Coordinately assisted distillation of quantum coherence in multipartite system

Abstract

We investigate the issue of assisted coherence distillation in the asymptotic limit, by coordinately performing the identical local operations on the auxiliary systems of each copy. When we further restrict the coordinate operations to projective measurements, the distillation process is branched into many sub-processes. Finally, a computable measure of the assisted distillable coherence is derived as the maximal average coherence of the residual states with the maximization taken over all the projective measurements on the auxiliary. The measure can be conveniently used to evaluate the assisted distillable coherence in experiments, especially suitable for the case that the system and its auxiliary are in mixed states. By using the measure, we for the first time study the assisted coherence distillation in multipartite systems. Monogamy-like inequalities are derived to constrain the distribution of the assisted distillable coherence in the subsystems. Taking three-qubit system for example, we experimentally prepare two types of tripartite correlated states, i.e., the W-type and GHZ-type states in a linear optical setup, and experimentally test the assisted distillable coherence. Theoretical and experimental results agree well to verify the distribution inequalities given by us. Three measures of multipartite quantum correlation are also considered. The close relationship between the assisted coherence distillation and the multipartite correlation is revealed.

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PAPER
Coordinately assisted distillation of quantum
coherence in multipartite system
To cite this article: Huang-Qiu-Chen Wang et al 2022 Quantum Sci. Technol. 7 045024
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Quantum Sci. Technol. 7(2022) 045024 https://doi.org/10.1088/2058-9565/ac87cc
RECEIVED
19 March 2022
REVISED
21 July 2022
ACCEPTED FOR PUBLICATION
8 August 2022
PUBLISHED
24 August 2022
PAPER
Coordinately assisted distillation of quantum coherence
in multipartite system
Huang-Qiu-Chen Wang, Qi Luo , Qi-Ping Su, Yong-Nan Sun, Nengji Zhou,
Li Yu and Zhe Sun∗
School of Physics, Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
∗Author to whom any correspondence should be addressed.
E-mail: sunzhe@hznu.edu.cn
Keywords: quantum information, quantum resource theory, quantum optics
Abstract
We investigate the issue of assisted coherence distillation in the asymptotic limit, by coordinately
performing the identical local operations on the auxiliary systems of each copy. When we further
restrict the coordinate operations to projective measurements, the distillation process branches
into many sub-processes. Finally, a computable measure of the assisted distillable coherence is
derived as the maximal average coherence of the residual states with the maximization taken over
all the projective measurements on the auxiliary. The measure can be conveniently used to evaluate
the assisted distillable coherence in experiments, especially suitable for the case that the system and
its auxiliary are in mixed states. By using the measure, we for the first time study the assisted
coherence distillation in multipartite systems. Monogamy-like inequalities are derived to constrain
the distribution of the assisted distillable coherence in the subsystems. Taking a three-qubit system
for example, we experimentally prepare two types of tripartite correlated states, i.e., the W-type
and GHZ-type states in a linear optical setup, and experimentally test the assisted distillable
coherence. Theoretical and experimental results agree well to verify the distribution inequalities
given by us. Three measures of multipartite quantum correlation are also considered. The close
relationship between the assisted coherence distillation and the multipartite correlation is revealed.
1. Introduction
Quantum coherence, as the fundamental feature of quantum mechanics and a kind of resource [1,2], is
widely used in quantum information processing [3], quantum computation, quantum algorithm [4,5],
quantum metrology [6–9], and quantum thermodynamics [10,11]. It is the main reason why the quantum
world is different from the classical world [12].
In order to quantify coherence [13], one needs a set of reference bases {|i,i=0, 1, 2, ...},basedon
which, the class of incoherent states Iis defined with diagonal density matrices, i.e., iρi|ii|∈I.
Following this, incoherent operations (IOs) act unchangeably on the assemblage of all incoherent states and
satisfy the map ΛIO (I)⊆I. Different types of IOs are proposed in [13–17]. A common measure of
coherence for a state ρis defined by the relative entropy [13], Cr(ρ):=minσ∈I S(ρσ), to characterize the
minimal distance of ρto the class of incoherent states I. One of the most operational measure of the
coherence is the distillable coherence which is similar to the framework of the distillable entanglement
[18,19], and was introduced in [20] at the asymptotic limit by considering infinite copies of the state. The
optimal rate of a state ρin a coherence distillation process, defined as the distillable coherence Cd(ρ), is
evaluated analytically Cd(ρ)=Cr(ρ)[20,21]. However, in experiments, it is a huge challenge to collectively
manipulate a large number of state copies, and to achieve the asymptotic limit. Therefore, a kind of
one-shot coherence distillation was proposed [22] which also provided the possibility of the experimental
study. This one-shot scenario was experimentally demonstrated based on a linear optical system [23], where
akindofN-dimensional (N2) IOs were realized.
© 2022 IOP Publishing Ltd

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
On the other hand, the asymptotic scenario of coherence distillation was developed into bipartite system
ρAB, where only the operations performed on the second party (B) are restricted to IOs, and the classical
communication is allowed between the two parties. These sets of operations are called local
quantum-incoherent operations and classical communications (LQICC). Following the LQICC, the concept
of the assisted coherence distillation was established in asymptotic settings [24]. The assisted distillation rate
Rof subsystem ρBis bounded by quantum-incoherence (QI) relative entropy CA|B
r(ρAB). For pure states
|ΨAB, the upper bound is accessible, while for mixed states ρAB, it is still an open question whether the
upper bound can be achieved. The experimental study of the assisted coherence distillation was reported in
[25], where the authors employed a one-copy scenario. To overcome the difficulty of the experimental
demonstration, the nonasymptotic settings of the assisted coherence distillation were proposed [26,27].
Different from the distillation framework above, in [28] the authors introduced a scenario of
steering-induced coherence, which is defined on the eigenvectors of the considered system, and has been
conveniently used in open systems [29].
Quantum coherence in multipartite and multilevel systems has been attracted much attention in the last
decade. A necessary and sufficient analytical criterion was reported in [30] to verify the presence of
multilevel coherence. The problems of the quantum coherence distribution among the constituent
subsystems were considered in [31,32]. The conversion between quantum coherence and quantum
correlation was studied in [33,34]. We find that an issue worthy of study is what distribution law of the
assisted distillable coherence obeys in the multipartite system when a subsystem is chosen as the assistance.
Furthermore, the relation between the multipartite correlation and the assisted coherence distillation is also
worth investigating.
In this work, we take into account a class of operations named as coordinate local quantum-incoherent
operations and classical communication (CoLQICC), which is a subset of LQICC. Based on the CoLQICC
operations, we introduce the coordinate asymptotic scenario of the assisted coherence distillation and derive
a computable measure to evaluate the assisted distillablecoherence.Byusingthemeasure,weforthefirst
time investigate the distribution of the assisted distillable coherence in multipartite systems. We develop a
monogamy-like inequality to reveal that the assisted distillable coherence of each subsystem should be
constrained by that of the remaining combined system. Taking a three-qubit system for example, we
experimentally measure the assisted distillable coherence and verify the inequality relation. By numerical
calculations, we show the relationship between the multipartite correlation and the assisted coherence
distillation.
2. Coordinately assisted distillation of coherence
For a bipartite system of Alice and Bob sharing the state ρAB, the aim of assisted coherence distillation is to
concentrate the coherence resource on Bob’s side by allowing Alice to perform arbitrary quantum
operations. To quantify the optimal rate of assisted distillable coherence, the definition is given as
follows [24]:
CA|B
d(ρ)=supR:lim
n→∞ inf
ΛΛ(ρ⊗n)−Φ⊗nR
21=0,(1)
where the infimum is taken over all LQICC operations Λand xreturns the maximum integer no larger
than x.O1=Tr √O†Ois the trace norm. In D-dimensional Hilbert space H, the maximal coherent
resource state is |ΦD≡D−1
i=0|i/√D,andΦ2≡|Φ2Φ2|denotes the density matrix of the
two-dimensional maximal coherent state. It has been proved that, there is an upper bound of the assisted
distillable coherence, i.e., CA|B
d(ρAB)CA|B
r(ρAB), with the definition CA|B
r(ρAB)=S(BρAB )−S(ρAB ), and
B(ρAB)≡iIA⊗|iBi|ρIA⊗|iBi|with Ibeing a identity matrix [24]. For a pure bipartite state,
the equal sign holds, however, for a mixed state, whether the upper bound can be achieved is still unknown.
In the conventional asymptotic scenario of the coherence distillation, many independent and identically
distributed (i.i.d.) copies of the initial resource are to be converted into many i.i.d. copies of the target state.
Collective operations are needed to be performed on all the state copies to reach the optimal
interconversion rate. Therefore, it is a huge challenge for the current technology to experimentally
demonstrate the coherence distillation process. Even for theoretical research, it is still difficult to prove the
reachability of the upper bound for a mixed state in the asymptotic scenario.
To derive an effective measure to evaluate the assisted distillable coherence in the asymptotic framework,
we introduce a new scenario by considering a type of operation named as CoLQICC. The operation of
CoLQICC proposed by us consists of two parts: (i) identical local measurements (operations) on Alice’s side
are coordinately and separably performed on each copy of the resource state. Let the mapping ΛA:CoQ
denote the operation on Alice, for many copies of the state, there should be Λ⊗n
A:CoQρ⊗n
AB =ΛA:CoQρAB⊗n.
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Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
A similar setting can be found in [35]; (ii) incoherence operations, denoted by the mapping ΛIO
B,working
on the Bob’s side. In our consideration, the operations of ΛIO
Bwill collectively act on the copies of
Bob’s residual states. Therefore, the CoLQICC can be described by a complete mapping, i.e., ΛCoQ ≡
ΛIO
B◦Λ⊗n
A:CoQ
.
Under the CoLQICC, we define the optimal rate of the coordinately assisted coherence distillation as:
CA|B
CoQ(ρ)=supR:lim
n→∞ inf
ΛCoQΛCoQ (ρ⊗n)−Φ⊗nR
21=0,(2)
where the infimum is taken over all the CoLQICC operators ΛCoQ.Obviously,whenthestateofAliceand
Bob is in a product form, i.e., ρAB =ρA⊗ρB, the assist system Ahas no effects, then the assisted distillable
coherence CA|B
d(ρ) transforms to the distillable coherence Cd(ρB). In order to obtain a computable measure,
we further simplify the operations on AlicetoorthogonalprojectivemeasurementsΞi
A, the operators
satisfy Tr(Ξi
AΞk
A)=δik,iΞi
A=IA,andΞi
A2=Ξ
i
A. In this setting, Λ⊗n
A:CoQ
ρ⊗n
AB =i(Ξi
A⊗IB)⊗n
ρ⊗n
AB (Ξi
A⊗IB)⊗n. Thus, in the following sections, we emphasize the projective measurements by restricting
the CoLQICC operations to the coordinate local projective-incoherent operations and classical
communication (CoLPICC).
Lemma 1. The optimal rate of the assisted coherence distillation under the proposed CoLPICC operations, can
be expressed as follows:
CA|B
CoP (ρAB)=max
{Ξi
A}
i
Pisup{Ri:lim
n→∞ inf
{IOB}ΛIO
Bρi
B⊗n−Φ2⊗nRi1=0},(3)
where Piis the probability distribution and ρi
Bis the residual density with the definitions:
Pi=TrΞi
A⊗IBρAB,
ρi
B=TrAΞi
A⊗IBρAB
Pi
.(4)
The maximum is taken over all the Alice’s projective measurements and the infimum is taken over the IOs on
Bob’s side. The rate of the assisted coherence distillation is a probabilistic sum of all the subprocesses. Finally, the
rate of coordinately assisted coherence distillation becomes R=maxΞi
AiPiRi(proof details are shown in
appendix A).
Our results reveal that different sets of operations performed on Alice will provide different final distillation
rates. We design a new scenario of coordinately performing identical projective measurements on each copy of
Alice, which will help us to derive a computable measure of the assisted distillable coherence available in the
asymptotic limit. The measurements on Alice’s side cause the coherence distillation process to branch into several
sub-processes, each of which corresponds to a distillation rate Ri.SinceCoLPICC⊂CoLQICC ⊂LQICC, one
has the relation C A|B
CoP (ρAB)CA|B
CoQ(ρAB )CA|B
d(ρAB)CA|B
r(ρAB).CA|B
CoP and C A|B
CoQ correspond to the different
sets CoLPICC and CoLQICC, respectively.
Theorem 1. With the proposed CoLPICC operators, the optimal rate of the coordinately assisted coherence
distillation has an explicit solution:
CA|B
CoP (ρAB)=max
{Ξi
A}
i
PiCrρi
B,(5)
with the definitions of Piand ρi
Bin equation (4) and the maximum is taken over all the projective measurements
Ξi
Aon Alice. Perhaps it seems quite restrictive that the local operations are set to projective measurements. The
reasons are that: (i) the identical local projective measurements satisfy our requirement well to derive a reachable
and computable solution of the assisted coherence distillation in the asymptotic limit, (ii) projective
measurements are particularly easy to operate and are widely used in theoretical and experimental studies. It
makes the choice more reasonable and easy to accept. Certainly, other types of measurements can be considered,
however, it will reduce the possibility of obtaining the computable formula.
Importantly, the measure in equation (5) is suitable for the case that ρAB is a mixed state, and which is
convenient to be tested in experiments. As is known that for mixed states, there is still a lack of the measure,
derived in the asymptotic framework, to evaluate the assisted distillable coherence. Whether the analytical
upper bound CA|B
r(ρAB) can be achieved for mixed states is still unknown. Now, we show that CA|B
CoP(ρAB )
can act as an affirmative and reachable measure for mixed states. In appendix B, we prove in detail that for a
mixed state, Bob can asymptotically obtain the distillable coherence with the rate of CA|B
CoP(ρAB )whenproper
3

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
projective measurements performed on Alice’s side. Our results provide an operational interpretation of the
average relative entropy of coherence, which should not be simply understood as a one-copy scenario to
approximately evaluate the assisted distillable coherence [25], but a determined reachable measure derived
in the asymptotic limit. The proof details of theorem 1can be found in appendix B, where we first prove the
upper bound of the distillation rate is the average of the relative entropy of coherence. Then we prove that
the upper bound can be reached by using the typical sequence technique.
Certainly, the CoLQICC operation introduced by us still includes a collective operation on the copies of
Bob, and thus this scenario does not overcome the difficulties in the experimental demonstration of the
asymptotic distillation process. However, the measure in equation (5), derived through the scenario of
CoLQICC, is suitable for the case that one focuses on evaluating the amount of the assisted distillable
coherence available in the asymptotic limit. He can prepare one-copy resource state and implement optimal
projective measurements on Alice’s side and perform tomography on Bob’s side to evaluate the final
distillable coherence.
For a pure state density ΨAB ≡|ΨAB ΨAB|, through the local measurements on Alice together with the
communications with Bob, any possible pure decomposition of ρBcan be obtained, i.e., ρB=ipiΨi
Bfor
any set of {pi}and the corresponding pure state density is Ψi
B≡Ψi
BΨi
B. Therefore, based on the
definition in equation (5), we have CA|B
CoP(ΨAB)=max{Ξi
A}iPiCrΨi
B=max{Ξi
A}iPiSΔΨi
B,whichis
identical to the concept of coherence of assistance (COA) Ca(ρB)[24]. Moreover, one can find
CA|B
CoP(ΨAB )CA|B
d(ΨAB)=CA|B
r(ΨAB)=S(BΨAB )=S(ρB). Now let us discuss two special cases of
pure states:
(a) The dimension of subsystem Bis dim(HB)=2, then one has Ca(ρB)=S(ρB)[24]. Consequently, we
have
CA|B
CoP(ΨAB )=CA|B
r(ΨAB)=S(ρB).(6)
(b) The dimension of auxiliary system (Alice) is dim(HA)=2 and that of Bob is dim(HB)=n(n>2).
For a set of reference basis {|i}, where the quantum coherence is defined. If the Schmidt
decomposition of |ΨABcan be written as follows:
|ΨAB=λ1|φ1
A⎛
⎝
j=i|jB⎞
⎠+λ2|φ2
A|iB,(7)
where φ2
A|φ1
A=0. Then by performing the projective measurement of {|φ1
A±|φ2
A/√2}on Alice,
one can easily obtain CA|B
CoP(ΨAB )=S(ρB). The expression in equation (7) also gives an answer to the
remaining issue in [24] that for which kind of high-dimensional pure states, the assisted of coherence
(COA), i.e., Ca(ρB)=CA|B
CoP for pure states in this work, is equal to the regularized COA for the infinite
copies of the state.
Let us expand to the cases of multipartite systems, and take a tripartite pure state for example. If the
Schmidt decomposition of a pure state |ΨABC with the condition dim(HA)=2, can be presented as:
|ΨABC=λ1|φ1
A⎛
⎝
mn|ij=0|mnBC⎞
⎠+λ2|φ2
A|ijBC ,(8)
where {|ij} denotes a set of reference basis and φ2
A|φ1
A=0, we also have the similar equality in
equation (6)that:
CA|BC
CoP (|ΨABC)=S(ρBC ).(9)
For example, the GHZ-type and W-type states satisfy the decomposition in equation (8), thus the above
equality holds.
Assisted coherence distillation in multipartite systems. In the following sections, we will discuss the
problems of coordinately assisted coherence distillation in multipartite systems. Let us start from the
tripartite case.
Theorem 2. In tripartite system, for a pure state |ΨABCsatisfying the condition in equation (8) and with the
dimension of the auxiliary system dim(HA)=2, the following inequality holds,
CA|BC
CoP |ΨABCCA|B
CoP |ΨABC +CA|C
CoP |ΨABC , (10)
4

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
where the first process C A|BC
CoP (ΨABC)=max{Ξi
A}iPi
BCCrρi
BCwith ρi
BC =TrAΞi
A⊗IBCρABC /Pi
BC,and
Pi
BC =TrΞi
A⊗IBCρABC .ThesecondprocessCA|B
CoP (ΨABC )=maxΓj
AjPj
BCrρj
Bwith
ρj
B=TrAC Γj
A⊗IBCρABC /Pj
Band P j
B=TrΓj
A⊗IBCρABC .Thethirdprocess
CA|C
CoP (ΨABC)=max{Θk
A}kPk
CCrρk
Cwith ρk
C=TrABΘj
A⊗IBCρABC /Pk
Cand Pk
C=TrΘk
A⊗IBCρABC .
Obviously, when the state is in a product form, i.e., |ΨABC=|ΨAB ⊗|ΨC,|ΨABC =|ΨAC ⊗|ΨB,or
|ΨABC=|ΨA⊗|ΨB⊗|ΨCthe equality holds.
Note that there are actually three optimization processes in the inequality of equation (10), which are realized
by choosing proper projective measurements Ξi
A,Γj
A,andΘk
Ato achieve the maximal values of C A|BC
CoP ,CA|B
CoP ,and
CA|C
CoP , respectively. The proof of theorem 2is shown in appendix C. The theorem reveals a distribution formula of
the coordinately assisted coherence distillation in a tripartite system. The monogamy-like inequality in
equation (10) implies that the process of distillating coherence on the combined subsystem BC with assistant A
cannot always be divided into two independent subprocesses, i.e., distillating coherence in each subsystem B and
C with assistant A. In other words, it points out that the assisted distillable coherence available to each subsystem
B (or C) should be constrained by that available to the remaining combined system BC.
When considering the multipartite case of N >3, for a pure state satisfying the condition by extending
equation (8) to the multipartite cases, also with dim(HA)=2, then the following inequality holds:
CA|B1B2···BN
CoP ρAB1B2···BN
N

α=1
CA|Bα
CoP ρAB1B2···BN, (11)
where each CA|Bα
CoP is obtained by performing the corresponding optimal measurement Ξα
A,opt on system A. The
inequality reveals that the assisted distillable coherence of each subsystems should be constrained by that of the
remaining combined part.
Theorem 3. For a general state ρAB1B2···BN(either pure or mixed), the following inequality holds:
CA|B1···BN
CoP (ρAB1···BN)max
{Ξi
A}
i
PiN

α=1
Cr(ρi
Bα).(12)
Note that the inequality above describes a different assist process from that in theorem 2. Here, Alice only
performs the optimal measurement Ξi
A,opt once to achieve the maximal average of the sum of the distillable
coherence of the residual states corresponding to each subsystem Bα. Obviously, when the state is in a product
form, e.g., ρAB1...BN=ρAB1⊗ρB2...⊗ρBN(i.e., at most a pair of subsystems are related) the equality holds.
ThedetailedproofcanbefoundinappendixD.Withthehelpofthemeasureinequation(5)andthe
monogamy-like inequalities in theorems 2and 3, we can experimentally test the distribution relationship of the
assisted distillable coherence based on a linear optical setup.
It is known that, the assisted coherence distillation is closely related to the quantum correlation. For a
bipartite system AB, it is easy to find that if they share a product state, the coherence resource of Bob will
not be changed, no matter what operations are performed on Alice, i.e., the assist system Aprovides no
effect. However, if there exists a proper quantum correlation between the subsystems, the operations on
Alice will induce an increase of the distillable coherence averagely available to Bob. It can be understood as a
nonlocal advantage of quantum coherence, which has been attributed to the quantum correlations, such as
quantum steering process [28,38], Bell nonlocality [39], and quantum deficit-like correlations [40].
In the following section, we first verify the distribution inequality of the assisted distillable coherence
numerically and experimentally. Then we intend to study the relation between the multipartite correlation
and the assisted coherence distillation. As is known, the monogamy relation [41–43] was discovered as a
fundamental perspective for revealing multipartite correlations. It directly motivates us to derive the
monogamy relation in the assisted coherence distillation as in theorem 2.
We define the distribution core τA≡CA|BC
CoP −CA|B
CoP −CA|C
CoP, and the symmetrized form τ≡min
(τA,τB,τC)withτBand τCcorresponding to the cases that B,Cacts as the auxiliary system, respectively.
For the cases of pure state, when the state is in a completely product form |ΨA⊗|ΨB⊗|ΨC, or partially
product state with bipartite correlations, i.e., |ΨAB⊗|ΨC,|ΨAC ⊗|ΨB,|ΨA⊗|ΨBC,thereisτ=0.
That is τ=0 implies that the state cannot be written as the product forms, which is equivalent to
indicating the existence of multipartite correlations. For the cases τ>0, different values of τreflect
different monogamy degrees of the assisted distillable coherence, and which will be found to have
relationship with the multipartite correlation. In the following section, several typical measures of
multipartite correlations are compared with the results of τby numerical calculations.
5

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
Figure 1. Experimental setups and the stages of the experimental implementation. (a) The photon pairs with wavelength 810 nm
were prepared by spontaneous parametric down-conversion of the 1.5 cm-long type-II PPKTP nonlinear crystal. The dichroic
mirror is to reflect photons of 405 nm and transmit the photons of 810 nm. (b) Experiment setups for preparing the state with
tripartite quantum correlation. One photon is sent to the upper experiment installationto act as the auxiliary system A.The
other one goes into the lower experimental setting where its polarization modes interact with the spatial modes. The partial CN
gate and conditional BF quantum channels are experimentally realized. Different angles of the half-wave plates (HWP) H1are
adjusted to simulate the superposition coefficient pin the tripartite state in equations (15)and(16). The angle of H0is set to
zero. The angle of H2is set to zero for simulating BF channel, while set to π/4 for CN channel. When the angle is set to π/4, the
HWP can perform the inversions between the polarization modes |H→|Vand |V→|H(in the text the horizontal (|H)
and vertical (|V) modes are denoted as |0and |1for simplicity). (c) The angles of H3and H4are set to π/4, together with the
BD between them, to realize an anti-BD, which has opposite effects of the ordinary BD, i.e., it transmits horizontally polarized
photons and reflects the vertical ones. (d1) Setup for the projective measurements performed on the subsystem A.The
quarter-wave plates, HWP, and polarizing beam splitters are employed to realize the measurement basis. (d2) Tomography
measurements on the polarization modes of the second photon and the coupled spatial modes. The residual densities can be
constructed based on the measurement probability of subsystem A. The other devices are interference filters.
3. Experimental demonstration distribution of coordinately assisted distillable
coherence
In order to prepare entangled photon pairs, the 405 nm pump laser (3 mW) outputs from the continuous
laser. The 810 nm photon pairs are generated by spontaneous parametric down conversion of the
1.5 cm-long type-II periodically poled potassium titanyl phosphate (PPKTP) nonlinear crystal in Sagnac
loop (shown in the module (a) in figure 1). The entangled state is encoded in the polarization modes, and
thus the two-qubit space is spanned by the basis vectors {|iA|jB}with i,j=0, 1. We obtain 45 000 s
entangled photon pairs with the concurrence being 0.982, and the fidelity to the maximally entangled pure
state |11AB +|00AB/√2, reaching 99.8%. In the module (b) of figure 1, by using the beam displacer
(BD), the polarization modes of photon B(|jB) is coupled to the spatial modes (|kCwith k=0, 1). Based
on the polarization–spatial interactions, we prepare the tripartite states [36]. Moreover, in this work, two
types of quantum channels are constructed to realize the polarization–spatial interactions, one is the partial
controlled-Not (CN) gate channel corresponding to the following map [37]:
|0B|0C→|0B|0C,
|1B|0C→1−p|1B|0C+√p|1B|1C(13)
and the other is the conditional bit-flip (BF) channel [37]:
|0B|0C→|0B|0C,
|1B|0C→1−p|1B|0C+√p|0B|1C.(14)
With the help of the two channels above, we prepare two types of three-qubit entangled states [36]. For the
initial state 1
√3|10AB +√2|01AB|0C, the CN channel produces the W-type state
|φ=1
√3|100+2
31−p|010+√p|001.(15)
6

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
Figure 2. Experimental and theoretical results of the distribution of the coordinately assisted distillation of quantum coherence
in the tripartite system. Three measures of multipartite correlations are considered. The blue solid line is the theoretical curve of
the distribution core τA≡CA|BC
CoP −CA|B
CoP −CA|C
CoP of the assisted coherence distillation defined from the inequality in
equation (10). The red triangle-errorbar denotes the experimental result of τA. The black dashed line displays the genuine
quantum correlation based on the multipartite discord D(3) [44]. The orange dot-dashed line describes multipartite
entanglement indicator based on the monogamy relation of the squared entanglement of formation ΔSEF,[43]. The green
dot-solid line denotes the three-tangle [36].
For p=1/2, the state becomes the Wstate. The subscripts A,B,Care omitted for simplicity. In the
experiment, the parameter pcan be simulated by the rotation angle θof HWP1with the relation
p=sin2(2θ).
For the initial state 1
√2|00AB +|11AB|0C, the BF channel produces the GHZ-type state
|φ=1
√2(|000+1−p|110+√p|111), (16)
which becomes the GHZ state for p=1. In the following section, we experimentally test the measure to
evaluate the assisted distillable coherence and verify the inequalities (10) based on the prepared tripartite
entangled states.
Experimental results. In the experiment, we perform optimal projective measurements on Alice to obtain
the assisted distillable coherence CA|BC
CoP in equation (5). One can find that the optimal measurement basis
should be |0±|1/√2, which is due to that both GHZ-type and W-type states satisfy the Schmidt
decomposition in equation (8). The tomography is performed on the residual state ρ±
BC (corresponding to
the probability of the measurements on Alice) to evaluate the assisted distillable coherence of BC.
To o b t a i n CA|B
CoP and CA|C
CoP , one should take into account the reduced density ρAB and ρAC .Inordertofind
the optimal measurement on Alice, we introduce a general set of projective measurement basis denoted by
cos θ|0±sin θeiϕ|1.First,letusstudytheW-type state, since the expression of the relative entropy is
complicate, another measure, i.e., l1norm of coherence [13], is employed. By numerically calculation, we
find the behavior of the l1norm of coherence is similar with the relative entropy of coherence in W-type
state. One can easily obtain the average l1norm of coherence of the subsystems Band Cand which is found
to be proportional to √1−psin θcos θ.Obviously,themeasurementofθ=π/4 (i.e., the measurement
basis |0±|1/√2) is optimal to help the system B(C) to capture the maximal average coherence. For
the GHZ-type state, the relative entropy of coherence is easy to obtained (details are shown in the
appendix). We find that the optimal measurement basis of Aare |0±|1/√2toobtainCA|B
CoP,while
|0,|1to obtain CA|C
CoP .
In figure 2(a), we prepare the W-type tripartite state, and perform the optimal measurement on photon
A. Then the residual states of the subsystem BC,B,andCcan be detected by tomography. Furthermore, one
obtains the coherence of the residual states, and thus the assisted distillable coherence, i.e., CA|BC
CoP ,CA|B
CoP,and
CA|C
CoP . For experimental simplicity, we only investigate τA≡CA|BC
CoP −CA|B
CoP −CA|C
CoP and show its theoretical
and experimental results versus the superposition parameter pin figure 2(a). One can find that τA0in
the whole parameter region, which verifies the inequality (10). Moreover, τAreaches its maximum at
7

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
Figure 3. (a) Numerical results of τand the measures of the multipartite correlation versus the superposition coefficient pin the
state |ψ1. The red solid line displays τ, the green line with circles denotes the measure ΔSEF, the blue line with triangles displays
the three-tangle, and the black dashed line describes the measure D(3). (b) Numerical results of the quantities mentioned above
in the state |ψ2.
p=1/2, where the tripartite state becomes the Wstate, i.e., |φW=|100+|010+|001/√3. While,
τAreaches zero at p=0 and 1, where the tripartite quantum correlations degenerate into the bipartite
correlations figure 2(a).
It is known that W-type state is rich in genuine tripartite quantum correlations [43,44]. We believe that
the values of τA, which reflects the degrees of the monogamous distribution of the assisted distillable
coherence, are in close relationship with the genuine quantum correlations. We numerically calculate the
genuine tripartite quantum entanglement ΔSEF [43] and the genuine tripartite quantum discord D(3) [44]
(the definitions can be found in the appendix). One can find that τA,ΔSEF ,andD(3) reach the zero values
for the same parameters p=0, 1, and reach their respective maximal values at the same position of
p=1/2. The increase (decrease) of τAis synchronized with the increase (decrease) of ΔSEF and D(3) .We
also consider another well-known measure, i.e., the three-tangle [36], which is found to be always zero in
the considered region of p. It implies that nonzero τAshould be connected with the multipartite correlation
that cannot be detected by three-tangle but can be characterized by ΔSEF and D(3).
In figure 2(b), the case of GHZ-type states is studied. One can see that the four quantities τA,ΔSEF,D(3),
and three-tangle all increase monotonously as pincreases, which is different from that in the case of W-type
states. More specially, τAand D(3) are completely coincident. The zero values of the four quantities are
found at p=0, where the genuine tripartite correlation disappears, instead, only bipartite correlation exists.
While, at p=1, the state becomes GHZ state, which displays the maximal genuine tripartite correlation.
Other types of states with multipartite correlations are also considered and some of the numerical
results are shown in the appendix. All the results show the nearly synchronous increase and decrease of the
quantity τand the multipartite correlation measure.
4. Conclusion
We have considered the issue of assisted coherence distillation in the asymptotic limit. Different types of
measurements on the auxiliary system were discussed. Then, we focused on coordinately performing
identical projective measurements on the auxiliary of each resource state copy. In this coordinate asymptotic
scenario, the coherence distillation process branches into many subprocesses, each of which has a
corresponding distillation rate. Finally, a simple measure of the assisted distillable coherence is obtained as
the maximal average coherence of the residual states with the maximum being taken over all the projective
measurements on the auxiliary. More importantly, the measure is applicable for the cases that the
considered system and its auxiliary are in a composite mixed state. In addition, it is convenient to be
experimentally tested in one-copy state to evaluate the assisted distillable coherence available in the
coordinate asymptotic scenario.
We for the first time investigated the assisted coherence distillation in multipartite systems.
Monogamy-like inequalities were given to constrain the distribution of the assisted distillable coherence in
the subsystems. We experimentally prepared two types of tripartite correlated states, i.e., the W-type and
8

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
GHZ-type states, and experimentally test the measure derived by us to evaluate the assisted distillable
coherence. Theoretical and experimental results agree well to verify the distribution inequality. We find that
the monogamy-like law of assisted coherence distillation can be used to detect the existence of multipartite
correlations. Three measures of multipartite correlation were considered. The numerical results reveal the
close relationship between the assisted coherence distillation and the multipartite quantum correlations.
Acknowledgments
The work is supported by the National Natural Science Foundation of China (NSFC) (11775065, 12175052,
11875120, 62105086); the NKRDP of China (Grant No. 2016YFA0301802).
Data availability statement
The data generated and/or analysed during the current study are not publicly available for legal/ethical
reasons but are available from the corresponding author on reasonable request.
Appendix A. Proof of lemma 1
In this appendix, we will show the proof of lemma 1. Based on our proposed CoLQICC, after coordinately
performing the projective measurements on the ncopiesoftheresourcestate,weobtainamixedformof
Alice’s post-measurement state and Bob’s residual state:
ΛA:CoPρ⊗n
AB =
i
Pi(Ξi
A⊗ρi
B)⊗n, (17)
with Pi=TrΞi
A⊗IBρAB,andρi
B=TrAΞi
A⊗IBρAB/Pi. Recalling the map ΛCoP ≡ΛIO
B◦ΛA:CoP and
substituting the equation above into the definition of the coordinately assisted coherence distillation,
we have:
CA|B
CoP(ρ)=sup{R :lim
n→∞ inf
ΛCoPΛCoP (ρ⊗n
AB )−Φ⊗nR
21=0}
=max
Ξi
A
supR:lim
n→∞ inf
IOB
i
PiΞi
A⊗n⊗ΛIO
Bρi
B⊗n−Φ⊗nR
21=0.(18)
We find that after the projective measurements, IOs are only performed on Bob’s side to finally realize
the goal of the coherence distillation. Therefore, focusing on the core part, the trace norm actually becomes:
DΞi
Aρi
B≡

i
Pi(Ξi
A)⊗n⊗ΛIO
B(ρi
B)⊗n−
i
Pi(Ξi
A)⊗n⊗Φ2⊗nRi1

i
PiΞi
A⊗n⊗ΛIO
B(ρi
B)⊗n−Ξi
A⊗n⊗Φ2⊗nRi1
=
i
PiΛIO
B(ρi
B)⊗n−(Φ2)⊗nRi1, (19)
where the first inequality is due to the convexity of trace norm, and the second equality comes from the fact
Ξ⊗M1=M1for a Hermitian matrix Mand a matrix Ξof rank 1. Now recalling the original concept
of coherence distillation in [20], when n→∞, proper IOs on Bob can be found to make (ρi
B)⊗napproach
(Φ2)⊗nRiasymptotically, i.e., existing an arbitrarily small εi→0 that the trace norm satisfies
inf
IOBΛIO
B(ρi
B)⊗n−(Φ2)⊗nRi1εi.(20)
Then one has
lim
n→∞ inf
IOB
DΞi
Aρi
Bε≡lim
n→∞
i
Piεi→0, (21)
which implies that the process of the coordinately assisted coherence distillation, i.e., the asymptotic
incoherent transformation ρ⊗n
AB
CoLPICC
→ 1−ε
≈Φ⊗nR
2is achievable as n→∞,ε→0. Subsequently, the rate of
coherence distillation in this assisted scenario is a probabilistic sum of all the parts: R=iPiRi,whose
9

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
maximum is taken over all the projective measurements {Ξi
A}. Finally, the rate of coordinately assisted
distillation of coherence becomes R=maxΞi
AiPiRi.
Appendix B. Proof of theorem 1
First let us prove the upper bound of the rate of the coordinately assisted coherence distillation, i.e.,
Rmax{Ξi
A}iPiCr(ρi
B).
Due to the continuity of the entropy, for two states ρAB,σAB, supposing the trace norm satisfies
ρAB −σAB1ε(with 0 ε1/2), the QI relative entropy (i.e., the relative entropy between a state and
a QI state) is proved to be continuous [24], i.e.,
|CA|B
r(ρAB)−CA|B
r(σAB)|εlog2dAB +2h(ε/2), (22)
where the QI relative entropy CA|B
r(ρAB)≡S(ΔBρAB )−S(ρAB )withS(ρ) being the von Neumann entropy,
and the function h(x)≡−xlog2(x)−(1 −x)log2(1 −x), and dAB is the dimension of the Hilbert space. For
a projective measurement Ξi
Aon Alice, when we have the trace
norm iPiΛIO
B(Ξi
A⊗ρi
B)⊗n−iPiΞi
A⊗n⊗Φ2⊗nRi1εat the limit of n→∞and taking the infimum
over ΛIO
B,onecanobtaintheasymptoticcontinuity
CA|B
r
i
PiΛIO
B(Ξi
A⊗ρi
B)⊗nCA|B
r
i
PiΞi
A⊗n⊗Φ2⊗nRi−f(ε), (23)
where f(ε)≡nεlog2dAB +2h(ε/2). The right-hand side (rhs) of the inequality
RHS =S
i
PiΞi
A⊗n⊗ΔΦ2⊗nRi−S
i
PiΞi
A⊗n⊗Φ2⊗nRi−f(ε)
=
i
PiS(Φ⊗nRi
2)+H{Pi}−
i
PiS(Φ⊗nRi
2)−H{Pi}−f(ε)
=
i
nPiRiS(Φ2)−f(ε)
=n
i
PiRi−1
nf(ε), (24)
In the second equality, we make use of the property of von Neumann entropy, i.e., Sipi|ii|⊗ρi=
H(pi)+ipiS(ρi)whereH(pi) is the Shannon entropy and |iare orthogonal states. When n→∞,thereis
ε→0. In addition, since the relative entropy cannot be increased by the IOs, one has
CA|B
r
i
Pi(Ξi
A⊗ρi
B)⊗nCA|B
r
i
PiΛIO
B(Ξi
A⊗ρi
B)⊗nn
i
PiRi.(25)
Based on the definition of the relative entropy in terms of entropy, one can easily have
CA|B
r[iPi(Ξi
A⊗ρi
B)⊗n]=niPiCrρi
B. Thus, the upper bound is given in the inequality
R=
i
PiRi
i
PiCr(ρi
B).(26)
Then one can obtain the maximum of Rby taking all the projective measurement {Ξi
A}, i.e.,
R=max{Ξi
A}iPiRimax{Ξi
A}iPiCr(ρi
B).
Now, we should prove that the upper bound of the distillation rate can be achieved. The typicality
technique will be employed to analysize the asymptotic limit case [20,45]. Let us start from the purification
of ρAB, i.e., ρAB =⇒
purification|ΨABE ΨABE|. We suppose that the optimal projective measurement performed on
system Ais {Πν
A}, and Alice sends the outcomes to Bob by a classical way. Then the post-measurement state,
corresponding to the projector Πν
A≡|Πν
AΠν
A|, becomes proportional to |Πν
A|ψν
BEwith the probability
Pν=TrΠν
A⊗IBEρABE .Then,tothencopies of |ΨABE , after coordinately and independently performing
the projective measurement Πν
A⊗n, the post-measurement state will be proportional to
|Φν
ABE⊗n∼|Πν
A⊗n|ψν
BE⊗n.(27)
10

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
Then, if we implement the type measurement MPon the subsystem B, i.e.,
MP=
in∈TB,ν
n(P)|in
νin
ν|, (28)
where |in
ν≡|i1,i2,...,inνdescribes the typical state sequence corresponding to the space of the
post-measurement states. Each group {|in}, corresponding to the nth copy, can be the reference basis on
which the quantum coherence is defined. The type measurement MP, consisting of the projectors, can help
us to choose all the typical sequences corresponding to the probability distribution P,whichderivesfrom
the considered state. Thus, Pis used to represent the type of strings |inwith length n.TB,ν
n(P)denotes the
type class of Pcorresponding to the measurement Πν
A,thenδ-typical (δ>0) class satisfies
c=|in
ν:−1
nlog pin
ν−H(P)
<δ
, (29)
where the probability sequence pin
ν=pν
i1pν
i2...pν
in, with the definition pi=i|ρν
B|ifor each set of basis {in}
and ρν
Bbeing the reduced density matrix of system Bafter the measurement Πν
A. The Shannon entropy
H(P)=−ipilog pi. The length of the δ-typical class |TB,ν
n(P)|should be
2n(S(Δρν
B)−δ)TB,ν
n(P)2n(S(Δρν
B)+δ), (30)
i.e., |TB,ν
n(P)|indicates the number of the typical sequences, and Δ(ρν
B)=i|ii|ρν
B|ii|with |ibeing the
reference basis vector in each copy, on which the quantum coherence is defined. Then the dimension of the
typical space holds
2n(S(Δρν
B)−δ)dimTB,ν
n(P)2n(S(Δρν
B)+δ).(31)
After the measurement Πν
Aand the type measurement, the state of Band Ecan be expressed as
|Φν
BE⊗n
Tn=1
TB,ν
n(P)
in∈TB,ν
n(P)|in
νϕin
E,ν.(32)
Due to the typical subspace theorem [45]andthefact|TB,ν
n(P)|=|Fν|·|Mν|, there is a partition of the type
class TB,ν
n(P)into subsets {f}with the length |Fν|, and the total number of the subsets is |Mν|,andeachof
the subset can be denoted by the vectors of {m}.
Since the property of the entropy
SΔρν
B=SΔB|ψν
BEψν
BE|
=IB:EΔB|ψν
BEψν
BE|+SB|EΔB|ψν
BEψν
BE|, (33)
where the post-measurement state |ψν
BEcomes from equation (27), and IB:Edenotes the mutual
information and SB|Eis the conditional entropy. Then the length of the set {m}is |Mν|≈2nIB:E.
By using the Schmidt decomposition form of |ψν
BE,onecansimplyobtain
ΔB|ψν
BEψν
BE|=
i
qν
i|iBi|⊗|ϕν
iEϕν
i|, (34)
where qν
i=kλν
k2|i|φk|2with λν
kbeing the Schmidt coefficient and |φkis the Schmidt basis of B.Then
the mutual information
IB:EΔB|ψν
BEψν
BE|=S
i
qν
i|ii|+S
i
qν
i|ϕν
iϕν
i|−S
i
qν
i|ii|⊗|ϕν
iϕν
i|
=Sρν
E=Sρν
B.(35)
Thus, taking δ→0 for simplicity, we have
|Fν|=TB,ν
n(P)/|Mν|
=2n[S(Δρν
B)−IB:E]
=2n[S(Δρν
B)−S(ρν
B)].(36)
11

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
Let us relabel (in)→f,m, then the post-measurement state can be expressed as:
|Φν
ABE⊗n
Tn=√Pν|Πν
A⊗n1
|Fν|·|Mν|
f∈Fν,m∈Mν|f|m⊗ϕfm
E,v
=√Pν|Πν
A⊗n1
|Fν|
f∈Fν|f1
|Mν|
m∈Mν|mϕfm
E,v, (37)
where Pν=TrΠν
A⊗IBCρABC .Whenwedefine|φf
ν≡1
√|Mν|m∈Mν|mϕfm
E,v, based on Uhlmann’s
theorem [46,47], there exists a unitary Uf
νon Esuch that Im⊗Uf
ν|φf
ν≈|φ0
νfor each state |φf
ν.It
implies that we can construct the incoherence operation described by a group of Kraus operators {Kν
r},
satisfying rKν
r†Kν
r=I,oneachensemblePν,|Φν
ABE⊗n
Tn[20], i.e.,
Kν
r=I⊗n
A⊗
f∈Fν|ff|⊗|0r|Uf
ν.(38)
We ob t ai n
Kν
r|Φν
ABE⊗n
Tn≈√Pν|Πν
A⊗n1
|Fν|
f∈Fν|f⊗|0r|φ0
ν.(39)
Now we approximately obtain the maximal coherent state |Φν
B|Fν|=1
√|Fν|f∈Fν|f.Withn→∞and the
majorization condition ΔΦν
B|Fν|≺ΔΦ⊗nR
2[20], where Φν
B|Fν|≡|Φν
B|Fν|Φν
B|,andΦ2being the
density of the two-dimensional maximal coherent state, one has Φν
B|Fν|
IO
−→ Φ⊗nR
2. There should be an
equality of the length of the typical sequences, i.e.,
|Fν|=2n[S(Δρν
B)−S(ρν
B)] =2nRνS(ΔΦ2), (40)
and thus
Rν=SΔρν
B−Sρν
B=Crρν
B, (41)
which means that with the assistance of the optimal coordinate measurement Πν
Awe can distillate Φ2by
rate Rν=Crρν
Bat the asymptotic limit. Finally, we have the total distillation rate
R=νPνCrρν
B=max{Ξi
A}iPiCrρi
B. The proof is completed.
Appendix C. Proof of theorem 2
Recalling the theorem 2, i.e.,
CA|BC
CoP (ΨABC)CA|B
CoP(ΨABC )+CA|C
CoP(ΨABC ).(42)
Let us give the detailed proof by defining the core: τA≡CA|BC
CoP (ΨABC)−CA|B
CoP(ΨABC )−CA|C
CoP(ΨABC ).Fora
general bipartite state, we have CA|B
CoP(ρAB )CA|B
d(ρAB)CA|B
r(ρAB), and for the type of states in
equation (8), we have CA|BC
CoP (ΨABC)=S(ΔρBC ), then there is an inequality:
τAS(ΔρBC)−CA|B
r(ρAB)−CA|C
r(ρAC ).(43)
It is known that relative entropy will increase by adding a subsystem [48], i.e., CAB|C
r(ΨABC)CA|C
r(ρAC ),
thus
τAS(ΔρBC)−CA|B
r(ρAB)−CAB|C
r(ΨABC).(44)
By using the conditional entropy SC|AB,wehaveCA|B
r(ρAB)+CAB|C
r(ΨABC)=SC|AB (CΨABC )+S(BρAB ).
Since relative entropy cannot be increased by performing CPTP operations, thus we have
S(CρABCρAB ⊗
CρC)S(BCρABC
BρAB ⊗
CρC). By expanding the relative entropy, one will
obtain the relationship between the conditional entropy, i.e., SC|AB(BCρABC )SC|AB(CρABC). Then the
rhs of the inequality (44) holds:
12

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
RHS S(ΔρBC)−SC|AB (BC ΨABC )−S(BρAB )=0, (45)
which is due to
SC|AB(BC ΨABC )+S(BρAB )=S(BCΨABC)−S(BρAB )+S(BρAB )
=S(BCΨABC )=S(ΔρBC ).(46)
Finally, we have
τA0.(47)
Now let us extend the proof to the multipartite cases of N>3. For a pure state |ΨAB1...BN,withthe
dimension of the auxiliary is dim(HA)=2, and its Schmidt decomposition can be presented as:
|ΨAB1...BN=λ1|φ1
A
{i
B}|i
B1...i
BN+λ2|φ2
A|iB1...iBN, (48)
where {i
B}denotes the subset consisting of the reference basis different from |iB1...iBN,andthetwostates
of auxiliary satisfy φ2
A|φ1
A=0, then we have CA|B1...BN
CoP (ΨAB1...BN)=SΔB1...BNΨAB1...BNwith the
definition ΨAB1...BN≡|ΨAB1...BNΨAB1...BN|. By using the tripartite inequality (44)and(45), we have
CA|B1...BN
CoP (ΨAB1...BN)CA|B1
r(ρAB1)+CAB1|B2...BN
r(ΨAB1...BN), (49)
where CAB1|B2...BN
r(ΨAB1...BN)=SΔB2...BNΨAB1...BN. Then the tripartite inequality is reused that
CAB1|B2...BN
r(ΨAB1...BN)CAB1|B2
r(ρAB1B2)+CAB1B2|B3...BN
r(ΨAB1...BN)
CA|B2
r(ρAB1B2)+CAB1B2|B3...BN
r(ΨAB1...BN), (50)
where we use the property of the relative entropy CAB1|B2
r(ρAB1B2)CA|B2
r(ρAB1B2). Then
CAB1B2|B3...BN
r(ΨAB1...BN)CAB1B2|B3
r(ρAB1B3)+CAB1B2B3|B4...BN
r(ΨAB1...BN)
CA|B3
r(ρAB1B2)+CAB1B2B3|B4...BN
r(ΨAB1...BN).(51)
By repeatedly using the inequalities above, one will finally have
CA|B1...BN
CoP (ΨAB1...BN)
N

α=1
CA|Bα
rρAB1B2···BN

N

α=1
CA|Bα
CoP ρAB1B2···BN.(52)
Then the proof is completed.
Appendix D. Proof of theorem 3
For a multipartite state ρAB1B2···BN, a set of projective measurements {Ξi
A}are performed on the subsystem A
and classical communications are allowed among the subsystems, then the residual states of each subsystem
Bjis ρi
Bj. Assuming that an optimal set of operations {˜
Ξi
A}help us to achieve the maximal average of the
coherence, i.e., i˜
PiN
j=1Cr˜ρi
Bj . Because of the supper additivity of coherence relative entropy, i.e.,
Cr(ρ1)+Cr(ρ2)Cr(ρ12), where the reduced density ρ1(2)=Tr 2(1)(ρ12 ).Onehas

i
˜
Pi⎡
⎣
N

j=1
Cr˜ρi
Bj⎤
⎦
i
˜
PiCr˜ρi
B1B2···BN.(53)
Obviously, {˜
Ξi
A}is the optimal measurement to obtain the maximum of the subsystem coherence
N
j=1Crρi
Bjand not the coherence of the composite system Crρi
B1B2···BN, thus when we take into account
all the projective measurements {Ξi
A}, there is the following inequality, i.e.,
13

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al

i
˜
PiCr˜ρi
B1B2···BNmax
{Ξi
A}
i
PiCrρi
B1B2···BN
=CA|B1B2···BN
CoP (ρAB1B2···BN).(54)
The proof is completed.
Appendix E. Selection of optimal measurement
In this appendix, we show the details of how to choose the optimal measurement performed on subsystem
A.ThecasesofGHZ-typeandW-type states are considered. To obtain the assisted coherence distillation
CA|BC
CoP , the optimal measurement basis should be |0±|1/√2, which is due to that both GHZ-type and
W-type states satisfy the Schmidt decomposition in equation (8). While, to obtain CA|B
CoP and CA|C
CoP,one
should take into account the reduced density ρAB and ρAC.
First, let us consider the case of W-type state. For the reduced state ρAB, we perform a general projective
measurement, with the basis |ϕ+=cos θ|0+sin θeiϕ|1and |ϕ−=sin θ|0−cos θeiϕ|1, on subsystem
A. Then the corresponding probability are P+=1+cos2θ/3andP−=1+sin2θ/3, and the residual
state of system Bis (the classical communications between Aand Bare followed):
ρ+,B=3
1+cos2θ%2cos
2θ
3p+sin2θ
3&|00|+2
3(1 −p)|11|
+sin θcos θe−iϕ√2
31−p|01|+sin θcos θeiϕ√2
31−p|10|.(55)
Then, we make use of l1norm (defined as Cl1=i=jρi,j,withρi,jbeing the off-diagonal elements) to
measure the quantum coherence. Through numerical calculation, we find that the behavior of the l1
norm of coherence is similar with the relative entropy of coherence, and the former is easy to calculate. For
the residual state ρ+,Band ρ−,B,wehavethat
Cl1(ρ+,B)∼1−p|sin θcos θ|
1+cos2θ, (56)
and which is same with Cl1(ρ−,B). Obviously, the average l1norm of coherence Cl1=i=+,−PiCl1(ρi,B)is
proportional to |sin θcos θ|, which means that Cl1reaches its maximal value at θ=π/4, i.e., the optimal
measurement basis on Ashould be |0±|1/√2.ThesameistrueforρAC .
To the case of the GHZ-type state, we first consider the reduced state ρAC also by introducing a general
form of the measurement basis |ϕ+=cos θ|0+sin θeiϕ|1and |ϕ−=sin θ|0−cos θeiϕ|1.Itiseasyto
obtain that P+=P−=1/2. After measurement, the residual states of Care ρ+,Cand ρ−,C:
ρ+,C=1−psin2θp(1 −p)sin2θ
p(1 −p)sin2θpsin2θ, (57)
ρ−,C=%1−pcos2θp(1 −p)cos2θ
p(1 −p)cos2θpcos2θ&.(58)
Then the eigenvalues of the two density are 1±1−p2sin22θ/2, and finally we have the average
relative entropy of coherence, i.e., the assisted distillable coherence of ρAC :
2CA|C
CoP(ρAC )=max
θF(p,θ), (59)
where
F(p,θ)=H1−psin2θ,psin2θ+H1−pcos2θ,pcos2θ−2H1
2%1±1−p2sin22θ&, (60)
with H{A,B}≡−(Alog A+Blog B) is the binary Shannon entropy. By calculating the first and second
order derivative of F(p,θ)withrespecttoθ, one can find that the minimum of F(p,θ)isatθ=π/4, while
themaximumcanbereachedatθ=0orπ, which implies that the best measurement basis are {|0,|1},
then CA|C
CoP(ρAC )=1
2H{1−p,p}.
14

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
By doing a similar analysis to ρAB, the maximal value of CA|B
CoP(ρAB) can be reached at θ=π
4. Then the
optimal measurement basis are |0+|1/√2, |0−|1/√2.
Appendix F. Measures of genuine tripartite quantum correlation ΔSEF,andD(3)
In this appendix we introduce the concept of two types of genuine tripartite quantum correlation. The first
one is based on the squared entanglement of formation, i.e., [43]
ΔSEF(ρABC )=E2
f(ρA|BC)−E2
f(ρA|B)−E2
f(ρA|C),
which detects that the multipartite entanglement is not stored in pairs of qubits. Ef(ρi|j) is the entanglement
offormationinthesubsystemρij with the definition Ef(ρi|j)=minpm,|φij
mmpmSTri(|φij
m),wherethe
minimum is taken over all the pure state decompositions 'pm,|φij
m(. In two-qubit quantum states, the
entanglement of formation has an analytical expression Ef(ρi|j)=H1±√1−C2(ρij)
2,where
H(x)=−xlog x−(1 −x)log(1 −x) is the binary entropy and Cρijis the concurrence of ρij .Moreover,in
a tripartite pure state |ψABC,wehavetherelationE2
f(ρA|BC)=S2(ρA)inwhichEf(A|BC)isthe
entanglement of formation in the partition A|BC [43]andS(ρ) is the von Neumann entropy.
Another concept is the multipartite discord with the definition (for the tripartite case) [44]:
D(3)(ρ):=D(ρ)−D(2)(ρ), (61)
where D(3)(ρ)describes the genuine tripartite quantum correlation. Genuine correlations should contain all
the contributions that cannot be accounted for considering any of the possible subsystems.
D(ρ)≡T(ρ)−J(ρ)is called the total quantum discord with the total information (or correlation
information) T(ρ)≡S(ρρi⊗ρj⊗ρk), and the total classical correlation
J(ρ)≡maxP{i,j,k}S(ρi)−Sρi|j+S(ρk)−Sρk|ijwith the maximum among the six indices
permutations of the probability Pi,j,k=Pi|j,kPj|kPk.NotethatSρi|j≡min{Ei
l}Si|{Ej
l}with respect to the
positive operator valued measure {Ej
l}, and the average entropy Si|{Ej
l}=kpkS(ρi|Ej
m)withthe
probability pk=Tr(Ej
l⊗Iρij) and the residual density ρi|Ej
m. Extending to the tripartite case, it becomes
Sρk|ij≡min{Ei
l,Ej
l}Sk|{Ei
l,Ej
l}. The minimum bipartite discord D(2) (ρ)is defined as
D(2)(ρ)≡max[Dρi,j,Dρk,j,Dρi,k]. The symmetrized quantum discord
Dρi,j≡min[Dρi:j,Dρj:i], where Dρi:j≡I(ρi,j)−max{Ej
m}[S(ρi)−S(ρi|{Ej
m})] is the quantum
discord, and I(ρi,j)≡S(ρi,jρi⊗ρj) is the mutual information. For the pure state |φijk, if the following
inequality is satisfies: I(ρij )I(ρik )I(ρjk ), there is a simple result that D(3) (ρ)=S(ρk)[44]. Therefore,
for the GHZ-type states, it is easy to check by numerical calculation that I(ρAB )I(ρAC )I(ρBC). Then we
have D(3)
GHZ =S(ρC)=H1−p
2,p
2, and the minimum value min D(3)
GHZ =H1−p
2,p
2p=0=0, while the
maximum value max D(3)
GHZ =H1−p
2,p
2p=1=1.
As for the assisted distillable coherence, one can analytically obtain the minimum value of τAat p=0,
where the distillable coherence CA|BC
CoP (ρABC)=1, CA|C
CoP(ρABC )=1, and CA|B
CoP(ρABC )=0, then τA=0. While,
at p=1, we have CA|BC
CoP (ρABC)=1, CA|C
CoP(ρABC )=0, and CA|B
CoP(ρABC )=0, which means τA=1. In
figure 2(b), the numerical and experimental results of τAshow that in the case of the GHZ-type state, the
behaviors of τAand D(3)
GHZ are the same.
For the W-type states, the behavior of D(3) is different from the GHZ-type states that when 0 p0.5,
there are I(ρAB)I(ρBC)I(ρAC ), then the tripartite quantum discord D(3) =S(ρC). When 0.5<p1,
there are I(ρAC)I(ρBC )I(ρAB), and thus D(3) =S(ρB). Obviously, on both sides of the point p=0.5,
D(3) behaves differently. We also discuss the relation between τAand D(3).Whenp=0and1,therewillbe
D(3) =0, where one can also find τA=0. While, at the special point of p=0.5, the two quantities reach
their maximum values D(3) 0.918 and τA0.848. More clearly, one can find the numerical and
experimental results in figure 2(a), where τAand D(3) display a similar behavior except the regions near the
maximal value.
15

Quantum Sci. Technol. 7(2022) 045024 H-Q-C Wang et al
Appendix G. Numerical results of τand the multipartite correlations in other types
of three-qubit states
In this section, we numerically calculate the quantities of τ=min(τA,τB,τC)andthemeasureof
multipartite correlation, i.e., three-tangle, ΔSEF,andD(3) .Wechoosetwotripartitepurestates,whichwill
show some different phenomena from those in figure 3. The states are
|ψ1=p
3|000+p
3|001+p
3|011+1−p|111, (62)
|ψ2=p
3|00++p
3|11−+p
3|10++1−p|11−, (63)
where |± =(|0±|1)/√2. From the numerical results, we find that the minimum values of the four
quantities appear at the same position, while the locations where the maximum values appear are slightly
different. There are a roughly consistent increase and decrease process of τand the measures of the
multipartite correlation. We also check many other states, and all the results support the conclusion that
larger values of τcorrespond to stronger multipartite correlations.
ORCID iDs
Zhe Sun https://orcid.org/0000-0002-0162-4160
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