Table Of ContentAssisted state discrimination without entanglement
Bo Li,1,2, Shao-Ming Fei,3,4 Zhi-Xi Wang,3 and Heng Fan1,
∗ †
1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
2Department of Mathematics and Computer, Shangrao Normal University, Shangrao 334001, China
3School of Mathematical Sciences, Capital Normal University, Beijing 100048, China
4Max-Planck-Institute for Mathematics in the Sciences, 04103 Leipzig, Germany
It is shown that the dissonance, a quantum correlation which is equal to quantum discord for
separable state, is required for assisted optimal state discrimination. We find that only one side
discordisrequiredintheoptimalprocessofassistedstatediscrimination,whileanothersidediscord
and entanglement is not necessary. We confirm that the quantum discord, which is asymmetric
dependingon local measurements, is a resource for assisted state discrimination. With theabsence
2 of entanglement, we give the necessary and sufficient condition for vanishing one side discord in
1 assisted state discrimination for a class of d nonorthogonal states. As a byproduct, we find that
0 the positive-partial-transposition (PPT) condition is the necessary and sufficient condition for the
2 separability of a class of 2×d states.
n
a PACSnumbers: 03.67.-a,03.65.Ud,03.65.Yz
J
7
2 I. INTRODUCTION Ekρ)/pAk, the supreme is taken over all the von Neu-
mann me|asurement sets {Ek} on system B. Similarly,
] D (ρ ) refers to the “left” discord and is given by
h Entanglement is regarded as a key resource in quan- A AB
p tum information processing such as teleportation and
D (ρ )=I(ρ ) sup S(ρ ) p S(ρ ) ,(2)
nt- ssuhpowerndetnhsaet caoddientger,meitnci.s[1ti]c. quOannttuhme cootmhepruthaatniodn, wititihs A AB AB − Ek{ B −Xk B|k B|k }
ua ognleemqeunbt.itW(DhQilCe1t)he[3q]ucaanntubme cdaisrcrioerddo[4u–t8w],iathnoouthteernttyapne- wNiottheptBh|akt=thTer(dEiffke⊗re11nBceρ)b,eρtBw|kee=nTthrAos(e(Etkw⊗o11dBisρc)o/rpdBs|kis.
q
of quantum correlation, might be the reason for the ad- that the measurement is performed on party A or on
[
vantage of this algorithm that surpasses the correspond- party B, respectively. It is thus expected that this defi-
3 ing classicalalgorithms. Besidesquantumentanglement, nition of quantum discordis not symmetric with respect
v there are also many quantum nonlocal properties which to A and B.
5
can be manifested by separable states or separableoper- Duetothesupremeinthedefinition(1),quantumdis-
4
6 ations[2]. Itisthus reasonableto assumethatthe quan- cord does not have analytic or operational expression in
2 tumnesscorrelationcanbeviewedfromdifferentaspects. general. However,explicitexpressionofdiscordhasbeen
. Recently much efforts have been devoting to studying found for cases like Bell-diagonal state [13, 14] and sub-
1
various measures of nonclassical correlation, see for ex- class of so-called X states [15–17]. Numerical results
1
1 ampleRefs.[27–35]. Quantumdiscord,whichisourmain for general two-qubit states are also presented in [22].
1 concern in this paper, is recently pointed out to have Moreover,the quantumdiscordcanbe relatedto the en-
: operational interpretations, for example in terms of the tanglement of formation (EOF) [23]. This relation can
v
quantumstatemergingprotocol[21]andasentanglement be used to calculate the quantum discord for the rank-
i
X by an activation protocol or by measurement[27, 28]. two states [26]. Nevertheless, zero discord can be eas-
r Quantum discord [4, 5] is measured by the difference ily found by a necessary and sufficient condition pointed
a
between the mutual information and the maximal con- out in Ref.[20]. If ρ is a quantum state of d d
AB A B
×
ditional mutual information obtained by local measure- dimensional bipartite quantum systems, from [20] ρ
AB
ment. Explicitly, let us consider a bipartite quantum canbe writtenindiagonalformρ = L c S F ,
stateρAB,the“right”quantumdiscordisgivenby[4,5], where {Sn},{Fm},(n = 1,··· ,dA2AB,m =Pn1=,·1··n,dn2B⊗) arne
the bases of the respective local spaces, L is the rank of
D (ρ )=I(ρ ) sup S(ρ ) p S(ρ ) ,(1) ρ[20]. Thenthenecessaryandsufficientconditionofzero
B AB AB A Ak Ak
− Ek{ − k | | } “left” discord is
X
[S ,S ]=0,i,j =1, ,L. (3)
where S(ρ) is the von Neumann entropy and I(ρ ) = i j
AB ···
S(ρ )+S(ρ ) S(ρ ) is the quantum mutual infor-
A B − AB Similarly, the necessary and sufficient condition of zero
mation, and pA|k = Tr(11A ⊗ Ekρ), ρA|k = TrB(11A ⊗ “right” discord is
[F ,F ]=0,i,j =1, ,L. (4)
i j
···
∗Electronicaddress: [email protected] It is shown that almost all quantum states have nonzero
†Electronicaddress: [email protected] discord [24, 25].
2
Aunifiedviewofquantumcorrelationbasedonrelative success probability is given by
entropy was introduced in [36]. Quantum dissonance is
athekincldosoefstqsutaantteutmo acosrerpelaartaiobnleosftasteepaρraabslemsetaastuerse.dAbys P(|α+|)=1−p− α|α+|22 −p+|α+|2. (7)
| |
relativeentropyisthestateρitself,thedissonanceisjust
the quantum discord in this case. For general α+ , the probability is neither 0 nor 1.
| |
It is remarkable that dissonance is found to be useful We shall use the PPT criterion [18] to characterize
idnisasossniastnecde sistattheedqisucarnimtuinmatdioisncorredceinntlyth[e11o]p.tiNmoatlectahsaet. tthhee sPePpaTracboinlidtyitioofnρi|sα+b|o.thFonre2ce×ssa2ryanadnd2×suffi3 csiyesnttemfosr,
In this paper, we further find that actually only one side the separability of quantum states [19]. From the PPT
qduisacnotrudmofdaisncootrhderapspideearcsaninbtehezeorpot.imHaelnpcreotcheses,rowlehiolef pcroisteedriomna,tρr|iαx+|ρTisAsephaarsabalenoifna-nnedgoantilvyeifspitescptraurtmia.lHtreanncse-
quantum discord played in assisted state discrimination thedeterminant|αD+|ofρTA shouldbenon-negativeeither.
might be quite different for different parties. We show By direct calculation w|αe+h|ave
clearly that the quantum discord is really a useful re-
source which can be used in assisted state discrimina- 1 α 2 1 α 2
tion. We also extend the study of discord for assisted D = (p+ −| +| +p −| −| ) (p+ α+ 2+
− 2 − 2 × | |
state discrimination to more general cases.
1 α 2 1 α 2
The paper is organized as follows. In Sec. II, we first p α 2) p+ −| +| α+ p −| −| α 2.
giveabriefreviewofthemodelinassistedstatediscrimi- −| −| ×| r 2 − −r 2 −|
nation. Wethenshowthatonlyonesidediscordisneces-
D 0 implies that
saryinthe optimalcase. InSec. III andSec. IV,wegen- ≥
eralize the model to discriminate a class of d nonorthog-
1 α 2 1 α 2
onalstates,in whichwe stillfind that one-sidediscordis p+ −| +| α+ =p −| −| α , (8)
required in the process. Sec. V is the summary. r 2 −r 2 −
that is,
II. UNAMBIGUOUS DISCRIMINATION OF
1 α 2 1 α 2
TWO NONORTHOGONAL STATES p+ −| +| α+ 2 =p −| −| α, (9)
r 2 | | −r 2
Followingthe modelofassistedstatediscriminationin where α = α α . From (9) we know that α must be a
∗+
Ref. [9–11], consider that a qubit is randomly prepared realnumber. Eq−.(9)isanecessaryconditionforthesepa-
ainpornioeroifptrhoebtawboilintioensopr+thaongodnpalswtaittehsp|ψ+++ipor|=ψ−1i.wOituhr rItabisiliitnyfoafctρ|tαh+e| asanmdeisaascttuhaellEyqa.(ls7o)ainsu[ffi11c]i.entcondition.
aim is to discriminate the two−states ψ+ or−ψ . The That the condition (8) is also a sufficient condition
system is coupled to an auxiliary qubi|t Aiby a|jo−init uni- for separability can be seen from the following separable
tary transformation U such that form of ρ ,
|α+|
UU||ψψ+−ii||kkiiaa==p11−−||αα+−||22||+−ii||00iiaa++αα+−||00ii||11iiaa,, (5) ρ|α+| = ((p1+−|αp++||2α++|p2−−|αp−−||2α)−|0|i2h)0ρ|S1⊗⊗ρ|A20,iah0|+(10)
where k is an apuxiliary state with orthonormal basis
| ia whereρS andρA arethedensitymatricesoftheprincipal
0 , 1 , (0 1 )/√2 are the orthonormal 1 2
{| ia | ia} |±i ≡ | i ± | i system and the auxiliary system respectively,
states of the system that can be discriminated. For con-
venience,weadoptthesymbolsusedinRef.[11]. Apriori 1
fiαx∗+edisotvheerlcaopmips,lehxψc+o|nψj−ugia=teαof=α+|α,|θeiiθs=theαp∗+hαa−s,eswohfeαre. ρS1 = +1p−p(1+|α+α|2−2)p−|α−|2),(p+(1−|α+|2)|+ih+|
The mixed state we consider in discrimination is given
− −| −| |−ih−|
by ρA = 1 ((p α 2+p α 2)1 1
2 p+ α+ 2+p α 2 +| +| −| −| | iah |
ρ|α+| = p+U(|ψ+ihψ+|⊗|kiahk|)U† +√|2p|α −1| −α| 2 1 0 +
+ + + a
+p U(ψ ψ k k )U . (6) −| | | i h |
a †
Byperformingavon−Neu|ma−ninh m−e|a⊗su|reimhen|tontheaux- √2p+α∗+ p1−|α+|2|0iah1|).
p
iliary system by basis, 0 0, 1 1 , the auxiliary From (10) and the necessary and sufficient condition of
a a
{| i h | | i h |}
state in (6) will collapse to either 0 0 or 1 1 . zero discord given by Eq.(3) and (4), we have that state
a a
{| i h |} {| i h |}
In case the system collapses to 0 0 , we will dis- ρ haszero“right”quantumdiscordwhenthecompo-
criminatesuccessfullythe original{s|taitaehs|i}nce wecandis- n|eαn+t|s of the “right” reduced density operators are com-
tinguish deterministically the states as in (5). The muting,[ρA, 0 0]=0. Thiscanbesatisfiedonlywhen
|±i 2 | iah |
3
α = 0 or α = 1. Since ψ and ψ are different randomly prepared in the d (d 2) nonorthogonal and
+ + +
nonorthogon|al|states, we do|notineed|to−cionsider those linearly independent states, ψ≥, ψ , , ψ , with a
1 2 d
| i | i ··· | i
two cases because they corresponds to either the same priori probabilities p ,p , ,p with p + +p =1.
1 2 d 1 d
··· ···
state or two orthogonal states. Hence as pointed out in The system is coupled to an auxiliary qubit A by a joint
Ref.[11], the “right” discord is always non-zero. On the unitary transformation U such that
1
other hand, ρ has zero “left” quantum discordif and
only if [ρS, 0|α0+|]=0, that is,
1 | ih |
1 + + d
p+(1−|α+|2)=p−(1−|α−|2). (11) U1|ψ1i|kia = 1−|α1|2|1i|0ia+α1| i √··d· | i|1ia,
p 1 + + d
Combining (9) and (11), we have U1 ψ2 k a = 1 α2 2 2 0 a+α2| i ··· | i 1 a,
| i| i −| | | i| i √d | i
α is a real number, and α 0; p ,
• ≥ ···················································
• p+ =p− = 21; U1|ψdi|kia = 1−|αd|2|di|0ia+αd|1i+√··d·+|di|1ia(,12)
α = α = α =√α. p
+
• | | | −| | |
It is interesting thatpthose three conditions coincide ex-
where 1 , 2 , , d is the orthogonal basis in the
actly with the optimal assisted state discrimination case {| i | i ··· | i}
state space. Note that the auxiliary state is still two-
in [11] with a priori probabilities and θ =0, in which it
dimensional. For d = 2, we can apply a Hadamard
is shown that the dissonance is required in the discrim-
gate on the first qubit, and the model returns to the
ination. Thus, we obtain one of our main results: The
one given in Section II. This model is a natural gener-
assistedstatediscriminationoftwononorthogonalstates
alization of discrimination of two nonorthogonal states.
canbeperformedwiththeabsenceofentanglement. The
To ensure that there exists the unitary transformation
“right”quantumdiscordisrequiredforassistedstatedis-
U , the inner product of vectors on the right hand side
1
crimination. However,the “left” discord is not necessar-
shouldbe equalto the overlapofthe correspondingorig-
ily to be non-zero. In particular, in the assisted opti-
inal states[12], that is α α = ψ ψ . For convenience,
mal state discrimination, the “left” discord is found to ∗i j h i| ji
we denote α = ψ ψ .
ij i j
be zero. Here we remark that, it is not surprising that h | i
the “left” discordis nonzero except for the optimal case, The mixedstatewe considerindiscriminationis given
since quantum discord in general is non-zero [24, 29]. now by
Recall that the motivation of quantum discord is to
find the difference between total correlations, including
bothquantumandclassicalcorrelationquantifiedbymu-
tual information,and the accessible classicalcorrelation, ρ = p1U1(|ψ1ihψ1|⊗|kiahk|)U1†
which is quantified by the maximal conditional entropy +p2U1(|ψ2ihψ2|⊗|kiahk|)U1†
obtained by local measurement [5]. While the mutual
information is symmetric, the asymmetry of the quan- +···+pdU1(|ψdihψd|⊗|kiahk|)U1†. (13)
tum discord is due to local measurements. The proto-
col of assisted state discrimination in this paper and in
Thesuccessprobabilitytodiscriminatethe stateisgiven
Ref.[11],isexactlyassistedbylocalmeasurementsonthe
by
“right”partysoastodistinguishnonorthogonalstatesof
the “left” party. Thus the corresponding “right” quan-
tum discord is necessary while the “left” discord, which
is quantified by a local measurementon the “left” party, P =1 p1 α1 2 p2 α2 2 pd αd 2. (14)
− | | − | | −···− | |
is useless. Thus we can find that in the optimal assisted
state discrimination, the “left” discord is zero. Indeed,
the deep reasonthat quantum discord is requiredfor as- In general, it is not zero. If ρ is a separable state, the
sisted state discrimination is that it is really used, as a partial transposed matrix must be positive. Then all
consuming resource in such quantum information pro- the principal minor determinants of ρTS is non-negative,
cessing. where ρTS is the partial transposed matrix with respect
to the qudit system. In the following, as a necessary
condition for separability, we calculate those 4 4 prin-
III. UNAMBIGUOUS DISCRIMINATION OF d ×
cipal minor determinants that should be non-negative.
NONORTHOGONAL STATES Let Mij be the principal minor matrix of ρTS by select-
ing the i 0, i 1, j 0, j 1 rows,and
a a a a
h |⊗ h | h |⊗ h | h |⊗ h | h |⊗ h |
In the following, we generalize the previous model to the i 0 , i 1 , j 0 , j 1 columns. By
a a a a
| i⊗| i | i⊗| i | i⊗| i | i⊗| i
d-dimensional system. Let us consider that a qudit is straightforwardcalculations we have
4
pi(1−|αi|2) piα∗i 1−|dαi|2 0 pjα∗j 1−|dαj|2
M = piαi 1−|dαi|2 1d(p1|α1|2+q···+pd|αd|2) piαi 1−|dαi|2 d1(p1|α1|2+q···+pd|αd|2) . (15)
ij
q0 piα∗i 1−|dαi|2 pj(1q−|αj|2) pjα∗j 1−|dαj|2
pjαj 1−|dαj|2 d1(p1|α1|2+q···+pd|αd|2) pjαj 1−|dαj|2 d1(p1|α1|2+q···+pd|αd|2)
q q
The determinant D of M is given by Weconsidernowthe“left”quantumdiscord. Itiseasy
ij ij
to find that ρ ,ρ is linearly independent. Hence ρ has
1 2
Dij = Det[Mij] vanishing “left” discord if and only if [ρ1,ρ2]= 0. With
1 this commuting condition, i.e., assuming that the “left”
= (p (1 α 2)+p (1 α 2)) (p α 2+
−d2 i −| i| j −| j| × 1| 1| ··· discord is zero, we should have
+pd αd 2) piαi 1 αi 2 pjαj 1 αj 2 2. p (1 α 2)= =p (1 α 2). (19)
| | ×| −| | − −| | | 1 1 d d
−| | ··· −| |
p q
D 0 implies that
ij It means that ρ is an identity and commutes with any
≥ 1
densityoperators. Combining Eq. (16)andEq. (19), we
p α 1 α 2 = =p α 1 α 2. (16)
1 1 1 d d d obtain
−| | ··· −| |
In the pprotocol of assisted sptate discrimination, 1
p = =p = , (20)
p1,p2, ,pd arethepriori probabilities. Sincetheover- 1 ··· d d
···
laalpityα,1wie=cahψn1r|eψwiiri=teα(1∗1α6)i aissfixed, without lose of gener- α1 =···=αd ≡γ. (21)
Ontheotherhand,fromtheunitarytransformation(12),
α α
p α 1 α 2 = p 12 1 12 2 = the priori fixed overlaps are equal and should be a real
1 1 1 2
−| | α∗1 r −| α1 | ··· number,
p
α α
= pd α1∗1dr1−| α11d|2. (17) hψi|ψji=α∗iαj =|γ|2. (22)
Thereisonlyonevariableα in(17)butd 1equalities. To conclude, the “right”discordis alwaysrequiredfor
1
By using the condition (16), we successfu−lly write ρ in the assisted state discrimination of d (d 2) nonorthog-
≥
the following separable form, onal states, though the quantum entanglement could be
absent, i.e. the condition (16) is fulfilled. This is a
ρ=ρ1 0 a 0 +ρ2 ρa, (18) generalization of assisted state discrimination for two
⊗| i h | ⊗
nonorthogonalstates [11]. On the other hand, the “left”
where discord is not necessarily required in this process since
conditions (20) and (21) can be fulfilled for some cases
ρ = p (1 α 2)1 1 + +p (1 α 2)d d;
1 1 −| 1| | ih | ··· d −| d| | ih | which lead also to the absence of entanglement.
1
ρ = (1 + + d )( 1 + + d); As a byproduct, one may notice that the PPT con-
2
d | i ··· | i h | ··· h | dition is a sufficient condition of separability for state
ρa = (p1 α1 2+ +pd αd 2)1 a 1 Eq.(13) since it can be written into the form of(18). On
| | ··· | | | i h |
+√dp1 1−|α1|2(α1|1iah0|+α∗1|0iah1|). tbhileitoy.thTerhuhsanPdP,TPPcrTiteisriaonneiscebsostahrythcoenndeicteiosnsafroyrasnepdatrhae-
Thus condition p(16) is actually also the sufficient condi- sufficientcondition forthe separabilityof a classof 2 d
×
tion for separability. states of the form (13).
Fromthenecessaryandsufficientconditionofzerodis-
cordproposedbyEq.(3)and(4), ρhasvanishing“right”
discordif and only if [0 0,ρ ]=0, which gives rise to IV. OPTIMAL UNAMBIGUOUS
a a
| i h | DISCRIMINATION OF d NONORTHOGONAL
α = 0 or α = 1. From (16) it means that the overlap
1 1
STATES
ψ ψ can only be either 0 or 1, namely ψ , ψ are
i j i j
h | i | i | i
either orthogonal or equal, which contradicts with our
assumption. Thereforeweconcludethatinassistedstate Next, we will try to determine the exact form of
discrimination,the“right”quantumdiscordisalwaysre- the optimal success probability given by (14). As
quired. This agrees with the case of two non-orthogonal p ,p , ,p arethe priori probabilitiesandtheoverlap
1 2 d
···
states. α = ψ ψ = α α is known, the success probability
1i h 1| ii ∗1 i
5
in ( 14) can be rewritten as Weseethattheassistedoptimalstatediscriminationcan
be accomplished with a vanishing “left” discord. Also
P =1−p1|α1|2− p2|α12|2+α···2+pd|α1d|2. (23) fbriolimty(i2s5m),onwoetohnaivcealtlyhadtecforeraasinnygdw,itthhethoepntiomna-nlepgraotbivae-
1
| |
parameter γ , i.e., the smaller the overlap is, the larger
| |
There is only one variable α1 in (23). In order to find probabilitywe candiscriminate. This is understandable,
| |
the optimal probability, we define when they are orthogonal, i.e. the overlap is zero, we
can discriminate them deterministically; when they are
p α 2+ +p α 2 close to each other, it is difficult to discriminate them.
α¯ = 4 2| 12| ··· d| 1d| . (24)
In short, we have shown that the optimal assisted state
s p1
discrimination of d (d 2) nonorthogonal states can be
≥
The optimal probability can be found by dealing with α¯ performed with the absence of entanglement and “left”
in three different regions: discord, while the “right” discord is always required.
1. Ifmax α , , α α¯ 1,thenwhen α =
12 1d 1
{| | ··· | |}≤ ≤ | |
α¯, we have the optimal probability V. SUMMARY AND DISCUSSIONS
P =1 2√p1 p2 α12 2+ +pd α1d 2.
− | | ··· | |
It is recently known that besides quantum entangle-
p
2. If α¯ max α , , α , then when α = ment, other quantum correlations such as the quantum
12 1d 1
max ≤α , {|, α| ··,·P|rea|c}hes its optimal|po|int, discord are also useful in quantum information process-
12 1d
{| | ··· | |} ing. The physicaloroperationalinterpretationsofquan-
P = 1 p max α 2, , α 2 tum discord are still under exploration from different
1 12 1d
− {| | ··· | | }−
p α 2+ +p α 2 points of view , see e.g. [27, 28]. On the other hand,
2 12 d 1d
| | ··· | | . in the assistedstate discrimination[11], it is knownthat
max α 2, , α 2
{| 12| ··· | 1d| } the quantum discord is required. Further in this paper,
we find that only the “right” quantum discord is neces-
3. If 1 α¯, when α =1, we can obtainthe optimal
1 sary,while the “left” quantum discordcanbe zero. This
≤ | |
probability
clarifiesthe roleofquantumdiscordinassistedstate dis-
crimination. In particular, we find that as a resource
P =1 p p α 2 p α 2.
− 1− 2| 12| −···− d| 1d| for quantum informationprocessing,the use ofquantum
discord depends on the specified processing task. Ex-
As we have found that the “right” discord is always plicitly in the process of assisted state discrimination,
required for assisted state discrimination, but “left” dis- the measurement on the right hand side is performed,
cordcan be zero. In the following, we will show that the so the “right” quantum discord is necessary. In the ab-
“left” discord vanishes in the optimal process. sence of entanglement, if one side quantum discord is
As we already show that (20) and (21) are the condi- zero, depending on this specified one side measurement,
tions for vanishing “left” discord, in assisted state dis- thequantumcorrelationofdiscordislikeaclassicalstate.
crimination, a qudit is randomly prepared in one of the If applied in the assisted state discrimination, it means
d nonorthogonal states ψi (i =1, ,d) with an equal thatthe processis like aclassicaltype. Thus the “right”
| i ···
priori probability pj and an equal priori and nonzero quantum discord is really a useful resource in this pro-
overlap ψi ψj = γ 2 for all i=j. Since ψi is selected cess.
h | i | | 6 | i
inddimensionHilbertspace,thiscanalwaysbeachieved.
In this case, the parameter α¯ = √4d 1γ , and we also
have,max α , , α = γ 2 −α¯, t|hu|s the optimal
12 1d
{| | ··· | |} | | ≤
probability can be expressed as
P = 1− 2√dd−1|γ|2, 0≤|γ|≤ √4d1 1, (25) A“9c7k3n”opwrolegrdagmm(e2n0t1s0CTBh9is22w9o0r4k)iasnsduPppHoRrt2e0d10b0y71N0S7F.C,
opt ( d−d1(1−|γ|4), √4d1 1 ≤|γ|≤−1.
−
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