from above the plane will remain above the plane in both measurement
situations. But then the two measurements will give opposite spin outcomes for
one and the same ontic state. Thus, measurement noncontextuality (13)-(14) is
violated.

There is, however, another meaning of contextuality often used with respect to
Bohmian mechanics. It is said that the spin (and all other observables except
position) is contextual—meaning that it depends not only on the state of the
system but also on the state of the measurement apparatus. In other words,
spin is not an inherent property of the system rather a kind of coarse-
graining over the final positions of the electron after the Stern-Gerlach
measurement. This is environmental contextuality in the sense of
(ii).252525See the end of Section 4.

Some Bohmians object against calling this latter fact contextuality. As Daumer
et al. put it:

> We thus believe that contextuality reflects little more than the rather
> obvious observation that the result of an experiment should depend upon how
> it is performed! (Daumer et al., 1996, p. 389)

I agree with these authors: measurement outcomes need not depend only on the
operator associated with a measurement but can also depend on the very
measurement. In this sense there is nothing extraordinary in that Bohmian
mechanics (or any other physical theory) is contextual. However, Bohmian
mechanics also provides a striking example for an ontological model in which
the ontic states respond differently to different measurements represented by
the same operator. That is Bohmian mechanics provides an example for a
measurement contextual ontological model. The example is striking since
measurement noncontextuality, as said in Section 3, is a kind of inference to
the best explanation for why different measurements can have the same
distribution of outcomes in every preparation. It is not an a priori truth
but, no doubt, a reasonable requirement on ontological models. And Bohmian
mechanics shows how an ontological model can still live without it.

##  8 Conclusions

Simultaneous noncontextuality and measurement noncontextuality are two
different and logically independent concepts. To prove that quantum mechanics
is contextual in the one sense or in the other requires different strategies
that is different interpretations of the operators in a Kochen-Specker
theorem. To prove that quantum mechanics is simultaneous contextual, one needs
to associate the Kochen-Specker operators with physical measurements in a one-
to-one way. To prove that quantum mechanics is measurement contextual, such a
one-to-one association is not needed: operators sitting in more than one
commuting subsets can be associated with more than one measurements. Different
authors in the literature using the same Kochen-Specker theorems provide
different Kochen-Specker arguments depending on whether they use the one
operator-one measurement interpretation or the one operator-many measurements
interpretation. Consequently they prove quantum contextuality in two different
senses.

In the paper I grouped some of the most prominent accounts on contextuality
into two categories: simultaneous contextuality and measurement contextuality.
I argued that the formal and the sheaf theoretic approaches are lacking a
physical interpretation and hence their contextuality concept is purely
mathematical. I also argued that the two seminal papers, namely Bell (1966)
and Kochen and Specker (1967) provide two different interpretations of a no-go
theorem and hence the former excludes simultaneous noncontextual, the latter
measurement noncontextual ontological models. Next, I argued that Spekkens’
and the Bohmian account of noncontextuality is measurement noncontextuality
(which in a non-disturbing theory like quantum theory includes simultaneous
noncontextuality). Finally, I argued that Van Fraassen’s and Redhead’s
ontological contextuality is a special case of measurement contextuality; that
Shimony’s algebraic contextuality is a special case of simultaneous
contextuality; and that Bohr’s conceptual contextuality and Redhead’s and
Shimony’s environmental contextuality are unrelated to either simultaneous or
measurement contextuality.

Acknowledgements. This work has been supported by the Alexander von Humboldt
Foundation, the National Research, Development and Innovation Office
(K-115593), and a Senior Research Scholarship of the Institute of Advanced
Studies Koszeg. I thank Márton Gömöri for his careful reading of the
manuscript.

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