Sequential state discrimination with quantum correlation

0.21 MB·

by  Jin-Hua Zhang

Noname manuscript No.
(will be inserted by the editor)
Sequential state discrimination with quantum
correlation
Jin-Hua Zhang · Fu-Lin Zhang⋆ ·
Mai-Lin Liang
7
Received: date/Accepted: date
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0
2
Abstract We study the sequential unambiguous state discrimination (SSD)
n
with an ancilla system for initial states prepared with arbitrary priori prob-
a
J abilities. The results are compared with some strategies that allow classical
communication. It’s found that the optimal success probability of SSD is en-
9
hanced by the deviation from equal probability.The proportionof left (right)

] discord in their total is positively (negatively) correlated with the informa-
h
tion extracted by Bob. Identifying unequal prior states calls for less quantum
p
- correlation than the equal prior case.
t
n
Keywords Sequential state discrimination Entanglement Discord
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· ·
u
q
[
1 Introduction

1
v
It is crucialto makeclear the rolesof quantumcorrelationsinquantum infor-
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0 mation processing. Such correlations have been widely investigated in various
1 perspectives.Manyofthemarerelatedtoquantumentanglement[1],Bellnon-
2 locality [2], and quantum discord [3,4]. The results in studies [5,6] show that
0
the algorithmfordeterministic quantumcomputationwithonequbit(DQC1)
.
1 can surpass the performance of the corresponding classical algorithm in the
0
absence of entanglement between the control qubit and a completely mixed
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state. Thus, the entanglement which had been regarded as the only resource
1
: for demonstrating the superiority of quantum information processing [1,7] is
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completely redundant [8]. The quantum discord, which gives a measurement
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X
r ⋆ Correspondingauthor: fl[email protected]
a
Jin-HuaZhang
DepartmentofPhysics,XinzhouTeacher’sUniversity,Xinzhou034000,China
Fu-LinZhang Mai-LinLiang
·
DepartmentofPhysics,SchoolofScience,TianjinUniversity,Tianjin300072, China
2 Jin-HuaZhangetal.
of the nonclassical correlations and can exist in a separable state, is consid-
ered to be the key resource in this quantum algorithm and has gained wide
attention in recent years [9,10]. Another type of quantum correlations called
dissonance and put forwardby [9] measuresthe nonclassicalcorrelationswith
entanglement being completely eliminated. For a separable state, its disso-
nance is exactly equal to the quantum discord which plays a key role in the
computational process.
Unambiguous discrimination among linearly independent nonorthogonal
quantum states is a fundamental subject in quantum information theory [11,
12,13,14,15] whichis the secondexample after DQC1 aidedonly by quantum
dissonance rather than entanglement[16,17]. The distinction of two unknown
nonorthogonal states of a qubit, Ψ and Ψ , prepared by Alice and sent
1 2
| i | i
to Bob, may sometimes fail as the price to pay for no error. Thus, the mea-
surement has three possible outcomes, 1, 2, 0 corresponding to Ψ , Ψ and
1 2
| i | i
failureoutcomerespectively.Thetopicofextractinginformationfromaquan-
tum system by multiple observers have been studied by [18,19,20].
Amongthemisthetheoryofnondestructivesequentialstatediscrimination
(SSD) putforwardbyBergouet al [20].Namely,afterhis discrimination,Bob
sendthe qubit to a next observerCharlie,who also performsanunambiguous
discrimination measurement to determine which state is prepared by Alice.
The success probability of Charlie’s measurement depends on the overlapbe-
tweenthetwopossiblestatesthatBob’smeasurementleavesinthequbit.Pang
et.al. [8]study the theory by using the languageof systemancilla andexplore
the roles of quantum correlations in such procedure. It is indicated that the
optimal success probability prepared for both Bob and Charlie to recognize
between the two states relies on the quantum dissonance which equals to dis-
cord as the entanglementis absent. However,only the special case with equal
priorisconcernedinthe twopublishedpapers.The aimofthe presentworkis
to investigate the influence of a priori probabilities and to check whether the
existing conclusions hold for the unequal prior cases.
The detailed discussions of SSD is presented in Sec. 2. We compare the
results of SSD with other three protocols that allows classicalcommunication
in Sec. 3. The effects of quantum correlations is discussed in Sec. 4. The final
section is a summary.
2 Sequential state discrimination
WenowconsiderSSDwitharbitrarypriori probabilities(theprotocolisshown
inFig.1).AqubitAsentbyAlicetoBobwaspreparedinoneofnonorthogonal
states Ψ or Ψ with their corresponding priori probabilities P , P (P +
1 2 1 2 1
| i | i
P = 1) respectively. After a joint unitary transformation U between A and
2 b
B is performed, the state of the composite system is obtained by Bob
P1 :Ub|Ψ1i|0bi=qq1b|χ1i|0ib+q1−q1b|φ1i|1ib,
P2 :Ub|Ψ2i|0bi=qq2b|χ2i|0ib+q1−q2b|φ2i|2ib, (1)
Sequential statediscriminationwithquantum correlation 3
Fig.1 ProtocolforSSD.AqubitAispreparedbyAliceinoneofthetwostates Ψ1 , Ψ2
| i | i
withpriori probabilitiesP1 ,P2 respectively.AfterthequbitissenttoBob,ajointunitary
operation is performed between the qubit A and an auxiliary qutrit B, followed by a von
Neumannmeasurementonthequtrit.Thestatediscriminationissuccessfuliftheoutcome
is1(for Ψ1 ) and2(for Ψ2 ), butunsuccessful ifthe outcome is 0.Then thequbit which
| i | i
isinthepostmeasurementstateissenttoCharlie.Charlieperformsasimilarjointunitary
operationUcbetweenitandhisqutritCandmakesanoptimalunambiguousdiscrimination
measurementonC.
with 0 , 1 , 2 as the basis of the ancilla and χ and φ as the pure
b b b 1,2 1,2
| i | i | i | i | i
states of A. If the state Ψ (i=1,2) is input, after his measurement, we say
i
| i
Bob succeeds in discrimination if the ancilla is collapses to i and the qubit
b
| i
to φ ; while he fails if the ancilla is onto 0 and A to χ . Because unitary
i b i
| i | i | i
operation doesn’t change the inner product of the two states Ψ Ψ = s,
1 2
h | i
so the failure state χ satisfies the constraint qbqb χ χ = s. Without
| ii p 1 2h 1| 2i
consideringthe lossofgenerality,we take the overlaps to be real(0 s 1).
≤ ≤
If we set χ χ =t, the success probability of Bob is obtained as
1 2
h | i
P =P (1 qb)+P (1 qb). (2)
b 1 − 1 2 − 2
Considering the constraint q q = s2 and q 1,q 1, the following
inequality should be satisfied q1 2 s2,tq2 s21. T≤hen t2he≤maximal success
1 ≥ t2 2 ≥ t2
probabilityP ofBobhastwo candidateswhichcanbe achievedfordiffer-
bmax
entvaluesofP .Thetwocasesare(i):0 P < s2 and(ii): s2 P 1
1 ≤ 1 s2+t2 s2+t2 ≤ 1 ≤ 2
with the maximal success probability
s2
(i): P =(1 P )(1 ); (3a)
bmax − 1 − t2
s
(ii): P =1 2 P P . (3b)
bmax − p 1 2t
P is attained for qb =1, P2s corresponding to the two cases respec-
bmax 1 qP1 t
tively. A special case (P = P = 1/2, P = 1 s) which is consistent
1 2 bmax − t
with the results in [8] corresponds to the minimum P of Bob. While the
bmax
maximum measurement probability in case (i) causes a fact that one of the
initial states (Ψ or Ψ ) corresponds to a result which is bound to fail is
1 2
| i | i
ignored. As P approaches 0 or 1, the maximum success probability of the
1
measurement occurs. Outwardly Alice has only one state (Ψ or Ψ ), but
1 2
| i | i
Bobdoesn’tknowthetruthbecausetheclassicalcommunicationisprohibited.
If the outcome is 0, he is still unsuccessful. Thus the success probability can’t
reach 1 despite only one state left by Alice.
4 Jin-HuaZhangetal.
Then, the qubit A is sent to the second observer Charlie. We assume that
Charlie knows Bob’s protocol. Because only pure states can be discriminated
by Charlie, which requires χ = φ , therefor the states in Eq.(1) become
i i
| i | i
P :U Ψ 0 = φ α ,
1 b 1 b 1 1 b
| i| i | i| i
P :U Ψ 0 = φ α , (4)
2 b 2 b 2 2 b
| i| i | i| i
where α = qb 0 + 1 qb i with i=1,2.After the qubit is received,
| iib p i| ib p − i| ib
Charlie makes a similar joint unitary operation U between the qubit A from
c
Boband his auxiliary qutrit C as is shown by the Eq.(1)with the parameters
qc and qc, followed by an optimal unambiguous measurement [20] on C. The
1 2
optimal success probability can be derived as
(i): P =(1 P )(1 t2), (5a)
cmax 1
− −
(ii): P =1 2 P P t, (5b)
cmax − p 1 2
corresponding to the two cases (i): 0 P < t2 and (ii): t2 P 1
≤ 1 1+t2 1+t2 ≤ 1 ≤ 2
respectively.
Forfixeds,it’sfoundthatP isnegativelyrelatedtoP .Theoverlap
cmax bmax
t (s t 1) is positively correlated with the information extracted by Bob
≤ ≤
afterhis measurement.Thuswhent increases,the informationleft forCharlie
to identify the states decreases on the contrary, that’s the reason why P
cmax
and P take on opposite relationship against the parameter t. t s leads
bmax
→
to the completely failure of discrimination for Bob while t 1 means the
→
extracting of all information for Bob and Charlie can’t use the same qubit to
discriminate the states.
The probability for both Bob and Charlie to identify the state can be
obtained as
PSSD =P (1 qb)(1 qc)+P (1 qb)(1 qc), (6)
1 − 1 − 1 2 − 2 − 2
which satisfies the constraint qbqb = s2/t2 and qcqc = t2. The optimal joint
1 2 1 2
success probability PSSD of Bob and Charlie has two candidates which can
S
be achievedfor different values of P . The two casesare (i): 0 P <P and
1 1 D
≤
(ii): P P 1 with the optimal success probability
D ≤ 1 ≤ 2
(i): PSSD =(1 P )(1 s)2; (7a)
S − 1 −
(ii): PSSD =1 4 P P s+s; (7b)
S − p 1 2
where P = P is the critical point corresponding to a continuous piecewise
1 D
function and the maximum PSSD is attained for qb = qc = 1, qb = qc = s,
S 1 1 2 2
t = √s and qb = qc = P2s, qb = qc = P1s, t = √s corresponding to the
1 1 qP1 2 2 qP2
twocasesrespectively.t=√sasanecessaryconditionforoptimalprobability
Sequential statediscriminationwithquantum correlation 5
(a)
1.0
0.8
0.6
SSDPS 0.4
0.2
0.0
0.0 0.1 0.2 0.3 0.4 0.5
P1
(b)
1.0
0.8
0.6
SSDPS 0.4
0.2
0.0
0.0 0.2 0.4 0.6 0.8 1.0
s
Fig. 2 Thejointoptimalsuccess probabilityPSSSD asafunctionoftheparameterP1 (for
s=0.04(solidline),0.36(dottedline)showninFig(a))ands(forP1=0.5(solidline),0.4
(dotted line),0.2(dashedline)asisshowninFig(b)).
of SSD is the same as the equal prior case in [8]. The result is presented in
Fig.2. We can find that the variation curve is symmetrical for P =1/2 (with
1
the success probability minimized for equal prior case while maximized for
extremely unequal prior case P = 1,0). This result shows that identifying
1
unequal prior initial states is much easier than the equal prior case. PSSD
S
equals to 1, 0 for s = 0,1 which means that we identifying two orthogonal
initialstates Ψ and Ψ isextremelyeasywhile distinguishingtwoidentical
1 2
| i | i
states is impossible. For larger values of s, the possibility of existence of the
result in the case (i) which means one of the initial states is ignored will be
enlarged and the impact of a priori probabilities on the outcome is weakened
whichisshowninFig.2(a).FordifferentvaluesofP ,themoreunequalapriori
1
probabilitiesoftheinitialstatesis,thetinierthepossibilityofexistenceofcase
(ii) will be and we are tend to be more probably to ignore one of the initial
states which is shown in Fig.2(b). One can draw a conclusion that symmetry
breaking in a priori probabilities of the prepared qubit leads to enhancement
of the success probability of SSD on the contrary.
3 Comparison with other protocols
Then we begin to consider the success probability of another three protocols
that allow Bob and Charlie to communicate classically with arbitrary priori
probabilities and compare the results with SSD.
(1)Bobperformsanoptimalunambiguousdiscriminationmeasurementon
the qubit he receives from Alice (which means all information of the qubit
6 Jin-HuaZhangetal.
extracted thus we should set t = φ φ = 1). No matter Bob succeeds or
1 2
h | i
fails he tells Charlie the results; but if he fails, the procedure ends at that
time. The optimal probability of both of them succeeding can be written as
(i): P(1) =(1 P )(1 s2); (8a)
S − 1 −
(ii): P(1) =1 2 P P s; (8b)
S − p 1 2
for the two cases 0 P < s2 and s2 P 1 respectively.
≤ 1 1+s2 1+s2 ≤ 1 ≤ 2
(2) After the qubit is received from Alice, Bob performs an optimal un-
ambiguous discrimination measurement on it. If he succeeds he will sends a
qubit which is in the same state as Alice has given him; but if he fails, the
procedureends. The probability ofboth of them succeedingcanbe written as
P(2) = [P (1 qb)+P (1 qb)][Pc(1 qc)+Pc(1 qc)] where Pc and Pc
1 − 1 2 − 2 1 − 1 2 − 2 1 2
are new priori probabilities of Charlie’s qubit corresponding to the optimal
discrimination of Bob with the constraints qcqc = qbqb = s2. The optimal
1 2 1 2
probability P(2) has two candidates which can be achieved for different val-
S
ues of P . The two cases are (i): 0 P < s2 where Pc = 0, Pc = 1
1 ≤ 1 1+s2 1 2
and (ii): s2 P 1 which contains two situations: (a) 0 Pc s2
1+s2 ≤ 1 ≤ 2 ≤ 1 ≤ 1+s2
and (b) 1+s2s2 < P1c ≤ 21 where P1c = P11−2√√PP11PP22ss and P2c = P12−2√√PP11PP22ss. The
corresponding optimal success probabilit−y P(2) can be written−as
S
(i): P(2) =(1 P )(1 s2)2; (9a)
S − 1 −
(iia): P(2) =(P P P s)(1 s2); (9b)
S 2−p 1 2 −
(iib): P(2) =(1 2 P P s)(1 2 PcPcs). (9c)
S − p 1 2 − p 1 2
(3) Bob performs a probabilistic unitary optimal clone operation on the
qubit he receives from Alice [21,22] and Bob’s cloning operation is shown as
U(Ψ )0 )=√γ Ψ Ψ α + 1 γ β β α ,i=1,2, (10)
| ii | i i| ii| ii| ii p − i| i| i| 0i
where 0 isainitializedstateoftheancillas. α and α areorthogonalstates
i 0
| i | i | i
of the flag corresponding to success or failure of the cloning respectively. γ
i
is the success probability of the cloning for the state Ψ and β is a genetic
i
| i | i
failurestate.TheaveragesuccesscloneprobabilityisP =P γ +P γ .The
clone 1 1 2 2
calculationdetailsforoptimalvalueofsuccesscloningprobabilityP can
m(clone)
be seen in Appendix.
If Bob succeeds in cloning, he keeps one of his clone qubits and sends
the other one to Charlie and both of them perform optimal unambiguous
discrimination to their qubits. If he fails he will inform Charlie and ends the
Sequential statediscriminationwithquantum correlation 7
1.0
0(cid:0)(cid:2)
0(cid:0)(cid:1)
PS
0.7
0.6
0.5
0.0 0.1 0.2 0.3 0.4 0.5
P1
Fig. 3 ThejointprobabilityPS asafunctionofP1 isshownfors=0.04correspondingto
the four strategies respectively. Solid line: PSSD, dotted line: P(1), dot-dashed line: P(2),
S S S
dashedline:P(3).
S
procedure. The probability of both of their succeeding is P(3) =P P P .
(clone) b c
Its optimal value can be derived
(3)
P =P P P , (11)
S m(clone) bmax cmax
where P and P equals to each other and they can be written as
bmax cmax
(i): P =(1 P0)(1 s2); (12a)
bmax − 1 −
(ii): Pbmax =1−2qP10P20s; (12b)
with P0 = P1γ1 , P0 = P2γ2 for the two cases 0 P0 < s2 and
1 P1γ1+P2γ2 2 P1γ1+P2γ2 ≤ 1 1+s2
s2 P0 1 respectively.
1+s2 ≤ 1 ≤ 2
It’sshowninFig.3thattheminimumsucceedingprobabilityofdiscrimina-
tioncorrespondsto the equalpriorstates identificationwhichis moredifficult
than the unequal case irrespective of which strategies we adopt. The protocol
(1) has the most optimal success probability (the results in protocol (2),(3)
differsonlyalittlefromprotocol(1))whiletheSSDstrategyattainstheweak-
est.Comparedwith the resultsforP =P =1/2[20],the successprobability
1 2
difference between SSD and the other three strategies is reduced for unequal
prior cases.
Then we consider the probability that at least one of the parties succeeds
in identifying the states (shown in Fig.4). For SSD protocol, we have known
thatPSSD =P (1 qbqc)+P (1 qbqc)withthesameconstraintsasEq.(6).
∗ 1 − 1 1 2 − 2 2
The optimal probability is attained PSSD = P(1). The results is the same
S ∗ S
for protocol (1), (2). But for the cloning protocol, P(3) = P [1 (P0qb +
∗ c − 1 1
8 Jin-HuaZhangetal.
(cid:3)(cid:4)(cid:7)(cid:3)
(cid:3)(cid:4)(cid:5)(cid:6)
(cid:3)(cid:4)(cid:5)(cid:3)
*P 0.75
0.70
0.65
0.60
0.0 0.1 0.2 0.3 0.4 0.5
P1
Fig. 4 TheprobabilityP∗ forone ofthe twoparties succeeds asafunction ofP1 forfour
strategies.Solidline:PSSD∗,P(1)∗,P(2)∗,dashedline:P(3)∗
S S S S
P0qb)(P0qc +P0qc)] with the constraint qbqb = qcqc = s2. Its optimal value
2 2 1 1 2 2 1 2 1 2
PS(3)∗ has two candidates which can be achieved for different values of P10.
The two cases are (i): 0 P0 < s2 and (ii): s2 P0 1 with the
≤ 1 1+s2 1+s2 ≤ 1 ≤ 2
maximal success probability
(i): PS(3)∗ =Pcmax[1−(P10+P20s2)2]; (13a)
(ii): PS(3)∗ =Pcmax(1−4P10P20s2). (13b)
The outcomes in [20] correspondingto the equal prior cases indicates that
PS(1)∗ =PS(2)∗ =PS(3)∗ =PSSSD∗ =1−s which is a special case for our results
(shown in Fig.4). The clone protocol presents some slight advantages in the
success probability of discrimination compared with the other three which is
equivalent to the results of P(1). When P approaches0, 1, 1/2,the results of
S 1
the four protocols equal to each other.
Then which protocol will do better? If we require both Bob and Charlie
succeeding, the protocol (1) will do best. But for unequal prior case, the dif-
ference between the protocol (1) and the other three shrinks a lot. While if
we require at least one of them succeeds, the situation is almost the same for
the fourstrategies.However,weshouldmentionafactthatSSDprotocoluses
onlyonequbitwhilethe otherthreestrategiesusetwo.Forthe overallconsid-
eration of economical use of quantum information source and high efficiency
of state discrimination, we take SSD protocol as our better choice.
Sequential statediscriminationwithquantum correlation 9
4 Correlations
The results in [8] indicates that the success probability of SSD is positively
correlatedwithquantumcorrelationsforthespecialequalpriorcase.Thenfor
the general unequal prior one, what is the relation of the success probability
of SSD, arbitrary priori probabilities P , P and the quantum correlation
1 2
betweenAandB?That’sthemainquestionwenowfocuson.Choosing φ =
i
| i
χ ,afterBob’sunitaryoperation,the stateofoursystemwhichcontainsthe
i
| i
ancilla can be written as
ρAB =Ub[P1|Ψ1ihΨ1|⊗|0ibh0|b+P2|Ψ2ihΨ2|⊗|0ibh0|b]Ub†
=P φ φ α α +P φ φ α α . (14)
1 1 1 1 b 1 b 2 2 2 2 b 2 b
| ih |⊗| i h | | ih |⊗| i h |
This is a mixed separable state obviously. A and B are completely disentan-
gled.Apartfromthis,we turnto akindofcorrelation-quantumdiscordwhich
contains the right discord D and the left discord D [8].
AB BA
Forthetwo-ranksystemρ ,byusingtheKoashi-Winteridentity[23],its
AB
discord can be can be obtained as a reduced state of the tripartite state
Ψ = P φ α 0 + P φ α 1 , (15)
| i p 1| 1i| 1ib| id p 2| 2i| 2ib| id
where 0 , 1 is the basis of a fictitious qubit D. The right discord can be
d d
| i | i
expressed as
D(ρ )=S(ρ ) S(ρ )+E(ρ ), (16)
AB B D AD
−
where S(ρ ),S(ρ ) and E(ρ ) correspond to the reduced entropy of B, D
B D AD
and the entanglement of formation between A and D respectively. Go to a
step further, it can be expressed explicitly as
D(ρ )=F(τ ) F(τ )+F(τ τ ), (17)
AB B D A ABD
− −
where
1+√1 x 1+√1 x 1 √1 x 1 √1 x
F(x)= − log − − − log − − . (18)
− 2 2 2 − 2 2 2
τ istheresidualtangleofthetripartitestate[24],τ (similarlyforτ and
ABD A B
τ ) is the tangle between A and B, D. They can be obtained by
D
τ =4P P (1 t2)(1 r2),τ =4P P (1 t2),
ABD 1 2 A 1 2
− − −
τ =4P P (1 r2),τ =4P P (1 t2r2), (19)
B 1 2 D 1 2
− −
wherewesetr =s/t= α α .WecanalsogettheleftdiscordD(ρ )easily
1 2 BA
h | i
by interchanging the subscript A and B in Eq.(17). To analyze their roles in
SSD, we define the proportion of left (right) discord in their total
D(ρ ) D(ρ )
BA AB
D = ,D = , (20)
left right
D(ρ )+D(ρ ) D(ρ )+D(ρ )
e AB BA e AB BA
10 Jin-HuaZhangetal.
and a symmetrized discord
D = D(ρ )D(ρ ). (21)
symm p BA AB
Itis obviouslyshowninFig.5(a)that the proportionof left (right)discord
in their total is positively (negatively) correlated with the parameter t which
correspondstothe informationextractedbyBobafterhismeasurement.Both
D and D change very little toward the priori probability P (shown
left right 1
ien Fig.5(b))e. The maximum symmetrized discord of initial state is obtained
at P = P = 1/2 which is shown in Fig.5(c) while the success probability
1 2
of SSD becomes weakest on the contrary. The deviation from the equal prior
case for the initial state enhances the success probability of quantum state
discriminationbutdestroysboththe rightandthe leftdiscordbetweenAand
B conversely. When P approaches 0 or 1, the system-ancilla state turns into
1
a pure product state with two parts have no quantum correlation with each
other and both left and right discord equals to 0 (optimal success probability
of SSD can de derived there according to the results in Fig.2). Hence, we can
drawa conclusionthat distinguishing the initial states Ψ and Ψ prepared
1 2
| i | i
in equal priors is most difficult and requires more quantum correlation than
the unequal prior case.
Othercalculationsshowthatbothleftandrightdiscordalsoequalto0for
s 0,1.Becauseidentifyingtwoorthogonalinitialstatesismucheasierwhile
→
distinguishing two identical states is impossible, these two procedures don’t
require any quantum correlation.
For a fixed value of s, we set t = s1/n. Through a simple calculation,
we have known that n = 1 (t = s) corresponding to a complete failure of
discrimination for Bob also makes A and B uncorrelated. When n = 2 (t =
√s), left discord equals to the right one. For arbitrary priori probabilities,
t=√sasanecessaryconditionforoptimalvalueofbothquantumcorrelation
andsuccessprobabilityofSSDisthesameastheresultsforspecialequalprior
case.Thus,theconclusionsin[8]aregeneralizedhere.Thefurtherincreaseofn
(tincreasesaswell)signifyingmoreinformationextractedinducesamuchmore
severe destroying effect on the right discord than the left one. When n
→ ∞
(t 1), we can easily obtain D = D = 0 which signifies the complete
BA AB
→
death of the quantum correlation between A and B. This result means that
Bob performs an optimal state discrimination and extracts all information
while Charlie coulddo nothing exceptresortingto classicalcommunicationas
is mentioned in the above three protocols if he wants to know the result.
So one can draw a conclusion that quantum correlation disappears in the
following three cases: (1) initial states prepared with extremely unequal prior
(P 0,1); (2) orthogonal initial states (s = 0) to be distinguished; (3) the
1
→
optimal state discrimination performed by Bob (t=1). The deviation for the
overlap ψ ψ , α α and the priori probability P from their maximum
1 2 1 2 1
h | i h | i
andminimummayavoidtheoccurrenceofquantumcorrelationdeathbetween
A and B.