Ordering states with coherence measures

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by  C. L. Liu

Ordering states with coherence measures
C. L. Liu, Xiao-Dong Yu, G. F. Xu, and D. M. Tong
∗
Department of Physics, Shandong University, Jinan 250100, China
(Dated: September20, 2016)
The quantification of quantum coherence has attracted a growing attention, and based on vari-
ous physical contexts, several coherence measures have been put forward. An interesting question
is whether these coherence measures give the same ordering when they are used to quantify the
coherence of quantum states. In this paper, we consider the two well-known coherence measures,
thel1 normofcoherenceandtherelativeentropyofcoherence,toshowthattherearethestatesfor
which thetwo measures give adifferent ordering. Ouranalysis can beextended toother coherence
measures, and as an illustration of the extension we further consider theformation of coherence to
6
1 show that the l1 norm of coherence and the formation of coherence, as well as the relative entropy
0 of coherence and thecoherence of formation, donot give thesame ordering too.
2
PACSnumbers: 03.65.Aa,03.67.Mn
p
e
S
I. INTRODUCTION well-knowncoherencemeasures,thel normofcoherence
1

9 and the relative entropy of coherence, which have been
1 widely used as a tool to investigate various aspects of
Quantumcoherenceisconsideredtobeoneofthemost
coherence, such as the freezing phenomena of quantum
important resources in quantum physics. It has many
]
coherence [17], the complementarity relations for coher-
h significantapplicationsinvarioussubjects,suchasquan-
p tum optics [1], quantum information [2], quantum biol- encemeasures[21,26,27],andthecoheringanddecoher-
- ogy[3,4],andthermodynamics[5–7]. However,ageneral ing power of quantum channels [25]. We will show that
t
n criterion to quantify the coherence of quantum states in there are the states for which the two measures do not
a give the same ordering. As an extension of our analyses,
information theoretic terms has only been proposed re-
u wewillalsobrieflydiscussthe othercoherencemeasures,
cently [8], although the topic has ever been attempted
q
[ in early papers [9–15]. The general criterion has trig- the l1 norm of coherence and the coherence of formation
gered the community’s great interest [16–30],and based as well as the relative entropy of coherence and the co-
3 on the criterion, various concrete measures of coherence herence of formation, showing that they do not give the
v same ordering too.
have been given. The l norm of coherence and the rela-
6 1
tiveentropyofcoherence,whichsatisfythegeneralcrite- Thepaperisorganizedasfollows. InSec. 2,wereview
3
9 rionofquantifyingthecoherenceofquantumstates,were some notions related to quantifying coherence and recall
3 first suggested as two coherence measures based on dis- the three coherence measures under considered, i.e., the
0 tance in Ref.[8]. The coherence measure based on skew l1 norm of coherence, the relative entropy of coherence,
1. information [16], and the coherence measure based on andthecoherenceofformation. InSec. 3,wepresentour
0 entanglement[18]werethen proposed. Moreexpressions main results, i.e., we show that there are the states for
6 were subsequently shown to fulfill the general criterion which the l1 norm of coherence and the relative entropy
1 of quantifying coherence, such as the coherence of for- ofcoherencedonotgivethe sameordering,andalsogive
v: mation, the distillable coherence,and the coherence cost examples to show that the l1 norm of coherence and the
i [23, 30]. coherence of formation,as well as the relative entropyof
X There havebeen severaldifferentexpressionsofcoher- coherence and the coherence of formation, do not give
r ence measures, as mentioned above. However, these dif- the same ordering too. Sec. 4 is our conclusions.
a
ferent expressions, which are based on various physical
contexts,resultindifferentvaluesofcoherenceforastate
in general. In this situation, an interesting question is II. THE QUANTIFICATION OF COHERENCE
whether these coherence measures give the same order-
ing of states. That is, for two coherence measures A We review some notions related to quantifying coher-
C
and B, does A(ρ1) A(ρ2) imply B(ρ1) B(ρ2) ence,suchasincoherentstatesandincoherentoperations,
C C ≤ C C ≤ C
for any two states ρ1 and ρ2? As two effective mea- and recall the three coherence measures, the l1 norm of
sures of coherence, one may expect A and B to give coherence, the relative entropy of coherence, and the co-
C C
thesameorderingforallthestatesevenifnumericalval- herence of formation.
ues of them are not quantitatively equal. In the paper,
Let be the Hilbert space for a d-dimensional quan-
we address this issue. Our discussions focus on the two H
tum system. A particular basis of is denoted as
H
i , i = 1,2, ,d , which is chosen according to the
{| i ··· }
physical problem under discussion. Coherence of a state
is then measured based on the basis chosen [8]. A state
∗ [email protected] is called an incoherent state if and only if its density
2
operator is diagonal in the basis, and the set of all the of distillation process, i.e., from the process that distills
incoherent states is usually denoted as . Therefore, a the maximally coherent states ϕd from ρ under in-
density operator δ is of the form δI= d δ i i. coherentoperations. It is the m|axmimaxail rate at which the
∈ I i=1 i| ih |
Allotherstates,whichcannotbewrittenasdiagonalma- maximallycoherentstatescanbeobtainedfromthegiven
trices in the basis, are called coherent statePs. Hereafter, state [30].
we use ρ to represent a general state, a coherent state The coherence of formation, is defined as
or an incoherent state, and use δ specially to denote an
incoherent state.
(ρ)= min p S((ϕ ϕ ) ), (3)
A completely positive trace preserving map, Φ(ρ) = Cf pi,ϕi i | iih i| diag
{ | i} i
iKiρKi†, is said to be an incoherent completely pos- X
itive trace preserving (ICPTP) map, or an incoherent
Poperation, if the Kraus operators Ki satisfy not only where ρ = ipi|ϕiihϕi| is any decomposition of ρ into
pure states ϕ with p 0. Expression (3), as the co-
iKi†Ki = 1 but also KiIKi† ⊆ I, i.e., each Ki maps herence of Pf|orimiation, iw≥as first given in Ref. [9], and
an incoherent state to an incoherent state.
P was proved to be a coherence measure, i.e., satisfying
Afunctional canbe takenas ameasureofcoherence
C the four postulates, in Ref. [23]. The coherence mea-
if it satisfies the four postulates [8]:
sure expressed by Eq.(3) can also be derived based on
(C1) (ρ) 0, and (ρ)=0 if and only if ρ ;
C ≥ C ∈I the notion of coherence cost, i.e., from the process that
(C2) (ρ) (Φ (ρ))forallICPTPmapsΦ ;
C ≥C ICPTP ICPTP prepares the state ρ by consuming maximally coherent
(C3) C(ρ) ≥ iTr(KiρKi†)C KiρKi†/Tr(KiρKi†) for states|ϕdmaxiunderincoherentoperations. Itisthemin-
all {Ki} with PnKi†Ki =I an(cid:16)d KiIKi† ⊆I; (cid:17) ibmeaclornastuemaetdwthoicphretphaermetahxeimgaivlleyncsothateereunntdsetratienscohhaevreentot
(C4) ( p ρ ) p (ρ ) for any set of states ρ
C i i Pi ≤ i iC i { i} operations [30].
and any p 0 with p =1.
Pi ≥ P i i
Postulate (C1) imposes coherence measures to be a non-
P
negative functional. (C2) and (C3) show the monotonic-
ity of coherence measures under incoherent operations
and that under selective incoherent operations. (C4) III. MAIN RESULTS
means that coherence measures are nonincreasing under
mixing of states. It is obvious that (C3) and (C4) imply A. Ordering states with coherence measures
(C2). These four postulates comprise a general criterion
of defining a coherence measure.
Before proceeding further, we first specify the notion
In accordance with the general criterion, several co-
of orderingstates with coherence measures. For a coher-
herence measures have been put forward. Out of them,
ence measure , the coherence of a state, (ρ), is always
there are the l1 norm of coherence, the relative entropy C C
a non-negative number, and therefore according to the
of coherence, and the coherence of formation, which are
numbers, all the states can be ranked in numerical or-
considered in this paper.
der. Fortwocoherencemeasures and ,wesaythat
The l1 norm of coherence is defined as they give the same ordering if theCfAollowinCBg relations are
Cl1(ρ)= |ρij|, (1) satisfied for all the states ρ1 and ρ2,
i=j
X6
(ρ ) (ρ ) (ρ ) (ρ ). (4)
whereρij areentriesofρ. Thecoherencemeasuredefined CA 1 ≤CA 2 ⇔CB 1 ≤CB 2
by the l norm is based on the minimal distances of ρ to
1
the set of incoherent states , (ρ) = min (ρ,δ) Otherwise, we say that they do not give the same order-
δ
uwpitpherDbobuenidngistahtetali1nendofromr,thI0em≤CaDxCilm1(aρl)ly≤codhe−∈rIe1nD.tsTtahtee iifng.BR(ρe1l)ationB((4ρ)2m)efoarnasltlhtahteCsAta(tρe1s)ρ≤1aCnAd(ρρ22),iofraenqduoanlllyy,
C ≤C
ϕd = 1 d i [8]. A(ρ1) A(ρ2) if and only if B(ρ1) B(ρ2). Similar
| maxi √d i=1| i Cnotions≥ofCorderingstateshavebCeenwid≥elCyusedinentan-
The relative entropy of coherence is defined as
P glementmeasures[31–44]andotherquantum-correlation
(ρ)=minS(ρ σ)=S(ρ ) S(ρ), (2) measures [45–54].
r diag
C σ k −
∈I We will show that the l1 norm of coherence and the
where S(ρ σ) = Tr(ρlog ρ ρlog σ) is the quantum relative entropy of coherence do not give the same or-
k 2 − 2
relative entropy, S(ρ) = Tr(ρlog ρ) is the von Neu- dering of states, i.e., there are the states for which the
− 2
mann entropy, and ρ = ρ i i is the diagonal two measures give a different ordering. To this end,
diag i ii| ih |
part of ρ. The coherence measure defined by the rela- we need to find the states for which (ρ ) < (ρ )
P Cl1 1 Cl1 2
tive entropy is based on the minimal distances of ρ to and (ρ ) > (ρ ), or (ρ ) > (ρ ) and (ρ ) <
Cr 1 Cr 2 Cl1 1 Cl1 2 Cr 1
, (ρ)=min (ρ,δ) with being the the relative (ρ ). Hereafter,we referto the pairofstatesforwhich
δ r 2
eIntCroDpy, 0 (ρ∈)IDlog d. TheDcoherence measure ex- tChe two coherence measures give a different ordering as
≤ Cr ≤ 2
pressedbyEq.(2)canalsobederivedbasedonthenotion the ordering-differentpair for convenience.
3
B. The ordering-different pairs in 2-dimensional Without loss of generality, we assume that t1 < t2, i.e.,
systems (ρ )< (ρ ). Thequestionthenturnstofindz and
Cl1 1 Cl1 2 1
z such that (ρ )> (ρ ), i.e.,
2 r 1 r 2
C C
Wefirstconsider2-dimensionalquantumsystems. The
density operators of a 2-dimensional system can be gen- H(1 z1) H(1 z12+t21)>H(1 z2) H(1 z22+t22).
erally written as 2− 2 − 2− 2 2− 2 − 2− 2
p p
(10)
ρ(x,y,z)= 1 1+z x−iy , (5) For a given pair of t1 < t2, z1 and z2 satisfying Eq.
2 x+iy 1 z (10) may not exist in general. For instance, at t = 3
(cid:18) − (cid:19) 1 5
withx2+y2+z2 1. ρ(x,y,z)canbe further expressed and t2 = 54, there do not exist z1 and z2 fulfilling Eq.
≤ (10). Since (ρ)is anincreasingfunctionoft andz,the
as ρ(x,y,z)=U ρ(t,z)U , where r
α α† C
maximal value of the left side of Eq. (10) is appearing
1 1+z t at z = 1 t2, and the minimal value of the right side
ρ(t,z)= 2 t 1 z , (6) is ap1pearing−at1z = 0. Hence, we obtain the necessary
(cid:18) − (cid:19) p 2
and sufficient condition of t and t ,
with t = x2+y2 and t2 + z2 1, and U = 1 2
α
diaagrct1a,neiyα. pIits ma edaiangsotnhaaltanρd(xu,yn,itza)ry≤anmdaρtr(itx,zw)itchanαb=e H(1− 1−t21)>1 H(1−t2), (11)
− (cid:8) x (cid:9) 2 − 2
transformed into each other by an incoherent operation, p
which further implies that they correspond to the same under which z and z exist. It means that z and z
1 2 1 2
coherence value for all the coherence measures due to satisfying the inequality (10) exist if and only if t and
1
postulate (C2). Therefore, we only need to consider the t (t t ) satisfy the relation (11).
2 1 2
family of density operators with the form ρ(t,z). ≤
By substituting Eq. (6) into Eqs. (1) and (2), we
obtain the l norm of coherence,
1
(ρ)=t, (7)
Cl1
0.2
and the relative entropy of coherence,
0.1
1 z 1 √z2+t2
r(ρ)=H( ) H( ), (8) 0
C 2 − 2 − 2 − 2 Cr
∆ −0.1
whereH(x)= xlog x (1 x)log (1 x)isthebinary
− 2 − − 2 −
Shannon entropy function. C (ρ) is only dependent on −0.2
l1
t,whileC (ρ)isdependentonbothtandz. SinceC (ρ)
r r −0.3
is an even function of z, we can further restrict our dis- 1
0.5 1
cussionto the range0 z 1withoutlossofgenerality.
≤ ≤ 0 0.5
Specially, when ρ is a pure state, there is (ρ) = 0
Cr −0.5 −0.5
H(12 − √12−t2) due to t2 +z2 = 1. In this case, Cr(ρ) z1 −1 −1 z2
is an increasing function of t just like (ρ), and there-
Cl1
fore the l norm of coherence and the relative entropy
1
FIG.1. (Coloronline). Anillustrationtoshowthatthereare
of coherence always give the same ordering for all the
2-dimensionalpure states. There is no ordering-different manysolutionsofz1andz2satisfying∆Cr =Cr(ρ1)−Cr(ρ2)>
pair in this case. 0. Here, we have taken t1 = 45 and t2 = √26. All the pairs
of states ρ1 and ρ2 defined by z1 and z2 in the domain for
To find the ordering-different pairs for which the l
norm of coherence and the relative entropy of coherenc1e ∆Cr >0 fulfill Cl1(ρ1)<Cl1(ρ2) and Cr(ρ1)>Cr(ρ2).
give a different ordering, we further analyze the expres-
Therefore,tofindρ andρ thatsatisfyboth (ρ )<
sionof r(ρ)withouttherestrictionofpurestates. From 1 2 Cl1 1
0aEnq≤d. z∂(C8∂≤rC)z(,ρ1)w,=ewhh12iaclvohegm2∂11eC+−a∂rtzzn(ρs)+C=r2(√ρt2)2z√+itsz2r2+alzno2gil2nocg11r−+2e√√1a1−+zzs22i√√n++zzgtt2222++fu≥tt22nc0≥tifoon0r Ctzt2a2l1k(((et0ρ12t≤)<1 a=ztn12,d)54zwC2arint(≤hdρ11tt)h)2>eb=yaCiudr√2(s6oρin,f2g)Ew,Eqho.qinc.(eh1(m11s)a0at)ayi.nsfifFdyrostrEtheqecxn.h(ao1fimo1ns)pde,leazt,11llifaatwnnhddee
of t as well as z. z and z in the domain with ∆ = (ρ ) (ρ )>0
1 2 r r 1 r 2
C C −C
With these knowledge, we may now try to find a pair are given in Fig. 1. Clearly, there are infinitely many
ofstatesρ andρ ,suchthatthe orderingof (ρ ) and pairsofstatessatisfying∆ >0. Anexplicitexpression
1 2 Cl1 1 Cr
(ρ ) are different from that of (ρ ) and (ρ ). For of ρ and ρ reads
Cl1 2 Cr 1 Cr 2 1 2
this, we let
4 2 1 1
ρ1 =12(cid:18)1+t1z1 1−t1z1 (cid:19), ρ2 = 21(cid:18)1+t2z2 1−t2z2((cid:19)9). ρ1 =(cid:18) 255 551 (cid:19), ρ2 = √216 √126 !, (12)
4
for which, (ρ ) = 4 < (ρ ) = 2 but D. Further discussions
Cl1 1 5 Cl1 2 √6
(ρ )=0.7219> (ρ )=0.5576.
r 1 r 2
C C
After having shown that the l normof coherence and
1
the relative entropy of coherence do not give the same
ordering of states, we now extend our discussion to one
C. The ordering-different pairs in high dimensional morecoherencemeasure,thecoherenceofformation. We
systems
wonder whether the l norm of coherence and the co-
1
herence of formation, as well as the relative entropy of
We now consider high dimensional quantum systems. coherence and the coherence of formation, give the same
The above discussion is only on 2-dimensional states. It orderingofstates. Wewillbrieflydiscusstheissueinthis
shows that there are the pairs of states for which the l1 subsection. Infact,withtheaidoftheabovecalculations,
norm and the relative entropy of coherence do not give it is very easy to confirm that there are infinitely many
the same ordering. This conclusion can be easily ex- ordering-differentpairsforthesecoherencemeasurestoo.
tendedtothecaseofhighdimensionaldensityoperators. We first address the l norm of coherence and the co-
1
In fact, any two d( 3)-dimensional states defined by herence of formation. Form Eqs. (2) and (3), we have
≥
(ϕ ) = (ϕ ) for all pure states ϕ , which implies
ρ(1d) = 12(δ1(d−2)⊕ρ1), ρ(2d) = 21(δ2(d−2)⊕ρ2), (13) Cthfa|t aill theCrpu|reistates that satisfy Cl1|(|iϕ1i)<Cl1(|ϕ2i)
and (ϕ )> (ϕ ) must fulfill (ϕ )< (ϕ )
ewnhtersetaδt1(eds−,2)neacnedssδa2(rdi−ly2)saatriesf(yd−Cl12()ρ-d(1di)m)e<nsiColn1a(lρ2(idn)c)ohaenrd- ddaniimmdeeCCnnrfssii(|oo|ϕnn11aaiill)sstt>aatCteeCrssf|(dd|eϕe2fifii2nni)ee.ddbTbyyheEErqeqf..o(r(1eC1,5l41)a)la|larnep1diatihrtesheooCrfdld1(te≥h|rein243gi)---
(ρ(d)) > (ρ(d)) as long as ρ and ρ do. This is
bCreca1use Cl1(Cρr(id)2) = 12Cl1(ρi) and1 Cr(ρ(id2)) = 12Cr(ρi), dcoiffheerreenntcepaoifrsfofromratthioenl1tonoo.rmNooftecothheartentcheeraendis tnhoe
i=1,2. ordering-different pair in the 2-dimensional case, since
It is worth noting that although the l1 norm of coher- both (ρ) = t and (ρ) = H(1 + √1 t2) [18, 23] are
ence and the relative entropy of coherence give the same Cl1 Cf 2 2−
increasingfunctionsoftandthereforetheygivethesame
ordering for all the 2-dimensional pure states, it is not
ordering for all the 2-dimensional states.
true for the d( 3)-dimensional case. There are pairs of
≥ We now address the relative entropy of coherence and
highdimensionalpurestatesforwhichthel normofco-
1 the coherence of formation. Since the coherence of for-
herenceandtherelativeentropyofcoherencedonotgive
mation and the l norm of coherence give the same or-
1
the same ordering. For example, in the 3-dimensional
dering for all the 2-dimensional states as shown above,
case,
i.e., (ρ ) < (ρ ) if and only if (ρ ) < (ρ ), all
Cf 1 Cf 2 Cl1 1 Cl1 2
the 2-dimensional states that satisfy (ρ ) < (ρ )
ϕ(3) = 12 1 + 12 2 + 1 3 , and (ρ ) > (ρ ) must fulfill (ρCl)1 <1 (ρC)l1an2d
| 1 i 25| i 25| i 25| i Cr 1 Cr 2 Cf 1 Cf 2
r r r (14) (ρ ) > (ρ ). Therefore, all the 2-dimensional
r 1 r 2
|ϕ(23)i= 170|1i+ 15|2i+ 110|3i, dCoriffdeerreinngt-dpiaffiCresrefnotrpairasnfdor Clt1ooa.ndACsrpoairnetethdeoourtdienritnhge-
r r r Cf Cr
case of the l norm of coherence and the relative en-
is a pair of states for which the l norm of coherence 1
1
tropyofcoherence,thereareinfinitelymanysuchpairsof
and the relative entropy of coherence do not give the
states,andonemayobtainthemwiththeaidofEqs. (10)
same ordering, since (ϕ ) = 1.5143 < (ϕ ) =
Cl1 | 1i Cl1 | 2i and (11). An explicit expression of ρ and ρ is given in
1.5603 but (ϕ )=1.2023> (ϕ )=1.1568. With 1 2
r 1 r 2
C | i C | i Eq. (12), for which (ρ ) = 0.7219 < (ρ ) = 0.7440
the aid of the 3-dimensional case, we can find pairs of Cf 1 Cf 2
and (ρ )=0.7219> (ρ )=0.5576.
higher dimensional pure states for which the l1 norm of Cr 1 Cr 2
Besides,justlikethe resultthatanytwoentanglement
coherence and the relative entropy of coherence do not
measurescoincidingonpurestatesareeitheridenticalor
give the same ordering. Indeed, any pairs of d( 4)-
≥ give a different ordering [33], we can also obtain a sim-
dimensional pure states defined by
ilar result for coherence measures. Let us consider two
d coherence measures and that coincide on all pure
|ϕ(1d)i=α|ϕ(13)i+ βi|ii, states. Note that foCrAany sCtBate ρ and any sufficiently
i=4 small ǫ > 0, there must exist pure states φ and ψ
X (15)
| i | i
d such that (ψ ) ǫ = (ρ) = (φ ) + ǫ. If
A A A A
|ϕ(2d)i=α|ϕ(23)i+ βi|ii, and CB givCe t|heisa−me ordCering on Call |stiates, we haCve
Xi=4 CB(|ψi) ≥ CB(ρ) ≥ CB(|φi). As CA and CB coincide on
wisfiythC|lα1|(2|ϕ+(1Pd)(idi3)=)4<|βCi|l21(=|ϕ(1(23d)a)in)da0n<d C|αr(||<ϕ(1d1)ni)ec>essCarr(i|lϕy2(sda)it)- pfinuugrrtehǫesrttoaltezeaesd,rsow,teowgeCeAtfi(nρCa)All+(y|ψǫoib)≥t≥aCinBC(BCρA()ρ()ρ≥)≥C=AC(ACρ(B)|φ(−ρi))ǫ,,.wwThhaiicckhh-
as long as ϕ and ϕ do (see the Appendix for the implies that any two coherence measures that coincide
| 1 i | 2 i
proof). on pure states are either identical or give a different or-
5
dering. This result can be used to determine whether with α 2 + β 2 = 1 and 0 < α < 1, satisfy
d d d
| | | | | |
tshomeseamcoeheorrednecreingm.eHasouwreevs,erf,oirtiinssntoatncaeppClricaabnldetCof,mgainvye Cl1W(|ϕe1(nd)oiw)<prCovl1e(|tϕh2(ed)tih)eaonredmC.r(|ϕ1(d)i)>Cr(|ϕ(2d)i).
other coherencemeasures,for example Cl1 andCr, which By the definition of the l1-norm of coherence,we have
do not coincide on pure states.
(d 1)
Cl1(|ϕ1 − i)=2 |aiaj|, (18)
1 i<j d 1
IV. CONCLUSIONS ≤ X≤ −
and
The issue of ordering states with coherence measures
is first discussed in this paper. We have shown that the (ϕ(d) )=2α 2 a a +2α β d−1a ,
twowell-knowncoherencemeasures,thel1normofcoher- Cl1 | 1 i | d| | i j| | d d| | i|
ence and the relative entropy of coherence, do not give 1≤i<Xj≤d−1 Xi=1
the same ordering of states. There are infinitely many
d 1
eonrdceerainngd-dthiffeerreelnattivpeaiernstrfoorpywohficchohtehreenlc1engoivrmeaodfiffcoerheenrt- =2|αd|21 i<j d 1|aiaj|+2|αdβd|vu(i=−1|ai|)2
ordering. Detailed calculations show that the ordering- ≤ X≤ − ut X
different states include the 2-dimensional mixed states, =2αd 2 aiaj
| | | |
the d(≥ 3)-dimensional mixed states and the d(≥ 3)- 1≤i<Xj≤d−1
dimensional pure states, but exclude 2-dimensional pure
+2α β 1+2 a a ,
states since allthe coherence measures give the same or- | d d| | i j|
dering for 2-dimensional pure states. s 1≤i<Xj≤d−1
surOeus,raanndalyassisacnanillbuesterxatteionndeodftothoetheexrtecnoshieornenwcee mhaevae- =|αd|2Cl1(|ϕ1(d−1)i)
shown that the l1 norm of coherence and the formation +2|αdβd| 1+Cl1(|ϕ1(d−1)i),
of coherence, as well as the relative entropy of coherence
q (19)
andthe coherenceofformation,do notgivethe same or-
deringtoo. Ourresultsindicatethatatleasteachpairof and similarly,
the three coherence measures, the l norm of coherence,
1
tfohremraetliaotniv,egievnetraodpiyffeorfenctohoerrdeenrcinegaonfdsttahteesc,oahltehreonucgehoitf Cl1(|ϕ2(d−1)i)=2 |bibj|, (20)
1 i<j d 1
remains an open question to examine whether all other ≤ X≤ −
coherence measures give a different ordering of states. and
Notingthatanytwostatesinanordering-differentpair
cearantnioontsb,eoutrranressfourltmmedayinbtoeehaeclhpfoutlhienrdbeyteinrmcoihneinregnwthoapt- Cl1(|ϕ2(d)i)=|βd|2Cl1(|ϕ2(d−1)i)+2|αdβd| 1+Cl1(|ϕ2(d−1)i).
q (21)
kinds ofstates cannot be transformedinto eachother by
incoherent operations.
(d) (d)
Eqs. (19) and (21) show that (φ ) < (φ ) if
Cl1 | 1 i Cl1 | 2 i
Cl1(|ϕ1(d−1)i)<Cl1(|ϕ2(d−1)i).
V. APPENDIX By the definition of the relative entropy of coherence,
we have
We show that any two d( 4)-dimensional pure states
≥ d 1
define(dd)by Eq. (15(d)) necessaril(yd)satisfy Cl1(|ϕ1(d)i) < Cr(|ϕ1(d−1)i)=− − |ai|2log2|ai|2, (22)
Cl1(|ϕ2 i)andCr(|ϕ1 i)>Cr(|ϕ2 i)aslongas|ϕ1iand Xi=1
ϕ do. Tothisend, weonlyneedtoprovethefollowing
2
| i and
theorem: If the two (d 1)-dimensional states expressed
−
as
(ϕ(d) )= β 2log β 2
d 1 d 1 Cr | 1 i −| d| 2| d|
|ϕ1(d−1)i= − ai|ii, |ϕ2(d−1)i= − bi|ii, (16) d−1α 2 a 2log (α 2 a 2)
Xi=1 Xi=1 − | d| | i| 2 | d| | i|
i=1
bCsayrt(i|sφf2(yd−C1l)1i()|ϕ, 1(tdh−e1n)it)h<e tCwl1o(|dϕ-2(ddi−m1e)in)siaonndalCsrt(a|tϕe1(sd−d1e)fii)ne>d =−X|αβd|22ldo−g12|aβd2|2lo−g|αad|22log2|αd|2 (23)
−| d| | i| 2| i|
|ϕ(1d)i=αd|ϕ1(d−1)i+βd|di, |ϕ(2d)i=αd|ϕ1(d−1)i+β(d1|d7i) =|αd|2Cr(Xi|=ϕ11(d−1)i)+H(|αd|2),
6
whereH(x)=−xlog2x−(1−x)log2(1−x)isthebinary Cr(|ϕ1(d−1)i)>Cr(|ϕ2(d−1)i). This completes the proof of
Shannon entropy function, and similarly, the theorem. With this theorem, it is easy to obtain the
conclusion related to Eq. (15).
d 1
Cr(|ϕ2(d−1)i)=− − |bi|2log2|bi|2, (24)
i=1
X
ACKNOWLEDGMENTS
and
Cr(|ϕ(2d)i)=|αd|2Cr(|ϕ2(d−1)i)+H(|αd|2). (25) ThisworkwassupportedbyNSFChinathroughGrant
No. 11575101and the NationalBasic ResearchProgram
Eqs. (24) and (25) show that (ϕ(d) ) > (ϕ(d) ) if of China through Grant No. 2015CB921004.
Cr | 1 i Cr | 2 i
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