Quasi-probability distributions for observ-
ables in dynamic systems
Patrick P. Hofer
Department of Applied Physics, University of Geneva, 1211 Geneva 4, Switzerland
October 12, 2017
We develop a general framework to investigate fluctuations of non-commuting ob-
servables. To this end, we consider the Keldysh quasi-probability distribution (KQPD).
This distribution provides a measurement-independent description of the observables
of interest and their time-evolution. Nevertheless, positive probability distributions for
measurement outcomes can be obtained from the KQPD by taking into account the
effect of measurement back-action and imprecision. Negativity in the KQPD can be
linked to an interference effect and acts as an indicator for non-classical behavior. No-
table examples of the KQPD are the Wigner function and the full counting statistics,
both of which have been used extensively to describe systems in the absence as well
as in the presence of a measurement apparatus. Here we discuss the KQPD and its
moments in detail and connect it to various time-dependent problems including weak
values, fluctuating work, and Leggett-Garg inequalities. Our results are illustrated
using the simple example of two subsequent, non-commuting spin measurements.
1 Introduction
Quasi-probability distributions such as the Wigner function have been an important tool in quan-
tum mechanics since the early days of the theory [1]. Indeed, the Wigner function lies at the heart
of the phase-space representation of quantum mechanics which is tightly connected to classical
statistical mechanics and provides an alternative route to quantization [2]. In classical statistical
mechanics, averages of observables can be described as averages taken with respect to some un-
derlying probability distribution. This probability distribution fully describes the system under
consideration and reflects the lack of knowledge of the observer. If one has perfect knowledge
of the behavior of all degrees of freedom, there would be no need for probability distributions
and measurement outcomes could be predicted in a deterministic manner. This is not possible
in quantum mechanics due to the probabilistic nature of the theory. Similarly to classical sta-
tistical mechanics, the fluctuations in quantum mechanics can be encoded in quasi-probability
distributions. These distributions can take on negative values which is ultimately a consequence
of the non-commutativity of operators, reflecting the fact that a measurement inevitably disturbs
all subsequent measurements of observables which do not commute with the first one. Negative
quasi-probability distributions therefore reflect the impossibility of accessing observables without
disturbing them, indicating a non-classical behavior of the system which requires to take into ac-
count the action of the measurement apparatus to predict its outputs. The concept of negative
quasi-probability distributions, or negativity in short, is therefore closely related to the concept
of contextuality which deals with the dependence of measurement outcomes on the experimental
setup [3–6].
So far, most quasi-probability distributions describe the quantum state of a system. This holds
true for the Wigner function and its generalizations to finite dimensional Hilbert spaces [6–8], as well
as a number of other distributions used in quantum optics [9, 10]. Negativity in these distributions
Patrick P. Hofer: patrick.hofer@unige.ch
Accepted in Quantum 2017-09-19, click title to verify 1
arXiv:1702.00998v4 [quant-ph] 11 Oct 2017
indicates non-classicality of the quantum state. However, a state that exhibits negativity in some
distribution might not do so in another one [11], implying that non-classicality is not necessarily
a general feature but depends on the way a state is interrogated (see also Refs. [3–5, 12]). A well
studied quasi-probability distribution which does not describe the quantum state of a system but
rather its dynamics is the full counting statistics (FCS) [13–19]. Negativity in the FCS has recently
been connected to a peculiar interference effect and it was shown that systems which do not show
any negativity in their Wigner function can still show negativity in the FCS [19]. This further
stresses the fact that systems might exhibit completely classical behavior in some aspects, e.g.
their instantaneous states, while exhibiting non-classicality in others, e.g. their dynamics which
determine the temporal fluctuations within the system. It would therefore be highly desirable
to have access to a quasi-probability distribution which encodes only the aspects of interest of a
system. This is in marked contrast to Refs. [3–5, 20] which consider a full description of states and
measurements in terms of quasi-probabilities.
The goal of the present paper is to introduce a general quasi-probability distribution that can be
tailored to the problem at hand and thus provides a general framework for investigating fluctuations
of non-commuting observables. Because this distribution is based on the Keldysh formalism, we
call it the Keldysh quasi-probability distribution (KQPD). When the observables of interest are
the position and momentum operators, the KQPD reduces to the Wigner function. Similarly, when
one is interested in the time-integral of an observable, the FCS of the corresponding operator is
obtained. Furthermore, the outcome of von Neumann type measurements [21] can be predicted
by convolving the corresponding KQPD with the Wigner function of the detectors [cf. Sec. 3].
This convolution takes into account the imprecision as well as the back-action of the performed
measurement. An example is provided by the Husimi Q-function which is obtained by convolving
the Wigner function with Gaussians and predicts the outcomes of simultaneous but imprecise
measurements of position and momentum [22].
The rest of the article is structured as follows. In Sec. 2, we introduce the KQPD, discuss
the Wigner function, the FCS, and two subsequent spin measurements as examples, and discuss
negativity of the KQPD as an indicator for non-classical behavior. In Sec. 3, we provide a more
physical motivation for investigating the KQPD by connecting it to measurement outcomes which
requires the introduction of measurement imprecision and back-action. Before concluding in Sec. 5,
we discuss applications of the KQPD in Sec. 4, including the distribution of thermodynamic work,
(anomalous) weak values, and Leggett-Garg inequalities. Throughout this paper, we set ~ = 1.
2 The Keldysh quasi-probability distribution
In this section we briefly introduce the Keldysh formalism, a powerful framework for tackling out-
of-equilibrium problems, where the KQPD arises quite naturally through its moment generating
function. Here we only touch upon the ideas in the Keldysh formalism which are relevant for our
discussion. For a detailed introduction to the subject we refer the reader to Ref. [23]. We will
start by considering the somewhat trivial example of the statistics of a single observable before we
generalize the results to the highly non-trivial scenario of multiple, non-commuting observables at
different times.
2.1 Single observable
The main idea that we will be concerned with is the evolution along the closed time contour
illustrated in Fig. 1, an idea that goes back to Schwinger [24]. In this section, we are interested in
the statistics of a single observable
ˆ
A at time τ . Its average value can be written as
hAi
τ
= Tr
n
ˆ
U(0, τ)
ˆ
A
ˆ
U(τ, 0)ˆρ
0
o
=
1
2
Tr
n
ˆ
U(0, T )
ˆ
U(T , τ)
ˆ
A
ˆ
U(τ, 0)ˆρ
0
o
+
1
2
Tr
n
ˆ
U(0, τ)
ˆ
A
ˆ
U(τ, T )
ˆ
U(T , 0)ˆρ
0
o
,
(1)
where
ˆ
U(t
0
, t) =
ˆ
U
†
(t, t
0
) is the time-evolution operator from time t to time t
0
and ˆρ
0
denotes
the density matrix at time t = 0. Throughout this article, we consider unitary time evolution.
Generalizing the results to non-unitary time evolution is in principle straightforward and discussed
Accepted in Quantum 2017-09-19, click title to verify 2
Figure 1: Illustration of the closed time contour. The average h
ˆ
Ai
τ
can be obtained by evolving the state (or
a mixture of states) up to time τ , where the operator
ˆ
A is evaluated, and back again. Alternatively, one can
extend this closed-time contour to the Keldysh contour (from t = 0 to t = T and back again) and evaluate the
operator either on the forward-in-time or on the backward-in-time branch.
in App. A. Writing the density matrix as a mixture of pure states, the first line of the last expression
can be interpreted as time evolving the states up to time τ , evaluating the operator
ˆ
A, and evolving
the states backward to the initial time t = 0 [cf. the left-hand side of Fig. 1]. On the second line,
we inserted identities 1 =
ˆ
U(τ, T )
ˆ
U(T , τ), extending the contour along which the state is time
evolved. We call the resulting contour, from t = 0 to t = T and back again, the Keldysh contour.
We can evaluate the operator
ˆ
A either along the forward-in-time or along the backward-in-time
branch of the contour. As discussed below, the most natural choice is to treat both contours on
an equal footing as illustrated graphically in Fig. 1.
More information on the observable
ˆ
A can be obtained by considering its moment generating
function given by
Λ(λ; τ) = he
−iλ
ˆ
A
i
τ
= Tr
n
ˆ
U(0, τ)e
−i
λ
2
ˆ
A
ˆ
U(τ, T )
ˆ
U(T , τ)e
−i
λ
2
ˆ
A
ˆ
U(τ, 0)ˆρ
0
o
. (2)
All higher moments can be obtained from this generating function by taking derivatives with
respect to λ
h
ˆ
A
k
i
τ
= i
k
∂
k
λ
Λ(λ; τ)
λ=0
. (3)
In anticipation of the discussion below, we wrote Eq. (2) symmetrically in terms of the Keldysh
contour. Taking the derivative with respect to λ at λ = 0, we recover Eq. (1). Since the moment
generating function is the average of the exponential of
ˆ
A, it corresponds to the Fourier transform
of a probability distribution associated to the fluctuations of
ˆ
A
Λ(λ; τ) =
Z
∞
−∞
dAP(A; τ)e
−iλA
, P(A; τ) =
X
A
0
δ(A − A
0
)hA
0
|
ˆ
U(τ, 0)ˆρ
0
ˆ
U
†
(τ, 0)|A
0
i, (4)
where the sum over A
0
goes over all eigenstates
ˆ
A|A
0
i = A
0
|A
0
i and has to be replaced by an
integral if
ˆ
A has a continuous spectrum. In the case of a single observable, the KQPD P(A; τ ) is
a positive probability distribution which predicts the outcome of projectively measuring
ˆ
A at time
τ.
The symmetric treatment with respect to the forward-in-time and the backward-in-time branches
of the Keldysh contour might seem like an unnecessary complication at this point. Below, we show
how such a treatment for multiple observables leads to a time-ordering prescription which is rel-
evant for von Neumann type measurements. Before discussing multiple observables in detail, we
express the moment generating function in terms of the classical and quantum fields obtained by
a Keldysh rotation [cf. Eq. (8)]. To this end, we assume that the observable
ˆ
A has a discrete
spectrum and that its eigenstates span the whole Hilbert space, i.e. 1 =
P
A
|AihA|. We then
divide the time evolution along the Keldysh contour in Eq. (2) into 2N steps (N for the forward-,
N for the backward-in-time branch) of infinitesimal length δt and insert identities in between the
steps. Writing the trace in terms of the eigenstates of
ˆ
A, we then find
Λ(λ; τ) =
X
A
+
,A
−
δ
A
+
T
,A
−
T
e
−i
λ
2
(A
+
τ
+A
−
τ
)
A(A
+
)A
∗
(A
−
)hA
+
0
|ˆρ|A
−
0
i. (5)
Here, A
+(−)
t
denotes the state that was inserted at time t on the forward-in-time (backward-in-time)
branch, the vectors A
±
gather all A
±
t
and the amplitudes along the branches read
A(A
±
) = hA
±
T
|
ˆ
U(T , T − δt)|A
±
T −δt
ihA
±
T −δt
|···|A
±
δt
ihA
±
δt
|
ˆ
U(δt, 0)|A
±
0
i. (6)
Accepted in Quantum 2017-09-19, click title to verify 3
The Kronecker delta in Eq. (5) is due to the fact that the point t = T connects the two branches
(cf. Fig. 1). We can write the moment generating function in the form of a path integral
Λ(λ; τ) =
Z
D[A
+
t
, A
−
t
]e
−i
λ
2
(A
+
τ
+A
−
τ
)
e
iS[A
+
t
,A
−
t
]
, (7)
where the action S is determined by exp(iS[A
+
t
, A
−
t
]) = A(A
+
)A
∗
(A
−
)hA
+
0
|ˆρ|A
−
0
i and the inte-
gration measure reads
R
D[A
+
t
, A
−
t
] =
P
A
+
,A
−
δ
A
+
T
,A
−
T
. Finally, we perform a Keldysh rotation
by introducing the new fields [23]
A
cl
t
=
1
2
(A
+
t
+ A
−
t
), A
q
t
= A
+
t
− A
−
t
, (8)
where the superscript stands for classical and quantum respectively. This nomenclature derives
from the fact that the forward- and backward-in-time evolutions are equal for classical processes.
Using these new integration variables, the path integral in Eq. (7) reduces to
Λ(λ; τ) =
Z
D[A
cl
t
, A
q
t
]e
−iλA
cl
τ
e
iS[A
cl
t
,A
q
t
]
. (9)
The corresponding probability distribution then reads
P(A; τ) =
1
2π
Z
∞
−∞
e
iλA
dλΛ(λ; τ) =
Z
D[A
cl
t
, A
q
t
]δ(A − A
cl
τ
)e
iS[A
cl
t
,A
q
t
]
. (10)
The moment generating function thus describes the fluctuations of A
cl
τ
and the probability density
P(A; τ) is given by the sum over all paths for which A
cl
τ
= A, weighted by the complex exponential
of the action in complete analogy to Feynman path integrals [25]. In the case of a single observable,
the quantum field is zero (i.e. A
+
τ
= A
−
τ
), making its introduction seem like an unnecessary
complication. However, when multiple observables are involved, the quantum fields are no longer
bound to be zero. Considering the fluctuations of the classical fields then corresponds to a given
time-ordering prescription. We now generalize the results of this section to an arbitrary number
of (possibly non-commuting) observables at arbitrary times.
2.2 Multiple observables
The KQPD, denoted by P(A), can be defined via its moment generating function which is obtained
by generalizing Eq. (2) and reads
Λ(λ) ≡
Z
dλe
−iλ·A
P(A)
≡ Tr
(
ˆ
T
+
exp
"
−
i
2
Z
T
0
dτ
X
l
f
l
(τ)λ
l
ˆ
A
l
(τ)
#
ˆρ
0
ˆ
T
−
exp
"
−
i
2
Z
T
0
dτ
X
l
f
l
(τ)λ
l
ˆ
A
l
(τ)
#)
= Tr
(
ˆ
T
K
exp
"
−
i
2
Z
K
dτ
X
l
f
l
(τ)λ
l
ˆ
A
l
(τ)
#
ˆρ
0
)
.
(11)
Here
ˆ
T
+(−)
is the (anti) time-ordering operator, the variables λ
l
and A
l
are grouped in vectors λ
and A, and the operators are given in the Heisenberg picture, i.e.
ˆ
A
l
(t) =
ˆ
U(0, t)
ˆ
A
l
ˆ
U(t, 0), (12)
where
ˆ
A denotes the operator in the Schrödinger picture (where all considered operators are time-
independent). In the last line of Eq. (11),
ˆ
T
K
denotes time-ordering along the Keldysh contour
(from t = 0 to t = T and back again). The functions f
l
(τ) take into account a possibly time-
dependent coupling to the observables
ˆ
A
l
. When using the Keldysh contour, the functions f
l
take
on the same values on the forward- and the backward-in-time branch. The case of N observables
Accepted in Quantum 2017-09-19, click title to verify 4
probed instantaneously at different times τ
l
is obtained by taking f
l
(τ
l
) = δ(t − τ
l
) (for l ≤ N)
resulting in the moment generating function
Λ(λ) = Tr
n
e
−i
λ
N
2
ˆ
A
N
(τ
N
)
···e
−i
λ
1
2
ˆ
A
1
(τ
1
)
ˆρ
0
e
−i
λ
1
2
ˆ
A
1
(τ
1
)
···e
−i
λ
N
2
ˆ
A
N
(τ
N
)
o
. (13)
The apparent symmetry between terms on the left and on the right of the density matrix is equiva-
lent to treating the forward- and the backward-in-time branch of the time evolution symmetrically
and determines the time-ordering of operators when evaluating higher moments (see below).
2.3 Examples
We now introduce two well-known examples of the KQPD: the Wigner function and the Full
counting statistics (FCS). To recover the Wigner function from the moment generating function
in Eq. (11), we set
X
l
f
l
(τ)λ
l
ˆ
A
l
(τ) = δ(τ ) [λ
x
ˆx(τ ) + λ
p
ˆp(τ)] , (14)
which results in
Λ(λ
x
, λ
p
) = Tr
n
e
−
i
2
(λ
x
ˆx+λ
p
ˆp)
ˆρ
0
e
−
i
2
(λ
x
ˆx+λ
p
ˆp)
o
, (15)
where we identified ˆx = ˆx(0) and similarly for ˆp. It is straightforward to verify that the Wigner
function is indeed given by
W(x, p) =
1
(2π)
2
Z
dλ
x
dλ
p
e
iλ
x
x+iλ
p
p
Λ(λ
x
, λ
p
) =
1
2π
Z
dλ
p
e
−iλ
p
p
x +
λ
p
2
ˆρ
0
x −
λ
p
2
. (16)
The Wigner function is thus recovered as the KQPD corresponding to a simultaneous observation
of ˆx and ˆp. In the next section, we discuss how the Wigner function can be used to predict a
simultaneous (but imprecise) measurement of position and momentum [26]. The moments obtained
from the Wigner function
hx
n
p
m
i
W
=
Z
dxdp x
n
p
m
W(x, p), (17)
are determined by the Weyl order [27] obtained by complete symmetrization, i.e. every possible
order is given the same weight, e.g.
hxpi
W
=
1
2
Tr {(ˆxˆp + ˆpˆx)ˆρ
0
}, hx
2
pi
W
=
1
3
Tr
(ˆx
2
ˆp + ˆxˆpˆx + ˆpˆx
2
)ˆρ
0
. (18)
This is a general result for the KQPD; when considering multiple operators at equal times, the
moments generated by the KQPD are determined by the quantum averages of the completely
symmetrized operators.
The next example we consider in this section is the FCS which is obtained from the moment
generating function in Eq. (11) upon setting
X
l
f
l
(τ)λ
l
ˆ
A
l
(τ) = λ
ˆ
A(τ), (19)
which results in
Λ(λ) = Tr
ˆ
T
+
e
−i
λ
2
R
T
0
dτ
ˆ
A(τ)
ˆρ
0
ˆ
T
−
e
−i
λ
2
R
T
0
dτ
ˆ
A(τ)
= Tr
ˆ
T
K
e
−i
λ
2
R
K
dτ
ˆ
A(τ)
ˆρ
0
. (20)
The FCS is given by the Fourier transform of the last expression
F(m) ≡
1
2π
Z
dλe
iλm
Λ(λ), (21)
and gives information on the fluctuations of the time integral of
ˆ
A. The FCS has been investigated
intensively in the context of electron transport [14], where
ˆ
A corresponds to the current operator
Accepted in Quantum 2017-09-19, click title to verify 5
and one is interested in the fluctuations of the charge that is transferred through a conductor
during a time interval [0, T ].
The moments of the FCS can conveniently be expressed using the Keldysh contour
Z
dm m
k
F(m) = i
k
∂
k
λ
Λ(λ)
λ=0
=
Z
K
dτ
1
···dτ
k
Tr
n
ˆ
T
K
ˆ
A(τ
1
) ···
ˆ
A(τ
k
)ˆρ
0
o
. (22)
We thus find that the time-ordering corresponding to the KQPD is determined by the Keldysh time-
ordering, implying that operators are ordered along the Keldysh contour. The Wigner function as
well as the FCS have been discussed extensively in the literature. Explicit evaluations exhibiting
negativity can be found in, e.g., Refs. [19, 28].
To further illustrate the KQPD, we consider the very simple example of two subsequent spin
measurements. The operators of interest are given by the rotated Pauli matrices ˆσ
1
and ˆσ
2
. For
simplicity, we consider a qubit in a pure state |+
0
i which is an eigenstate of ˆσ
0
with eigenvalue
+1. The KQPD then reads
P(Σ
1
, Σ
2
) =
1
(2π)
2
Z
dλ
1
dλ
2
e
iλ
1
Σ
1
+iλ
2
Σ
2
Tr
n
e
−i
λ
2
2
ˆσ
2
e
−i
λ
1
2
ˆσ
1
|+
0
ih+
0
|e
−i
λ
1
2
ˆσ
1
e
−i
λ
2
2
ˆσ
2
o
. (23)
Writing the eigenstates of ˆσ
j
in the eigenbasis of ˆσ
1
|+
0
i = α|+
1
i + β|−
1
i |+
2
i = γ|+
1
i + δ|−
1
i, (24)
we can evaluate Eq. (23), which can be written as
P(Σ
1
, Σ
2
) =
X
σ
1
=0,±1
X
σ
2
=±1
˜
P(σ
1
, σ
2
)δ(Σ
1
− σ
1
)δ(Σ
2
− σ
2
), (25)
with the discrete probability distribution
˜
P(+1, +1) = |α|
2
|γ|
2
,
˜
P(−1, +1) = |β|
2
|δ|
2
,
˜
P(0, +1) = 2Re {αβ
∗
γ
∗
δ},
˜
P(+1, −1) = |α|
2
|δ|
2
,
˜
P(−1, −1) = |β|
2
|γ|
2
,
˜
P(0, −1) = −2Re {αβ
∗
γ
∗
δ}.
(26)
Note that although the operators ˆσ
j
have eigenvalues ±1, the KQPD is non-zero also for σ
1
= 0.
Furthermore, these terms can become negative. We will discuss the origin of these terms and
their negativity in more detail in the next section. For now, we only mention that the negativity
reflects the fact that a measurement of ˆσ
1
disturbs a subsequent measurement of ˆσ
2
. The negative
terms thus vanish if these two operators are simultaneously measurable, [ˆσ
1
, ˆσ
2
] = 0, if the first
measurement does not influence the initial state, [ˆσ
0
, ˆσ
1
] = 0, or if a measurement of ˆσ
2
has a
completely random outcome no matter if ˆσ
1
is measured or not. The latter is the case if ˆσ
0
= ˆσ
x
,
ˆσ
1
= ˆσ
y
, and ˆσ
2
= σ
z
. In all these cases, a measurement of ˆσ
2
yields outcomes which are
independent of the presence or absence of a previous measurement of ˆσ
1
. The KQPD for two
subsequent spin measurements is illustrated in Fig. 2.
2.4 Negativity
As mentioned above, negativity in the KQPD indicates that measuring the considered observables
necessarily influences the outcomes (i.e. measuring all of the operators results in different statistics
than measuring each of them in a different realization). Following Ref. [19], we corroborate this
claim by connecting negative values in the KQPD to coherent superpositions of different eigenstates
of the observables. In the presence of such superpositions, any measurement (partly) collapses the
state, possibly influencing subsequent measurement outcomes. For clarity, we consider the case of
N observables at subsequent times as described by the moment generating function in Eq. (13).
Assuming that the eigenstates of each operator
ˆ
A
l
span the whole Hilbert space, we replace
e
−i
λ
l
2
ˆ
A
l
(τ
l
)
=
X
A
α
l
e
−i
λ
l
2
A
α
l
ˆ
U(0, τ
l
)|A
α
l
ihA
α
l
|
ˆ
U(τ
l
, 0), (27)
where
ˆ
A
l
|A
α
l
i = A
α
l
|A
α
l
i and the superscript α = L, R labels if the replacement happened on the
left (L) or on the right (R) of the density matrix in Eq. (13), i.e. α labels the branch (forward- or
Accepted in Quantum 2017-09-19, click title to verify 6
Figure 2: Keldysh quasi-probability distribution for two subsequent spin measurements on a pure qubit state.
While the second argument is restricted to the eigenvalues of ˆσ
2
, the first argument can also be zero. (a)
ˆσ
0
= ˆσ
2
= ˆσ
x
and ˆσ
1
= ˆσ
z
. In this case, a measurement of ˆσ
1
influences a subsequent measurement of
ˆσ
2
which results in a negative value for
˜
P(0, −1). (b) ˆσ
0
= ˆσ
x
, ˆσ
1
= ˆσ
y
, and ˆσ
2
= σ
z
. In this case, the
measurement outcome of ˆσ
2
is completely random, no matter if the first measurement is performed or not.
A measurement of ˆσ
1
does therefore not influence a subsequent measurement of ˆσ
2
and the KQPD is purely
positive.
backward-in-time). This allows us to take the λ-dependent factors out of the trace and obtain the
KQPD by a Fourier transformation
P(A) =
X
A
L
,A
R
δ
A
L
N
,A
R
N
δ
A −
A
L
+ A
R
2
A(A
L
)A
∗
(A
R
)hA
L
1
|ˆρ
τ
1
|A
R
1
i. (28)
Here the vectors A
L/R
have as their entries A
L/R
l
and thus denote a particular manifestation of
eigenvalues that we call a trajectory. The amplitudes associated with such a trajectory read
A(A
α
) = hA
α
N
|
ˆ
U(τ
N
, τ
N−1
)|A
α
N−1
ihA
α
N−1
|···|A
α
2
ihA
α
2
|
ˆ
U(τ
2
, τ
1
)|A
α
1
i, (29)
the Dirac delta for vectors is defined as
δ
A −
A
L
+ A
R
2
=
Y
l
δ
A
l
−
A
R
l
+ A
L
l
2
, (30)
and ˆρ
τ
1
=
ˆ
U(τ
1
, 0)ˆρ
0
ˆ
U(0, τ
1
).
Equation (28) expresses the KQPD as a sum over pairs of trajectories. These pairs, here
denoted left (L) and right (R) trajectory, are equivalent to a single trajectory along the Keldysh
contour. A given pair of trajectories contributes with a weight that is given by the interference
term A(A
L
)A
∗
(A
R
) times a density matrix element hA
L
1
|ˆρ
τ
1
|A
R
1
i corresponding to the “starting
point” of the trajectories. Each pair contributes to the KQPD at the argument which corresponds
to the mean value of the observable on the two trajectories, again underlining the fact that the
KQPD considers the fluctuations of the classical field in the Keldysh language (see below). Finally,
the Kronecker delta in Eq. (28) results from the trace and reflects the fact that the disturbance of
the last measurement is irrelevant.
From Eq. (28), we see that as long as A
L
= A
R
, the KQPD is strictly greater or equal to
zero. It is therefore the interference terms of trajectories that differ from each other which allow
the KQPD to become negative. This is a genuine quantum effect since these interference terms
are a direct result of a coherent superposition between different eigenstates of the observables
ˆ
A
l
.
Although superposition can be encountered in classical (wave) theories, a coherent superposition
between two states which correspond to two different values for an observable does not exist in
classical theories (indeed, the presence of such superpositions violates the assumptions made by
Leggett and Garg to obtain their inequalities [29], see Sec. 4.3).
In analogy to Eqs. (8) and (10), we can introduce a Keldysh rotation
A
cl
=
1
2
(A
L
+ A
R
), A
q
= A
L
− A
R
, (31)
Accepted in Quantum 2017-09-19, click title to verify 7
and write the KQPD in Eq. (28) as a path integral
P(A) =
Z
D[A
cl
, A
q
]δ(A − A
cl
)e
iS[A
cl
,A
q
]
, (32)
with the action
e
iS[A
cl
,A
q
]
= A(A
cl
+ A
q
/2)A
∗
(A
cl
− A
q
/2)hA
cl
1
+ A
q
1
/2|ˆρ
τ
1
|A
cl
1
− A
q
1
/2i, (33)
and the integration measure
R
D[A
cl
, A
q
] =
P
A
cl
,A
q
δ
A
q
N
,0
. In this language, the KQPD is given
by the sum over all paths for which A = A
cl
, weighted by the complex exponential of the action.
From Eqs. (32) and (33), we see that any negativity in the KQPD necessarily implies the presence
of a non-zero quantum field A
q
6= 0. In contrast to Eq. (10), which describes a single observable,
the quantum field in Eq. (32) is not necessarily zero implying that the statistics of A
cl
, determined
by the Keldysh time-ordering, are not necessarily equal to the statistics of A
L
or A
R
.
We now illustrate these ideas with the KQPD for two subsequent qubit measurements intro-
duced in the last section. In terms of trajectories, the discrete KQPD in Eq. (26) can be expressed
as
˜
P(σ
1
, σ
2
) =
X
σ
L/R
1
=±
δ
σ
1
,(σ
L
1
+σ
R
1
)/2
hσ
2
|σ
L
1
ihσ
L
1
|+
0
ih+
0
|σ
R
1
ihσ
R
1
|σ
2
i. (34)
We thus see that the terms where σ
L
1
6= σ
R
1
are responsible for the terms
˜
P(0, σ
2
). Note that these
terms are only non-zero if the initial state is in a coherent superposition of the two eigenstates of
ˆσ
1
. One can further show that
X
σ
1
,σ
2
=±1
˜
P(σ
1
, σ
2
) = 1, (35)
which directly implies that
˜
P(0, +1) = −
˜
P(0, −1). Thus any contribution to Eq. (34) from pairs
of trajectories with σ
L
1
6= σ
R
1
results in negativity in the KQPD as illustrated in Fig. 2.
We close this section with a brief comparison to other non-classicality indicators based on
negative quasi-probabilities. As mentioned above, the Wigner function (at time t) is the KQPD for
the observables ˆx(t) and ˆp(t). A negative Wigner function implies an impossibility of describing
the corresponding state classically in the canonical position-momentum phase space (i.e. as a
mixture of states with well defined positions and momenta). The KQPD generalizes this concept
to arbitrary observables. Negativity in the KQPD implies that it is impossible to describe the
system and its time evolution as a mixture of states with well defined values for the observable
outcomes. We note that negative values in a discrete analogue of the Wigner function have recently
been connected to the computational speed-up of a quantum computer [30–32].
In quantum optics, one often considers negativity, or strong singularity, in the Glauber-Sudarshan
P function as an indicator of non-classicality [33, 34]. This criterion captures all states with non-
positive Wigner functions as well as squeezed states and entangled states [35]. A non-classical P
function implies that the state can not be interpreted as a mixture of coherent states which indi-
cates that the state can not be fully described by classical optics [34]. We note that the P function
has recently been extended to multiple times [36–38]. In contrast to the KQPD, this distribution
is relevant for photo-detection and its moments correspond to normal ordering and regular time
ordering.
While these QPDs only contain information on the states and possibly their time-evolution,
quasi-probability representations also describe measurements with QPDs [3–6]. It has been shown
that a positive representation (i.e. a representation that only uses positive probability distributions)
of all states and measurements is impossible. Restricting the states and measurements to a given
experimental procedure, a procedure is then said to require non-classical resources if no positive
representation can be found [4]. It is an interesting question if negativity in the KQPD implies the
absence of a positive representation for a corresponding experimental procedure.
3 von Neumann measurements
In this section, we consider von Neumann type measurements and we show that the corresponding
probability distributions can be obtained by convolving the KQPD with the back-action and impre-
cision associated with the measurement. We start by considering the instantaneous measurement
Accepted in Quantum 2017-09-19, click title to verify 8
of a single observable before moving on to subsequent instantaneous measurements, simultaneous
measurements, and measurements of finite duration.
3.1 Instantaneous measurements
Following von Neumann [21], we model the measurement of an observable
ˆ
A at time τ by coupling
it to a detector described by the canonically conjugate observables ˆr and ˆπ via the interaction
Hamiltonian
ˆ
H
m
= δ(t − τ)χ
ˆ
Aˆπ, (36)
where χ denotes the coupling strength. This interaction induces the time-evolution
ˆ
U
m
= exp(−iχ
ˆ
Aˆπ),
which displaces the detector coordinate ˆr by an amount that depends on the state of the system.
Projectively measuring the position of the detector then yields the measurement of
ˆ
A. Measuring
a single observable in this way yields the probability distribution (see App. B)
P (A) =
Z
χdA
0
hχ(A − A
0
)|ˆρ
det
|χ(A − A
0
)iP(A
0
; τ), (37)
where ˆρ
det
is the density matrix of the detector, |χ(A − A
0
)i is an eigenstate of the operator
ˆr, and P(A
0
; τ) is the KQPD for a single observable given in Eq. (4). Note that the position
uncertainty in the initial detector state directly translates into a measurement uncertainty. A
projective measurement can be obtained by choosing a position eigenstate as the initial state for
the detector. In this case, the measured distribution coincides with the KQPD. This is due to the
fact that the back-action of the measurement is irrelevant for a single observable since we are not
interested in the post-measured state.
For multiple subsequent (instantaneous) measurements, we couple each observable to a different
detector. The measurement Hamiltonian then reads
ˆ
H
m
=
N
X
l=1
δ(t − τ
l
)χ
l
ˆ
A
l
ˆπ
l
. (38)
In contrast to a single observable, each measurement can now influence all subsequent measure-
ments. To capture this back-action effect, we evaluate the KQPD with the additional term
P
l
δ(t − τ
l
)γ
l
ˆ
A
l
in the Hamiltonian, where the γ
j
are random variables which are distributed
according to the detector states (see below). This yields the modified moment generating function
Λ(λ; γ) = Tr
e
−i
λ
N
2
+γ
N
ˆ
A
N
(τ
N
)
···e
−i
(
λ
1
2
+γ
1
)
ˆ
A
1
(τ
1
)
ˆρ
0
e
−i
(
λ
1
2
−γ
1
)
ˆ
A
1
(τ
1
)
···e
−i
λ
N
2
−γ
N
ˆ
A
N
(τ
N
)
,
(39)
which can be obtained by replacing λ
l
→ λ
l
± 2γ
l
in Eq. (13), where the + (−) sign corresponds
to terms on the left (right) side of the density matrix.
The probability distribution which describes subsequent von Neumann measurements can then
be shown to read (for a derivation, see App. B)
P (A) =
Z
dA
0
dγ
"
Y
l
W
l
(χ
l
[A
l
− A
0
l
], γ
l
/χ
l
)
#
P(A
0
; γ), (40)
where W
l
(r, γ) denotes the Wigner function of detector l and P(A
0
; γ) is the KQPD defined as the
Fourier transform of the moment generating function in Eq. (39). The measurement outcomes are
determined by a convolution of the KQPD and the Wigner functions of the detectors. We can thus
interpret the KQPD as describing the intrinsic fluctuations of the observables of interest which is
being influenced by the effects of the measurement [17]. The position distributions of the detectors
determine the precision of the measurement while the momentum distributions determine the back-
action via the γ
l
-dependent terms in the KQPD. The Heisenberg uncertainty relation implies a
trade-off between these two effects, indicating that we can never eliminate both influences.
As an example, we consider a measurement of position ˆx at τ = 0 followed immediately by a
momentum ˆp measurement. The probability distribution describing such a measurement reads
W
x→p
(x, p) =
Z
dx
0
dp
0
dγ
x
dγ
p
W
p
(χ
p
[p − p
0
], γ
p
/χ
p
)W
x
(χ
x
[x − x
0
], γ
x
/χ
x
)W(x
0
, p
0
+ γ
x
). (41)
Accepted in Quantum 2017-09-19, click title to verify 9
Here W
x/p
denotes the Wigner function of the detector measuring x/p and W without a subscript
denotes the KQPD which in this case is given by the Wigner function of the system. The position
measurement disturbs the momentum of the system as reflected by the second argument of the
KQPD. Note that it is the momentum distribution of the x-detector which determines the effect
of this disturbance. This reflects the trade-off between the precision of the position measurement
(which requires a peaked position distribution for the x-detector) and the disturbance of the sub-
sequent momentum distribution (which is minimized for a peaked momentum distribution for the
x-detector). Note that the last expression is independent of the momentum distribution of the
p-detector (this can be seen by performing the integral over γ
p
), reflecting the fact that the back-
action of the momentum measurement is irrelevant since it is the last measurement. Inverting the
order of the measurements yields a probability distribution which is given by the last expression
upon exchanging W(x
0
, p
0
+ γ
x
) → W(x
0
− γ
p
, p
0
).
Returning to arbitrary observables and choosing Gaussian Wigner functions for the detectors
W
l
(r, γ) =
1
2πσ
r
l
σ
γ
l
e
−
r
2
2σ
2
r
l
e
−
γ
2
2σ
2
γ
l
, (42)
we can write the probability distribution in Eq. (40) in terms of trajectories [cf. Eq. (28)] yielding
P (A) =
X
A
L
,A
R
δ
A
L
N
,A
R
N
Y
l
1
√
2πσ
imp,l
e
−
1
2σ
2
imp,l
A
l
−
A
L
l
+A
R
l
2
2
e
−
σ
2
BA,l
2
(
A
L
l
−A
R
l
)
2
× A(A
L
)A
∗
(A
R
)hA
L
1
|ˆρ
τ
1
|A
R
1
i.
, (43)
where σ
imp,l
= σ
r
l
/χ
l
and σ
BA,l
= σ
γ
l
χ
l
. Comparing the last expression with the KQPD in
Eq. (28), we can clearly identify the effects of the measurement imprecision and back-action. The
first Gaussian in the last expression replaces the Dirac delta in Eq. (28). It implies that even terms
with A
l
6= (A
L
l
+ A
R
l
)/2 contribute to the probability of measuring A
l
. This effect captures the
imprecision of the measurement which becomes more pronounced as the uncertainty in the position
of detector l grows. The second Gaussian in the last expression reduces the weight of terms which
have A
L
l
6= A
R
l
, i.e. a finite quantum field in the Keldysh language. This effect is due to the
measurement back-action which (partly) collapses the coherent superpositions between eigenstates
of
ˆ
A
l
. The coupling Hamiltonian in Eq. (38) entangles the system with the detector leading to
dephasing in the reduced state of the system. As the uncertainty in the detector momentum grows,
this effect becomes more pronounced. From the Heisenberg uncertainty relation, we find
σ
imp,l
σ
BA,l
= σ
r
l
σ
γ
l
≥
1
2
, (44)
reflecting the trade-off between measurement imprecision and back-action (see also Ref. [39]) which
ensures that the measured distribution is positive even if the KQPD exhibits negativities. We note
that coupling the system to additional degrees of freedom, such as a thermal reservoir, introduces
dephasing analogously to the measurement back-action [19].
To illustrate the effect of the measurement, we return to our example of two subsequent spin
measurements [cf. Eq. (23)]. Since the back-action of the second measurement is irrelevant, we
consider a measurement of intermediate strength of ˆσ
1
followed by a projective measurement of
ˆσ
2
. In terms of the discrete KQPD given in Eq. (26), we find for the distribution describing a von
Neumann measurement (with Gaussian detectors)
P (σ
1
, σ
2
) =
1
√
2πσ
imp
X
σ
0
1
=0,±1
e
−
1
2σ
2
imp
(σ
1
−σ
0
1
)
2
e
−2σ
2
BA
δ
σ
0
1
,0
˜
P(σ
0
1
, σ
2
). (45)
The effect of the measurement is illustrated in Fig. 3. We note that the distribution of measure-
ment outcomes for the first measurement depends very strongly on the outcome of the second
measurement. Specifically, we see from Fig. 3 that only considering events where the second (pro-
jective) measurement yields −1 allows one to distinguish the outcomes of the first measurement
much better (for a given measurement strength). This effect, which arises from the negativity of
the KQPD, is responsible for the occurrence of anomalous weak values as discussed in Sec. 4.2.
Note that subsequent von Neumann measurements can also be used to model continuous mea-
surements, a subject which has received a lot of attention recently, see e.g. [40–44].
Accepted in Quantum 2017-09-19, click title to verify 10
Figure 3: Effect of measurement imprecision and back-action on the KQPD. Here ˆσ
0
= ˆσ
2
= ˆσ
x
and ˆσ
1
= ˆσ
z
to
maximize the negativity in the KQPD. The measurement imprecision and back-action modify the KQPD resulting
in a positive probability distribution. Here σ
imp
σ
BA
= 1/2, minimizing the influence of the measurement.
Depending on the measurement outcome of the second (projective) measurement, the distribution for the first
measurement results in two clearly distinguishable outcomes or not. This is directly related to the terms
˜
P(0, σ
2
)
which are responsible for the negative values in the KQPD. Note that the peaks are shifted with respect to the
eigenvalues of ˆσ
1
(dashed lines). This effect allows for the occurrence of anomalous weak values as discussed
in Sec. 4.2.
3.2 Simultaneous measurements of position and momentum
We now turn to the simultaneous measurement of position and momentum. In analogy to Eq. (36)
(and following Arthurs and Kelly [26]), we model the measurement by the simultaneous coupling
of two detectors to the position and momentum of the system respectively
ˆ
H
m
= δ(t) (χ
x
ˆxˆπ
x
+ χ
p
ˆpˆπ
p
) . (46)
It can be shown that this measurement is described by the probability distribution [45]
W (x, p) =
Z
dx
0
dp
0
dγ
x
dγ
p
W
p
(χ
p
[p−p
0
], γ
p
/χ
p
)W
x
(χ
x
[x−x
0
], γ
x
/χ
x
)W(x
0
−γ
p
/2, p
0
+γ
x
/2). (47)
The arguments of the last Wigner function reflect the fact that in a simultaneous measurement,
both position and momentum are being disturbed. Note the close similarity between the last
expression and the probability distribution describing subsequent measurements of position and
momentum [cf. Eq. (41)]. Notably, the KQPD is in both cases given by the Wigner function.
This is a consequence of the commutation relation [ˆx, ˆp] = i and does not hold in general for
non-commuting observables.
For a detailed discussion on the von Neumann type simultaneous measurement of position and
momentum, we refer the reader to Ref. [22]. Here we only discuss the case of Gaussian states of
minimal uncertainty for the detectors (k = x, p)
W
k
(r, γ) =
1
π
e
−r
2
/σ
2
k
e
−γ
2
σ
2
k
, (48)
with the additional constraint σ
x
σ
p
/(χ
x
χ
p
) = 1/2. This constrained minimizes the disturbance
of the measurement and yields the Husimi Q-function as the measured probability distribution
[22, 46]
W (x, p) =
1
π
Z
dx
0
dp
0
e
−
(x−x
0
)
2
2σ
2
e
−2σ
2
(p−p
0
)
2
W(x
0
, p
0
), (49)
where σ = σ
x
/χ
x
. In addition to this von Neumann type measurement, the Husimi Q-function
also describes heterodyne measurements of two field quadratures [47].
3.3 Measuring the FCS
In this section, we consider a von Neumann type measurement of the time-integral of an observable
[15]. To this end, we couple a detector to the observable of interest for a finite amount of time
ˆ
H
m
= Θ(t)Θ(T − t)χ
ˆ
Aˆπ. (50)
Accepted in Quantum 2017-09-19, click title to verify 11
This measurement can be shown to be described by the probability distribution [15]
F (m) =
Z
dγW(χ[m − m
0
], γ/χ)F(m
0
; γ), (51)
where F(m; γ) is obtained through Eqs. (20) and (21) upon adding the term Θ(t)Θ(T − t)γ
ˆ
A
to the Hamiltonian. Again, we find a trade-off between measurement imprecision, depending on
the spread in the detector position, and back-action which depends on the spread in the detector
momentum. As discussed in Ref. [19], the last equation can be expressed in terms of trajectories
yielding (for a Gaussian detector)
F (m) =
1
√
2πσ
imp
X
A
L
,A
R
δ
A
L
T
,A
R
T
e
−
(
m−
1
2
m
L
−
1
2
m
R
)
2
2σ
2
imp
e
−
σ
2
BA
2
(m
L
−m
R
)
2
hA
L
0
|ˆρ|A
R
0
iA(A
L
)A
∗
(A
R
),
(52)
where the trajectory amplitudes are obtained by dividing the time evolution into N time-slices of
infinitesimal duration δt and inserting identities resolved in the eigenstates of
ˆ
A after each slice
A(A
α
) = hA
α
T
|
ˆ
U(T , T − δt)|A
α
T −δt
ihA
α
τ−δt
|···|A
α
δt
ihA
α
δt
|
ˆ
U(δt, 0)|A
α
0
i. (53)
The discrete time-integral over such a trajectory is denoted by m
α
=
P
l
A
α
l
δt. The Heisenberg
uncertainty relation applied to the detector variables again enforces σ
imp
σ
BA
≥ 1/2, reflecting
the trade-off between smearing out the negativities in the FCS by measurement imprecision and
reducing the weight of negative terms (which necessarily have m
L
6= m
R
) by measurement back-
action in complete analogy to Eq. (43).
This concludes our treatment on von Neumann measurements. Expressing the measured prob-
ability distributions in terms of the KQPD suggests that the latter should be interpreted as de-
scribing the intrinsic fluctuations of the observables of interest. In the next section, we will build on
this interpretation and demonstrate the utility of studying the KQPD to understand the quantum
behavior of dynamic systems.
4 Applications of the KQPD
We now turn to situations where it is beneficial to study the KQPD in the absence of any mea-
surement device. We are especially interested in situations where the KQPD fails to be positive.
As discussed in Sec. 2.4, this indicates non-classical behavior. The most prominent KQPD is by
far the Wigner function, which is a representation of the density matrix and is used extensively to
illustrate the state of a quantum system and to capture its non-classicality [28, 48]. An example
for a KQPD which goes beyond instantaneous states is provided by the FCS which describes the
fluctuations of an observable over a certain time-interval. While the FCS has mostly been em-
ployed to describe electronic transport in systems where it is a purely positive function, its utility
for detecting non-classical behavior has been recognized [16–19, 39, 49]. Here we discuss three ad-
ditional applications of the KQPD, stressing its versatility and its utility in capturing non-classical
behavior in the dynamics of quantum systems.
4.1 Work fluctuations
Since work is not a state function, it is in general not sufficient to interrogate a system at a single
time in order to determine the work performed by the system. As we have seen above, performing
measurements at multiple times generally results in a probability distribution which is influenced
by the measurement itself. As a consequence, the observed amount of work will in general depend
on the way it is measured [50]. Here we use the KQPD to describe work as a measurement-
independent, fluctuating quantity. We note that the KQPD has been used before to describe work
fluctuations [51–53].
In classical statistical mechanics, work is defined as the energy change in the macroscopic,
accessible degrees of freedom, while heat is the energy change in the microscopic, inaccessible
degrees of freedom. In quantum systems, all degrees of freedom are in principle microscopic. We
Accepted in Quantum 2017-09-19, click title to verify 12
can however still define work as being the energy change mediated by the accessible degrees of
freedom. Here we consider two different scenarios: In the first, one has access to the energy stored
in a part of the system that we call the work storage device. The energy increase of this device is
then interpreted as the work performed by the system. This is the analog of determining the work
performed on a suspended weight by measuring its height. In the second scenario, one has access
to the power produced by the system and the work is defined as the time-integral of the power.
This captures the situation in thermoelectric heat engines, where one can access the power through
a measurement of the electrical current. In terms of the suspended weight, this corresponds to the
analog of measuring the power by measuring the velocity of the weight.
For the first scenario, we write the total Hamiltonian as
ˆ
H =
ˆ
H
w
+
ˆ
H
rest
, where
ˆ
H
w
denotes the
Hamiltonian of the work storage device. Work fluctuations can then be described by the KQPD
corresponding to the difference of two subsequent energy measurements on the work storage device
P(w; τ) =
1
2π
Z
dλe
iλw
Tr
n
e
−i
λ
2
ˆ
H
w
ˆ
U(τ, 0)e
i
λ
2
ˆ
H
w
ˆρ
0
e
i
λ
2
ˆ
H
w
ˆ
U(0, τ)e
−i
λ
2
ˆ
H
w
o
=
X
E
τ
,E
L
,E
R
δ
w −
E
τ
−
E
L
+ E
R
2
Tr
n
hE
τ
|
ˆ
U(τ, 0)|E
L
ihE
L
|ˆρ
0
|E
R
ihE
R
|
ˆ
U(0, τ)|E
τ
i
o
.
(54)
Here
ˆ
H
w
|E
j
i = E
j
|E
j
i and τ denotes the time interval between the two measurements. For initial
states which commute with
ˆ
H
w
, we find E
L
= E
R
and we recover the probability for obtaining w as
the difference of two projective energy measurements [54]. Any negativity in the above distribution
is thus due to the coherence in the initial state. We note that the probability distribution obtained
by performing two Gaussian energy measurements, as discussed in Ref. [55], is obtained from
Eq. (54) by including the imprecision and back-action of the measurement as discussed in Sec. 3.
The moments generated by the KQPD are given by
hw
k
i =
k
X
l=0
1
2
k
k
l
Tr
ˆ
T
−
ˆ
H
w
(τ) −
ˆ
H
w
l
ˆ
T
+
ˆ
H
w
(τ) −
ˆ
H
w
(k−l)
ˆρ
0
, (55)
where
ˆ
H
w
(τ) =
ˆ
U(0, τ)
ˆ
H
w
ˆ
U(τ, 0). For the first two moments, i.e. k = 1, 2 in the last expression,
this reduces to
hw
k
i = Tr
ˆ
H
w
(τ) −
ˆ
H
w
k
ˆρ
0
, for k = 1, 2. (56)
The mean work obtained through the KQPD is thus just given by the difference of the average
energy in the work storage device at t = τ and t = 0. This is in contrast to the probability
distribution obtained by performing two projective energy measurements, where the mean value
only coincides with Eq. (56) for initial states that commute with
ˆ
H
w
. For an explicit evaluation
of Eq. (54), we refer to Ref. [53]. See Ref. [56] for an experimental reconstruction of the work
distribution by measuring its moment generating function (for an initial state that commutes with
the Hamiltonian).
In case one has access to a power operator
ˆ
P , the KQPD describing work fluctuations is given
by the FCS of the power operator. The KQPD and its moments are then given by Eqs. (20-22)
upon replacing
ˆ
A with
ˆ
P . This scenario is relevant for thermoelectric devices, where the power
operator is given by the electrical current operator times the voltage across the device
ˆ
P =
ˆ
IV .
In this case, the work fluctuations reduce to the problem of charge FCS [14]. We now return to
systems described by
ˆ
H =
ˆ
H
w
+
ˆ
H
rest
, where we take the Hamiltonians to be time-independent for
simplicity (see Ref. [57] for a discussion on the power operator for a time-dependent Hamiltonian).
In this case, the power operator is given by the energy change of the work storage device per
unit time, i.e.
ˆ
P = i[
ˆ
H,
ˆ
H
w
]. If the power operator commutes with the total Hamiltonian, i.e.
[
ˆ
P ,
ˆ
H] = 0, we find
e
−i
λ
2
ˆ
H
w
ˆ
U(τ, 0)e
i
λ
2
ˆ
H
w
=
ˆ
T
+
e
−i
λ
2
R
τ
0
dt
ˆ
P (t)
. (57)
This implies that the FCS of the power operator is equal to the KQPD for two energy measurements
given in Eq. (54) [51]. The difference of the two scenarios considered here is then only relevant when
taking into account the disturbance from an actual measurement. For a thermoelectric device, the
Accepted in Quantum 2017-09-19, click title to verify 13
energy in the work storage device corresponds to the electric potential energy stored in an electronic
contact, i.e.
ˆ
H
w
= V
ˆ
Q, where
ˆ
Q denotes the charge operator in the contact with
ˆ
I = i[
ˆ
H,
ˆ
Q].
The work provided by a thermoelectric device can thus either be determined by measuring the
electrical current (determining the power) or by measuring the number of electrons in an electric
contact (determining the energy in the work storage device). We note that both approaches have
been considered in the theory of charge FCS [14].
4.2 Weak values
The weak value of an observable
ˆ
A is obtained by preparing the system in some state |Ii, performing
a weak measurement of
ˆ
A, and post-selecting the measurement outcomes on the final state |F i
[58]. Describing the weak measurement as a von Neumann measurement, the post-measured state
of the detector is determined by the weak value
A
w
=
hF |
ˆ
A|Ii
hF |Ii
, (58)
and the average of the measurement outcome is given by the real part of the last expression [58, 59].
Any possible time-evolution between pre-selection, measurement and post-selection is absorbed in
the definition of the initial and final states. In the KQPD framework, we can describe the weak
value measurement using the distribution describing two subsequent measurements, one for the
weak measurement and one for the post-selection. For simplicity, we do not treat the pre-selection
as an additional measurement but impose it by setting ˆρ
0
= |IihI|. The relevant KQPD then reads
P(A, B) =
1
(2π)
2
Z
dλ
A
dλ
B
e
iλ
A
A
e
iλ
B
B
Tr
n
e
−i
λ
B
2
ˆ
B
e
−i
λ
A
2
ˆ
A
|IihI|e
−i
λ
A
2
ˆ
A
e
−i
λ
B
2
ˆ
B
o
=
X
B
F
,A
L
,A
R
δ (B − B
F
) δ
A −
A
L
+ A
R
2
hB
F
|A
L
ihA
L
|IihI|A
R
ihA
R
|B
F
i,
(59)
where |B
F
i are eigenstates of
ˆ
B. The KQPD corresponding to the measurement of the weak value
is then obtained by taking into account the post-selection, i.e. fixing the argument corresponding
to the second measurement
P
w
(A) =
P(A, F )
R
dAP(A, F )
=
1
|hF |Ii|
2
X
A
L
,A
R
δ
A −
A
L
+ A
R
2
hF |A
L
ihA
L
|IihI|A
R
ihA
R
|F i. (60)
The statistics of weak value measurements (with post-selection performed by a projective mea-
surement) can be described using the last expression by including measurement imprecision and
back-action as discussed in Sec. 3. In the weak measurement limit, the disturbance of the mea-
surement does not alter the average value of the KQPD, which is equal to the real part of Eq. (58)
hAi
w
=
Z
dA AP
w
(A) = Re
(
hF |
ˆ
A|Ii
hF |Ii
)
. (61)
From Eq. (60), we see that the KQPD vanishes for arguments that lie outside the spectrum
of
ˆ
A. However Eq. (61) can take on values which lie outside the spectrum of
ˆ
A. In such cases,
the weak value is called anomalous. We now show that an anomalous weak value implies that the
distribution P
w
(A) takes on negative values. To see this, assume that P
w
(A) ≥ 0. In that case,
we find
hAi
w
=
Z
A
max
A
min
dA AP
w
(A) ≤ A
max
Z
A
max
A
min
dAP
w
(A) = A
max
, (62)
where A
min (max)
is the smallest (largest) eigenvalue of
ˆ
A and a similar inequality restricts hAi
w
≥
A
min
. Note that the inequality in Eq. (62) is only valid for a strictly positive KQPD. An anomalous
weak value thus necessarily implies that the KQPD P
w
(A) [and also P(A, B), cf. Eq. (60)] exhibits
negativity indicating non-classical behavior. This is an agreement with Ref. [60], where anomalous
weak values have been shown to imply contextuality. We note that weak values have been connected
Accepted in Quantum 2017-09-19, click title to verify 14
to quasi-probability distributions before [59]. In particular, the average momentum at a given
position that is obtained from the Wigner function has been shown to be determined by the weak
value of the momentum
R
dp pW(x, p)
R
dpW(x, p)
= Re
hx|ˆp|Ii
hx|Ii
, (63)
which follows from our approach when taking
ˆ
A = ˆp and
ˆ
B = ˆx.
Finally, we note that the distribution in Eq. (60) is relevant for a measurement of
ˆ
A with an
arbitrary strength. It has the variance
h
ˆ
A
2
i
w
−
h
ˆ
Ai
w
2
=
1
2
Re
(
hF |
ˆ
A
2
|Ii
hF |Ii
)
−
1
2
Re
(
hF |
ˆ
A|Ii
hF |Ii
)!
2
. (64)
For
ˆ
A = ˆp and |F i = |xi, the last expression is proportional to the quantum potential energy [59], a
central object in Bohmian mechanics [61]. See also Ref. [62] for a connection between weak values
and von Neumann measurements of arbitrary strength and post-selection.
For our example of two subsequent spin measurements, we find [using Eqs. (23), (26), and (60)]
P
w
(σ
1
) =
|α|
2
|δ|
2
δ(σ
1
− 1) + |β|
2
|γ|
2
δ(σ
1
+ 1) − 2Re{αβ
∗
γ
∗
δ}δ(σ
1
)
|αδ − βγ|
2
, (65)
where we chose |−
2
i as the final state |F i. We see that for a sufficiently small denominator, the
probability density associated with σ
1
= ±1 can become very large as long as the presence of
the negative term ensures the normalization of P
w
(σ
1
). In this way, the weak value, determined
by P
w
(1) − P
w
(−1) can be amplified resulting in values far outside the range [−1, +1]. This can
however only occur for a sufficiently large negative term in the KQPD as can be seen from Eq. (65)
which implies
hσ
1
i
w
> 1 ⇔ Re{αβ
∗
γ
∗
δ} > |β|
2
|γ|
2
. (66)
4.3 Leggett-Garg inequalities
In Ref. [29], Leggett and Garg derived inequalities which hold under the assumption of macroscopic
realism [i.e., a (macroscopic) system will at all times be in one (and only one) of the states
available to it] and non-invasive measurability. Clearly quantum systems do not fulfill either of
these assumptions and can thus violate Leggett-Garg inequalities. In contrast to Bell inequalities,
where multiple spatially separated parties apply local operations, Leggett-Garg inequalities are
obtained by probing the same system at multiple times. We note that even if a Leggett-Garg
inequality is violated, one can always argue that this happened because the measurement did
influence the system, and therefore all subsequent measurements, in an unexpected (but in principle
avoidable) way. This is known as the clumsiness loophole. Since it is not possible to certify the
non-invasiveness of a measurement, the clumsiness loophole can never be closed [63].
We will now show that the violation of a particular Leggett-Garg inequality directly implies
the occurence of negative values in the corresponding KQPD. The considered inequality reads
K = C
21
+ C
32
− C
31
≤ 1, (67)
where the correlators are given by
C
ij
=
X
Q
i
,Q
j
=±1
Q
i
Q
j
P
ij
(Q
i
, Q
j
). (68)
Here P
ij
(Q
i
, Q
j
) is the probability of obtaining the outcome Q
i
at time t
i
and Q
j
at t
j
if the
system is measured at these two times only. The measurement outcomes are restricted to Q
i
= ±1.
Violation of the inequality in Eq. (67) implies that the correlators C
ij
can not be obtained from a
positive probability distribution which describes all three measurements and is independent of the
choice of measurements that are performed [29, 63], i.e.
K > 1 ⇒ C
ij
6=
X
Q
1
,Q
2
,Q
3
=±1
Q
i
Q
j
P (Q
3
, Q
2
, Q
1
), (69)
Accepted in Quantum 2017-09-19, click title to verify 15
with P (Q
3
, Q
2
, Q
1
) ≥ 0.
For projective measurements of a dichotomic observable
ˆ
Q, the correlator to be used in Eq. (67)
is given by [64]
C
ij
= Tr
1
2
n
ˆ
Q(t
i
),
ˆ
Q(t
j
)
o
ˆρ
0
. (70)
We now introduce the KQPD for three subsequent measurements of
ˆ
Q
P(Q
3
, Q
2
, Q
1
) =
1
(2π)
3
Z
dλ
3
dλ
2
λ
1
e
iλ
3
Q
3
+iλ
2
Q
2
+iλ
1
Q
1
× Tr
n
e
−i
λ
3
2
ˆ
Q(t
3
)
e
−i
λ
2
2
ˆ
Q(t
2
)
e
−i
λ
1
2
ˆ
Q(t
1
)
ˆρ
0
e
−i
λ
1
2
ˆ
Q(t
1
)
e
−i
λ
2
2
ˆ
Q(t
2
)
e
−i
λ
3
2
ˆ
Q(t
3
)
o
.
(71)
It is straightforward to show that the correlators in Eq. (70) can directly be obtained from the
KQPD
C
ij
=
Z
dQ
1
dQ
2
dQ
3
Q
i
Q
j
P(Q
3
, Q
2
, Q
1
). (72)
If the KQPD was strictly positive, it would thus provide a positive probability distribution which
describes all three measurements and is independent of the choice of measurements that are per-
formed. The existence of such a probability distribution prevents the Leggett-Garg inequality to
be violated [63] (see also App. C). Inverting the argument, we conclude that a violation of the
Leggett-Garg inequality in Eq. (67) implies negative values for the KQPD in Eq. (71) and thus
non-classical behavior in the time-evolution of
ˆ
Q.
While we discussed projective measurements, the KQPD could also find applications when
considering Leggett-Garg inequalities obtained from weak continuous measurements [65, 66].
5 Conclusions
The KQPD provides a generalization of the Wigner function and the FCS to describe arbitrary
observables within a quasi-probability formalism. This provides a unified framework for the de-
scription of fluctuations in dynamic quantum systems. Negative values of the KQPD can be directly
linked to an interference between paths through Hilbert space which visit different eigenstates of
the observables of interest. This requires coherent superpositions of states that correspond to
different measurement outcomes, implying that negativity in the KQPD indicates non-classical
behavior. As we have shown, this negativity is witnessed by anomalous weak values and violated
Leggett-Garg inequalities (see also Refs. [67, 68] for a connection between anomalous weak values
and Leggett-Garg inequalities). The versatility of the KQPD allows to describe the problem at
hand with a tailor-made quasi-probability distribution which focuses on the observables of inter-
est. When measuring these observables in <