of the KD distribution $Q_{j,k}(\lvert a_{i}\rangle\\!\langle
a_{i}\rvert)\equiv\braket{a_{i}}{c_{k}}\braket{c_{k}}{b_{j}}\braket{b_{j}}{a_{i}}$.
For simplicity, we suppose that the quantum system begins in an $\hat{A}$
eigenstate $|a_{i}\rangle$. The first strong measurement yields the eigenvalue
$a_{i}$. It is followed by a weak measurement of ${\hat{B}}$ and a strong
measurement of ${\hat{C}}$. In this weak-measurement limit, we can replace the
correlation function with the approximation

${\cal L}\approx{\rm Re}\left[\langle
a_{i}|{\hat{A}}{\hat{B}}+{\hat{B}}{\hat{C}}-{\hat{A}}{\hat{C}}|a_{i}\rangle\right].$
(71)

This is the form of the correlation function considered by Leggett and Garg
Leggett and Garg (1985). We stress that the ${\hat{B}}$ measurement’s weakness
allows us to neglect that measurement’s influence on the ${\hat{A}}{\hat{C}}$
correlator. If the intermediate measurement is of finite strength, this
assumption breaks down. Reference Williams and Jordan (2008) describes the
measurement’s influence on the corrrelator. Let us insert the operators’
spectral expansions into Eq. (71). We obtain an expression dependent on the KD
distribution:

${\cal L}\approx\sum_{j,k}{\rm Re}[Q_{j,k}(\lvert a_{i}\rangle\\!\langle
a_{i}\rvert)]\left(a_{i}b_{j}+b_{j}c_{k}-a_{i}c_{k}\right).$ (72)

The rightmost parenthetical factor is upper-bounded by 1. Hence we see that,
for the right-hand side to exceed the upper bound of 1, some joint KD quasi-
probability $Q_{j,k}(\ket{a_{i}}\bra{a_{i}})$ must have a negative real part.
This bound violation was observed experimentally. Furthermore, this connection
between Leggett–Garg-type inequalities and quasi-probability negativity was
observed also in Refs. Bednorz  _et al._ (2012); Dressel  _et al._ (2011).

The Leggett–Garg inequality further underscores the connection between weak
values and the KD distribution. Weakly measuring the intermediate observable
$\hat{B}$, given strong pre- and post-selective measurements, is equivalent to
measuring the weak values
$B_{\mathrm{w}}(a_{i},c_{k})\equiv\frac{\braket{c_{k}}{\hat{B}}{a_{i}}}{\braket{c_{k}}{a_{i}}}$
[Eq. 40]. Thus, in the weak measurement limit, ${\cal L}$ can be expressed as

$\displaystyle{\cal L}\approx\sum_{k}P(c_{k}|a_{i})\Big{\\{}(a_{i}+c_{k}){\rm
Re}[B_{\mathrm{w}}(a_{i},c_{k})]-a_{i}c_{k}\Big{\\}}.$ (73)

The $P(c_{k}|a_{i})=|\langle c_{k}|a_{i}\rangle|^{2}$ denotes the conditional
probability that, if the first measurement yields $a_{i}$ and the $\hat{B}$
measurement is weak, the final measurement yields $c_{k}$.

The right-hand side of Eq. (73) is a convex sum. Therefore, for the right-hand
side to exceed the classical upper bound of $1$, the factor in curly brackets
must exceed $1$ for some $a_{i}$ and $c_{k}$. When
$\text{Re}\left[B_{\mathrm{w}}\right]>0$, this condition can be met only when
$a_{i}=c_{k}=+1$ (when $i=k=1$). Otherwise, the factor in curly brackets is
negative. In such a case, the bound-violation condition simplifies to
$2\text{Re}\left[B_{\mathrm{w}}(a_{1},c_{1})-1\right]>1$, or
$\text{Re}\left[B_{\mathrm{w}}(a_{1},c_{1})\right]>1$. That is, the weak
value’s real part must exceed the greatest $\hat{B}$ eigenvalue. If, instead,
$\text{Re}[B_{\mathrm{w}}]<0$, the classical bound $\mathcal{L}\leq 1$ can be
violated only when $a_{i}=c_{k}=-1$ (when $i=k=2$). In this case, the
condition necessary for violating the Leggett–Garg inequality simplifies to
$-2\text{Re}\left[B_{\mathrm{w}}(a_{2},c_{2})-1\right]>1$, or
$\text{Re}\left[B_{\mathrm{w}}(a_{2},c_{2})\right]<-1$. Again, the weak value
must lie outside the $\hat{B}$ spectrum.

These relations establish a one-to-one correspondence between a Leggett–Garg-
inequality violation and an anomalous weak value. Expanding the conditions in
terms of KD quasi-probabilities yields
$\sum_{j}b_{j}\,\text{Re}\left[Q_{j,k=1}(\ket{a_{1}}\bra{a_{1}})\right]>\sum_{j}\text{Re}\left[Q_{j,k=1}(\ket{a_{1}}\bra{a_{1}})\right]$,
implying $\text{Re}\left[Q_{j=2,k=1}(\ket{a_{1}}\bra{a_{1}})\right]<0$, or
$\sum_{j}b_{j}\,Q_{j,k=2}(\ket{a_{2}}\bra{a_{2}}))<-\sum_{j}\text{Re}\left[Q_{j,k=2}(\ket{a_{2}}\bra{a_{2}})\right]$,
implying $\text{Re}\left[Q_{j,k=2}(\ket{a_{2}}\bra{a_{2}})\right]<0$. We can
now interpret, in terms of KD quasi-probabilities, the conditions necessary
for violating the Leggett–Garg inequality’s classical upper bound of 1, as has
been observed experimentally. At least one of two KD quasi-probabilities,
$Q_{j=2,k=1}(\ket{a_{1}}\bra{a_{1}})$ or
$Q_{j=1,k=2}(\ket{a_{2}}\bra{a_{2}})$, must have a negative real part. Each
such negative real part causes a corresponding weak value, ${\rm
Re}[B_{\mathrm{w}}(a_{1},c_{1})]$ or ${\rm Re}[B_{\mathrm{w}}(a_{2},c_{2})]$,
to lie outside the $\hat{B}$ spectrum.

###  VIII.2 The KD distribution and the consistent-histories interpretation of
quantum mechanics

The mathematical framework of quantum mechanics is well-established but not a
complete physical theory. The role of an interpretation of quantum mechanics
is to append to the mathematical formalism a theory of real underlying (ontic)
physics that results in the quantum phenomena we observe in laboratories. Such
interpretations include the many-worlds interpretation Everett (1973); Dewitt
and Graham (2015), de Broglie–Bohm mechanics Bohm (1952) and the consistent-
histories interpretation Griffiths (1984). The latter has a deep connection
with the KD distribution. We review this connection below.

The consistent-histories interpretation of quantum mechanics predicts the same
measurement probabilities as textbook quantum mechanics. The Copenhagen
interpretation, however, suggests that quantum states have well-defined
properties only when measured. In contrast, the consistent-histories
interpretation provides a framework that (sometimes) describes properties of
quantum particles between observations. The interpretation was developed by
several scholars Gell-Mann and Hartle (1993); Omnes (1999); Hartle (2007,
2010), most notably by Griffiths Griffiths (1984); Hohenberg (2010).

In the consistent-histories interpretation, properties (or physical
attributes) of quantum states are defined in accordance with von Neumann’s
theory Von Neumann (2018); Griffiths (1984). A property is associated with a
projective operator onto a subspace of a Hilbert space.444The theory
accommodates more-general positive-operator-valued measures, however. Via
Steinspring dilation, a positive-operator-valued measure defined one one
Hilbert space is equivalent to a projector-valued measure on a larger Hilbert
space. (Non-unique dilations are problematic for ontic interpretations.) For
example, ‘position’ is not a property. However, ‘the particle is at position
$x$’ is. Consider two sets of projective measurement operators,
$\mathcal{A}=\\{\hat{\Pi}^{(1)}_{i}\\}$ and
$\mathcal{B}=\\{\hat{\Pi}^{(2)}_{j}\\}$. Quantum theory struggles to describe
quantum states’ underlying because the question _does the state have
properties $\hat{\Pi}^{(1)}_{1}$ and $\hat{\Pi}^{(2)}_{3}$?_ seems difficult
to answer when $[\hat{\Pi}^{(1)}_{1},\hat{\Pi}^{(2)}_{3}]\neq 0$. Under
certain consistency conditions, the consistent-histories interpretation
facilitates a description of underlying physical properties of a quantum
state’s time-evolution through Hilbert space.

Next, we review the technical framework for the consistent-histories
interpretation. Afterwards, we connect the consistency criterion to a KD
distribution. Finally, we provide an example.

Consider a pure initial quantum state $\hat{\rho}_{\mathrm{i}}$. Consider also
a set of times $t_{0}<t_{1}<\ldots<t_{k}<t_{\mathrm{f}}$. We work in the
Heisenberg picture, in which quantum states are constant in time and
observables evolve unitarily. Define $k$ sets of projector-valued measures
$\mathcal{A}^{(l)}=\\{\hat{\Pi}^{(l)}_{i_{l}}(t_{l})\\}$, where
$\sum_{i_{l}}\hat{\Pi}^{(l)}_{i_{l}}(t_{l})=\hat{\mathbbm{1}}$ and
$l=1,2,\ldots,k$. We convert a Schrödinger-picture observable $\hat{X}$ to a
Heisenberg-picture observable through
$\hat{X}(t_{i})=\hat{U}^{\dagger}(t_{0},t_{i})\hat{X}\hat{U}(t_{0},t_{i})$.
Finally, consider a final rank-$1$ observable
$\hat{\rho}_{\mathrm{f}}(t_{\mathrm{f}})$.555Generalisations of this setting
are described in Griffiths (1984); Hohenberg (2010). A history is defined as

$H\left(\hat{\rho}_{\mathrm{i}},\hat{\Pi}^{(1)}_{i_{1}},\hat{\Pi}^{(2)}_{i_{2}},\ldots,\hat{\Pi}^{(k)}_{i_{k}},\hat{\rho}_{\mathrm{f}}\right)\equiv\hat{\rho}_{\mathrm{i}}\rightarrow\hat{\Pi}^{(1)}_{i_{1}}(t_{1})\rightarrow\hat{\Pi}^{(2)}_{i_{2}}(t_{2})\rightarrow\cdots\rightarrow\hat{\Pi}^{(k)}_{i_{k}}(t_{k})\rightarrow\hat{\rho}_{\mathrm{f}}(t_{\mathrm{f}}).$
(74)

The history means that a quantum system is initialised in a state
$\hat{\rho}_{\mathrm{i}}$ at time $t_{0}$; at time $t_{1}$, the system has a
property corresponding to $\hat{\Pi}^{(1)}_{i_{1}}(t_{1})$; at time $t_{2}$,
the system has a property corresponding to $\hat{\Pi}^{(2)}_{i_{2}}(t_{2})$;
and so on, until, at the final time $t_{\mathrm{f}}$, the system has a
property corresponding to $\hat{\rho}_{\mathrm{f}}$. Given an initial state
$\hat{\rho}_{\mathrm{i}}$ and a final observable
$\hat{\rho}_{\mathrm{f}}(t_{\mathrm{f}})$, one can write down several
potential histories. Two histories,
$H\left(\hat{\rho}_{\mathrm{i}},\hat{\Pi}^{(1)}_{i_{1}},\hat{\Pi}^{(2)}_{i_{2}},\ldots,\hat{\Pi}^{(k)}_{i_{k}},\hat{\rho}_{\mathrm{f}}\right)$
and
$H^{\star}\left(\hat{\rho}_{\mathrm{i}},\hat{\Pi}^{(1)}_{i_{1}^{\star}},\hat{\Pi}^{(2)}_{i_{2}^{\star}},\ldots,\hat{\Pi}^{(k)}_{i_{k}^{\star}},\hat{\rho}_{\mathrm{f}}\right)$,
are said to be consistent with respect to one another if, and only if, their
(normalised) Hilbert–Schmidt inner product vanishes:

$\tilde{Q}(H,H^{\star})\equiv\frac{1}{\operatorname{Tr}\bm{\left[}\hat{\rho}_{\mathrm{f}}(t_{\mathrm{f}})\hat{\rho}_{\mathrm{i}}\bm{}}\operatorname{Tr}\left[\hat{\rho}_{\mathrm{f}}(t_{\mathrm{f}})\hat{\Pi}^{(k)}_{i_{k}}(t_{k})\cdots\hat{\Pi}^{(1)}_{i_{1}}(t_{1})\hat{\rho}_{\mathrm{i}}\hat{\Pi}^{(1)}_{i_{1}^{\star}}(t_{1})\cdots\hat{\Pi}^{(k)}_{i_{k}^{\star}}(t_{k})\hat{\rho}_{\mathrm{f}}(t_{\mathrm{f}})\right]=0.$
(75)

$\tilde{Q}(H,H^{\star})$ is a KD quasi-probability representing the strength
of the quantum interference between two paths (histories) in Hilbert space.

A _family_ $f=\\{H_{i}\\}$ of histories consists of all the histories

$f\equiv\left\\{\hat{\rho}_{\mathrm{i}}\rightarrow\mathcal{C}\left(\mathcal{A}^{(1)}\right)\rightarrow\mathcal{C}\left(\mathcal{A}^{(2)}\right)\rightarrow\cdots\rightarrow\mathcal{C}\left(\mathcal{A}^{(k)}\right)\rightarrow\hat{\rho}_{\mathrm{f}}\right\\},$
(76)

where $\mathcal{C}\left(\mathcal{A}^{(l)}\right)$ represents any projector in
$\mathcal{A}^{(l)}$. A family $f=\\{H_{i}\\}$ of histories is consistent if,
and only if,

$Q(H_{i},H_{j})=0\;\mathrm{and}\;H_{i},H_{j}\in f,\;\;\forall i\neq j.$ (77)

If $f$ is consistent, then $\tilde{Q}(H_{i},H_{i})$ is a classical conditional
joint probability distribution for all $H_{i}\in f$. Within the context of a
consistent family, one can ask if an individual history has happened. Given
consistency, one can regard $\tilde{Q}(H_{i},H_{i})$ as the probability that
$H_{i}$ happened.

Often, one is interested only in whether a single history happened. In this
case, one need only consider the minimal family of which that history can be a
member. The minimal family of
$H\left(\hat{\rho}_{\mathrm{i}},\hat{\Pi}^{(1)}_{i_{1}},\hat{\Pi}^{(2)}_{i_{2}},\ldots,\hat{\Pi}^{(k)}_{i_{k}},\hat{\rho}_{\mathrm{f}}\right)$
is

$f_{\mathrm{min}}\left(H\right)\equiv\left\\{\hat{\rho}_{\mathrm{i}}\rightarrow\mathcal{C}\left(\left\\{\hat{\Pi}^{(1)}_{i_{1}},\,\hat{\mathbbm{1}}-\hat{\Pi}^{(1)}_{i_{1}}\right\\}\right)\rightarrow\mathcal{C}\left(\left\\{\hat{\Pi}^{(2)}_{i_{2}},\,\hat{\mathbbm{1}}-\hat{\Pi}^{(2)}_{i_{1}}\right\\}\right)\rightarrow\cdots\rightarrow\mathcal{C}\left(\left\\{\hat{\Pi}^{(k)}_{i_{k}},\,\hat{\mathbbm{1}}-\hat{\Pi}^{(k)}_{i_{k}}\right\\}\right)\rightarrow\hat{\rho}_{\mathrm{f}}\right\\}.$
(78)

If $f_{\mathrm{min}}\left(H\right)$ is consistent, then the history $H$ is
also said to be consistent. In this case, one can meaningfully ask, _did $H$
occur, or not?_

Figure 11: Mach–Zehnder histories. A Mach–Zehnder interferometer. The quantum
particle is initialised in the lower path. Hollow rectangles represent 50–50
beam-splitters; shaded rectangles represent mirrors.

To illustrate the consistent-histories interpretation, we provide an example.
Consider a Mach–Zehnder interferometer Zehnder (1891); Mach (1892) as shown in
Fig. 11 . We associate the lower paths (green) with the state $\ket{0}$ and
the upper paths (blue) with the state $\ket{1}$. At time $t_{0}$, a quantum
particle is prepared in the state $\rho_{\mathrm{i}}=\lvert 0\rangle\\!\langle
0\rvert$. Just before time $t_{1}$, a 50–50 beam-splitter implements the
unitary operator
$\hat{U}_{\mathrm{BS}}(t_{0},t_{1})=e^{-i\frac{\pi}{4}\hat{\sigma}_{y}}$. Just
before time $t_{3}$, another 50–50 beam-splitter implements
$\hat{U}_{\mathrm{BS}}(t_{2},t_{3})=e^{-i\frac{\pi}{4}\hat{\sigma}_{y}}$. In a
final event, the particle has the property $\rho_{f}(t_{4})=\lvert
1\rangle\\!\langle 1\rvert$ at time $t_{4}$.

We wish to say something about the particle’s position property at time
$t_{2}$, between the two beam-splitters. According to the Copenhagen
interpretation, we cannot say anything about this property at $t_{2}$, because
no measurement was made. Let us now analyse the problem from the consistent-
histories perspective. We want to ask, at $t_{2}$, if the particle had the
property $\hat{\Pi}^{0}\equiv\lvert 0\rangle\\!\langle 0\rvert$ (was on the
lower path) or had the property $\hat{\Pi}^{1}\equiv\lvert 1\rangle\\!\langle
1\rvert=\hat{\mathbbm{1}}-\hat{\Pi}^{0}$ (was on the upper path). Define the
histories

$\displaystyle H_{0}$
$\displaystyle\equiv\hat{\rho}_{\mathrm{i}}\rightarrow\hat{\Pi}^{0}(t_{2})\rightarrow\hat{\rho}_{\mathrm{f}}(t_{3})\;\;\mathrm{and}$
(79) $\displaystyle H_{1}$
$\displaystyle\equiv\hat{\rho}_{\mathrm{i}}\rightarrow\hat{\Pi}^{1}(t_{2})\rightarrow\hat{\rho}_{\mathrm{f}}(t_{3}).$
(80)

$\\{H_{0},H_{1}\\}$ forms the minimal family for $H_{0}$ and for $H_{1}$; we
call this family $f^{(0/1)}$. We can calculate
$\tilde{Q}(H_{0},H_{1})=\tilde{Q}(H_{1},H_{0})=\frac{1}{4}$. Thus, according
to the consistent-histories interpretation, one cannot ask, in a meaningful
way, if the particle was in the upper or lower path at $t_{2}$.

Instead, let us define
$\ket{\pm}=\frac{1}{\sqrt{2}}\left(\ket{0}\pm\ket{1}\right)$. We now ask
whether, at $t_{2}$, the particle had the property
$\hat{\Pi}^{+}\equiv\lvert+\rangle\\!\langle+\rvert$ or the property
$\hat{\Pi}^{-}\equiv\lvert-\rangle\\!\langle-\rvert=\hat{\mathbbm{1}}-\hat{\Pi}^{+}$.
Each of the histories

$\displaystyle H_{+}$
$\displaystyle\equiv\hat{\rho}_{\mathrm{i}}\rightarrow\hat{\Pi}^{+}(t_{2})\rightarrow\hat{\rho}_{\mathrm{f}}(t_{3})\;\;\mathrm{and}$
(81) $\displaystyle H_{-}$
$\displaystyle\equiv\hat{\rho}_{\mathrm{i}}\rightarrow\hat{\Pi}^{-}(t_{2})\rightarrow\hat{\rho}_{\mathrm{f}}(t_{3})$
(82)

has the minimal family $f^{(+/-)}\equiv\\{H_{+},H_{-}\\}$. We can calculate
$\tilde{Q}(H_{+},H_{-})=\tilde{Q}(H_{-},H_{+})=0$. Thus, according to the
consistent-histories interpretation, one can ask meaningfully if the particle
had the property of a $\ket{+}$-type or a $\ket{-}$-type superposition at
$t_{2}$. In fact, $\tilde{Q}(H_{+},H_{+})=1$. Therefore, the consistent-
histories interpretation implies that a particle initially in
$\rho_{\mathrm{i}}=\lvert 0\rangle\\!\langle 0\rvert$ and finally in
$\rho_{\mathrm{f}}=\lvert 1\rangle\\!\langle 1\rvert$ definitely had the
property $\hat{\Pi}^{+}=\lvert+\rangle\\!\langle+\rvert$ at $t_{2}$.

###  VIII.3 Generalised contextuality

According to a useful notion of non-classicality, an experiment is called non-
classical when it provably requires explanations involving radical departures
from the classical world-view Jennings and Leifer (2016). Bell non-locality
Brunner  _et al._ (2014) is arguably the strongest notion of non-classicality.
However, it can be provably identified only in experiments involving space-
like-separated systems. A more widely applicable notion is (generalised)
contextuality Leifer (2014a); Spekkens (2005).

Below, we introduce generalised contextuality. We then explain that, if a
particular protocol for measuring the KD distribution yields non-positive
values, then the protocol is contextual. Afterwards, we discuss how KD
distributions witness contextuality in enhanced work extraction and in the
breaking of bounds on thermodynamic currents.

Non-contextuality, in its generalised form Spekkens (2005); Leifer (2014a), is
the property that a physical theory does not introduce distinct mathematical
representations of operationally indistinguishable phenomena within a class of
theories known as _ontological models_. In other words, if two things cannot
be distinguished, they are the same thing within the ontological model. For
example, Einstein, in his ‘On the Electrodynamics of Moving Bodies’ Einstein
_et al._ (1905), criticised the way in which electrodynamics (as understood at
the time via the notion of ether) assigns distinct physical descriptions to a
metallic ring moving towards a magnet and to a magnet moving towards a ring.
After all, the two experiments are impossible to tell apart by any
measurement. One can argue that the theory-selection principle being applied
is non-contextuality Spekkens (2019).

The word _ontological_ labels a theory of an underlying physical reality. The
underlying physical reality may be inaccessible (hidden). An experiment
consists of preparation procedures $P$, transformation procedures $T$, and
measurement procedures $M$. These generate a set $\\{p(k|P,T,M)\\}_{P,T,M}$ of
outcomes statistics. $p(k|P,T,M)$ denotes the probability of observing outcome
$k$, given that we prepared the system according to $P$, transformed it
according to $T$ and measured it according to $M$. An ontological model
associates to operational procedures (preparations, transformations and
measurements) probabilistic representations over a measurable state space
$\Lambda$ Leifer (2014a), the set of _physical states_. We model a preparation
procedure $P$ by sampling $\lambda\in\Lambda$ according to a probability
distribution $p_{P}(\lambda)$. Each transformation $T$ is modelled by a
matrix; each element $p_{T}(\lambda^{\prime}|\lambda)$ equals the probability
of jumping from $\lambda$ to $\lambda^{\prime}$. Each measurement procedure
$M$ is modelled by another matrix; each element $p_{M}(k|\lambda^{\prime})$
equals the probability of obtaining outcome $k$, given the ontic state
$\lambda^{\prime}$.

The maps $P\mapsto p_{P}$, $T\mapsto p_{T}$ and $M\mapsto p_{M}$ must be
convex-linear. This property ensures a reasonable probabilistic
interpretation. For example, imagine tossing a fair coin to decide whether we
follow the preparation procedure $P_{1}$ or $P_{2}$. The ontological model
must assign the probability $[p_{P_{1}}(\lambda)+p_{P_{2}}(\lambda)]/2$ to the
physical state $\lambda$. From the propagation of probabilities, an
ontological model predicts that outcome $k$ will occur with a probability

$p(k|P,T,M)=\int_{\Lambda}d\lambda\int_{\Lambda}d\lambda^{\prime}\,p_{M}(k|\lambda^{\prime})\,p_{T}(\lambda^{\prime}|\lambda)\,p_{P}(\lambda)\,.$
(83)

See Ref. Leifer (2014b) for an introduction to ontological models.

Ontological models, unless otherwise restricted, can reproduce nearly
anything. For example, consider a quantum experiment. In the quantum
formalism, $P$ is represented by a pure state $|\psi\rangle\in\mathbb{C}^{d}$,
$T$ by a $d\times d$ unitary $\hat{U}$ and $\hat{M}$ by a projective
measurement $\\{\ket{k}\bra{k}\\}_{k=1}^{d}$. By the Born rule,
$p(k|P,T,M)=|\langle k|\hat{U}|\psi\rangle|^{2}$. The right-hand side of Eq.
(83) can reproduce this function if $\Lambda$ consists of normalised pure
states. In this model, whose ontic states are normalised pure states, the
following assignments are made: $p_{\psi}(\lambda)=\delta(\lambda-\psi)$,
$p_{T}(\lambda^{\prime}|\lambda)=\delta(\lambda^{\prime}-\hat{U}\psi)$ and
$p_{M}(k|\lambda^{\prime})=|\langle k|\lambda^{\prime}\rangle|^{2}$
Beltrametti and Bugajski (1995). On can represent, within the ontological
model, a general preparation procedure $P$ with a probability measure on
$\Lambda$:

$p_{P}(\lambda)=\sum_{n}p_{n}\delta(\lambda-\psi_{n})=\sum_{n}p_{n}p_{\psi_{n}}(\lambda),\quad\text{such
that}\quad\sum_{n}p_{n}=1.$ (84)

Hence a general ontological model can model any quantum experiment.
Ontological models need not be deterministic. For example, in the above model,
$\lambda$ determines the measurement outcome probabilistically.

Non-contextuality can restrict ontological models to satisfy a notion of
classicality. Imagine preparing two copies of a system according to two
different procedures, $P$ and $P^{\prime}$. Suppose that we obtain the same
measurement statistics, no matter which measurements $M$ we conduct:
$p(k|P,M)=p(k|P^{\prime},M)$ for all $k$ and $M$. This condition, known as an
_operational equivalence_ , is denoted by $P\sim P^{\prime}$ Spekkens (2005).
Non-contextuality requires that the ontological model assign the same
distribution over $\lambda$ to both preparations:
$p_{P}(\lambda)=p_{P^{\prime}}(\lambda)$ for all $\lambda$. For example,
consider forming a non-contextual description of the preparation of a density
operator $\hat{\rho}$. We must represent $\hat{\rho}$ by the same
$p_{\hat{\rho}}(\lambda)$, independently of the pure-state mixture used to
prepare $\hat{\rho}$. The previous example of an ontological model for quantum
mechanics is contextual (is not non-contextual). Indeed, $p_{P}$ can differ
from $p_{P^{\prime}}$ [see Eq. (84)], while

$\rho_{P}=\sum_{n}p_{n}\lvert\psi_{n}\rangle\\!\langle\psi_{n}\rvert=\rho_{P^{\prime}}=\sum_{n}p^{\prime}_{n}\lvert\psi^{\prime}_{n}\rangle\\!\langle\psi^{\prime}_{n}\rvert.$
(85)

In other words, in contextual models, different procedures $P$ can yield
different $p_{P}$, while nevertheless yielding the same density matrix [see
Eq. (84)] and hence the same experimental outcomes. In non-contextual models
$P\sim P^{\prime}$ implies $p_{P}(\lambda)=p_{P^{\prime}}(\lambda)$.
Similarly, consider two transformations that realise the same quantum channel
$\mathcal{C}$. Non-contextuality requires that the transformations be assigned
the same $p_{\mathcal{C}}(\lambda^{\prime}|\lambda)$. A similar rule governs
two measurements associated to the same positive-operator-valued measure
$\mathcal{M}$. The measurements are assigned the same
$p_{\mathcal{M}}(k|\lambda)$ in any non-contextual model. In short, non-
contextuality is the property that operational equivalence implies ontological
identity.

From an experimental viewpoint, operational equivalence need not be checked
exactly. In fact, they cannot be. Fortunately, contextuality proofs can
accommodate experimental imperfections, e.g., Schmid and Spekkens (2018);
Lostaglio and Senno (2020); Kunjwal  _et al._ (2019). Experimental
contextuality tests must check operational equivalences for a tomographically
complete set of preparations and measurements. To achieve this complete
testing one can rely on quantum mechanics to tell us what these
tomographically complete sets are. Alternatively, we can perform theory-
agnostic generalised-probabilistic-theory tomography Mazurek  _et al._ (2016,
2021). For other questions related to generalised contextuality and its
experimental testing, see Ref. Schmid  _et al._ (2023).

If an experiment $\\{p(k|P,T,M)\\}_{P,T,M}$ is _contextual_ , no non-
contextual ontological model satisfies Eq. (83). Contextuality has been
identified in (specific aspects of) experiments that involve state
discrimination Schmid and Spekkens (2018); Flatt  _et al._ (2022), approximate
cloning Lostaglio and Senno (2020), uncertainty relations Catani  _et al._
(2022), interference Flatt  _et al._ (2022), randomness certification i
Carceller  _et al._ (2022) and more. However, we focus on the link between
contextuality and quasi-probabilities—specifically, the KD distribution.

A positive quasi-probability representation of quantum mechanics (or of a
subtheory of quantum mechanics) assigns to each density operator $\hat{\rho}$,
quantum channel $\mathcal{C}$ and positive-operator-valued measure
$\mathcal{M}=\\{\hat{M}_{k}\\}$ probability distributions
$p_{\hat{\rho}}(\lambda)$, $p_{\mathcal{C}}(\lambda^{\prime}|\lambda)$ and
$p_{\mathcal{M}}(k|\lambda)$ over a measurable space $\Lambda$ Ferrie (2011).
Density operators, channels and positive-operator-valued measures can be
understood as equivalence classes of preparation procedures, transformations
and measurements. (The _equivalence_ refers to the above sense of operational
equivalence.) Hence, two conditions are equivalent: the existence of a non-
contextual ontological model for an experiment and the existence of a positive
quasi-probability representation of the experiment Spekkens (2008). In other
words, contextuality is equivalent to the negativity or non-reality of _every_
quasi-probability representation of the experiment. A particular quasi-
probability’s negativity does not suffice for proving contextuality,
generally. (However, the condition can suffice in some subtheories of quantum
mechanics Schmid  _et al._ (2022).) Nevertheless, there exist surprisingly
direct connections between the KD distribution and generalised contextuality.

Suppose that a KD distribution has negative real part:

$\textrm{Re}\,\bm{(}Q_{j,k}(\hat{\rho})\bm{)}=\textrm{Re}\bm{(}\operatorname{Tr}(\hat{\Pi}_{k}^{(B)}\hat{\Pi}_{j}^{(A)}\hat{\rho})\bm{)}<0.$
(86)

The following sequential-measurement scheme (the weak-value experiment
described in Sec. IV.1) is therefore contextual Pusey (2014); Kunjwal  _et
al._ (2019): weakly measure $\\{\hat{\Pi}_{j}^{(A)}\\}$ (measuring the meter’s
position eigenbasis), then strongly measure $\\{\hat{\Pi}_{k}^{(B)}\\}$. No
non-contextual ontological model can explain the set
$\\{p(k|P,T,M)\\}_{P,T,M}$ of outcome statistics (including the statistics
that support the operational equivalences) Pusey (2014). Therefore, KD
negativity implies the negativity of every quasi-probability representation of
this experiment. Similarly, suppose that
$\textrm{Im}\,\bm{(}Q_{j,k}(\hat{\rho})\bm{)}\neq 0$. An analogous scheme, in
which the meter’s momentum eigenbasis is measured, is contextual Kunjwal  _et
al._ (2019). Even if a continuous meter’s outcomes are coarse-grained (even if
a qubit replaces the meter), the results hold Kunjwal  _et al._ (2019). In
conclusion, KD non-positivity and contextuality characterise an experiment
that reports an anomalous weak value Pusey (2014); Kunjwal  _et al._ (2019).

The KD distribution also witnesses contextuality in the context of work
extraction (and injection), in the linear-response regime discussed in Sec. VI
Lostaglio (2020). We have already mentioned that the work scales as $O(g)$
only when a the imaginary part of an average with respect to a KD distribution
is non-zero. Otherwise, the work scales as $O(g^{2})$. More precisely,

$\langle
W\rangle=\frac{2g\tau}{\hbar}\mathrm{Im}\left[\operatorname{Tr}\bm{(}H_{0}\bar{V_{I}}(\tau)\hat{\rho}\bm{)}\right]+O\left(g^{2}\right)\,.$
(87)

This expression connects the work’s $O(g)$ behaviour with a KD quantity’s
being non-zero:
$\mathrm{Im}\left[\operatorname{Tr}\bm{(}H_{0}\bar{V_{I}}(\tau)\hat{\rho}\bm{)}\right]\neq
0$. This result says nothing directly about contextuality. However, a work-
extraction transformation $T$ (a unitary in quantum mechanics), in the
presence of an extra condition called _stochastic reversibility_ , can
generate $O(g)$ behaviour only if the experiment is contextual Lostaglio
(2020). Stochastic reversibility is the condition under which one can reverse
a transformation, up to first order in $g$, via probabilistic mixture with
another operation $T^{*}$. That is, there exist transformations $T^{*}$ and
$T^{\prime}$ such that

$\frac{1}{2}T+\frac{1}{2}T^{*}\sim(1-p_{d})I+p_{d}T^{\prime},$ (88)

where $I$ is the trivial (identity) transformation and $p_{d}=O(g^{2})$. Qubit
systems always satisfy the above operational equivalence, according to quantum
mechanics.

Complementary to work is heat. As discussed in Section VI, KD distributions
satisfy many desirable criteria for distributions characterising measurements
of fluctuating work, heat and entropy. Moreover, we have seen that KD non-
positivity witnesses contextuality in experiments that report anomalous weak
values. The connection between such experiments and contextuality has led to
theoretical bounds on the average heat Levy and Lostaglio (2020) and average
work Hernández-Gómez  _et al._ (2022) that can flow during a sequential-
measurement protocol admitting of a non-contextual ontological model. These
bounds can be broken only if the sequential-measurement process is contextual.
Like heat and work, entropy production is a thermodynamic quantity that
fluctuates from trial to trial. A notion of stochastic entropy production can
be defined via KD distributions Upadhyaya  _et al._ (2023). A non-real
stochastic entropy production signals the contextuality of a thermodynamic
process that involves sequential measurements Upadhyaya  _et al._ (2023).

##  IX Mathematical structure of the KD-positive states and properties of KD
non-positivity

If $Q\left(\hat{\rho}\right)$ has neither negative nor non-real components, it
is a classical probability distribution. A KD distribution’s ability to assume
negative and non-real values allows the distribution to describe quantum
experiments that cannot be described and analysed with classical probability
theories. Recall that we refer to negative or non-real values simply as _non-
positive_ , and that we call $Q\left(\hat{\rho}\right)$ _positive_ if all its
entries are positive or zero. Consider a KD distribution defined with respect
to two operators’s eigenbases. For the KD distribution to contain non-positive
values, a necessary but insufficient condition is that the operators do not
commute Arvidsson-Shukur  _et al._ (2021). Thus, KD non-positivity can be seen
as a property related to, but stricter than, non-commutation.

As we saw in previous sections, experiments described by non-positive KD
distributions can often generate data that cannot be generated via sampling
from classical joint probability distributions. It is, therefore, important to
understand when a KD distribution is non-positive. In this section, we
summarise the literature on this topic.

###  IX.1 KD positivity

Consider a $d$-dimensional Hilbert space ${\mathcal{H}}$, as well as $k$
projective positive-operator-valued measures
$\mathcal{M}^{(l)}\equiv\\{\hat{M}_{i_{l}}^{l}\\}$, where $l=1,2,\ldots,k$. We
call a state $\hat{\rho}$ KD-positive if, for all indices,
$Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\geq 0$. In this case, $Q(\hat{\rho})$
defines a probability distribution over the set of indices. KD positivity of
$\hat{\rho}$ depends on the choice of positive-operator-valued measures
$\mathcal{M}^{(l)}$. In what follows, we will not indicate this dependence,
the choice being clear from the context.

We denote the set of all KD-positive states by ${\mathcal{E}}_{\mathrm{KD+}}$.
${\mathcal{E}}_{\mathrm{KD+}}$ is a convex, bounded, closed set. We denote by
${\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{ext}}$ the set of extreme points666A
convex set’s extreme points are the points that do not lie on any open line
segment that connects two points in the set. of
${\mathcal{E}}_{\mathrm{KD+}}$. Further, we denote the set of pure KD-positive
states by ${\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}$. We denote by
$\mathrm{conv}\left(\mathcal{E}\right)$ the set of all convex combinations of
the elements of a set $\mathcal{E}$. A convex combination is defined as a
weighted finite sum of elements $\hat{\rho}_{j}\in\mathcal{E}$, of the form
$\sum_{j}\alpha_{j}\hat{\rho}_{i}$, such that $\sum_{j}\alpha_{j}=1$ and
$\alpha_{j}\geq 0$. One has that
${\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}\subseteq{\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{ext}}$,
and thus

$\mathrm{conv}\left({\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}\right)\subseteq\mathrm{conv}\left({\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{ext}}\right)={\mathcal{E}}_{\mathrm{KD+}}.$
(89)

The last equality follows from the Krein-Milman theorem Hiriart-Urruty and
Lemaréchal (2001).

The following lemma links positivity of the KD distribution of all states to
commutativity of the positive-operator-valued measures
$\mathcal{M}^{(l)}=\\{\hat{M}_{i_{l}}^{(l)}\\}$ used in the distribution’s
construction.

######  Lemma IX.1.

The two following statements are equivalent:

  * •

For all quantum states $\hat{\rho}$, the KD distribution is positive:
$Q(\hat{\rho})\geq 0$.

  * •

All positive-operator-valued-measure operators pairwise commute:
$\left[\hat{M}^{(l)}_{i_{l}},\,\hat{M}^{(l^{\prime})}_{i_{l^{\prime}}}\right]=0$
for all $l,l^{\prime}$ and all ${i_{l}},{i_{l^{\prime}}}$.

The lemma excludes the occurrence of negative or non-real complex values of
the KD distribution of any state $\hat{\rho}$ when all the positive operators
$\hat{M}_{i_{l}}^{(l)}$ commute. By contraposition, the lemma also guarantees
the existence of a quantum state $\hat{\rho}$ for which $Q(\hat{\rho})$ is not
a probability distribution, when there exists at least one pair
$(l,l^{\prime})$ and one pair $(i_{l},i_{l^{\prime}})$ such that
$\hat{M}_{i_{l}}^{(l)}$ and $\hat{M}_{i_{l^{\prime}}}^{(l^{\prime})}$ do not
commute. So non-commutativity of measurement operators, a typical quantum
feature linked to incompatibility Designolle  _et al._ (2019), is a
prerequisite for the existence of at least one state $\hat{\rho}$ that is not
KD-positive. Further links between incompatibility and non-commutativity of
observables, uncertainty of states with respect to observables, and non-
positivity of the KD distribution are elaborated upon in Sec. IX.4 (see
Arvidsson-Shukur  _et al._ (2021); De Bièvre (2021, 2023)).

###### Proof.

First, we prove that the lemma’s first point implies the second point. For
ease of notation, and without loss of generality, we focus on $l=1$ and
$l^{\prime}=2$. Since $Q(\hat{\rho})\geqslant 0$ for all quantum states
$\hat{\rho}$,

$\forall(i_{1},i_{2}),\;\mathrm{Tr}(\hat{M}_{i_{2}}^{(2)}\hat{M}_{i_{1}}^{(1)}\hat{\rho})\geqslant
0.$ (90)

This statement governs all pure states, in particular. Therefore,
$\hat{M}_{i_{2}}^{(2)}\hat{M}_{i_{1}}^{(1)}\geqslant 0$ for all
$(i_{1},i_{2})$. Consequently, $\hat{M}_{i_{2}}^{(2)}\hat{M}_{i_{1}}^{(1)}$ is
self-adjoint; and thus,

$\hat{M}_{i_{1}}^{(1)}\hat{M}_{i_{2}}^{(2)}=\hat{M}_{i_{2}}^{(2)}\hat{M}_{i_{1}}^{(1)},$
(91)

which concludes the proof.

Now, we prove that the lemma’s second point implies the first. Recall that, if
$\hat{X}$ and $\hat{Y}$ are non-negative operators (if $\hat{X}\geqslant 0$
and $\hat{Y}\geqslant 0$) and if they commute, then $\hat{X}\hat{Y}\geqslant
0$. Therefore, under the hypothesis of the lemma’s second point, for all
$(i_{1},\dots,i_{k})$,
$\hat{M}_{i_{k}}^{(k)}\cdots\hat{M}_{i_{1}}^{(1)}\geqslant 0$. ∎

###  IX.2 Characterising the set of KD-positive states

A precise and simple characterisation of the KD-positive states is not easy to
come by. Most efforts have focused on the standard KD distribution [Eq. (13)],
for which $k=2$. We now discuss these efforts in some detail. We further
suppose that the measurement operators $\hat{M}_{i_{l}}^{(l)}$, with $l=1,2$,
form complete sets $\left\\{\hat{\Pi}_{i_{l}}^{(l)}\right\\}$ of 1-dimensional
orthogonal projectors in a $d$-dimensional Hilbert space. In other words,
there exist two orthonormal bases, $\\{|a_{i}\rangle\\}$
and$\\{|b_{j}\rangle\\}$, such that $\hat{\Pi}_{i}^{(1)}=\lvert
a_{i}\rangle\\!\langle a_{i}\rvert$ and $\hat{\Pi}_{j}^{(2)}=\lvert
b_{j}\rangle\\!\langle b_{j}\rvert$. For simplicity, we set $i\equiv i_{1}$
and $j\equiv i_{2}$. We introduce

${\mathcal{A}}=\\{\lvert a_{i}\rangle\\!\langle
a_{i}\rvert\\}\quad\mathrm{and}\quad{\mathcal{B}}=\\{\lvert
b_{j}\rangle\\!\langle b_{j}\rvert\\}.$ (92)

The union of the projectors onto the basis vectors forms a subset of the pure
KD-positive states, which form a subset of the extremal KD-positive states:

${\mathcal{A}}\cup{\mathcal{B}}\subseteq{\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}\subseteq{\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{ext}}.$
(93)

Indeed, one can straightforwardly check that all the basis states are KD-
positive. In view of Eq. (93), in the simplest situation,

${\mathcal{A}}\cup{\mathcal{B}}={\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}={\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{ext}}.$
(94)

In this situation,
${\mathcal{E}}_{\mathrm{KD+}}=\mathrm{conv}\left({\mathcal{A}}\cup{\mathcal{B}}\right)$,
and ${\mathcal{E}}_{\mathrm{KD+}}$ forms a polytope.

We define a unitary transition matrix $\hat{U}$ whose entries are
$\hat{U}_{i,j}\equiv\langle a_{i}|b_{j}\rangle$. The following result sums up
multiple in which Eq. (94) holds Langrenez  _et al._ (2023).

######  Theorem IX.2.

Suppose that $m_{{\mathcal{A}},{\mathcal{B}}}\equiv\min_{i,j}\left|\langle
a_{i}|b_{j}\rangle\right|>0$. Equation (94) is true under any one of the
following conditions:

  1. 1.

$d=2$.

  2. 2.

$d$ is a prime number, and the transition matrix $\hat{U}$ equals the
discrete-Fourier-transform matrix.

  3. 3.

In any dimension $d$, for a set of transition matrices $\hat{U}$ that has
probability $1$ in the set of all unitary matrices.

A more detailed statement, as well as proofs of parts (1) and (2), appear in
Langrenez  _et al._ (2023). There, part (3) of the theorem is conjectured. The
proof of part (3) appears in Arvidsson-Shukur  _et al._ (2024b). Part (3) of
the theorem also implies that, in all dimensions $d\geq 2$, the property

${\mathcal{A}}\cup{\mathcal{B}}={\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}$
(95)

holds on a set of transition matrices $\hat{U}$ that has probability $1$ in
the set of all unitary matrices. In other words, in the three scenarios
covered by the theorem, the only pure KD-positive states are the basis states
belonging to ${\mathcal{A}}$ or ${\mathcal{B}}$.

The theorem’s condition $m_{{\mathcal{A}},{\mathcal{B}}}>0$ can be interpreted
as a weak form of incompatibility for the measurements associated with
${\mathcal{A}}$ and ${\mathcal{B}}$ De Bièvre (2023). As mentioned above,
$m_{{\mathcal{A}},{\mathcal{B}}}>0$ also implies that the KD distribution is
informationally complete: it determines $\hat{\rho}$ uniquely [see Eq. (14)].

In specific cases, the structure of ${\mathcal{E}}_{\mathrm{KD+}}$ as a convex
set can be considerably more complicated than when the equalities of Eq. (94)
hold. Figure 12 shows a schematic representation of the situation where

${\mathcal{A}}\cup{\mathcal{B}}\subsetneq{\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}\subsetneq{\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{ext}}\,.$
(96)

In this case, some pure KD-positive states are not basis states, and some
extreme KD-positive states are mixed. Such states cannot be expressed as
convex combinations of KD-positive pure states. This situation arises in
dimension $d=3$, e.g., for spin-1 systems: one can choose for
$\\{|a_{i}\rangle\\}$ the eigenbasis of the angular-momentum operator
$\hat{J}_{z}$ in the $z$-direction and, for $\\{|b_{j}\rangle\\}$, the
eigenbasis of the angular-momentum operator $\hat{J}_{\hat{n}}$ for an
appropriate rotation axis $\hat{n}$ Langrenez  _et al._ (2023). More
information about ${\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}$, when the
transition matrix is a Hadamard matrix ($\left|\langle
a_{i}|b_{j}\rangle\right|^{2}=\frac{1}{d}=m_{{\mathcal{A}},{\mathcal{B}}}$ for
all $i,j$), appears in De Bièvre (2021, 2023); Xu (2022a).

$\mathrm{conv}\left({\mathcal{A}}\cup{\mathcal{B}}\right)$$|a_{1}\rangle\bra{a_{1}}$$|a_{2}\rangle\bra{a_{2}}$$|a_{3}\rangle\bra{a_{3}}$$|b_{1}\rangle\bra{b_{1}}$$|b_{2}\rangle\bra{b_{2}}$$|b_{3}\rangle\bra{b_{3}}$$D$$C$

$\mathrm{conv}\left({\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}\right)$$|a_{1}\rangle\bra{a_{1}}$$|a_{2}\rangle\bra{a_{2}}$$|a_{3}\rangle\bra{a_{3}}$$|b_{1}\rangle\bra{b_{1}}$$|b_{2}\rangle\bra{b_{2}}$$|b_{3}\rangle\bra{b_{3}}$$D$$C$

${\mathcal{E}}_{\mathrm{KD+}}$$|a_{1}\rangle\bra{a_{1}}$$|a_{2}\rangle\bra{a_{2}}$$|a_{3}\rangle\bra{a_{3}}$$|b_{1}\rangle\bra{b_{1}}$$|b_{2}\rangle\bra{b_{2}}$$|b_{3}\rangle\bra{b_{3}}$$D$$C$

Figure 12:  Geometry of the set of KD-positive states. Schematic
representation of the situation in which
${\mathcal{A}}\cup{\mathcal{B}}\subsetneq{\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}\subsetneq{\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{ext}}$
and the Hilbert space is 3-dimensional. The black circle represents the set of
pure quantum states. The point $C$ represents a pure KD-positive state
different from the basis states. The point $D$ represents a mixed extreme
state of ${\mathcal{E}}_{\mathrm{KD+}}$. The (grey) shaded area represents the
set $\mathrm{conv}\left({\mathcal{A}}\cup{\mathcal{B}}\right)$. The (orange)
horizontally hatched area represents the set
$\mathrm{conv}\left({\mathcal{E}}_{\mathrm{KD+}}^{\mathrm{pure}}\right)$. The
(green) dotted area represents the set ${\mathcal{E}}_{\mathrm{KD+}}$.

###  IX.3 Measure of KD non-positivity

If, and only if, a KD distribution is a classical joint probability
distribution,
$\sum_{i_{1},\ldots,i_{k}}\left|Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right|=1$.
Thus, a popular measure of a KD distribution’s total non-positivity is

$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]=\sum_{i_{1},\ldots,i_{k}}\left|Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right|\geq
1,$ (97)

as defined in González Alonso  _et al._ (2019). We now list a few properties
of this non-positivity measure. Details appear in Arvidsson-Shukur  _et al._
(2021).

  1. 1.

Positivity: A KD distribution is a classical probability distribution if, and
only if, $\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]=1$.

  2. 2.

Stricter than non-commutation: If
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]>1$, then
$[\hat{A},\hat{B}]\neq 0$, $[\hat{A},\hat{\rho}]\neq 0$ and
$[\hat{B},\hat{\rho}]\neq 0$. However, even if $[\hat{A},\hat{B}]\neq 0$,
$[\hat{A},\hat{\rho}]\neq 0$ and $[\hat{B},\hat{\rho}]\neq 0$, the
corresponding KD distribution may be positive, such that
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]=1$.

  3. 3.

Convexity: The total non-positivity is a convex function with respect to mixed
states:
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}\left(\sum_{t}p_{t}\hat{\rho}_{t}\right)\right]\leq\sum_{t}p_{t}\,\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}\left(\hat{\rho}_{t}\right)\right]$.

  4. 4.

Extension dependence: A KD distribution $Q_{i_{1},\ldots,i_{l}}(\hat{\rho})$
may follow from the marginalisation of another KD distribution,
$Q_{i_{1},\ldots,i_{k}}(\hat{\rho})$, where $k>l$. In this case,
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{l}}(\hat{\rho})\right]>1$ implies that
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]\geq\mathcal{N}\left[Q_{i_{1},\ldots,i_{l}}(\hat{\rho})\right]>1$.

  5. 5.

Coarse-graining: A KD distribution $Q_{i_{1},\ldots,i_{k}}(\hat{\rho})$ may
represent a state $\hat{\rho}$ in terms of measurement operators that result
from coarse-graining the operators in another KD distribution,
$Q_{i_{1}^{\prime},\ldots,i_{k}^{\prime}}^{\prime}(\hat{\rho})$, where
$k^{\prime}>k$. In this case,
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]\leq\mathcal{N}\left[Q_{i_{1}^{\prime},\ldots,i_{k^{\prime}}}^{\prime}(\hat{\rho})\right]$.

  6. 6.

Maximum value: Consider the KD distributions defined in terms of projective
positive-operator-valued measures $\mathcal{M}^{(1)},\ldots,\mathcal{M}^{(k)}$
(see Sec. II.4). For such KD distributions, the maximum value of $\mathcal{N}$
depends on both the Hilbert-space dimension $d$ and the extendedness $k$:

$\max_{\hat{\rho},\mathcal{M}^{(1)},\ldots,\mathcal{M}^{(k)}}\left\\{\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]\right\\}=d^{(k-1)/2}.$
(98)

To elucidate when the maximum, Eq. (98), attains, we review _mutually unbiased
bases._ Bases $\\{\ket{a_{i}}\\}$ and $\\{\ket{b_{j}}\\}$ are _mutually
unbiased_ if

$|\braket{a_{i}}{b_{j}}|=1/\sqrt{d}\,,$ (99)

for all $i,j$. For details, see Ref. Durt  _et al._ (2010). Suppose that a
basis $\\{\ket{a^{(l)}_{i_{l}}}\\}$ is mutually unbiased with respect to
$\\{\ket{a^{(l\oplus 1)}_{i_{l\oplus 1}}}\\}$, for all $l=1,2,\ldots,k$. The
$\oplus$ represents addition modulo $k$. Suppose, further, that $\hat{\rho}$
is pure and that
$\braket{a^{(1)}_{i_{1}}}{\hat{\rho}}{a^{(1)}_{i_{1}}}=\braket{a^{(k)}_{i_{k}}}{\hat{\rho}}{a^{(k)}_{i_{k}}}=1/d$
for all $i_{1}$ and $i_{k}$. This situation achieves the maximum Eq. (98).
Conceptually, the neighbouring vectors in Eq. (19) should be mutually
unbiased.

###  IX.4 KD-positivity witnesses, uncertainty and (complete) incompatibility

As pointed out in Sec. II.5, whether a KD distribution is positive or non-
positive can be of operational importance. A witness of this property is the
total non-positivity, defined in Eq. (97). A state $\hat{\rho}$ is KD-positive
($\hat{\rho}\in{\mathcal{E}}_{\mathrm{KD+}}$) if and only if
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]=1$. Thus,
$\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]$ faithfully
witnesses KD non-positivity, much as the Wigner negative volume faithfully
witnesses Wigner non-positivity Kenfack and Życzkowski (2004). However, to
compute $\mathcal{N}\left[Q_{i_{1},\ldots,i_{k}}(\hat{\rho})\right]$, one must
know the full KD distribution. One might therefore seek a witness whose
computation requires less information. Below, we review such witnesses.

From a theoretical perspective, an interesting witness is a state’s support
uncertainty, which we now introduce and study further. We, again, restrict our
analysis to the standard KD distribution with respect to orthonormal bases
$\\{\ket{a_{i}}\\}$ and $\\{\ket{b_{j}}\\}$. Let $\sharp X$ denote the
cardinality of the set $X$. Let $|\psi\rangle\in{\mathcal{H}}$ denote a pure
state. We define as follows the support uncertainties of $|\psi\rangle$ in the
bases $\\{\ket{a_{i}}\\}$ and $\\{\ket{b_{j}}\\}$:

$n_{{\mathcal{A}}}(\psi)\equiv\sharp\left\\{i\in[1,d]\mid\langle
a_{i}|\psi\rangle\neq 0\right\\},\quad\mathrm{and}\quad
n_{{\mathcal{B}}}(\psi)\equiv\sharp\left\\{j\in[1,d]\mid\langle
b_{j}|\psi\rangle\neq 0\right\\}.$ (100)

To interpret these definitions, we imagine expanding $|\psi\rangle$ in the
basis $\\{|a_{i}\rangle\\}$ [$\\{|b_{j}\rangle\\}$]. The number of non-
vanishing expansion coefficients is $n_{\mathcal{A}}(\psi)$
[$n_{\mathcal{B}}(\psi)$]. $n_{\mathcal{A}}(\psi)$ and $n_{\mathcal{B}}(\psi)$
enable the construction of an inequality Donoho and Stark (1989) that can be
viewed as an uncertainty relation:

$n_{{\mathcal{A}}}(\psi)\,n_{{\mathcal{B}}}(\psi)\geq
M^{-2},\quad\text{where}\quad M\equiv\max_{i,j}|\langle a_{i}|b_{j}\rangle|.$
(101)

The more $|\psi\rangle$ is localised in the $\\{\ket{a_{i}}\\}$ basis, the
less the state must be localised in the $\\{\ket{b_{j}}\\}$ basis, and vice
versa.

Figure 13 illustrates the lower bound Ineq. (101) with three examples of
mutually unbiased bases. The set of points
$\bm{(}n_{{\mathcal{A}}}(\psi),n_{{\mathcal{B}}}(\psi)\bm{)}$ in the
$n_{{\mathcal{A}}}$-$n_{{\mathcal{B}}}$ plane maps out an uncertainty diagram
for the two bases. It is known that, for any bases $\\{|a_{i}\rangle\\}$ and
$\\{|b_{j}\rangle\\}$, any state $|\psi\rangle$ that saturates Ineq. (101) is
KD-positive De Bièvre (2021). The reverse implication is not generally true,
however De Bièvre (2021). If the bases are mutually unbiased
($M=m_{{\mathcal{A}},{\mathcal{B}}}=1/\sqrt{d}$), nevertheless, saturating
Ineq. (101) is equivalent to $|\psi\rangle$’s being KD-positive Xu  _et al._
(2021). The three panels of Fig. 13 reflect this relation. As a result of this
relation, for mutually unbiased bases in prime dimension, the only pure KD-
positive states are the bases’ elements. For further results about the
geometric structure of the set of KD-positive states, see Sec. IX.2 and
Langrenez  _et al._ (2023).

Let us further explore the links amongst incompatibility, uncertainty and KD
non-positivity in the general case—when the bases are not necessarily mutually
unbiased bases. A different, additive uncertainty relation is more pertinent
than Ineq. (101). The support uncertainty De Bièvre (2021) is defined by

$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\equiv
n_{{\mathcal{A}}}(\psi)+n_{{\mathcal{B}}}(\psi).$ (102)

This quantity is linked to KD positivity through the following theorem De
Bièvre (2021).

######  Theorem IX.3.

Suppose that $m_{{\mathcal{A}},{\mathcal{B}}}\equiv\min_{i,j}\left|\langle
a_{i}|b_{j}\rangle\right|>0$. If
$\lvert\psi\rangle\\!\langle\psi\rvert\in{\mathcal{E}}_{\mathrm{KD+}}$, then
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\leqslant d+1$.

The theorem implies an uncertainty relation: if the uncertainty
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)>d+1$, then the KD distribution must be
non-positive. The bound of Theorem IX.3 holds even when
$m_{{\mathcal{A}},{\mathcal{B}}}=0$, such that the matrix with elements
$\langle a_{i}|b_{j}\rangle$ has zeroes, if there are not too many zeroes De
Bièvre (2021). In the absence of constraints on the two bases, other than that
none of the $|b_{j}\rangle$s are parallel to any of the $|a_{i}\rangle$s, the
upper bound becomes
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\leqslant\frac{3}{2}d$ Arvidsson-Shukur
_et al._ (2021). Intermediate estimates of the type
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\leqslant d+s$, with $1\leq
s\leq\frac{d}{2}\,,$ were recently proven in Xu (2022b) under suitable
conditions on the structure of the transition matrix $\hat{U}$. Reference
Arvidsson-Shukur  _et al._ (2021) encompasses also scenarios in which the KD
distribution is expressed in terms of projectors other than rank-$1$
projectors.

Theorem IX.3 implies that the support uncertainty is a KD-positivity witness
for pure states: if $n_{{\mathcal{A}},{\mathcal{B}}}(\psi)$ is large, then
$\ket{\psi}$ is not KD-positive. However,
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)$ is not always a faithful witness:
there may exist states $|\psi\rangle$ for which
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\leq d+1$ but that are nevertheless KD-
non-positive. See the three panels of Fig. 13 for an example. The support
uncertainty nevertheless has an advantage over the total non-positivity:
limited information about the pure state $|\psi\rangle$ suffices to determine
the support uncertainty. An extension of this result to mixed states appears
in Arvidsson-Shukur  _et al._ (2024c). The extension involves the convex roof
of the support uncertainty.

Theorem IX.3 states that pure KD-positive states have low support uncertainty
$n_{{\mathcal{A}},{\mathcal{B}}}$: no pure KD-positive state can lie above the
line $n_{{\mathcal{A}},{\mathcal{B}}}=d+1$. Figure 13 illustrates this
observation. Further examples appear in De Bièvre (2021, 2023). Theorem IX.3,
as such, evokes quantum optics. There, the Glauber–Sudarshan function of pure
states is positive only for coherent states, which minimise $(\Delta
x)^{2}+(\Delta p)^{2}$ (the sum of two conjugate quadratures’ squared
uncertainties). Unlike in quantum optics, however $d+1$ does not necessarily
_minimise_ the support uncertainty $n_{{\mathcal{A}},{\mathcal{B}}}$. Figure
13 evidences this fact. Nevertheless, the analogy with quantum optics can be
sharpened through _complete incompatibility_ , introduced in De Bièvre (2021,
2023), to which we now turn our attention.

Two observables are said to be incompatible when they do not commute. This
notion is weak: in the context of Eq. (1), it means that at least one of the
elements in $\\{\ket{a_{i}}\\}$ and one of the elements in $\\{\ket{b_{j}}\\}$
satisfy $0<|\langle a_{i}|b_{j}\rangle|<1$. This property is satisfied if the
bases are not permutations of one another. A slightly stronger requirement is
that $M_{{\mathcal{A}},{\mathcal{B}}}\equiv\max_{ij}|\langle
a_{i}|b_{j}\rangle|<1,$ which is implied by
$m_{{\mathcal{A}}{\mathcal{B}}}\equiv\min_{ij}|\langle a_{i}|b_{j}\rangle|>0.$
This latter statement guarantees that all $|a_{i}\rangle$s are distinct from
all $|b_{j}\rangle$s. The latter statement also entails that, if a measurement
of $\\{\ket{a_{i}}\\}$ yields the outcome $a_{i}$, then a subsequent
measurement of $\\{\ket{b_{j}}\\}$ can yield any value $b_{j}$. The
uncertainty in this outcome is maximal for mutually unbiased bases, for which
$M_{{\mathcal{A}},{\mathcal{B}}}=m_{{\mathcal{A}},{\mathcal{B}}}=\frac{1}{\sqrt{d}}$
[see Eq. (99)]. For this reason, mutually unbiased bases are sometimes called
_maximally incompatible_. To introduce complete incompatibility of the bases
$\\{\ket{a_{i}}\\}$ and $\\{\ket{b_{j}}\\}$, we proceed as follows.

false

Figure 13:  Uncertainty diagrams for three mutually unbiased bases. Solid
line: $n_{{\mathcal{A}}}(\psi)+n_{{\mathcal{B}}}(\psi)=d+1$. Dotted curve:
$n_{{\mathcal{A}}}(\psi)n_{{\mathcal{B}}}(\psi)=d$. Blue dots correspond to
KD-non-positive states, and green stars correspond to KD-positive states. The
left diagram is for the Tao matrix Tao (2004) in dimension 6; the middle
diagram is for the discrete-Fourier-transform matrix in dimension 7; and the
right diagram is for the discrete Fourier-transform matrix in dimension 6.

We consider measurements of ${\hat{\Pi}_{{\mathcal{A}}}}(S)$ and
${\hat{\Pi}_{{\mathcal{B}}}}(S)$. We regard these measurements as coarse-
grained measurements of the observables $\hat{A}$ and $\hat{B}$. For every
$S,T\subset\\{1,2,\dots,d\\}$, we let

${\hat{\Pi}_{{\mathcal{A}}}}(S)=\sum_{i\in S}|a_{i}\rangle\langle
a_{i}|\quad\mathrm{and}\quad{\hat{\Pi}_{{\mathcal{B}}}}(T)=\sum_{j\in
T}|b_{j}\rangle\langle b_{j}|.$ (103)

Suppose that a system is prepared in a state $|\psi\rangle$, a measurement of
the projector ${\hat{\Pi}_{{\mathcal{A}}}}(S)$ yields the outcome $1$, and a
subsequent measurement of ${\hat{\Pi}_{{\mathcal{B}}}}(T)$ also yields $1$.
The system is then in the non-normalised state
${\hat{\Pi}_{{\mathcal{B}}}}(T)\,{\hat{\Pi}_{{\mathcal{A}}}}(S)|\psi\rangle$.
Suppose, now, that a subsequent measurement of
${\hat{\Pi}_{{\mathcal{A}}}}(S)$ yields the outcome $1$ with probability $1$.
In other words, suppose that the measurement of
${\hat{\Pi}_{{\mathcal{B}}}}(T)$ has not disturbed the outcome of the previous
measurement of ${\hat{\Pi}_{{\mathcal{A}}}}(S)$. This condition is met only if
${\hat{\Pi}_{{\mathcal{B}}}}(T){\hat{\Pi}_{{\mathcal{A}}}}(S)\ket{\psi}\in{\hat{\Pi}_{{\mathcal{A}}}}(S){\mathcal{H}}$—in
other words, provided that

${\hat{\Pi}_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\hat{\Pi}_{{\mathcal{B}}}}(T){\mathcal{H}}\not=\\{0\\}.$
(104)

${\hat{\Pi}_{{\mathcal{A}}}}(S){\mathcal{H}}$ denotes the
${\hat{\Pi}_{{\mathcal{A}}}}(S)$ eigenspace associated with the eigenvalue
$1$. ${\hat{\Pi}_{{\mathcal{B}}}}(T){\mathcal{H}}$ is defined similarly.
Subsequent measurements of ${\hat{\Pi}_{{\mathcal{A}}}}(S)$ and
${\hat{\Pi}_{{\mathcal{B}}}}(T)$ will then consistently yield the outcome $1$.
Two successive measurements can be compatible in this way even if
${\hat{\Pi}_{{\mathcal{A}}}}(S)$ and ${\hat{\Pi}_{{\mathcal{B}}}}(T)$ do not
commute; see De Bièvre (2023) for examples.

Bases ${\mathcal{A}}$ and ${\mathcal{B}}$ are said to be completely
incompatible when the above-described situation [Eq. (104)] is never realised
De Bièvre (2021, 2023). That is, ${\mathcal{A}}$ and ${\mathcal{B}}$ are are
completely compatible when, for all $S,T$ with $|S|+|T|\leq d$,

${\hat{\Pi}_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\hat{\Pi}_{{\mathcal{B}}}}(T){\mathcal{H}}=\\{0\\}.$
(105)

The restriction $|S|+|T|\leq d$ is needed since, whenever $|S|+|T|>d$,
${\hat{\Pi}_{{\mathcal{A}}}}(S){\mathcal{H}}\cap{\hat{\Pi}_{{\mathcal{B}}}}(T){\mathcal{H}}\not=\\{0\\}$
for dimensional reasons: sufficiently coarse-grained measurements are always
compatible in the above sense [Eq. (104)].

For completely incompatible bases, for all $|\psi\rangle\in{\mathcal{H}}$,

$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)\geq d+1.$ (106)

Therefore, if the bases ${\mathcal{A}}$ and ${\mathcal{B}}$ are completely
incompatible and if
$\lvert\psi\rangle\\!\langle\psi\rvert\in{\mathcal{E}}_{\mathrm{KD+}}$, then
$n_{{\mathcal{A}},{\mathcal{B}}}(\psi)=d+1$. In other words, when the bases
are completely incompatible, the KD-positive states have minimal support
uncertainties. The middle panel of Fig. 13 realises this situation, but the
other two panels do not.

There is an open, dense set of bases that are completely incompatible De
Bièvre (2023). However, it is not generally straightforward to determine if
two given bases are completely incompatible. For example, consider two bases
whose transition matrix is the discrete Fourier transform. These bases are
completely incompatible only when the dimension $d$ is prime Langrenez  _et
al._ (2023). Consequently, not all mutually unbiased bases are completely
incompatible. In other words, maximal incompatibility does not imply complete
incompatibility. Figure 13 illustrates this phenomenon. Reference Xu (2022b)
proposes a further extension of the notion of complete incompatibility. In
summary, the KD distribution can be a useful tool for designing and studying
notions of incompatibility that extend beyond non-commutation.

##  X Conclusion and outlook

Throughout this Article, we have provided a comprehensive review of use cases
of the KD distribution. In Sec. II, we defined the KD distribution and showed
that it obeys a quasi-probabilistic version of Bayes’ theorem. In Sec.III.1,
we showed that non-real KD quasi-probabilities underlie the disturbance of
measurement-outcome probabilities. Similarly, as we outlined in Sec. III.2,
one’s ability to conduct quantum metrology, to learn unknown parameters
encoded in a quantum state, hinges on non-real KD quasi-probabilities. In
post-selected quantum metrology, one passes several quantum particles through
a filter that distils their metrological information into the particles which
pass the filter. In Sec.III.3, we showed that the rate of information
distillation could be arbitrarily large if a KD distribution has negative
components. In Sec. IV, we reviewed weak values, pre- and post-selected
observable averages of quantum states. We showed that non-positive KD
distributions can lead to a weak value that lies outside the measured
observable’s spectrum. We also reviewed how such anomalous weak values can
amplify metrological sensitivity to small unknown parameters. In Sec. V, we
introduced the continuous-variable KD distribution. Then, we reviewed how
measurements of continuous-variable KD distributions have been used to
directly measure quantum wavefunctions.

In classical thermodynamics, probability distributions describe statistic work
and heat exchanges. In Sec. VI, we showed how KD distributions can describe
statistic exchanges in quantum thermodynamics. In Sec. VII, we introduced the
out-of-time-ordered correlator (OTOC), a popular witness of many-body quantum
chaos. First, we showed that the OTOC equals the average over a KD
distribution. Second, whilst the OTOC struggles to distinguish information
scrambling from decoherence, the KD distribution’s non-positivity can witness
scrambling more reliably.

We also summarised the KD distribution’s importance in the foundations of
quantum mechanics. In Sec. VIII.1, we showed how non-positive KD quasi-
probabilities are required to violate Leggett–Garg inequalities (temporal Bell
inequalities). In Sec. VIII.2, we described how the KD distribution is used in
the consistent-histories interpretation of quantum mechanics. In Sec. VIII.3,
we reviewed a rigorous notion of non-classicality: contextuality. Non-positive
KD quasi-probabilities can enable non-classical advantages in the operation of
engines and weak measurements.

Given the KD distribution’s diverse use cases, it is unsurprising that the
distribution has been subject to growing mathematical research. We summarised,
in Sec. IX, the current knowledge about the KD distribution’s mathematical
properties. A basic necessary, but insufficient, condition for KD non-
positivity is that ${\mathcal{A}}=\\{\lvert a_{i}\rangle\\!\langle
a_{i}\rvert\\}$ differ from ${\mathcal{B}}=\\{\lvert b_{j}\rangle\\!\langle
b_{j}\rvert\\}$. One could further ask: When is a quantum state $\hat{\rho}$
KD-positive? As outlined in Sec. IX.2, there are several scenarios in which
$\hat{\rho}$ is KD-positive if, and only if, $\hat{\rho}$ is a convex
combination of $\lvert a_{i}\rangle\\!\langle a_{i}\rvert$s and $\lvert
b_{j}\rangle\\!\langle b_{j}\rvert$s. For some pairs of bases, however, KD-
positivity is not equivalent to this convex-combination property. Often, KD
non-positivity underlies non-classical advantages in quantum experiments. In
Sec. IX.3, we summarised properties of a KD distribution’s total non-
positivity. In Sec. IX.4, we showed how to construct KD non-positivity
witnesses that do not require full knowledge of the KD distribution. Also, we
discussed how the KD distribution relates to a quantum state’s uncertainties
with respect to two observables.

Research on the KD distribution is ongoing. We conclude this review by listing
a few promising outlook directions.

  * •

Quantum metrology: We described the relationships between discrete-variable
quantum metrology and non-positive KD distributions. Such relations have
barely been explored in continuous-variable systems. Negative Wigner quasi-
probabilities can underlie advantages in continuous-variable metrology. An
open question is whether the negativity requirement extends to KD
representations of the experiments.

  * •

Weak values: The KD distribution offers a means of understanding weak values’
anomalous behaviours. A negative KD quasi-probability is required for the
conditioned average of a weakly measured observable to lie outside the
observable’s spectrum. When is the KD distribution, or one of its extensions,
the most appropriate quasi-probability distribution for characterising more-
general conditioned sequences of weakened measurements? For example, as the
weak-measurement strength increases, the KD quasi-probabilities will transform
smoothly into projective-measurement probabilities. However, the intermediate-
measurement-strength regime requires more study.

  * •

Direct measurement: Direct measurement provides an effective characterisation
of quantum states, KD distributions, processes and detectors. An experimental
opportunity is to capitalise on direct measurement’s advantage over standard
quantum tomography in applications to complex quantum systems. Possible
targets include molecules and molecular processes, the multi-particle
entangled states prepared by quantum circuits and quantum materials.

  * •

Quantum thermodynamics: Opportunities for future work include the marriage of
KD distributions with conserved quantities (charges) that fail to commute with
each other Lostaglio  _et al._ (2017); Guryanova  _et al._ (2016); Yunger
Halpern  _et al._ (2016). Such charges were overlooked for decades but
engendered a growing subfield recently Majidy  _et al._ (2023); Yunger Halpern
and Majidy (2022); Kranzl  _et al._ (2023). Example charges include the $x-$,
$y-$, and $z-$components of spin Yunger Halpern  _et al._ (2020). The charges’
noncommutation suggests them as observables whose eigenbases can define KD
distributions naturally suited to quantum thermodynamics. Initial work on
applying KD distributions to noncommuting thermodynamic charges has begun
(Sections VI and VIII.3) but merits expansion beyond currents.

  * •

Foundations of quantum mechanics: One of the most rigorous notions of quantum
phenomena is contextuality. If all quasi-probabilistic representation of an
experiment are non-positive, the experiment is contextual Spekkens (2008).
Investigating infinitely many quasi-probability distributions is a formidable
task. Could some distributions be more important than others? Often, the KD
distribution is tailored to the operations that form an experiment. One could
imagine that KD non-positivity may propagate to other quasi-probability
representations.

  * •

Mathematical properties: As reviewed above, the geometric structure of the
convex set of KD-positive states is known in many cases. Nevertheless, for
several operational tasks, we lack figures of merit for determining which KD-
non-positive states are the most useful. Such metrics call for development.

##  XI Acknowledgements

The authors thank Holger Hofmann for useful discussions, in particular for
pointing out the relation between imaginary components of the KD distribution
and measurement disturbance. The authors also thank Noah Lupu-Gladstein and
Batuhan Yilmaz for sharing their data and preparing Fig. 3. Further, the
authors thank Crispin Barnes for input and guidance about the consistent-
histories interpretation of quantum mechanics and Wilfred Salmon for
suggestions regarding gate-based measurements of the KD distribution.

This work was supported in part by the Agence Nationale de la Recherche under
grant ANR-11-LABX-0007-01 (Labex CEMPI), by the Nord-Pas de Calais Regional
Council and the European Regional Development Fund through the Contrat de
Projets État-Région (CPER), by the CNRS through the MITI interdisciplinary
programs, and by the John Templeton Foundation (award no. 62422).

JSL acknowledges the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC), Canada Research Chairs, the Transformative Quantum
Technologies Canada First Excellence Research Fund.

DRMAS was supported by Girton College.

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