Distribution of quantum coherence in multipartite systems

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by  R. Chandrashekar

Distribution of quantum coherence in multipartite systems
Chandrashekar Radhakrishnan
New York University, 1555 Century Avenue, Pudong, Shanghai 200122, China and
NYU-ECNU Institute of Physics at NYU Shanghai,
3663 Zhongshan Road North, Shanghai 200062, China
Manikandan Parthasarathy and Segar Jambulingam
Department of Physics, Ramakrishna Mission Vivekananda College, Mylapore, Chennai 600004, India
Tim Byrnes
6 New York University, 1555 Century Avenue, Pudong, Shanghai 200122, China
1 NYU-ECNU Institute of Physics at NYU Shanghai,
0 3663 Zhongshan Road North, Shanghai 200062, China
2 Department of Physics, New York University, New York 10003, USA and
National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo 101-8430, Japan
r
p (Dated: April26, 2016)
A
Thedistributionofcoherenceinmultipartitesystemsisexamined. Weuseanewcoherencemea-
3 surewithentropicnatureandmetricproperties,basedonthequantumJensen-Shannondivergence.
2 The metric property allows for the coherence to be decomposed into various contributions, which
arise from local and intrinsic coherences. We find that there are trade-off relations between the

] various contributions of coherence, as a function of parameters of the quantum state. In bipartite
h systems the coherence resides on individual sites or distributed among the sites, which contribute
p
inacomplementaryway. Inmorecomplexsystems,thecharacteristics ofthecoherencecandisplay
-
moresubtlechangeswith respect totheparameters of thequantumstate. Inthecase of theXXZ
t
n Heisenbergmodel,thecoherencechangesfromamonogamoustoapolygamousnature. Thisallows
a usto definetheshareability of coherence, leading to monogamy relations for coherence.
u
q
PACSnumbers: 03.65.Ta,03.67.Mn
[

2
v The concept of wave particle duality introduced the (a) C (b) I(b)
6 importanceofquantumcoherenceinphysicalphenomena S
C
28 sthuechrmaosdlyonwamteimcsp[e2r–a4t]u,renatnhoesrcmaoledypnhaymsicicss[[51],],bqiuoalongtiucmal CI CL ρd CL σSmin CI ρ
0 I
systems [6, 7], and is one of the most basic aspects of
0 S
. quantum informationscience [8]. For this reason,under-
2
standing quantum coherence has a long history and is of FIG.1. Quantumcoherenceinmultipartitesystems. (a)The
0
6 fundamentalimportance to manyfields. Inquantumop- total coherence C has contributions from local coherence CL
1 tics [9, 10], the approach has been typically to examine on subsystemsand collective coherenceCI. (b)Definitionsof
various coherences according to the distance between states.
: quantities such as phase space distributions and higher
v order correlation functions [11]. While this method dis- IS is the set of separable states, while IS(b) is the set of sep-
i
X tinguishes between quantum and classical coherence, it arable states in a fixed basis b. ρd is the solution of (2) and
σSmin is the solution of (4).
r doesnotquantifycoherenceinarigoroussense. Morere-
a
cently, a procedure to quantify coherence using methods
of quantum information science was developed [12–15].
ence, (0 1 )(0 1 ) and 0 0 1 1 . In the for-
In the seminal work of Ref. [12], basic quantities such | i−| i | i−| i | i| i−| i| i
mer, the coherence lies on each qubit, while the latter
as incoherent states, incoherent operations, maximally
has a kind ofcollective coherence,i.e. entanglement. An
coherent states were defined and the set of properties a
interesting aspect of this is that the types of coherence
functional should satisfy to be consideredas a coherence
are complementary to each other – an increase in one
measure were listed.
type leads to a corresponding decrease in the other. In
One fundamental task that is desirable is to pinpoint orderto havemaximumcoherence ona particular qubit,
whatpartofaquantumsystemisresponsibleforanyco- itisoptimaltocreateasuperpositiononeachone,which
herence that is present. To understand the possibilities, excludes entanglement. On the other hand, for the Bell
let us consider a two qubit system as an example. Co- state, tracing out one of the qubits leaves a completely
herence is a basis-dependent quantity [15, 16], and the mixed (incoherent) state on the other qubit.
reference incoherent states are chosen as 0 , 1 . We can Thiscomplementarybehaviorisreminiscentofanother
| i | i
consider then two types of states which possess coher- quantumfeature,monogamyofentanglement,whichhas
2
attracted a lot of attention recently [17–21]. Monogamy where is a distance measure and (b) are the set of
D I
is a concept related to the shareability of entanglement incoherent states in a particular basis b. The functional
between different constituents in a multipartite system. isaquantumcoherencemeasureifitobeystheproper-
C
For example, in a tripartite system, if Alice and Bob ties [12]: (i) (ρ) 0 and (ρ) =0 iff ρ (b); (ii) (ρ)
C ≥ C ∈I C
have a maximally entangledstate then this rules out en- is invariant under unitary transformations; (iii) (ρ) is
C
tanglementtoCharlie. The monogamyrelationforthree monotonic under a ICPTP (incoherent completely posi-
qubits was introduced in Ref. [17], has also been gener- tive and trace preserving map); (iv) (ρ) monotonic un-
C
alized to multipartite systems [19]. Both these examples der selective incoherent measurements on average; and
illustratethetrade-offnatureofquantummechanicalfea- (v) (ρ) non-increasing under mixing of quantum states
C
tures, where increasing one imposes restrictions on the (convexity).
other. Another fundamental question which this raises Eq. (2)states thatthe amountofcoherenceina given
is the relationship between coherence and entanglement state is the distance to the closestincoherentstate. This
[15, 16]. The framework outlined in Ref. [12] closely definition clearly depends on what we deem to be an in-
followed the format of entanglement quantification de- coherentstate,andisresponsibleforthebasis-dependent
veloped in [22–24]. While entanglementis clearly a form nature of . Most generally, one may assume a form for
C
of coherence, the converse is not necessarily not true. In an incoherent state σ = p b b where the b
k k| kih k| {| ki}
thispaperweexplorethequestionofhowwecanquantify are a fixed particular basis choice b, and p are prob-
k
P
various types of coherence, and examine their trade-off abilities. Without the constraint of the fixed basis, it
relations within a multipartite system. By understand- is always possible to write σ = ρ by taking b to be
k
| i
ingthedistributionofcoherenceinamultipartitesystem, eigenvectorsofρ,whichimmediatelygives =0. Inthis
C
this leads us to find the relation between concepts such paper we are interested in how the overall coherence is
as coherence, entanglement, and monogamy. distributed in a multipartite system. For this reason it
One of the tools that we will use in this study is a co- will be most interesting to choose a local basis choice
herence measure which has both entropic and geometric
properties. In Ref. [12], two different functionals, one σ = p τ(b) τ(b) , (3)
k k,1⊗···⊗ k,N
basedonthe relativeentropyandthe otherbasedonthe k
X
ℓ -normwerefoundtosatisfythenecessarypropertiesas
1
a coherence measure. Of these, the former is anentropic where τ(b) is the incoherent state on the subsystem n
k,n
measurewhiletheotherisageometricmeasurewhichcan i.e., τ(b) = p b b . The setofstates thatare
be usedasa formaldistance measure. Any measure is k,n k k,n| k,nih k,n|
D separable and in a basis b are called (b).
consideredasaformaldistance overthe setX if ρ,σ P IS
∀ ∈ This gives a natural way to study various coherence
X itsatisfiesthe followingproperties: (i) (ρ,σ)>0for
D contributionswithinamultipartitesystem. As discussed
ρ = σ and (ρ,ρ) = 0, (ii) (ρ,σ) = (σ,ρ) (symme-
6 D D D above, we can distinguish between coherence that is lo-
try). If satisfies (iii) (ρ,σ)+ (σ,τ) (ρ,τ) (the
D D D ≥ D calizedonthesubsystemsn,andandcollectivecoherence
triangle inequality) in addition to the properties given
whichcannotbeattributedtoparticularsubsystems(see
above, then is a metric for the space X. The relative
D Fig. 1(a)). To remove the contribution from the subsys-
entropy S(ρ σ) Trρlog(ρ/σ) is not a distance since it
k ≡ tems, we may relax the basis constraint b and minimize
isasymmetricandfurtheritiswelldefinedonlywhenthe
over the set of states (3). This contribution is indepen-
supportofσ isequaltoorlargerthanthatofρ. Towards
dent of the basis choice, and is the coherence which is
this end we introduce here an alternative, the quantum
intrinsic within the system. We thus define the intrinsic
version of the Jensen-Shannon divergence (QJSD):
coherence
1
(ρ,σ)= [S(ρ (ρ+σ)/2)+S(σ (ρ+σ)/2)]. (1)
J 2 k k (ρ) min (ρ,σ ), (4)
I S
C ≡σS∈ISD
The QJSD is known to be a distance measure, be
bounded 0 1, and is well defined irrespective of where is the set of states of the form as given in (3),
≤ J ≤ IS
the nature of the support of ρ and σ [25–27]. The QJSD but is not necessarily in the basis b. Thus the only con-
does not obey the triangle inequality,but its squareroot straint here is the general form of the basis, that it is
obeys it for all pure states. In the case of mixed states separable, but the particular basis is not specified. Eq.
there is no general proof of the triangle inequality, but (4) is in fact equal to the entanglement, which is rea-
numerical studies up to five qubits [27] strongly indicate sonable from the point of view that entanglement must
its validity. contributetocoherence[15]. Theremainingcontribution
Quantum coherence trade-offs. The quantum coher- thenoriginatesfromcoherencethatexistsonthesubsys-
ence is defined as [12] tems, and we can write the local coherence as
(ρ) min (ρ,σ), (2)
C ≡σ∈I(b)D CL(ρ)≡D(σSmin,ρd), (5)
3
where σmin andρd arethe minimum solutions of(4) and 0.8 0.6
(2) respeSctively, and are implicit functions of ρ. 0.6 (a) C µ=0.99 (b)
C 0.4
We may visualize the two different contributions ac- C µ=0.5
0.4
cording to Fig. 1(b). According to the metric properties
0.2
of ,andthetriangleinequalityweimmediatelyseethat 0.2 C C
D L I µ=0.1
L+ I, (6) 0 0 0.5 1 1.5 2 00 π/4 π/2 3π/4 π
C ≤C C J/λ φ
1
For a product state σmin, the coherence measure is sub- C C
S 0.6 1:2:3 12:3 0.8 C (d)
additive which leads to . We thus have
CL ≤ nCL,n C 0.4 C2:3 C1:3 C 0.6 C1:2...N
N P 0.2 C M 0.4 C1:3 C 23 M
L,n+ I, (7) 1:2 0.2 1:2 1
C ≤nX=1C C 0 (c) 0 C1:6 0 0 4 8∆
0 π 2π -2 0 2 4 6 8 10
where is the coherence on each subsystem n sepa-
CL,n φ ∆
rately.
Anillustrativeexampleofthecoherencedecomposition
FIG. 2. Coherence as measured by the quantum Jensen-
is given by the ground state of the N = 2 Ising model Shannon divergence for various states. Coherence of (a) the
described by the Hamiltonian N =2 site Ising model with ǫ=0.2; (b) Werner GHZ state;
(c)Wstatewithθ=π/4;(d)TheN =10siteXXZ Heisen-
H =λσxσx+J(σx+σx)+ǫλ(σz +σz), (8) berg model ground state with J = 1. Inset: Monogamy for
1 2 1 2 1 2
theXXZ Heisenberg model as defined in (14).
whereJ,λcouplingparametersandǫisasmallsymmetry
breaking term. The numerically estimated values of ,
L
C states on the tripartite system. We note that as is
I,and aregiveninFig. 2(a),whereweusethe square C1:2:3
C C anintrinsic coherence,it does not containany coherence
root of the Jensen-Shannon divergence as our distance
located on the sites, but contains all coherences between
measure
the sites.
We can decompose a tripartite system in a bipartite
ρ+σ S(ρ) S(σ)
(ρ,σ)= (ρ,σ)= S , fashion, leading to the relation
D J s 2 − 2 − 2
(cid:18) (cid:19)
p + + + + , (10)
123 1 2 3 2:3 1:23
C ≤C C C C C
where S(ρ) = Trρlogρ is the von Neumann entropy.
− wherewe firstfindthe intrinsic coherencebetween2and
Taking the 0 , 1 basis as the reference state (this
n n
{| i | i } 3,thenweestimatetheintrinsiccoherencebetween1and
willbethecasethroughoutthispaper),weseethatthere
the bipartite subsystem 23. Similar decompositions can
is a crossover between coherence contributions from in-
be carried out with respect to the bipartitions 2:13 and
trinsicwhenJ λ,tolocalasJ λ. Thisisduetothe
≪ ≫ 3:12 as well. From (9) and (10) and the other possible
fact that for J = 0,ǫ 0 the ground state approaches
→ bipartitions suggested above we may deduce that
a Bell state 00 11 , and for λ = 0 the ground state
| i−| i
is (0 1 )(0 1 ), with intermediate J/λ giving an + + + . (11)
| i−| i | i−| i C1:2:3 ≃C2:3 C1:23 ≃C1:2 C12:3 ≃C1:3 C13:2
interpolation to these limits. The total coherence is less
Toillustrate the variouscontributions,firstletus con-
thanthesumofthelocalandintrinsiccontributions,fol-
sider the mixed GHZ states defined as ρ = 1−µ1ˆ+
lowing (6). GHZ 8
µ GHZ GHZ with GHZ = cosφ000 + sinφ111 ,
Multipartite coherence. The bipartite case studied | ih | | i | i | i
φ [0,2π), and 0 µ 1. The coherence is plotted in
above is the simplest case of more general trade-off re- ∈ ≤ ≤
Fig. 2(b). Forthisclassofstateswefindthatthevarious
lations in multipartite systems. One of the fundamental
contributions due to one and two sites are always zero:
properties we investigate is the shareability of coherence
= =0. This means that the only coherencecon-
betweensubsystems. Forexample, ina tripartite system Cn Cm:n
tributions originates from the intrinsic coherence where
ρ we may decompose the coherence using (6) accord-
123
all three sites are involved. The total coherence is thus
ing to
identical to the tripartite coherence = , which is
1:2:3
C C
+ + + . (9) verified numerically. It is also equal to the bipartitioned
123 1 2 3 1:2:3
C ≤C C C C intrinsic coherence = , where l,m,n are all per-
l:mn
C C
where we have introduced a shorthand for the local co- mutations of the sites. This verifies the relation (11) for
herence on subsystem n as = (ρ ), and ρ is this class of states.
n L,n n n
C C
the reduced density matrix. For the product states σmin In contrast to the GHZ state where there is only
S
the Eqn. (9) holds exactly. The intrinsic coherence one coherence contribution, the W states have a trade-
= (ρ ) is minimized over the set of separable off relation similar to that seen in the transverse Ising
1:2:3 I 123
C C
4
model. These are defined W = sinθcosφ100 + with2and3. ThisisimmediatelyevidentfromEq. (10),
| i | i
sinθsinφ010 + cosθ 001 with 0 φ < 2π and 0 wherethecoherenceisdecomposedintothesetwocontri-
| i | i ≤ ≤
θ π. The GHZ and W states are two classes of states butions. Ifsubsystem3iscoherentlyconnectedto1and2
≤
which are unrelated under local operations and classi- thenthetripartitesystemisdescribedtobepolygamous,
cal communications. From Fig. 2(c) we see that the andotherwiseismonogamous. Thecoherencemonogamy
calculated coherence can be attributed to several contri- relations may be identified from (11), where we observe
butions. Firstly the coherence is always constant that the tripartite coherence can be decomposed
12:3 1:2:3
C C
as the state for the choice θ = π/4 can be written as into several bipartite coherences. The genuine tripartite
W =[(cosφ10 +sinφ01 )0 + 00 1 ]/√2,thusthere coherence can be estimated by subtracting pairwise bi-
| i | i | i | i | i| i
is always intrinsic coherence between the bipartition of partite terms giving
sites12and3. Thecoherences and showcomple-
1:3 2:3
C C . (13)
mentarybehaviorasthe systemoscillatesbetweenaBell 1:2:3 1:2 2:3 1:3 1:23 1:2 1:3
C −C −C −C ≃C −C −C
statebetweensites13(φ=nπ)and23(φ=(n+1/2)π),
Foramultipartitesystemthemonogamyinequalityreads
with the remaining site being decoupled. There is co- N
. Thus, we define the multipartite
herencebetweenthe sites12betweenthesetwoextrema, C1:2...N ≥ n=2C1:n
monogamy of coherence with respect to a measure as:
giving with twice the oscillatory frequency. P
1:2
C
Thesameideascanbeequallyappliedtomorecomplex N
multipartite systems. The various coherence contribu- M = 1:n 1:2...N, (14)
C −C
tions can be used to understand the nature of the quan- nX=2
tum states in quantum many-body systems. We illus- which is monogamousfor M 0 due to the multipartite
≤
trate this by analyzing the one-dimensional Heisenberg coherence that is present. For M > 0 it is polygamous
XXZ model, one of the fundamental models in mag- sincethe dominantcoherenceisdistributedinapairwise
netism. The Hamiltonian of this model is fashion.
InFig. 2(c)wecalculate(14)fortheWstates. Wefind
H =J (σxσx +σyσy +∆σzσz ), (12)
n n+1 n n+1 n n+1 thatM 0forallθ,φ,hencethestateisstrictlypolyga-
n ≥
X mous. For the GHZ states as shownin Fig. 2(b) there is
where J is the nearest neighbor spin coupling and ∆ only one coherence contributionwith =0, which re-
1:n
C
is the anisotropy parameter. For an antiferromagnetic sultsinM = ,meaningthatitisstrictlymonogamous.
−C
coupling J > 0, the system has a phase transition from ThisisasexpectedsincetheGHZstatesaretripartiteen-
the ferromagnetic axial regime to the antiferromagnetic tangled,whereastheW statehasabipartitenature[28].
planar regime at ∆ = 1. Using exact diagonaliza- Forthe Heisenbergspinchainwe findbothmonogamous
−
tion techniques we estimate various types of coherence (∆ > 2.9) and polygamous behavior ( 1 < ∆ < 2.9)
−
as shown in Fig. 2(d). In the ferromagnetic phase with (see Fig. 2(d) inset). For ∆ 1 region when the
≫
∆ < 1, all coherences vanish due to spontaneous sym- ground state is a N´eel state, where the two-site coher-
−
metry breaking selecting a unique ferromagnetic ground ences vanish 0. Then the coherence is entirely
1:n
C →
state with all spins aligned in the σz basis. In the op- due to the 1:2...N bipartition, resulting in a monoga-
posite limit ∆ 1, the state is a superposition of N´eel mous state. This can be understood to be due to the
≫
states, due to the two-fold degeneracy of these states: fact that the N´eel state superposition is essentially the
(0101...01 + 1010...10 )/√2. Thecoherencethusap- same as a GHZ state up to a redefinition of state labels.
| i | i
proaches the Bell state value = 0.56, with Forsmall∆, there is alargereffect fromthe off-diagonal
1:2...N
all other coherence contributioCns vaCnishing.≈Due to the terms σxσx +σyσy = 2(σ+σ− +σ−σ+ ). This
n n+1 n n+1 n n+1 n n+1
spin flip symmetry, coherence on eachsite is always zero term tends to create coherence on nearby sites, which is
=0,and canalwaysbe written ina Bell state more characteristic of a polygamous behavior. In this
n 1:2...N
C C
form, resulting in a constant value. The coherence con- way the parameter ∆ switches the nature of the coher-
tributions between two sites decrease with distance as encebetweenmonogamyandpolygamybyredistribution
expected , due to the reduced correlations between it between relatively local sites to a genuinely multipar-
1:n
C
thesesites. Interestingly,at∆= 1thetwo-sitecorrela- tite form.
−
tions all converge to the same value, which we attribute Conclusions. Multipartite coherence is decomposed
to the fact that this is close to the antiferromagnetic- into local and intrinsic parts and quantified using a en-
ferromagnetic phase transition, which has the effect of tropic measure with metric nature. This decomposition
increasing the overall coherence in the system. into various contributions can be used not only to char-
Monogamy of coherence. From our coherence decom- acterize a given state but also to locate the origin of
positions,wearrivenaturallyatthenotionofmonogamy the coherence. In many cases there is a crossover be-
of coherence. In a tripartite system, if subsystems 2 and haviorbetween the coherences of different origins,which
3 are maximally coherent with respect each other, this depends upon the type of the state examined. In the
limits on the amount of coherence that subsystem 1 has transverseIsingmodel,thecoherencetransitionsbetween
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