Upper bound on the success probability for unambiguous discrimination

Physics Letters A 303 (2002) 140–146 www.elsevier.com/locate/pla

Upper bound on the success probability for unambiguous discrimination Daowen Qiu

a,b

a Department of Computer Science, Zhongshan University, Guangzhou 510275, PR China b Department of Computer Science and Technology, Tsinghua University, Beijing 100084, PR China

Received 22 August 2002; received in revised form 6 September 2002; accepted 6 September 2002 Communicated by P.R. Holland

Abstract With generalized measurement transformations, we derive an upper bound on the success probability for unambiguous discrimination among n states with respective a priori probabilities, which improves the existing bound in the literature. Interestingly, by exploiting the no-signaling condition independent of any measurement transformation, we can also obtain the same bound as above.  2002 Elsevier Science B.V. All rights reserved. PACS: 03.67.-a; 03.65.Ta; 03.65.Ud Keywords: Unambiguous state discrimination; Quantum measurement

In quantum information processing, distinguishing states is a fascinating and important issue [1], which is also related closely to quantum cloning [2–6] with the common features of incompletabilities. Discrimination of states has close connection to quantum measurement [5,7]. Roughly speaking, distinguishing states means that by considering a quantum system whose unknown state belongs to a finite, known set, then we try to devise a measurement yielding the most information from the initial state [5,7]. As is known, orthogonal states can be reliably distinguished with von Neumann measurement, but nonorthogonal ones are only discriminated in approximate fashions. In general, there are maybe two main strategies for discriminating quantum states (for the details we refer to [5]): one is by assuming the measurement gives never inconclusive results, but will incorrectly identify the states with certain probability, wherefore it follows the absolute maximum probability for discrimination expressed by the well-known Helstrom limit [7], and one may refer to [8] on the recent work; contrarily, the other is so-called unambiguous state discrimination [5,9–12] that means though the measurement gives without incorrect results, inconclusive ones arise likely. Of course, in some of the existing literature, for example, in [13] the discussion to a certain extent combines the above two strategies for discrimination. More interestingly, by exploiting the no-signaling condition, without reference to any measurement transformation,

E-mail addresses: [email protected], [email protected] (D. Qiu). 0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 1 2 3 6 - 7

D. Qiu / Physics Letters A 303 (2002) 140–146

141

Barnett and Andersson [14] derived a tight bound on unambiguously discriminating two nonorthogonal states, which exactly agrees with that given by Jaeger and Shimony [12]. Also, discriminating quantum states has been experimentally performed by Hutterner et al. [15] and then by Clarke et al. [16]. Unambiguous state discrimination originated from the works of Ivanovic [9], Dieks [10] and Peres [11], who initially distinguished two nonorthogonal states |φ and |ψ with the same prior probabilities p and q, i.e., p = q = 1/2, and derived the maximum probability of success called IDP limit as 1 − |φ|ψ|. Subsequently, Jaeger and Shimony [12] generalized it to the case of unequal a priori probabilities and obtained the results as √ 1 − 2 pq |φ|ψ| or p(1 − |φ|ψ|) in case p
q. As pointed out above, recently, the authors in [14] calculated these results by exploiting the no-signal condition, independent of any quantum measurement transformation. Actually, in recent years, unambiguous state discrimination has undergone intriguing extensions and further development [17–22]. Peres and Terno [17] discussed in detail the problem of optimal distinction of three states having arbitrary a priori probabilities. Chefles [18] showed that a set {|ψi } of states is amenable to unambiguous state discrimination, if and only if they are linearly independent; and Chefles in [19] dealt with unambiguous state discrimination between linearly dependent states with multiple copies. The optimal unambiguous discrimination among linearly independent symmetric states was solved in [20]. More recently, with Lagrange multiplier, Ref. [21] presented a scheme for calculating the optimum probabilities of unambiguous discrimination among linearly independent, nonorthogonal states, but the concrete procedure of calculation would be considerably complicated for the situation of more than three states, so the optimal solution is unknown yet. However, deriving some upper bounds on the success probability for unambiguous discrimination among states is always possible [6,22]. Notably, Chefles and Barnett in Ref. [4] proposed a more general operation than unambiguous discrimination, called quantum state separation, for the case of two states with equal prior probability 1/2. That is to say, considering a quantum system prepared in one of the two states |ϕ 1  and |ψ 1  with equal a priori probabilities, we aim to transform the two states into |ϕ 2  and |ψ 2 , respectively, such that
 2
2 
2
 1
1 
2
ϕ
ψ

ϕ
ψ
, making them more distinct. However, such operation cannot be always successful, so an upper bound on the probability PS of the state separation being successfully implemented was derived in [4] as PS 

1 − |ϕ 1 |ψ 1 | , 1 − |ϕ 2 |ψ 2 |

and they analyzed that the IDP limit (when |ϕ 2 |ψ 2 | = 0) and the bound on the success probability for the probabilistic cloning machine [3] are exactly its special cases. In [22], we generalized it to the situation of n nonorthogonal states having respective a priori probabilities, and derived an upper bound on the success probability for separating nonorthogonal states from |ψ1(1) , |ψ2(1) , . . . , |ψn(1)  to |ψ1(2) , |ψ2(2) , . . . , |ψn(2)  as follows: √ 1 1 pi + pj 1  pi pj |ψi |ψj | 1  − , (1) n−1 1 − |ψi2 |ψj2 | n − 1 i =j 1 − |ψi2 |ψj2 | i
which is also a main result in [22]. It is easy to see that if n = 2 then the above bound exactly agrees with IDP limit [9–11]. In this Letter, by exploiting the method of measurement transformations for state separation technique built

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D. Qiu / Physics Letters A 303 (2002) 140–146

on the work of Ref. [4] we derive an upper bound for unambiguously distinguishing nonorthogonal states that is not bigger than (2) and thus improve the previous result to a certain extent. Furthermore, by utilizing the no-signaling condition, we generalize the results in Ref. [14] on unambiguous state discrimination from two states to general n states, and obtain the same bound as that derived by measurement transformations. Let a quantum system be described by one of the finite states |ψ1 , |ψ2 , . . . , |ψn  with probability distribution p1 , p2 , . . . , pn . Assume that Aˆ Sk and Aˆ F k represent some linear transformation operators, where Aˆ Sk stand for the successful transformations, while Aˆ F k mean the failure ones. They satisfy the identity equation:  †  ˆ (3) Aˆ Sk Aˆ Sk + Aˆ †F k Aˆ F k = 1. k

These operators act as follows:
 Aˆ Sk
ψi1 = ski |χi ,
 Aˆ F k
ψi1 = fki |φi ,

(4) (5)

for each i ∈ {1, 2, . . . , n} with some complex coefficients ski and fki and normalized states |φi , where |χ1 , |χ2 , . . . , |χn  are orthonormal. First note that with Eqs. (3)–(5) we have   (6) |ski |2 + |fki |2 = 1, k

for each i = 1, 2, . . . , n. Then the success and failure probabilities are clearly defined as, respectively, P=

n 

Q=

i=1 n 

pi



|ski |2 ,

(7)

|fki |2 ,

(8)

k

pi

 k

i=1

which obviously satisfy P + Q = 1. In what follows, we aim to estimate their bounds. First note Q=

n 

pi ψi |



Aˆ †F k Aˆ F k |ψi .

k

i=1

With Cauchy–Schwarz inequality, we have (n − 1)

n 

pi2

i=1

=



2   † ˆ ˆ ψi | AF k AF k |ψi 



k

  2 2

 †  † 2 2 ˆ ˆ ˆ ˆ pi ψi | AF k AF k |ψi  + pj ψj | AF k AF k |ψj 

1
i





k

2pi pj ψi |

1
i
=

 i =j

pi pj ψi |

 k



Aˆ †F k Aˆ F k |ψi ψj |

k

Aˆ †F k Aˆ F k |ψi ψj |

 k



k

Aˆ †F k Aˆ F k |ψj 

k

Aˆ †F k Aˆ F k |ψj ,

D. Qiu / Physics Letters A 303 (2002) 140–146

and therefore 2  n  †  Aˆ F k Aˆ F k |ψi 
pi2 ψi | i=1

So

 Q = 2

n 

pi ψi |

i=1

=

n 


= =



 Aˆ †F k Aˆ F k |ψi 

n 

pj ψj |

j =1

k

k



(9)

k

Aˆ †F k Aˆ F k |ψj 

k

 2   †  †  † pi2 ψi | pi pj ψi | Aˆ F k Aˆ F k |ψi  + Aˆ F k Aˆ F k |ψi ψj | Aˆ F k Aˆ F k |ψj 

i=1




 †  † 1  Aˆ F k Aˆ F k |ψi ψj | Aˆ F k Aˆ F k |ψj . pi pj ψi | n−1 i =j

k

143

i =j

k

k

 †  † n  pi pj ψi | Aˆ F k Aˆ F k |ψi ψj | Aˆ F k Aˆ F k |ψj  n−1 i =j k k

2  

n Aˆ †F k Aˆ F k |ψj 

pi pj

ψi | n−1 i =j k

2  † 

n
ˆ ˆ ˆ pi pj
ψi |1 − ASk ASk |ψj 

n−1 i =j k

2 n  pi pj
ψi |ψj 
, n−1

k

i =j

and therefore


2
n  Q
pi pj
ψi |ψj 
, n−1

(10)

i =j

where with Cauchy–Schwarz inequality we have utilized

2  †  †  †


ψi | ˆ Aˆ F k |ψj 
 ψi | A Aˆ F k Aˆ F k |ψi ψj | Aˆ F k Aˆ F k |ψj . Fk

k

k

k

Equivalently to inequality (10)


2
n  P 1− pi pj
ψi |ψj 
. n−1

(11)

i =j

By using Cauchy–Schwarz inequality again we have  


2
2 n  1 √



pi pj ψi |ψj 
pi pj ψi |ψj  , n−1 n−1 i =j

and hence P 1−

(12)

i =j


2


n  1 √ pi pj
ψi |ψj 
 1 − pi pj
ψi |ψj 
. n−1 n−1 i =j

(13)

i =j

Since the equalities in (12) and (13) hold if and only if the terms pi pj |ψi |ψj |2 with i = j are equivalent to one another, it is seen that to a certain extent the derived bound is more precise than (2) in case of n
3. Nonetheless,

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D. Qiu / Physics Letters A 303 (2002) 140–146

the optimum solution is unknown yet and it may be intractable to solve this problem by the usual method of quantum measurement transformations, so it remains to be studied in future. As mentioned above, basing on the no-signaling condition, without reference to any measurement transformation completely, the authors in Ref. [14] also derived the same bound on unambiguous state discrimination between two nonorthogonal states as that obtained by Jaeger and Shimony [12]. Interestingly, by checking their procedure of derivation step by step, we find it can positively be generalized to the situation of distinguishing n states, and likewise results in the same bound as (10) we just obtain. For the sake of completeness, we outline a succinct process in the following and refer to [14] for the details. Suppose two separated particles are prepared in the entangled quantum state |Ψ  =

n  √

pi |χi R |ψi L ,

(14)

i=1

where |χ1 R , |χ2 R , . . . , |χn R are orthogonal basis states. In terms of the no-signaling condition, measurement of the left system will not change the reduced density matrix of the right system, so √ √  p1 p2 p1 ψ2 |ψ1  · · · p p ψ |ψ   √ √ n 1 n 1 p2 ··· pn p2 ψn |ψ2      p1 p2 ψ1 |ψ2  , ρR = trL |Ψ Ψ | =  (15) . . .. ..   .. .. . . √ √ p1 pn ψ1 |ψn  p2 pn ψ2 |ψn  · · · pn where ψi | = |ψi L (i = 1, 2, . . . , n) and these representations are valid in the sequel. Indeed, after the measurement of the left particle, the density matrix of the right particle can be expressed as     x11 x12 · · · x1n P1 0 · · · 0  x12 x22 · · · x2n   0 P2 · · · 0  + Q , ρR =  (16) ..  .. .. . . ..  ..   ...  ... . . . . .  . 0

0

···

Pn

xn1

xn2

···

xnn

where Pi mean the success probabilities of the result of measurement being |ψi L (i = 1, 2, . . . , n), and Q represents the failure one, together with a density matrix (xij ) to be determined. By the no-signaling condition, Eq. (15) and Eq. (16) are equivalent, so we have Qxii + Pi = pi , i = 1, 2, . . . , n, √ Qxij = pi pj ψj |ψi , i = j.

(17) (18)

From Eq. (17) it follows that Q

n 

xii +

i=1

n  i=1

Pi =

n 

pi = 1,

(19)

i=1

and therefore n 

xii = 1, i=1 n i=1 Pi is the

(20)

since success probability and Q + P = 1. Also, Eq. (20) can be directly yielded with the property of density matrix [1], i.e., tr(xij ) = 1. Similarly, another property of density matrix [1], that is, tr(xij )2  1, results in n  n  i=1 j =1

xij xj i = tr(xij )2  1.

(21)

D. Qiu / Physics Letters A 303 (2002) 140–146

With Cauchy–Schwarz inequality we have 2  n n  1  2 xii
xii . n i=1

145

(22)

i=1

From Eq. (21) it follows that n  n 

xij xj i =

i=1 j =1

 i =j

xij xj i +

n 

xii2  1.

(23)

i=1

By combining Eqs. (23) and (22) with (20) we obtain that  n−1 . xij xj i  n

(24)

i =j

As a consequence, with Eq. (18) and inequality (24) we have that  2
2
n  i =j pi pj |ψj |ψi | 2  Q =
pi pj
ψj |ψi 
, n−1 i =j xij xj i

(25)

i =j

which is consistent absolutely with the above bound described by inequality (10). To conclude, we have obtained a more superior bound on unambiguous discrimination among n states with respective a priori probabilities. However, the concrete optimum solution has not yet been calculated and it is worth being further studied in future. On the other, as the improved bound in this Letter, a natural question inspired is whether the bound on quantum state separation [4,6] among n nonorthogonal states can also be ameliorated correspondingly. We shall consider these problems elsewhere.

Acknowledgements I am grateful to the anonymous referee for valuable comments and suggestions. Thanks are also to those comrades in the seminar of quantum computation. This work was supported by the National Key Project for Basic Research and the Natural Science Foundation of Guangdong Province (Grant No. 020146) of China.

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