
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