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Incoherent witnessing of quantum coherence
Sahar Basiri-Esfahani1, ∗and Farid Shahandeh1, †
1Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
(Dated: August 10, 2021)
Theoretical and experimental studies have suggested the relevance of quantum coherence to the performance
of photovoltaic and light-harvesting complex molecular systems. However, there are ambiguities regarding
the validity of statements we can make about the coherence in such systems. Here we analyze the general
procedure for coherence detection in quantum systems and show the counterintuitive phenomenon of detecting
a quantum system’s initial coherence when both the input and output probe states are completely incoherent.
Our analysis yields the necessary and sufficient conditions for valid claims regarding the coherence of directly
inaccessible systems. We further provide a proof-of-principle protocol that uses entangled probes to detect
quantum coherence satisfying these conditions, and discuss its potency for detecting coherence.
Quantum coherence—that quantum systems occupy multi-
ple states simultaneously and hence exhibit interference—is
the distinctive feature of quantum systems compared to clas-
sical ones. In recent years, quantum coherence has become
a critical element of developing quantum technologies that
aim at improving over classical approaches [1–5]. Of partic-
ular importance is the potential role of quantum coherence in
the light-harvesting efficiency of biochemical processes, e.g.,
photosynthesis [6–9], as well as enhancing the performance
of molecular systems such as organic solar cells [10–14]. This
has stimulated studies of the effects of quantum coherence in
such systems, at the heart of which lies schemes to certify the
presence of quantum coherent mechanisms in molecular sys-
tems [15–19].
The main tool to examine quantum coherence in complex
systems including photosynthetic complexes is the ultrafast
spectroscopy [15–19]. Spectroscopic observations, however,
have led to debates [20–22] mainly because the dynamics
of systems in nature, as opposed to the spectroscopic tech-
niques, is initiated by incoherent inputs such as sunlight [22–
24]. These arguments indicate the need for further investi-
gating the existence of quantum coherence in systems operat-
ing under ambient conditions and proposals of new protocols
to detect quantum coherence using incoherent light sources
rather than coherent lasers [25]. Furthermore, it is intuitive
to assume that when quantum channels suffer from significant
decohereing noise the coherence of the system is untraceable.
We approach these arguments from a quantum informational
perspective and pose the following question (Fig. 1): Is it pos-
sible to make deductions about the coherence properties of a
system when both the input and output probe states are inco-
herent?
Here, we answer to the above question in the affirmative.
We show via a counter-example that quantum coherence of a
system can be observed even if both input and output to the
process are fully incoherent. We also provide rigorous neces-
sary and sufficient conditions for this observation to be possi-
ble. We propose a proof-of-principle protocol to detect the ex-
istence of quantum coherence within a system using incoher-
∗Electronic address: sahar.basiriesfahani@swansea.ac.uk
†Electronic address: shahandeh.f@gmail.com
Incoherent
input Λ
Coherence
generation?
Incoherent
output
FIG. 1. A coherence-generating quantum channel with incoherent
input and output. In such circumstances, the quantum coherence gen-
erated in a directly inaccessible system seems undetectable.
ent light sources and discuss the physics behind this counter-
intuitive phenomenon. Let us begin with the description of
quantum coherence from the perspective of quantum informa-
tion science [5]. In this picture, quantum coherence comprises
of the following ingredients. First, given a system described
in a finite-dimensional Hilbert space, an orthonormal basis
Binc={|ii} is defined as the incoherent basis. These vectors
represent the basis with respect to which we would like to con-
sider the quantum coherence properties of the system. Since
basis vectors |iiare also pure quantum states, they define all
incoherent quantum states as probabilistic mixtures of the in-
coherent basis states. In other words, every density opera-
tor %inc that can be written as %inc=Pipi|iihi|, for |ii∈Binc,
pi≥0, and Pipi=1, is an incoherent state. The terminology
is indeed justified by noting that every %inc is diagonal in the
basis Binc with no off-diagonal matrix elements. We denote
the collection of incoherent density operators by Sinc.
The second ingredient of the theory is the set of incoherent
channels denoted by Cinc. A quantum channel is a transforma-
tion that converts any input density operator to an output den-
sity operator. Most generally, a channel is called incoherent
if upon receiving an incoherent input quantum state outputs
an incoherent quantum state. Several classes of such trans-
formations have been studied within the quantum information
literature so far [5,26].
In order to better understand how these channels look like,
it is useful to introduce a particular transformation called the
fully dephasing channel ∆that acts on any quantum state as
∆[%] := Pihi|%|ii|iihi|=Pi%i|iihi|. Here, we have intro-
duced the shorthand %i:=hi|%|iifor the diagonal matrix ele-
ments of the density operator %in the incoherent basis Binc,
or simply the population of each incoherent state of the sys-
tem. It is thus evident that the action of the fully dephasing
arXiv:2108.04070v1 [physics.chem-ph] 6 Aug 2021
2
channel is to completely suppress the coherence of the input
quantum state. Incoherent channels can now be defined as all
channels Λthat satisfy [26,27]
∆◦Λ[%inc] = Λ[%inc ](1)
for all incoherent input quantum states %. Here, ◦denotes the
composition of quantum channels, meaning that each channel
is consecutively applied to the output of the previous channel
to its right. Equation (1) has a very intuitive physical inter-
pretation: a channel is incoherent if and only if the further
application of a fully dephasing channel on its output is re-
dundant for every incoherent input. Equation (1) can also be
rewritten in terms of all input quantum states as
∆◦Λ◦∆[%]=Λ◦∆[%],(2)
wherein a fully dephasing channel has been used to transform
the arbitrary input quantum state into an incoherent one.
The third and final ingredient of the theory of quantum co-
herence are incoherent measurements. It is known that ev-
ery quantum measurement is described by a set of effects
M={Mk}. An effect Mkis positive, i.e., it is Hermitian with
nonnegative eigenvalues, and corresponds to the outcome kof
the measurement M. The collection Msuch that PkMk=1is
called a positive operator-valued measure (POVM) [28]. Fur-
thermore, according to the Born rule, given the quantum state
%the probability of outcome kin measurement Mis given by
p(k|%, M) = Tr Mk%.
Now, a measurement Minc is called incoherent if the ma-
trix representation of each of its effects is diagonal with re-
spect to the incoherent basis Binc [5]. The simplest example of
an incoherent measurement is indeed the projective measure-
ment onto the incoherent basis, i.e. Minc={Pi=|iihi|}:=Πinc.
We denote the collection of all incoherent measurements by
Minc.
Channels and the probe-system interaction.— The concept
of a channel is very versatile. Consider the scenario in which
a probe interacts with a system and is then measured. It is
implicit that the system’s degree of freedom of interest cannot
directly be accessed without the mediation of the probe. As a
result, all we can speak of is the initial and the final quantum
state of the probe. In other words, the system—including its
initial and final quantum states—and its interaction with the
probe are subsumed by the quantum channel that transforms
the probe.
A generic channel Λtransforming an input state to the out-
put can be written as [28]
%(out) = Λ[%(in)] = X
i
Ki%(in)K†
i.(3)
Here, the operators Kiare called Kraus operators of the chan-
nel and satisfy PiK†
iKi=1. This way, it is guaranteed that
the output is a valid normalized density operator; see Support-
ing Information-1 (SI-1). The simplest example of a quantum
channel is indeed a unitary one that acts on the input as
%(out) = Υ[%(in)] = U %(in) U†.(4)
An important question following Eq. (3) is that what hap-
pens to the unitarity and reversibility of quantum mechanics.
A fundamental theorem of quantum mechanics implies that
every generic quantum channel as in Eq. (3) can be purified to
a unitary channel acting on the input state and some inacces-
sible auxiliary system [28], that is,
%(out) = Λ[%(in)] = TraΥ[%(in) ⊗π(in)
a].(5)
In Eq. (5), πais a suitably chosen state of the ancillary subsys-
tem a,Υis a suitably chosen interaction unitary channel act-
ing on the joint input-ancilla compound, and Trais the partial
trace taken over the ancilla. We note that the partial trace rep-
resents the inaccessibility of the auxilliary subsystem to the
experimenter.
Equation (5) can be brought to the context of probe-system
interaction by interpreting the input subsystem as the probe
(from now on denoted as %p) and the ancilla subsystem as the
system of interest (from now on denoted as πs). Neglecting
the interactions with the environment for the moment, Υcan
be interpreted as the unitary interaction between the probe and
the system, denoted by Υps. The inaccessibility of the ancilla
thus translates to the fact that direct measurements on the sys-
tem of interest are out of rich. Eq. (5) can thus be rewritten
as
%(out)
p= Λ[%(in)
p] = TrsΥps[%(in)
p⊗π(in)
s],(6)
that allows us to connect the properties of the inaccessible sys-
tem to the properties of the channel acting on the probe alone.
In the present work we are interested in the initial coher-
ence properties of the system π(in)
sby merely observing the
output probe state %(out)
p. It is evident from Eq. (6) that there
are three elements that can cause observable coherent effects
in the probe, namely, the initial system state π(in)
s, the ini-
tial probe state %(in)
p, and the probe-system interaction Υps.
Thus, to make valid statements about the coherence of π(in)
s,
we must make sure the latter two potential causes are not in
effect by making two assumptions:
(i) The input state of the probe is incoherent;
(ii) The channel Λis incoherent for all incoherent input
states of the system and probe.
We note that assumption (i) necessarily prevents objections
of the kind associated with spectroscopic techniques [20–24].
Now, let us posit the appropriate incoherent bases for the
probe and the system to be Binc;p and Binc;s, respectively. As-
sumption (ii) can then be expressed as
X
φ
ps
hk, φ|Ups|i0, j0ips hi0, j0|U†
ps|l, φips = 0,(7)
for all |i0ip,|kip,|lip∈ Binc;p with k6=land all |j0is∈
Binc;s. Furthermore, {|φis}is an arbitrary basis for the Hilbert
space of the system. Equation (7) gives us a general restric-
tion on the unitaries for which we can safely draw conclusions
about the coherence properties of the system based merely on
the observation of the probe.
3
Coherent scenario.—We can easily verify that Eq. (7) en-
sures that the output in Eq. (6) remains incoherent for initially
incoherent states of both system and probe. It is now evident
that whenever the output probe state is verified to be in a co-
herent superposition of states in Binc;p, that must be due to
the initial state of the system being coherent with respect to
the incoherent basis Binc;s. In other words, the unitary inter-
action Υps transfers the coherence of the system to the probe.
Indeed, this scenario is straightforward: it corresponds to a co-
herence generating map Λin Eq. (6) that transforms an inco-
herent input (probe) state into a coherent output (probe) state.
We are thus mainly interested in the challenging case in which
the map Λis incoherent.
Incoherent scenario.—Whenever the output probe state of
the process in Eq. (6) is incoherent, the naive conclusion is
that no signature of the system’s initial coherence survives the
incoherent process Λ. In the following, we show that this con-
clusion is not correct.
Schr¨
odinger versus Heisenber pictures.—According to the
standard quantum mechanics, the probability of outcome kin
the measurement Mon the output of the channel Λfor the
input state %(in) is given by the Born rule, that is,
p(k|%(in),Λ,M) = Tr(MkΛ[%(in)]) = Tr(Mk%(out)),(8)
where the map Λis defined most generally through its Kraus
operators as introduced in Eq.(3). For the special case of a
unitary channel as in Eq. (4), this reduces to
p(k|%(in),Υ,M) = Tr(MkΥ[%(in)]) = Tr(MkU%(in) U†).
(9)
We now recall from elementary quantum mechanics that
Eq. (9) describes the Born rule in the Schr¨
odinger picture
wherein the state of the system undergoes the dynamical evo-
lution according to the unitary Υand the effect Mkremains
stationary.
We know, however, that the unitary quantum evolution can
also be expressed in the Heisenberg picture by using the rule
of permutation-under-the-trace as
p(k|%(in),Υ,M) = Tr(U†MkU%(in)),(10)
wherein the dynamical evolution is associated with the ob-
servable rather than the input state of the system.
Similarly to Eq. (10), the case of a general channel of
Eq. (8) can also be recast in the Heisenberg picture as
p(k|%(in),Λ,M) = Tr(MkΛ[%(in)]) = Tr(Λd[Mk]%(in)).
(11)
Here, Λdis called the dual channel to Λ. One can easily work-
out the relation between Λand its dual Λdin the above equa-
tion (see SI-2) to find
Λd[Mk] := X
i
K†
iMkKi.(12)
It can be easily checked that Eq. (12) reduces to the usual
passage from the Schr¨
odinger to the Heisenberg picture for a
unitary channel.
p
m
S
FIG. 2. Schematic of our coherence detection protocol. The probe
and monitor are initially prepared in the maximally entangled state
%(in)
mp . The probe mode then interacts with the directly inaccessible
system which is part of the channel Λ. We observe the coherence
emerging in the state of the monitor conditioned on the outcomes of
an incoherent measurement on the probe.
Despite the similarities between a unitary and a generic
channel in Schrodinger and Heisenberg representations, the
two have a very sharp contrast. Suppose Υis an incoherent
unitary channel with respect to the incoherent basis Binc. Be-
cause Υis invertible, it must transform pure states in Binc
to only pure states in Binc. The latter must also hold for
the inverse channel Υ−1[·] = U−1·U. Using the fact that
U−1=U†we find Υ−1[·] = U†·U= Υd[·]. Thus, Υd[·]
must also transform pure states in Binc to only pure states in
Binc, i.e., the dual of an incoherent unitary channel is also
an incoherent unitary channel. This simple correspondence
between the coherence properties of a unitary channel and
its dual, however, does not hold for a generic channel [26].
In other words, there are incoherent channels whose dual is
not incoherent. We now move on to show that this asym-
metry can be exploited to prove the coherence of the ini-
tial state of an inaccessible system while the input and out-
put probe are both incoherent. The protocol.—A twin-mode
probe is initially prepared in the maximally entangled state
%(in)
mp =|Φ+imphΦ+|where
|Φ+imp =n−1/2
n−1
X
i=0 |iim|iip.(13)
Here, |iix∈Binc;x with x=m,p,Binc;m=Binc;p and nis any
natural number between two and infinity. The first mode is
called the monitor and the second mode is the probe to be
interacted with the system. Then, the probe mode is sent
through the channel Λto interact with the system. It is im-
portant to note that the quantum state of the probe mode given
by the marginal density operator Trm%(in)
mp =Pn−1
i=0 |iiphi|/n
is incoherent, as required.
After the interaction between the probe and the system, an
incoherent measurement on the probe pis carried out. In the
postprocessing stage, the outcomes of the measurement on the
monitor are sorted conditioned on the outcomes of the mea-
surement on the probe. Any coherence within the conditional
data, i.e., coherence within the conditional output state %(out)
m|k
for the outcome kof the probe, indicates that the initial state
of the system π(in)
swas in a coherent quantum superposition;
see Fig. 2.
In order to understand the working principles of our proto-
col, suppose that the channel Λpinduced by the probe-system
4
interaction as in Eq. (6) is asymmetric with respect to the
Schr¨
odinger and Heisenberg pictures, i.e., its dual Λd
pis not
incoherent. We proceed step by step according to the proto-
col. We have
%(out)
mp =Im⊗Λp[%(in)
mp ] = 1
nX
i,j,l |iimhj| ⊗ Kl|iiphj|K†
l
=1
nX
i,j,l
KT
l|iimhj|K∗
l⊗ |iiphj|.
(14)
In the last step we have used the trick I⊗Λ[Φ+]=ΛT⊗I[Φ+]
where Φ+is the shorthand for |Φ+ihΦ+|and Tis the trans-
position operation (see SI-3 for the proof). By postselecting
on the outcome kof an incoherent measurement on the probe
we obtain the conditional state
%(out)
m|k=hk|%(out)
mp |kip=1
nX
l
KT
l|kimhk|K∗
l.(15)
Finally, we examine the off-diagonal elements of the moni-
tor’s conditional state, that is,
hi|%(out)
m|k|jim=1
nX
lhi|KT
l|kimhk|K∗
l|ji
=1
nX
lhj|K†
l|kimhk|Kl|ii,
(16)
where we have simply transposed each matrix element and
rearranged the terms. If all the off-diagonal elements
hi|%(out)
m|k|jiof the conditional state are zero, that is, we are un-
able to observe any conditional coherences within the monitor,
this is equivalent to stating that Plhj|K†
l|kimhk|Kl|ii= 0
for all kand all i6=j. This readily means that the dual chan-
nel Λd[·] = PlK†
l·Klis incoherent (see Ref. [26] or SI-4
for a proof) which contradicts our assumption. Therefore, at
least one of the conditional states %(out)
m|kmust be coherent.
As we see, the detection power of our protocol is indepen-
dent of the channel being incoherent, rather it depends on the
coherence properties of the dual of the channel. Now, since (i)
the probe mode is incoherent—observe that Trm|Φ+ihΦ+|=
I/n, (ii) the channel Λis also incoherent, and (iii) the condi-
tional input monitor states hk|Φ+ihΦ+|kip=|kimhk|/n are
incoherent, we must have that the observed conditional coher-
ence is due to the initial coherence of the system’s initial state
π(in)
s.
Example.— Let us demonstrate our findings via a simple
physical example. Suppose the system, the probe, and the
monitor are two-level systems. We assume there are two reser-
voirs inducing decoherence, one of which is coupled to the
system and the other couples to the probe, and investigate two
regimes.
First, we consider the case where the probe-system interac-
tion is fast enough so that system’s decoherence during the in-
teraction is negligible. However, we assume that probe mode
fully dephases due to its interaction with its reservoir before
p
m
S
FIG. 3. Schematic of our example. Two two-level systems, p and
m, are initially prepared in the maximally entangled state %(in)
mp . The
probe mode exchanges excitation with the system, s, which is be-
lieved to be initially prepared in a superposition of its energy eigen-
states, at the rate gps. While the system and probe may suffer from
dephasings with rates γsand γp, respectively, the monitor is iso-
lated and decoherence-free. Observing coherence within the state
of the monitor mode conditioned on the outcomes of an incoherent
measurement on the probe certifies the coherence of system’s initial
state.
we measure it. In this case, Λp[%(in)
p]=∆p◦Ωp[%(in)
p]in
which Ωp[%(in)
p] = TrsΥps[%(in)
p⊗π(in)
s].
Second, we analyze the case where either of the system
or the probe can decohere during the interaction such that
Λp[%(in)
p] = Trs,rΥpsr[%(in)
p⊗π(in)
s⊗π(in)
r]. Here, the sub-
script rstands for the reservoirs acting on the system and
probe. Let |0iand |1ibe the eigenstates of an arbitrary de-
gree of freedom for the system, the probe, and the monitor.
We note that the degree of freedom may be different for each
of the three subsystems. The total Hamiltonian (~= 1) is
given by
H=Hm+Hp+Hs+Hint,(17)
in which Hj=ωjσ+
jσ−
jis the bare Hamiltonian of the mode
j = m,p,s.σ+
j(σ−
j) is the raising (lowering) operator for
mode jand ωjis the excitation energy. The probe-system in-
teraction is described by the Hamiltonian [29,30]
Hint =gps(σ+
pσ−
s+σ−
pσ+
s),(18)
where gps is the rate of the excitation exchange between the
system and the probe (see Fig. 3). The unitary interaction gen-
erated by this Hamiltonian satisfies the condition in Eq. (7)
which allows us to make valid statements regarding the sys-
tem’s initial coherence. We further assume that there are no
decay processes in the protocol that transfer the excitation of
mode jto the environment, that is the collective dynamics of
the probe-system preserves the total excitation number. This
is essential if we post-select excited states of the output probe.
However, the probe-system dynamics is susceptible to a pure
dephasing process that eliminates the phase coherence of local
excitations of mode i=p,sat a rate of γi. The total dynam-
ics of the probe-system can be modelled by a Born-Markov
master equation of the form [31–33]
d%
dt =−i[H, %] + Ldeph[σzs]%+Ldeph[σzp]%, (19)
where σziis the Pauli operator along the zaxis of mode i =
s,p. The action of the super-operator describing the dephasing
5
(a)
(b)
|
|
FIG. 4. The evolution of quantum coherence within the state of
the system in the first regime. The magnitude of the monitor’s off-
diagonal density matrix element conditioned on detecting a single
excitation in the probe is depicted (a) versus the degree of the sys-
tem’s initial coherence, , at gpst= 0.75; (b) versus the normalized
detection time, gpst, for different values of .
processes of mode iis
Ldeph[σzi]%=γi
2(σzi%σzi−%).(20)
First regime.— The system is initially in an arbitrary pure
state of the form |φ(0)is=|1is+√1−2|0iswhere deter-
mines its degree of coherence. We simulate the performance
of our protocol in this scenario in two steps. First, we let the
probe interact with the system via the interaction Hamiltonian
(18). Next, we allow the probe to fully dephase after the in-
teraction, followed by a measurement of the coherence in the
monitor conditioned on finding the probe in the excited state
|1ip. The details of the calculations for this simulation are
provided in SI-5.
Figure 4(a) shows the magnitude of the monitor’s off-
diagonal element of the density matrix, |h1|%(out)
m|1|0im|, versus
system’s degree of coherence . For = 0 and = 1, where
the system is in an incoherent state, the monitor remains in an
incoherent state as expected. However, for any other non-zero
, where the system contains initial coherence, a measurement
of the off-diagonal element of the monitor results in a non-
zero value with its maximum occurring at = 1/√2, i.e.
for the maximally-coherent initial state of the system. Fig-
ure 4(b) shows the magnitude of the monitor’s off-diagonal
element of the density matrix versus the detection time for dif-
ferent values of . These results clearly show that in situations
where the probe is prone to dephasing before we measure-
(a)
(b)
|
|
FIG. 5. The evolution of quantum coherence within the state of the
system in the second regime. The magnitude of the monitor’s off-
diagonal density matrix element conditioned on detecting a single
excitation in the probe is depicted (a) versus the normalised probe-
system interaction duration, gps τps, for the cases where the probe
alone decoheres during the interaction, i.e., γp= 0.1and γs= 0
(solid blue line), and where both the probe and the system deco-
here while interacting, i.e., γp= 0.1and γs= 0.1(dashed red
line); (b) versus the normalized evolution time of the monitor for
γs=γp= 0.1and two different values of interaction duration. In
both graphs the probe continues to dephase after τps until the mea-
surement is performed at an arbitrary time t.
ment it, our protocol successfully detects the initial coherence
within the system.
The effect of probe’s dephasing before the interaction with
the system can be seen as a reduction in the initial quantum
correlations between the probe and monitor. It is both natu-
ral and correct that such a reduction in correlations reduces
the power of the monitor to accumulate the coherence exhib-
ited by the probe. We provide the detail of this analysis in
SI-6. Second regime.— Let us now consider the more real-
istic regime in which the system and probe decohere during
their interaction due to their coupling to the environment. We
are particularly interested in the relative time scales of the co-
herent and dephasing evolutions which allow for detecting the
initial coherence of the system.
We consider the same initial states of |φ(0)isand |Φ+imp
for the system and monitor-probe, respectively. We let the
system and probe interact for the duration τps in which both
system and probe are susceptible to decoherence. The dynam-
ical evolution of the entire system is simulated by solving the
master equation (19) for the total state %mps(t). Then, we let
the probe alone decohere further for the duration τmeas until
it is being measured. Finally, we trace over the state of the
6
system which is inaccessible and calculate the monitor’s off-
diagonal element h1|%(out)
m|1|0im, i.e., conditioned on finding
the probe in the excited state |1ip.
In Figure 5(a), we depict monitor’s coherence as a function
of the probe-system interaction time. It can clearly be seen
that, for fast enough probe-system interactions such that τps <
min(1/γp,1/γs), the initial coherence of the system will be
successfully observed in the monitor even if the probe fully
dephases during its free evolution time τmeas. However, if τps
is of the order of the dephasing time of either system or probe,
that is τps ≈min(1/γp,1/γs), then the monitor will not retain
the coherence information. Figure 5(b) illustrates the time
evolution of the induced coherence within the monitor for this
case.
We observe that decoherence occurring during the probe-
system interaction causes loss of coherence information in the
monitor manifested in the reduction of |h1|%(out)
m|1|0im|. This
is better understood if we think of the probe as an infinites-
imally small part of the bath coupled to the system that is
accessible to us. Upon the system’s interaction with its sur-
roundings, its coherence information spreads to the entire en-
vironment. As a result, the longer the probe-system interac-
tion is, the smaller the share of the probe from that informa-
tion will be. Hence, for the monitor to preserve the informa-
tion on the initial coherence of the system, the probe-system
interaction time should be shorter than the coherence lifetime
of the system. Counter-intuitively, further decoherence of the
probe does not affect the monitoring of the system’s coher-
ence because, with the help of quantum correlations between
the probe and monitor, a copy of this information is stored in
the monitor, which is decoherence-free.
In conclusion, we have shown the counterintuitive phe-
nomenon of detecting a quantum system’s initial coherence
when the input and output probe states are completely inco-
herent. We achieved this through rigorously examining the
elements of coherence detection schemes from a quantum in-
formation perspective. Our analysis yields the necessary and
sufficient conditions to enable valid claims regarding the co-
herence of a directly inaccessible system. We provided a pro-
tocol for witnessing the quantum coherence of such systems
that satisfies these conditions. This protocol is then used in a
proof-of-principle demonstration of our results, highlighting
its power and limitations. Our results confirm the necessity
of fast measurements when it comes to the duration of the
probe’s interaction with the system.
We believe our analysis inspires novel protocols to detect
coherent channels using entangled probes. The use of a quan-
tum entangled twin-mode probe also opens up an avenue for
experimental schemes in which the probe might undergo de-
coherence before its information is retrieved. In particular,
we think that, in the near future, entangled light will play a
significant role in probing all sorts of quantum phenomena,
including quantum coherence, in complex systems.
ACKNOWLEDGMENTS
The authors gratefully acknowledge Eric Chitambar, Eric
Bittner, Martin Ringbauer, Ivan Kassal, and Eric Gauger for
helpful discussions. S.B-E. acknowledges funding from the
European Union Horizon 2020 research and innovation pro-
gramme under the Marie Skłodowska-Curie grant agreement
No 663830. F.S. was supported by the Royal Commission
for the Exhibition of 1851 Research Fellowship. This project
was also supported in part through the Sˆ
er SAM Project at
Swansea University, an initiative funded through the Welsh
Government’s Sˆ
er Cymru II Program (European Regional De-
velopment Fund).
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Supporting Information: Incoherent witnessing of quantum coherence
Sahar Basiri-Esfahani1, ∗and Farid Shahandeh1, †
1Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
SI-1: KRAUS DECOMPOSITION OF QUANTUM CHANNELS
A generic channel Λtransforming an input state to the output can be written as [1]
%(out) = Λ[%(in)] = X
i
Ki%(in)K†
i.(S1)
Here, the operators Kiare called Kraus operators of the channel and satisfy PiK†
iKi=1. To see that such a transformation
maps density operators to density operators, we first show that the output operator is positive. For an arbitrary quantum state
|ψiwe have
hψ|%(out)|ψi=X
ihψ|Ki%(in)K†
i|ψi.(S2)
It is clear that hφ|:= hψ|Kiis the Hermitian conjugate of |φi:= K†
i|ψi. Since %(in) is a positive operator, it follows that
hφ|%(in)|φi ≥ 0. Thus, hψ|%(out) |ψi ≥ 0which proves our claim.
Now we show that channel Λpreserves the normality of the density operators:
Tr %(out) =X
i
Tr(Ki%(in)K†
i) = X
i
Tr(K†
iKi%(in)) = Tr(X
i
K†
iKi%(in)) = Tr(1%(in))=1.(S3)
Hence, the output remains a normalized positive operator, i.e., a valid density operator.
SI-2: RELATION BETWEEN A QUANTUM CHANNEL AND ITS DUAL CHANNEL
Here, we show that the dual of a channel Λwith Kraus decomposition Λ[·] = PlKi[·]K†
iis given by Λ(d)[·] = PlK†
i[·]Ki.
Consider the following sequence of operations:
Tr( ˆ
MkΛ[%(in)]) = Tr ˆ
MkX
i
ˆ
Ki%(in) ˆ
K†
i!
=X
i
Tr ˆ
Mkˆ
Ki%(in) ˆ
K†
i
=X
i
Tr ˆ
K†
iˆ
Mkˆ
Ki%(in)
= Tr X
i
ˆ
K†
iˆ
Mkˆ
Ki%(in)!
= Tr(Λd[ˆ
Mk]%(in)).
(S4)
Since this is true for any input state, it follows from the last equality that
Λ(d)[·] = X
l
K†
i[·]Ki.(S5)
∗Electronic address: sahar.basiriesfahani@swansea.ac.uk
†Electronic address: shahandeh.f@gmail.com
arXiv:2108.04070v1 [physics.chem-ph] 6 Aug 2021
2
SI-3: PROOF OF I⊗Λ[Φ+] = ΛT⊗I[Φ+]
Given an operator ˆ
Kand a basis {|ii},
ˆ
I⊗ˆ
K|iiA|iiB=|iiAX
j|jiBhj|ˆ
K|iiB.(S6)
Then, we use the facts that (i) the two Hilbert spaces of A and B are assumed to be isomorphic, HA∼
=HB, and (ii) Bhj|ˆ
K|iiB
is a c-number and thus we can safely replace it with Ahj|ˆ
K|iiA, to write
ˆ
I⊗ˆ
K|iiA|iiB=X
j|iiAhj|ˆ
K|iiA|jiB
=X
j|iiAhi|ˆ
KT|jiA|jiB
=n1
2|iiAhi|ˆ
KT⊗ˆ
I|Φ+iAB,
(S7)
where |Φ+iAB possesses the computational basis as its Schmidt vectors. It is now easy to sum over ion both sides of Eq. (S7)
to obtain the desired result,
ˆ
I⊗ˆ
KX
i|iiA|iiB=n1
2X
i|iiAhi|ˆ
KT⊗ˆ
I|Φ+iAB
⇒ˆ
I⊗ˆ
K|Φ+iAB =ˆ
KT⊗ˆ
I|Φ+iAB.
(S8)
SI-4: NECESSARY AND SUFFICIENT CONDITION FOR INCOHERENT CHANNELS
For a quantum channel Λwith Kraus decomposition Λ[·] = PlKl[·]K†
l, the necessary and sufficient condition to be incoher-
ent is given by [2]
X
lhj|Kl|iiphi|K†
l|ki= 0,(S9)
for all iand j6=k.
To show this, we use a representation of quantum channel known as the Choi-jamiołkowski isomorphism [3–5]. For our
purpose, it is enough to recall that this isomorphism is equivalent to applying the channel to the maximally entangled state
|Φ+ihΦ+|, represented as Φ+for short. Now, using the definition of an incoherent channel given in the main text, that is
Λ◦∆[·] = ∆ ◦Λ◦∆[·], we find the Choi-jamiołkowski representation of both sides as
Λ◦∆[Φ+] = X
i,l |iimhi| ⊗ Kl|iiphi|K†
l(S10)
and
∆◦Λ◦∆[Φ+] = X
i,j,l |iimhi|⊗|jiphj|hj|Kl|iiphi|K†
l|ji.(S11)
Note that, the fully dephasing channel and Λonly act on the probe mode in Φ+. Equating the two relations, it follows that
X
lhj|Kl|iiphi|K†
l|ki= 0,(S12)
as claimed.
SI-5: DETECTING COHERENT STATE OF A TWO-LEVEL SYSTEM: FIRST REGIME
Here, we calculate the conditional state of the monitor conditioned on finding the probe in the excited state |1ip. To do this,
we first calculate the total state of the probe-system after the interaction. This will be %mps (t) = |Φ(t)impshΦ(t)|, where
|Φ(t)imps =Im⊗Ups(t)|Φ+imp |φ(0)is
=1
√2
1
X
i=0 |iim|χiips,(S13)
3
|
FIG. 1. Detected coherence for a probe which is partially dephased before its interaction with the system. The magnitude of monitor’s off-
diagonal element of the density matrix conditioned on detecting a single excitation in the probe is plotted versus the normalized detection time,
gpst, for = 1/√2and different values of η.
in which Ups(t) = e−iHint tis the evolution operator in the interaction picture and
|χiips =
1
X
j=0 |jip|αij is,
|α00is=p1−ε2|0is+εcos(gps t)|1is,
|α01is=−iε sin(gps t)|0is,
|α01is=−ip1−ε2sin(gps t)|1is,
|α11is=p1−ε2cos(gps t)|0is+ε|1is.
(S14)
Hence, the state of the probe is %mp(t) = Trs(%mps(t)). Now, the action of a fully dephasing operation on the probe p-mode
results in the output state
%(out)
mp = ∆[%mp(t)]
=1
2
1
X
j=0 |jiphj| ⊗
1
X
i,l=0hαlj |αij i|iimhl|.(S15)
Next, we postselect on outcome k(k= 0 or 1) of a projective measurement on the probe p-mode. The conditional state of the
probe m-mode becomes
%(out)
m|k=hk|%(out)
mp |kip
=1
2
1
X
i,l=0hαlk|αik i|iimhl|.(S16)
Finally, we measure the off diagonal elements of the conditional state of the monitor mode as
hi|%(out)
m|k|li=1
2hαlk|αik i.(S17)
Figure 4(b) in the main text shows the plot of the time evolution of the absolute value of this off-diagonal element for i= 1 and
l= 0.
SI-6: PROBE’S DECOHERENCE BEFORE THE INTERACTION
We mentioned within the main text that the effect of probe being dephased before the interaction with the system can be
seen as a reduction in the initial quantum correlations between the probe and monitor. To see this, let us introduce the partial
dephasing channel defined as
∆η[%] := η I[%] + (1 −η) ∆[%],(S18)
4
in which 0≤η≤1and ∆[%] = Pihi|%|ii|iihi|=Pi%i|iihi|is the fully dephasing channel. Equation (S18) has a simple
physical meaning: it is equivalent to applying no dephasing to the state with probability ηand fully dephasing it with probability
of 1−η.
A two-mode probe state that undergoes decoherence before interacting with the system can be thought of as going through
the partial dephasing channel ∆η. We thus find for the input probe-monitor state that
∆η[Φ+
mp] = 1
2
1
X
i=0 |iimhi|⊗|iiphi|+η
2
1
X
i,j=0 |iimhj|⊗|iiphj|.(S19)
It is evident that η= 1 represents no dephasing and the maximum correlations between the probe and monitor, while η= 0
represents a fully dephased initial state with no correlations between the two modes. It is both natural and correct that such a
reduction in correlations reduces the power of the monitor to accumulate the coherence exhibited by the probe. Having an initial
partial probe dephasing (a non-unity η) can be seen as a reduction in the maximum value of the magnitude of the monitor’s
off-diagonal element of the density matrix as shown in Fig. 1.
[1] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University
Press, 2010).
[2] E. Chitambar and G. Gour, Comparison of incoherent operations and measures of coherence, Phys. Rev. A 94, 052336 (2016).
[3] A. Jamiołkowski, Linear transformations which preserve trace and positive semidefiniteness of operators, Reports on Mathematical
Physics 3, 275 (1972).
[4] M.-D. Choi, Completely positive linear maps on complex matrices, Linear Algebra and its Applications 10, 285 (1975).
[5] M. Jiang, S. Luo, and S. Fu, Channel-state duality, Physical Review A 87, 022310 (2013).
