Accessible coherence in open quantum system dynamics
María García Díaz
1,*
, Benjamin Desef
2,*
, Matteo Rosati
1
, Dario Egloﬀ
2,3
, John Calsamiglia
1
, Andrea
Smirne
2,4
, Michalis Skotiniotis
1
, and Susana F. Huelga
2
1
Física Trica: Informació i Fenòmens Quàntics, Departament de Física, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona),
Spain
2
Institute of Theoretical Physics and IQST, Universität Ulm, Albert-Einstein-Allee 11, D-89069 Ulm, Germany
3
Institute of Theoretical Physics, Technical University Dresden, D-01062 Dresden, Germany
4
Dipartimento di Fisica Aldo Pontremoli, Università degli Studi di Milano, via Celoria 16, 20133 Milan, Italy
Quantum coherence generated in a physical
process can only be cast as a potentially use-
ful resource if its eﬀects can be detected at a
later time. Recently, the notion of non-coher-
ence-generating-and-detecting (NCGD) dynam-
ics has been introduced and related to the classi-
cality of the statistics associated with sequential
measurements at diﬀerent times. However, in or-
der for a dynamics to be NCGD, its propagators
need to satisfy a given set of conditions for al l
triples of consecutive times. We reduce this to
a ﬁnite set of d
(
d
1)
conditions, where d is the
dimension of the quantum system, provided that
the generator is time-independent. Further con-
ditions are derived for the more general time-
dependent case. The application of this result to
the case of a qubit dynamics allows us to eluci-
date which kind of noise gives rise to non-coher-
ence-generation-and-detection.
1 Introduction
Much experimental eﬀort in quantum physics focuses
on the creation, maintenance, and subsequent detec-
tion of coherent superpositions of quantum states [1
3], a distinctive feature of the quantum formalism that
furnishes signiﬁcant advantage in many communica-
tion [4], computation [5, 6], and metrological tasks [79].
Recently, resource theories for coherent superpositions
have been developed in order to provide a cohesive and
quantitative description of quantum coherence, both for
states [1014] as well as operations [1519].
Assessing the coherence capabilities of quantum dy-
namical maps is a subtle task. For example, the mere
ability of a quantum dynamics to generate or detect
coherence is of no practical advantage unless this coher-
ence can be harnessed in a beneﬁcial way for some task.
In this sense, a prerequisite for a quantum dynamical
*
These authors contributed equally.
evolution to generate resourceful coherence for a given
task is that the coherence it generates can be detected in
terms of discriminable statistics of subsequent measure-
[20] it has been shown that, under proper conditions, dy-
namics which are non-coherence-generating-and-detect-
ing (NCGD) are strictly related to the classicality of the
statistics associated with sequential measurements.
In this work we consider an open-system dynamics
and propose deﬁnite criteria to assess whether it is
able to generate and detect coherences. Our main result
shows that for dynamical maps stemming from time-
independent generators, the—in principle countless—
conditions of [20] reduce to a ﬁnite set of necessary
and suﬃcient conditions. In addition, those are given
in terms of the generator of the quantum dynamical
evolution. We then extend the notion of NCGD to time-
dependent dynamics. In this case, we can also derive re-
lations between NCGD and the generator, though now
in general we need uncountably inﬁnitely many condi-
tions.
The article is structured as follows: We review the nec-
essary background on open-system dynamics (Sec. 2.1)
as well as the concept of NCGD (Sec. 2.2) and its con-
nection with the statistics of sequential measurements.
Sec. 3 contains our main results elaborating on when
dynamical evolutions are NCGD based on their gener-
ators, and in Sec. 4 we exemplify our criteria in the
context of a Ramsey protocol widely used in precision
spectroscopy. For ease of exposition, we defer the proofs
of all theorems and propositions to the appendices.
2 Background
2.1 Open quantum systems
A realistic description of a quantum system must take
into account that every system is open, i.e., it interacts
with the surrounding environment [21]. In many circum-
stances, it is possible to provide such a description by a
Accepted in Quantum 2020-03-27, click title to verify. Published under CC-BY 4.0. 1
arXiv:1910.05089v2 [quant-ph] 27 Mar 2020
time-local quantum master equation (QME):
d
dt
ρ
s
(t) = L(t)[ρ
s
(t)], (1)
where L
(
t
)
is the dynamical generator of the evolution
and ρ
s
the reduced state of the system. Any L
(
t
)
that is
both trace- and hermiticity-preserving can be uniquely
decomposed as [22]
L(t)[ρ
s
] = i[H(t), ρ
s
]
+
d
2
1
X
i,j=1
D
ij
(t)
F
i
ρ
s
F
j
1
2
{F
j
F
i
, ρ
s
}
,
(2)
where d < is the dimension of the Hilbert space of the
system H, H
(
t
)
a Hermitian operator, D
(
t
)
a Hermitian
matrix, and {F
i
}
d
2
i=1
is an orthonormal operator basis
with F
d
2
= /
d and Tr(F
i
F
j
) = δ
ij
.
Upon integration, the QME leads to a family of trace-
preserving (TP) propagators E
t
2
,t
1
satisfying ρ
s
(
t
2
) =
E
t
2
,t
1
[ρ
s
(t
1
)] t
2
t
1
0.
In particular, we will ﬁrst address dynamics for which
H and D are time-independent. This class gives rise to
semigroups of the form E
t
= e
tL
, where L
(
t
)
L is time-
independent. Its most prominent representative is the
0
, which enforces
complete positivity (CP) on all propagators. Despite the
fact that a derivation of this form via a microscopic
model involves several approximations [21]—which is by
no means the only route to a GKSL QME [21, 2427]—
it has shown remarkable success and applicability, in
particular in the ﬁelds of quantum optics [28] and in
relevant problems of interest in solid-state physics, like
non-equilibrium transport of charge and energy [29].
For even more generality, we will however also con-
sider the case of a time-dependent generator.
Finally we shall refer to rank-k noise as those dynam-
ical generators L in Eq. (2) for which D has at most k
non-zero eigenvalues.
2.2
Non-coherence-generating-and-detecting dy-
namics
The realization that quantum coherence underpins the
performance of many quantum information and com-
munication tasks elevates it to a bona ﬁde resource.
Over the past decade, a great deal of eﬀort has gone
into developing resource theories of coherence in an
attempt to quantify and better leverage its use, both
for states [1014] as well as operations [1519]. Within
these frameworks, the set of free states, I, consists
of all states that are diagonal in some ﬁxed basis
{|ii
:
i
= 1
, . . . , dim
(
H
)
}, while free operations are
those that do not generate coherence out of incoherent
states. A plethora of non-coherence-generating opera-
tions has been proposed [30], among which maximally
incoherent operations (MIOs), deﬁned as all CPTP
maps M: B
(
H
)
B
(
H
)
such that M
(
I
)
I, con-
stitute the largest class [10]. Here, B
(
·
)
denotes the set
of bounded operators.
From the above deﬁnition of free operations it would
seem that the only desideratum for a resourceful quan-
tum operation is its ability to generate coherence. How-
ever, coherence in itself is of no value unless we are
able to subsequently harness its presence in a beneﬁ-
cial manner [17]. In order to do so, a dynamical map
E
t
1
,0
that generates coherence at some time t
1
must, at
some later time t
2
> t
1
, be able to detect coherence
(note that sometimes the word ‘activate’ [31] is used
as a synonym for ‘detect’). Therefore, what we are in-
terested in are the coherence-generating-and-detecting
properties of the propagators {E
t
2
,t
1
:
t
2
t
1
0
} as-
sociated with the dynamics. To that end we deﬁne non-
coherence-generating-and-detecting (NCGD) dynamics
as follows:
Deﬁnition 1.
[
20
] A dynamics with propagator
E
t
2
,t
1
,
t
2
t
1
0, is NCGD iﬀ the condition
E
t
3
,t
2
E
t
2
,t
1
∆ = ∆ E
t
3
,t
1
(3)
holds for all times
t
3
t
2
t
1
0, where
denotes
composition of maps and =
P
d
i=1
|iihi| · |iihi|
is the
complete dephasing map in the incoherent basis
{|ii}
d
i=1
.
Otherwise the dynamics is coherence-generating-and-de-
tecting (CGD).
By looking at the statistics of measurement outcomes
in the incoherent basis, it can be veriﬁed that CGD
evolutions produce accessible coherence, i.e., coherence
that aﬀects the statistics of sequential measurements.
In fact, under the assumptions that the dynamics of a
quantum system is given by a CP semigroup and that
the Quantum Regression Theorem holds [21, 3235], the
joint probability distribution arising from sequentially
measuring a non-degenerate observable of the quantum
system is compatible with a classical stochastic process
if and only if the dynamics is NCGD [20]. The creation
and subsequent detection of quantum coherence is thus,
in this case, the distinctive feature of any genuine non-
classicality in the process. Note that here by classical
process we mean any process whose statistics satisﬁes
the Kolmogorov consistency conditions [21, 36], moti-
vated by the fact that—at least in principle—classical
physics allows for noninvasive measurability. Violations
of Leggett–Garg-type inequalities [3739] can only be
observed if a dynamics is CGD [20], giving additional
justiﬁcation to this identiﬁcation. Importantly, the no-
tion of (N)CGD dynamics provides a theoretical frame-
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work that is amenable to deriving (quantitative) ex-
perimental benchmarks of coherence and its connection
with non-classicality, as exempliﬁed by the assessment
of the time-multiplexed optical quantum walk in [40].
The direct connection between the dynamics of quan-
tum coherence and non-classicality can be extended be-
yond the case of CP semigroups, but does not hold for
general evolutions [41, 42].
Finally, we stress that the class of NCGD dynamical
maps is not closed under composition: indeed, it is very
easy to obtain CGD dynamics by composing two NCGD
dynamical maps; the ﬁrst being coherence-generating
but not detecting while the second one being coherence-
detecting but not generating.
3 Characterizing NCGD dynamics
Given a quantum dynamics, Deﬁnition 1 can be used to
assess whether detectable coherences are generated. In
principle, this would require performing three separate
map tomography protocols for any three given instants
of time t
1
, t
2
, and t
3
, and reconstruct the propagators
involved in Eq. (3), which would involve an uncountably
inﬁnite number of measurements.
In this section we provide a ﬁnite set of necessary
and suﬃcient conditions certifying the CGD properties
of time-independent dynamical evolutions. Importantly,
these conditions pertain directly to the generator of the
dynamics in Eq. (1). As the latter is the central tool
to yield an explicit description, microscopically or phe-
nomenologically motivated, of the open-system dynam-
ics of concrete physical settings [21], our result is rel-
evant to certify the use of coherence in an open quan-
tum system dynamics. For time-dependent generators,
we provide an inﬁnite number of necessary and suﬃcient
conditions on the generator.
Recall that L
(
t
)
: B
(
H
)
B
(
H
)
for all times t. Hav-
ing ﬁxed the complete dephasing superoperator
, we
can decompose B(H) into two orthogonal subspaces
B(H) = B
p
(H) B
c
(H), (4a)
where
B
p
(H) = Image(∆),
B
c
(H) = Kernel(∆),
(4b)
are the subspaces associated with the population and
coherence basis elements respectively. In this basis, the
matrix representation of the generator L(t) is given by
L(t) =
L
pp
(t) L
pc
(t)
L
cp
(t) L
cc
(t)
, (4c)
where, for example, L
pc
: B
c
(H) B
p
(H).
We are now ready to formulate our ﬁrst result, which
is a complete characterization of NCGD based solely on
its time-independent generator.
Theorem 2.
For any time-independent generator
L
of
a quantum dynamics it holds that
NCGD
L
pc
L
j
cc
L
cp
= 0 j {0, . . . , d
2
d 1}
,
where d = dim H.
The proof in Appendix A makes use of the decompo-
sition in Eq. (4c). It then employs the Cayley-Hamilton
theorem [43] to reduce the uncountably inﬁnitely many
conditions in Eq. (3) to d
(
d
1)
conditions on the gen-
erator.
As a direct generalization of Theorem 2, we can state
the following Theorem for the more general case of a
time-dependent generator, under the regularity condi-
tion that the latter is analytic for all times considered.
Its proof can be found in Appendix B.
Theorem 3. For a suﬃciently regular L(t), we have
NCGD t
n
··· t
1
0, n 2 :
L
pc
(t
n
) L
cc
(t
n1
) ··· L
cc
(t
2
) L
cp
(t
1
) = 0.
Even though this characterization of NCGD, in con-
trast to the previous case, now consists of inﬁnitely
many conditions, in certain special cases some simpler
conditions already guarantee that a dynamics is NCGD.
This is captured in the following Corollary.
Corollary 4.
The following conditions on the generator
L
(
t
) of a quantum dynamics individually imply NCGD.
L
pc
(t) = 0 t 0
L
cp
(t) = 0 t 0
L(t
2
) L(t
1
) = L(t
2
) L(t
1
) t
2
t
1
0
L(t),
= 0 t 0.
It is suﬃcient to ﬁnd a single set of times for which
the equality in Theorem 3 does not hold in order to guar-
antee that the dynamics is CGD. The simplest instance
of this is captured in the following Corollary.
Corollary 5. For a suﬃciently regular L(t), we have
NCGD L
pc
(t
2
) L
cp
(t
1
) = 0 t
2
t
1
0.
4 (N)CGD dynamics for qubits
In this section, we will apply Theorem 2 to the special
case of a GKSL qubit dynamics. This will allow us to
explicitly give the structure of an NCGD dynamics.
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Eq. (2) in the normalized Pauli operator basis {σ
i
:
i = 0, . . . , 3} can be easily rewritten as
L[ρ
s
] =
1
2
3
X
i,j=0
L
ij
[σ
i
ρ
s
, σ
j
] + [σ
i
, ρ
s
σ
j
]
, (5)
where
L
C
4×4
is a Hermitian matrix. We will choose
(
σ
0
, σ
3
)
as our incoherent basis and
(
σ
1
, σ
2
)
as the co-
herent one. With this choice, the matrix representation
of Eq. (5) in the basis of Eq. (4c) is explicitly given by
L
pp
=
0 0
2 Im L
12
L
11
+ L
22
L
pc
=
0 0
Re L
13
Im L
02
Re L
23
+ Im L
01
L
cp
=
2 Im L
23
Re L
13
+ Im L
02
2 Im L
13
Re L
23
Im L
01
L
cc
=
L
22
+ L
33
Im L
03
Re L
12
Re L
12
Im L
03
L
11
+ L
33
.
(6)
Theorem 2 states that NCGD is equivalent to
L
pc
L
cp
= L
pc
L
cc
L
cp
= 0. (7)
In particular, the dynamics is coherence non-activating,
i.e., L
pc
= 0, when
Re L
13
= Im L
02
Re L
23
= Im L
01
, (8)
while it is coherence non-generating, i.e., L
cp
= 0
, when
Re L
13
= Im L
02
Re L
23
= Im L
01
Im L
13
= Im L
23
= 0.
(9)
Observe that both coherence non-activating and coher-
ence non-generating dynamics can arise from the sim-
plest open-systems dynamics, namely rank-one Pauli
noise. For example, assuming that all contributions
L
0i
arise solely from the Hamiltonian of the system, the fol-
lowing rank-one dissipators
¯
L C
3×3
,
¯
L
non-act.
= rr
>
, r =
Im L
02
Im L
01
1
>
;
¯
L
non-gen.
= ss
>
, s =
Im L
02
Im L
01
1
>
,
(10)
give rise to coherence non-activating and coherence non-
generating dynamics respectively.
Note, however, that one can have dynamical evolu-
tions that are capable of both generating and detecting
coherence, and yet are still NCGD. This occurs when-
ever coherence is generated in an orthogonal subspace to
the one where it is detected. In the case of qubits this
happens precisely when (assuming for simplicity that
the denominators involved are diﬀerent from 0)
Im L
13
Im L
23
=
Re L
13
Im L
02
Re L
23
+ Im L
01
=
Im L
01
Re L
23
Re L
13
+ Im L
02
(11)
and
L
11
L
22
=
(Im L
02
Re L
13
)(Im L
03
Re L
12
)
Im L
01
+ Re L
23
+
(Im L
01
+ Re L
23
)(Im L
03
+ Re L
12
)
Im L
02
Re L
13
.
(12)
Eq. (11) is equivalent to the ﬁrst condition in Eq. (7),
L
pc
L
cp
= 0
; the precise relationship among several coef-
ﬁcients of the dynamical map ensures that coherence is
generated in a subspace orthogonal to that of coherence
detection. Likewise, Eq. (12) rules out the second-order
coupling, L
pc
L
cc
L
cp
= 0.
Let us illustrate our ﬁndings with a concrete, and
practically relevant physical example; the Ramsey
scheme deployed in interferometry, spectroscopy and
atomic clocks. The simplest, non-trivial case of such
a scheme is that of rank-one Pauli noise in the same
direction as the Hamiltonian evolution—assumed with-
out loss of generality to be H
=
σ
3
—whose dynamics is
given by
L[ρ
s
] = iω[σ
3
, ρ
s
] + γ(σ
3
ρ
s
σ
3
ρ
s
/2), (13)
where ω is the detuning from the reference ﬁeld. Note
that due to the normalization of the Pauli matri-
ces, σ
2
3
=
2
. In the Ramsey scheme, the atoms—
approximated as qubits—are ﬁrst prepared in eigen-
states of σ
1
, then subjected to the evolution generated
by Eq. (13), and subsequently measured in the eigenba-
sis of σ
1
. Choosing B
p
(
H
) = (
σ
0
, σ
1
)
as our incoherent
basis, B
c
(
H
) = (
σ
2
, σ
3
)
, and using the matrix represen-
tation introduced in Eq. (4c), the generator of Eq. (13)
can be written as
L =
0 0 0 0
0 γ
2ω 0
0
2ω γ 0
0 0 0 0
. (14)
We can assess the CGD properties of such a setup by
looking at the distance between the left- and right-hand
sides of Eq. (3), as measured via the trace distance.
Deﬁning
p
±
(t
3
) = Tr
|±ih±|E
t
3
,t
1
∆[ρ
s
]
q
±
(t
3
, t
2
) = Tr
|±ih±|E
t
3
,t
2
E
t
2
,t
1
∆[ρ
s
]
,
(15)
Figure 1 shows the trace distance max
ρ
s
∈I
kp
(
t
3
)
q
(
t
3
, t
2
)
k as a function of the intermediate time t
2
for
various values of the ratio γ. The presence of co-
herence in the dynamics is most prominent half-way
through the evolution and, indeed, it is suppressed by
a stronger rate γ.
Let us now investigate a complementary scenario in
which we also include components orthogonal to the
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0
0.5 1 1.5 2 2.5 3
0.000
0.005
0.010
0.015
ωt
2
Trace distance
γ
ω
= 1
γ
ω
= 2
Figure 1: Coherence generated and detected, as measured via the
trace distance between the probability distributions of Eq.
(15)
,
maximized over
ρ
s
, for the open-system evolution described by
Eq.
(13)
. The total evolution time is ﬁxed to
ωt
3
= 3 and the
trace distance is plotted as a function of the intermediate time
0 t
2
t
3
.
Hamiltonian in the noise. Speciﬁcally, consider the open-
system dynamics
L[ρ
s
] = iω[σ
3
, ρ
s
] +
1
2
3
X
i,j=1
γ
ij
[σ
i
ρ
s
, σ
j
] + [σ
i
, ρ
s
σ
j
]
,
(16)
where γ
ij
=
γ
ji
are the damping rates; still, our inco-
herent basis is (σ
0
, σ
1
).
Coherence non-generating dynamics corresponds to
γ
12
=
2ω and γ
13
= 0
, whereas coherence non-
activating dynamics is given by Re γ
12
=
2ω and
Re γ
13
= 0.
To investigate the more general notion of NCGD dy-
namics, we look at the matrix representation for the
corresponding generator; assuming for the sake of sim-
plicity that all γ
ij
be real, it reduces to
L =
0 0 0 0
0 γ
22
γ
33
2ω + γ
12
γ
13
0
2ω + γ
12
γ
11
γ
33
γ
23
0 γ
13
γ
23
γ
11
γ
22
.
(17)
It can be veriﬁed that
L
pc
L
cp
= L
pc
L
cc
L
cp
= 0
2ω
2
= γ
2
12
+ γ
2
13
γ
2
13
(γ
22
γ
33
) = 2γ
12
γ
13
γ
23
(18)
so that indeed, the CGD capabilities of the dynamical
evolution depend on the damping rates γ
12
and γ
13
that
mix coherent with incoherent components.
0 0.5 1 1.5 2 2.5 3
0.000
0.005
0.010
0.015
0.020
ωt
2
Trace distance
(1, 0; 0, 0, 0) (1, 0; 1, 0, 0)
(2, 0; 2, 0, 0) (0.65, 0.65; 1, 2.1, 1)
Figure 2: Coherence generated and detected, as measured via
the trace distance between the two probability distributions
of Eq.
(15)
, maximized over
ρ
s
, for an open-system evolution
described by Eq. (16).
The legend gives, in units of
ω
, (
γ
11
=
γ
22
, γ
33
;
γ
12
, γ
13
, γ
23
);
in particular, all but the gray dash-dotted line represent cases
of purely orthogonal noise [
44
], since the non-zero rates are
associated in Eq.
(16)
only to Pauli operators in a direction
orthogonal to the Hamiltonian
ωσ
3
. The total evolution time is
ﬁxed to
ωt
3
= 3 and the trace distance is plotted as a function
of the intermediate time 0 t
2
t
3
.
Diﬀerent behaviors of the CGD capability for various
parameter choices are illustrated in Figure 2. On the
one hand, changing the weights of the noise components
can result in even qualitatively diﬀerent features of the
coherences generated and detected along the evolution,
characterized, for example, by diﬀerent locations and
number of maxima as a function of the intermediate
time t
2
.
On the other hand, rather diﬀerent kinds of noise
might exhibit a similar behavior. In fact, compare the
case of pure dephasing, see Figure 1, and the purely or-
thogonal noise represented by the solid blue curve in
Figure 2. The qualitative and even quantitative evolu-
tion of the coherences generated and detected is very
similar in the two cases. This is particularly relevant
since it is well known that, if we want to estimate the
value of the frequency ω via the Ramsey scheme, pure
dephasing and orthogonal noise will limit the optimal
achievable precision in a radically diﬀerent way. Pure de-
phasing enforces the shot-noise limit [4548], which is
typical of the classical estimation strategies [49]. Note
that this is the case even if error-correction techniques
are applied [5052]. Orthogonal noise, instead, allows for
Accepted in Quantum 2020-03-27, click title to verify. Published under CC-BY 4.0. 5
super-classical precision [44], which can be even raised
to the ultimate Heisenberg limit by means of error cor-
rection [5052]. This provides us with an example of
how the capability to generate coherences (in the rele-
vant basis) and later convert them to populations has to
be understood as a prerequisite to perform tasks which
rely on the advantage given by the use of quantum fea-
tures. However, CGD in itself does not guarantee that
such an advantage over any possible classical counter-
part is actually achieved.
5 Conclusion
In this work we have shown that fulﬁlling a ﬁnite num-
ber of criteria is necessary and suﬃcient to ensure that
a given quantum dynamics with time-independent gen-
erator cannot generate and subsequently detect coher-
ence. Importantly, these conditions are given in terms
of the generator of the dynamics itself, which makes
them even more convenient when one wants to char-
acterize the evolution of a certain open system. In the
more general case of a time-dependent generator, an un-
countably inﬁnite number of conditions arises. We have
exempliﬁed our results for the case of a GKSL qubit
dynamics, providing the deﬁning properties of genera-
tors that give rise to coherence non-generating as well
as coherence non-activating maps, and applied our ﬁnd-
ings to analyze the coherence-generating-and-detecting
capabilities of the open-system dynamics describing a
typical Ramsey protocol.
Our method provides a way of assessing the intercon-
version between coherence and population which repre-
sents a prerequisite for the potential use of coherence as
a resource in quantum information technology.
Acknowledgements
The authors thank Andreu Riera and Philipp Stras-
berg for interesting discussions on various aspects of
the present work. The authors acknowledge support
from Spanish MINECO, project FIS2016-80681-P with
the support of AEI/FEDER funds; the Generalitat de
Catalunya, project CIRIT 2017-SGR-1127; the ERC
Synergy Grant BioQ. MGD is supported by a doc-
toral studies fellowship of the Fundación “la Caixa,”
grant LCF/BQ/DE16/11570017. MS is supported by
the Spanish MINECO, project IJCI-2015-24643. MR
acknowledges partial ﬁnancial support by the Baidu-
UAB collaborative project ‘Learning of Quantum Hid-
den Markov Models.’
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A Proof of Theorem 2
A.1 General remarks
In the time-independent case, E
t
= e
tL
, we can write NCGD as
E
t
E
τ
∆ = 0 t, τ 0, (19)
where we deﬁned
:
= . Expanding the exponential, this is equivalent to
X
n=0
L
n
t
n
n!
X
n
0
=0
L
n
0
τ
n
0
n
0
!
∆ = 0 t, τ 0
L
n
L
n
0
∆ = 0 n, n
0
N
0
. (20)
Note that for ﬁnite-dimensional H the following equivalence holds:
L
pc
L
j
cc
L
cp
= 0 j
0, . . . ,  1
L
pc
L
j
cc
L
cp
= 0 j N
0
, (21)
where
=
dim B
c
(
H
) =
dim
2
H dim H
=
d
2
d. While is trivial, by Cayley–Hamilton [43], there are
coeﬃcients {α
i
}

i=0
C, such that

X
i=0
α
i
L
i
cc
= 0. (22)
Hence any power of L
cc
greater or equal to  can be expressed as a linear combination of powers from
0
to
1
.
All condition formulated in the following that rely on inﬁnite matrix powers are therefore already fulﬁlled if they
hold up to the (` 1)
th
power.
A.2 Structure lemma
Lemma 6. If L
pc
L
j
cc
L
cp
= 0 for all j < n,
L
n
=
L
n
pp
P
n
j=1
L
j1
pp
L
pc
L
nj
cc
0 0
and L
n
∆ =
L
n
pp
0
P
n
j=1
L
j1
cc
L
cp
L
nj
pp
0
. (23)
Proof. We will prove the ﬁrst statement, the second one follows analogously.
For n = 0, this is trivially true. Now let n 7→ n + 1.
L
n+1
=
L
n
pp
P
n
j=1
L
j1
pp
L
pc
L
nj
cc
0 0
L
pp
L
pc
L
cp
L
cc
=
L
n+1
pp
+
P
n
j=1
L
j1
pp
L
pc
L
nj
cc
L
cp
L
n
pp
L
pc
+
P
n
j=1
L
j1
pp
L
pc
L
nj+1
cc
0 0
!
By assumption, the highlighted sum is zero. Note that in the second column, the separate term is precisely the one
arising for j = n + 1.
L
n+1
=
L
n+1
pp
P
n+1
j=1
L
j1
pp
L
pc
L
n+1j
cc
0 0
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A.3 Forwards direction
We will prove
L
pc
L
j
cc
L
cp
= 0 j N
0
NCGD. (24)
Proof.
We directly apply the structure lemma, Eq. (23), to Eq. (20). The intermediate
removes the population-
to-population entry. Hence, we obtain
L
n
L
n
0
∆ =
0
P
n
j=1
L
j1
pp
L
pc
L
nj
cc
0 0
0 0
P
n
0
j
0
=1
L
j
0
1
cc
L
cp
L
n
0
j
0
pp
0
!
=
P
n
j=1
P
n
0
j
0
=1
L
j1
pp
L
pc
L
nj
cc
L
j
0
1
cc
L
cp
L
n
0
j
0
pp
0
0 0
!
, (25)
and we clearly see another L
pc
L
···
cc
L
cp
combination, which is zero by the assumption, Eq. (24).
A.4 Backwards direction
We will prove
NCGD
L
pc
L
j
cc
L
cp
= 0 j N
0
. (26)
Proof. By assumption,
L
n
L∆ = 0 n N
0
, (27)
where we ﬁxed n
0
= 1.
Using induction we will show that if
L
pc
L
j
cc
L
cp
= 0 holds for all
j < n
, then it holds for
j n
. Since this can be
done for any n N
0
and the case for j = 0, L
pc
L
cp
= 0, is implied by Eq. (27) with n = 1, the statement follows.
By hypothesis, L
pc
L
j
cc
L
cp
= 0 j < n. The structure lemma, Eq. (23), therefore applies and hence
2
L
n+1
=
0
P
n+1
j=1
L
j1
pp
L
pc
L
n+1j
cc
0 0
,
so that
L
n+1
L∆ =
0
P
n+1
j=1
L
j1
pp
L
pc
L
n+1j
cc
0 0
0 0
L
cp
0
=
P
n+1
j=1
L
j1
pp
L
pc
L
n+1j
cc
L
cp
0
0 0
,
and we can again insert the hypothesis to eliminate all terms except j = 1.
L
n+1
L∆ =
L
pc
L
n
cc
L
cp
0
0 0
;
but this, by the assumption of NCGD must be zero, verifying the hypothesis.
2
The ﬁrst identity directly uses the proof of the structure lemma; since we only regard the right column, the lemma also holds for
n + 1.
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A.5 Example: NCGD up to third order
Consider a 5-level system where coherence is generated, but not detected, between levels 1 and 2 (L
pc
= 0
, L
cp
6
= 0
),
and where the opposite occurs between levels 4 and 5 (L
cp
= 0
, L
pc
6
= 0
). At a ﬁrst instant, coherence is transferred
to levels 1 and 3, and, at the next step, to levels 4 and 5, where it is eventually detected. Such a system is described
by a rank-3 noise generator with Hamiltonian
H =
1
2
1 1 0 0 0
1 1 0 0 0
0 0 0 0 0
0 0 0 1 1
0 0 0 1 1
(28)
and jump operators
J
1
=
1
2
1 i 0 0 0
i 1 0 0 0
0 0 0 0 0
0 0 0 1 i
0 0 0 i 1
, J
2
=
1 0 0 0 0
0 0 0 0 0
0 1 0 0 0
0 0 0 0 0
0 0 0 0 0
, and J
3
=
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
1 0 0 0 0
0 0 1 0 0
. (29)
As expected, such a dynamics is NCGD only up to third order in time, since it can be checked that L
pc
L
cp
= 0
and
L
pc
L
cc
L
cp
= 0, but L
pc
L
2
cc
L
cp
6= 0.
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B Proof of Theorem 3
Proof. ”:
We ﬁrst show that if NCGD holds, then
E
pc
(t
n
, t
n1
)E
cc
(t
n1
, t
n2
) ···E
cc
(t
2
, t
1
)E
cp
(t
1
, t
0
) = 0 t
n
··· t
0
. (30)
If we assume we can expand L at the initial time of propagation,
L(t) = L(t
0
) + O(t t
0
), (31)
we immediately obtain
E(t
1
, t
0
) = + L(t
0
)(t t
0
) + O
(t t
0
)
2
. (32)
Expanding at the corresponding initial times and using linear independence, we can conclude that Eq.
(30)
then
implies the assertion,
L
pc
(t
n
)L
cc
(t
n1
) ···L
cc
(t
2
)L
cp
(t
1
) = 0. (33)
The proof of Eq. (30) is by induction:
Our hypothesis is that, under the assumption of NCGD, we can conclude
E
pc
(t
k
, t
k1
)E
cc
(t
k1
, t
k2
) ···E
cc
(t
2
, t
1
)E
cp
(t
1
, t
0
) = 0 t
k
··· t
0
2 k n 1. (34)
The case n = 2 is exactly the NCGD statement.
To perform the induction step, note that iterating the NCGD condition and using
2
= ∆ provides us with
E(t
n
, t
n1
) ···E(t
1
, t
0
)∆ = ∆E(t
n
, t
n1
)∆ ···E(t
1
, t
0
)∆ (35)
E(t
n
, t
n1
)
E(t
n1
, t
n2
) ···E(t
2
, t
1
) E
pp
(t
n1
, t
n2
) ···E
pp
(t
2
, t
1
)
E(t
2
, t
1
)∆ = 0 (36)
Now using E E
pp
+ E
pc
+ E
cp
+ E
cc
(composition of incompatible domains is deﬁned to be zero) we obtain
E(t
n
, t
n1
)
E
pc
(t
n1
, t
n2
)E
cc
(t
n2
, t
n3
) ···E
cc
(t
2
, t
1
) +
E
cc
(t
n1
, t
n2
)E
cc
(t
n2
, t
n3
) ···E
cp
(t
2
, t
1
) +
E
cc
(t
n1
, t
n2
)E
cc
(t
n2
, t
n3
) ···E
cc
(t
2
, t
1
)
E(t
1
, t
0
)∆ = 0
(37)
Here, we used the hypothesis to eliminate all cross-terms that contain terms of the form
E
pc
E
cc
···E
cc
E
cp
. The
ﬁrst term, combined with the preceding time evolution
E
(
t
1
, t
0
)∆, and also the second term, combined with the
succeeding time evolution E(t
n
, t
n1
), also vanish by the same argument. Hence, all that remains is
E(t
n
, t
n1
)E
cc
(t
n1
, t
n2
)E
cc
(t
n2
, t
n3
) ···E
cc
(t
2
, t
1
)E(t
1
, t
0
)∆ = 0, (38)
closing the induction.
”:
We look at the evolution given by
E
(
t
1
, t
0
)∆. Let us divide this evolution into small parts (of size d
s
= (
t
1
t
0
)
/n
)
E
(
t
1
, t
1
d
s
)
···E
(
t
0
+ d
s, t
0
)∆. By linearly approximating each (which becomes exact in the limit
n
), we get
+ L
pp
(t
1
ds) + L
pc
(t
1
ds) + L
cp
(t
1
ds) + L
cc
(t
1
ds)
···
+ L
pp
(t
0
+ ds) + L
pc
(t
0
+ ds) + L
cp
(t
0
+ ds) + L
cc
(t
0
+ ds)
=
+ L
pp
(t
1
ds) + L
pc
(t
1
ds)

+ L
pp
(t
1
2 ds) + L
pc
(t
1
2 ds) + L
cp
(t
1
2 ds) + L
cc
(t
1
2 ds)
···
+ L
pp
(t
0
+ 2 ds) + L
pc
(t
0
+ 2 ds) + L
cp
(t
0
+ 2 ds) + L
cc
(t
0
+ 2 ds)

+ L
pp
(t
0
+ ds) + L
cp
(t
0
+ ds)
(39)
=
+ L
pp
(t
1
ds)
···
+ L
pp
(t
0
+ ds)
, (40)
where we used that by assumption any term of the form
L
pc
(
r
j
)
L
cc
(
r
j1
)
···L
cc
(
r
k+1
)
L
cp
(
r
k
) is zero. This equality
means that we get the same total population transfer whether we dephase at every point in time or we do not.
Inserting a dephasing at the wanted time and using the equality backwards we get NCGD.
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