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A generalized programmable unambiguous state discriminator for unknown qubit systems
Physics Letters A 359 (2006) 103–109 www.elsevier.com/locate/pla
A generalized programmable unambiguous state discriminator for unknown qubit systems Bing He ∗ , János A. Bergou Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10021, USA Received 25 February 2006; received in revised form 7 April 2006; accepted 2 May 2006 Available online 11 May 2006 Communicated by P.R. Holland
Abstract We discuss a state discriminator that unambiguously distinguishes between two quantum registers prepared with multiple copies of two unknown qubits. This device achieves the optimal performance by von Neumann measurement and general POVM in different ranges of the input preparation probabilities, respectively, and in the limit of very large program registers the optimal measurement reduces to only von Neumann measurements. © 2006 Elsevier B.V. All rights reserved. PACS: 03.65.Bz; 03.67.-a; 03.67.Hk
The discrimination between a couple of linearly independent pure states |ψ1 and |ψ2 is possible only in two situations: We have complete classical knowledge of them, or the states carry some symmetry. Even in these situations, their identification is achieved only with a certain probability if these states are non-orthogonal. There are strategies to achieve the optimal result: First, the strategy that determines the state with minimum probability for the error [1,2] and, second, unambiguous discrimination (UD) that allows the possibility of an inconclusive result (with a minimal probability in the optimal case) [3–7]. In the recent years, the minimax approach to the discrimination of quantum states [8] is also proposed, where the situation with no information about the a priori probabilities of inputs can be dealt with. For practical applications, we need to look for the devices that achieve the optimal discrimination of a couple of quantum states in a versatile way, i.e. first of all, it works universally with any couple of the target states for discrimination and, secondly, it should perform optimally in the whole range of their a priori probabilities, or preparation probabilities, possibly arising in the preparation period. To realize the unknown quantum states (no classical information available) discrimination, we rely on the symmetry from having the available copies of the states we want to identify [9–12]. Here we only consider the UD for the unknown states. In [8] the authors proposed a state discriminator, in which the information about two unknown qubits is supplied in the form of the program in a programmable device [13] using 1 copy of these qubits as the program or reference digits. This problem was generalized to the situation of multiple copies used as the reference [11,12]. In [11], by means of the representation theory of U (n) group, Hayashi et al. obtained the optimal UD solution for the averages of two uniformly distributed unknown qudit systems prepared with n copies of reference plus 1 copy of data, while in [12] we studied by Jordan basis method the corresponding problem for two uniformly distributed unknown qubit systems prepared with n copies of reference plus m copies of data and with arbitrary a priori probabilities. All these results are about the optimal UD for the averages of systems. In this Letter, we construct a device that works on the exact input states without first taking the average of the systems; each time only one prepared input state is fed into the machine and, with a non-zero success probability, the measurement in the machine tells us which one it is.
* Corresponding author.
E-mail address: [email protected] (B. He). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.05.002
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We start with the generalization of the inputs in [9]. Here we store arbitrary n copies of qubits |ψ1 1 and |ψ2 2 in the program part of the quantum registers prepared as follows: in in Ψ = |ψ1 1 |ψ2 2 · · · |ψ1 2n−1 |ψ2 2n |ψ2 2n+1 , Ψ = |ψ1 1 |ψ2 2 · · · |ψ1 2n−1 |ψ2 2n |ψ1 2n+1 , (1) 1 2 where the data, either |ψ1 or |ψ2 , at the tail is appended to the program, a string of tensor product by copies of two unknown qubits. Without loss of generality in the preparation of the program parts, we place |ψ1 in the odd positions and |ψ2 in the even positions of the inputs.1 We need to find an optimal measurement to unambiguously distinguish between these inputs, bearing in mind the data qubits, |ψi = cos(θi /2)|0 + sin(θi /2)eiφi |1,
(2)
where i = 1, 2, are randomly (uniformly or non-uniformly) distributed on Bloch sphere with no available information about θi and φi . For generality, these quantum registers are assumed to be prepared with a priori probabilities η1 and η2 , respectively. The optimal measurement performed by our device is generally achieved by a positive operator value measure (POVM), and we denote its element of unambiguously detecting |Ψ1in as Π1 , that of unambiguously detecting |Ψ2in as Π2 , and that corresponding to failure as Π0 . They satisfy unit decomposition: I = Π1 + Π2 + Π0 .
(3)
The probabilities of successfully identifying these two possible input states are given by in in in in Ψ2 Π2 Ψ2 = p2 , Ψ1 Π1 Ψ1 = p1 ,
(4)
and the condition of no error implies that Π1 Ψ2in = 0. Π2 Ψ1in = 0,
(5)
Without any knowledge about |ψ1 and |ψ2 , what we can make use of in determining the inputs is their symmetry, i.e. |Ψ1in is invariant under the action of symmetric group on the odd positions and the tail position, while |Ψ2in invariant under the corresponding action on the even positions and the tail. To construct Π1 and Π2 that perform the UD of the inputs, we define the following orthonormal basis: 1 |e0 = |0, 0, . . . , 0, Cn0 1 |1, 0, . . . , 0 + |0, 1, . . . , 0 + · · · + |0, 0, . . . , 1 , |e1 = Cn1 .. . 1 |ek = |1, 1, . . . , 0, 0 +|0, 1, 1, . . . , 0 + · · · + |0, 0, . . . , 1, 1 ,
Ck n
k’s 1 in n digits
summation of Cnk terms
.. . 1 |en = |1, 1, . . . , 1, Cnn
(6)
where Cnk is the number of ways to choose k objects from a group of n objects without regard to order. In terms of this basis, the tensor product of n copies of a qubit is expanded as follows: n
⊗n cos(θ/2)|0 + sin(θ/2)eiφ |1 (7) = cosn−k (θ/2) sink (θ/2)eikφ Cnk |ek . k=0
Thus we construct the POVM elements satisfying the unambiguity (no error) in measurement: Π2 = c2 (IO,T − PO,T ) ⊗ IE , Π1 = c1 (IE,T − PE,T ) ⊗ IO , (8) n+1 n+1 where PE,T = k=0 |ek E,T E,T ek | (respectively PO,T = k=0 |ek O,T O,T ek |) is the projection operator onto the totally symmetric subspace with respect to the n even (respectively odd) positions and the tail position, IE,T (respectively IO,T ) the direct 1 The program registers can be set up with the provided 2n qubit copies in any specified order, and then for the POVM elements in Eq. (8) we only need to choose the corresponding projectors onto the totally symmetric spaces of the data and the program registers.
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product of unit matrices in these positions, and c1 , c2 the non-negative parameters arising from the unequal a priori probabilities of |Ψ1in and |Ψ2in . IO and IE are the direct products of the unit matrices in the odd and the even positions of the input states, respectively. Using these Πi , where i = 1, 2, we find their successfully detecting probabilities pi = Ψiin Πi Ψiin = ci − ci Ψiin PR,T ⊗ IS Ψiin , (9) where R = E, O but S = O, E. We use two tricks to calculate the average of the projection operators more conveniently. First, by the expansion k−1 k Cn Cn |e |0 + |ek−1 R |1T , |ek R,T = (10) k R T k k Cn+1 Cn+1 for k = 1, 2, . . . , n, we rewrite the projection operators as PR,T ⊗ IS = |e0 R |0T R e0 |T 0|
+
l−1 l−1 l Cn Cn C n |el R |0T + l |el−1 R |1T R el |T 0| + R el−1 |T 1| l l l Cn+1 Cn+1 Cn+1 Cn+1
n Cnl l=1
+ |en R |1T R en |T 1| ⊗ IS .
(11)
Second, we apply Eq. (7) to expand the input states by two parts: n
in n−l l ilφj Ψ = cos (θ /2) sin (θ /2)e Cnl |el R cos(θi /2)|0T + sin(θi /2)eiφi |1T j j i l=0
×
|Ψi
n
cosn−k (θi /2) sink (θi /2)eikφi Cnk |ek S ,
k=0
|Ψi
(12)
where i = 1, 2 but j = 2, 1. Putting the expectation values from these parts together, we obtain n in in k+1 2 n−k+1 2 Ψi PR,T ⊗ IS Ψi = cos (θi /2) + sin (θi /2) Cnk cos2(n−k) (θj /2) sin2k (θj /2) n+1 n+1 k=0
n 2k + C k cos2(n−k) (θj /2) sin2k (θj /2) cos(θj /2) cos(θi /2) sin(θi /2) cos(φi − φj ). n+1 n
(13)
k=1
The total success probability P to discriminate between the two unknown states, assuming |Ψ1in is produced with a priori probability η1 and |Ψ2in with η2 , is given by P = η 1 p1 + η 2 p2 .
(14)
Since we have no knowledge about the data states, what we use to indicate how well the device performs is the average of this success probability: 2 1 n dφ dθj sin θj η1 Ψ1in Π1 Ψ1in + η2 Ψ2in Π2 Ψ2in = (η1 c1 + η2 c2 ) . j 2 2(n + 1) (4π) 2π
P¯ =
j =1 0
π
(15)
0
We want to maximize this expression subject to the constraint that Π0 is a positive operator. Let H be the Hilbert space of the input states in given in Eq. (1), which, as it is seen from Eq. (10), is spanned by the orthonormal basis {|el O |em E |0T , |el O |em E |1T }, where l, m = 0, 1, . . . , n. To have a nice matrix representation of the inconclusive operator, Π0 = I − Π1 − Π2 = (1 − c1 − c2 )I + c1 PE,T ⊗ IO + c2 PO,T ⊗ IE ,
(16)
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we apply the following orthogonal transformations: Cnm Cnm−1 |e |e |0 + |ηlm = l O m E T m m |el O |em−1 E |1T , Cn+1 Cn+1 Cnm−1 Cnm |χlm = |el O |em E |0T − m m |el O |em−1 E |1T , Cn+1 Cn+1
(17)
where l = 0, 1, . . . , n, and m = 1, 2, . . . , n. The transformed vectors |ηlm and |χlm satisfy ηlm |χl m = 0, ηlm |ηl m = δl,l δm,m , and χlm |χl m = δl,l δm,m . If we put them together with the untransformed |el O |e0 E |0T and |el O |en E |1T , for l = 0, 1, . . . , n, all these unit vectors form an orthonormal basis, {|ηlm , |χlm , |el O |e0 E |0T , |el O |en E |1T }, of H . With this basis, the operator Π0 is represented by the direct sum of two series of block diagonal matrices: (1)
(2)
Π0 = Π0 ⊕ Π0 , where
⎛
⎜ ⎜ (1) Π0 = ⎜ ⎜ ⎝
(18) ⎞
J0
⎟ ⎟ ⎟, ⎟ ⎠
J1 J2 ..
.
⎛ ⎜ ⎜ (2) Π0 = ⎜ ⎜ ⎝
⎞
K0
⎟ ⎟ ⎟. ⎟ ⎠
K1 K2 ..
.
Jn
(19)
Kn
These block diagonal matrices are generated regularly; J0 and K0 are 1 × 1 matrix 1, and J1 is a 3 × 3 matrix: √ ⎞ ⎛ n 1 n c2 c − c 1 − n+1 2 2 3/2 3/2 (n+1) (n+1) ⎟ ⎜ √ ⎟ ⎜ n n3/2 n J1 = ⎜ (n+1)3/2 c2 1 − (n+1)2 c2 c ⎟, 2 2 (n+1) ⎠ ⎝ 3/2 2 n n n − (n+1)3/2 c2 c 1 − c1 − (1+n)2 c2 (n+1)2 2 and then the size of them grows up with the general Jl given as a (2l + 1) × (2l + 1) matrix: √ √ ⎛ l(n+1−l) ln(n+1−l) l 1 − n+1 c2 c − c2 ··· 0 2 3/2 (n+1) (n+1)3/2 ⎜ √ √ ⎜ l(n+1−l) (n+2−2l) n 1 + l−nl−1 c c2 ··· 0 ⎜ (n+1)3/2 c2 (n+1)2 2 (n+1)2 ⎜ √ √ ⎜ 2 Jl = ⎜ − ln(n+1−l) c2 (n+2−2l) n c2 1 − c1 + (l−1)n+1−l−n c2 · · · 0 ⎜ (n+1)3/2 (n+1)2 (n+1)2 ⎜ .. .. .. .. .. ⎜ . . . . . ⎝ 0
0
···
0
1 − c1 +
(20)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠
(21)
ln−n−n2 c (n+1)2 2
The matrices Kl , for l = 1, 2, . . . , n, differ from the corresponding Jl only with the signs of some off-diagonal elements: √ √ ⎛ ⎞ l(n+1−l) ln(n+1−l) l 1 − n+1 c2 c c2 ··· 0 (n+1)3/2 2 (n+1)3/2 ⎜ √ ⎟ √ ⎜ l(n+1−l) ⎟ (n+2−2l) n 1 + l−nl−1 c − c · · · 0 ⎜ (n+1)3/2 c2 ⎟ 2 2 2 2 (n+1) (n+1) ⎜√ ⎟ √ ⎜ ln(n+1−l) ⎟ 2 (n+2−2l) n (l−1)n+1−l−n Kl = ⎜ ⎟. c2 · · · 0 ⎜ (n+1)3/2 c2 − (n+1)2 c2 1 − c1 + ⎟ (n+1)2 ⎜ ⎟ .. .. .. .. .. ⎜ ⎟ . . . . . ⎝ ⎠ 2 ln−n−n 0 0 0 · · · 1 − c1 + (n+1)2 c2
(22)
All these real symmetric matrices can be diagonalized by finding their eigenvalues. To guarantee the positivity of Π0 , we just need to let all its eigenvalues, which are real due to the symmetry in the above matrices, be non-negative. For any fixed n, we can analytically solve the eigenvalue problem of every Jl or Kl with M ATHEMATICA or M ATHELAB. Through the induction on n, we obtained the eigenvalues of Jl or Kl in the diagonal of the following diagonalized matrices: ! ⎞ ⎛ c12 +c22 (n2 −2n−1)c1 c2 2−c1 −c2 − + 0 · · · 0 2 4 ⎟ ⎜ 2(n+1)2 ⎟ ⎜ ! ⎟ ⎜ 2 2 2 −2n−1)c c c +c (n 2−c −c ⎜ 1 2 1 2 0 + 14 2 + ··· 0⎟ Jl = Kl = ⎜ (23) ⎟. 2 2 2(n+1) ⎟ ⎜ ⎟ ⎜ . . . .. .. .. ⎝ . .. ⎠ 0
0
···
1
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The constant eigenvalue 1 exists for all Jl ’s and Kl ’s, where l = 0, 1, . . . , n. All the rest eigenvalues exist in couples, and the sum of each is 2 − c1 − c2 . After arranging the obtained couples in all Jl ’s and Kl ’s by the members with the minus sign before the square
2 −2n−1 root in an ascending order, we see that the least couple, 1 − 12 c1 − 12 c2 ± 14 c12 + 14 c22 + n2(n+1) 2 c1 c2 , are the common eigenvalues
2 −6n+1 of Jl and Kl for l = 1, 2, . . . , n, and the second least, 1 − 12 c1 − 12 c2 ± 14 c12 + 14 c22 + n2(n+1) 2 c1 c2 , the common eigenvalues of Jl and Kl for l = 2, 3, . . . , n, and so on. Therefore, if we require that the least common eigenvalue of all matrices except for J0 and K0 , the eigenvalue of which is the constant 1, be non-negative, i.e. 1 2 1 2 n2 − 2n − 1 1 1 c1 c2 0, c + c + 1 − c1 − c2 − (24) 2 2 4 1 4 2 2(n + 1)2 the inconclusive operator Π0 will be guaranteed to be positive. Since φ|Πi |φ 1 for ∀|φ ∈ H and all n, we have ci 1 for i = 1, 2 and a non-negative number 1 − 12 c1 − 12 c2 . Then, from the above inequality, we express c2 in terms of c1 : c2
1 − c1 1−
2n+1 c (n+1)2 1
(25)
.
To achieve the maximum success probability, we choose the equal sign, i.e. let the least eigenvalue of Π0 be 0. Inserting the resulting expression into Eq. (15) gives 1 − c1 n ¯ P = η1 c1 + η2 (26) . 2n+1 2(n + 1) 1 − (n+1)2 c1 We can easily find c1,opt where the right-hand side of this expression is maximized and the corresponding c2,opt : ! 1 − η1 η1 (n + 1)2 (n + 1)2 n n 1− 1− c1,opt = , c2,opt = . 2n + 1 n+1 η1 2n + 1 n + 1 1 − η1 Substituting these optimum values into (15) gives the optimum result for POVM: n n + 1 − 2n η1 (1 − η1 ) . P¯POVM (n, η1 ) = 4n + 2 Then we have the following optimum POVM elements: 1 − η1 (n + 1)2 n Π1,opt (n, η1 ) = 1− (IE,T − PE,T ) ⊗ IO , 2n + 1 n+1 η1 ! η1 (n + 1)2 n Π2,opt (n, η1 ) = 1− (IO,T − PO,T ) ⊗ IE . 2n + 1 n + 1 1 − η1
(27)
(28)
(29)
Here we have assumed that a priori probabilities of the inputs satisfy η1 + η2 = 1, i.e. no failure in the preparation period. This optimal POVM, however, holds valid only within the range of η1 or η2 where both 0 c1,opt 1 and 0 c2,opt 1 are satisfied simultaneously. From Eq. (27) we can find the range: (n + 1)2 n2 η , η . 1 2 n2 + (n + 1)2 n2 + (n + 1)2 If η1 =
(n+1)2 n2 +(n+1)2
(η2 =
n2 ), n2 +(n+1)2
(30)
Π1,opt = (IE,T − PE,T ) ⊗ IO and Π2,opt = 0. The continuity to the outside of the POVM’s 2
validity domain requires this structure remain for 1 η1 n2(n+1) . In other words, when the preparation is dominated by the first +(n+1)2 input, the optimal measurement is realized by standard von Neumann measurement, which projects the input onto the compliment of the totally symmetric subspace with respect to all the even digits in the program and the data digit. Then, we get the success probability for each operator: p1,opt = Ψ1in |(IE,T − PE,T ) ⊗ IO |Ψ1in and p2,opt = 0, indicating the second input is sacrificed completely. These results yield the average success probability, n P¯1 (n, η1 ) = η1 (31) , 2(n + 1) for η1
(n+1)2 . n2 +(n+1)2
Conversely, for η1 =
n2 n2 +(n+1)2
(η2 =
(n+1)2 ), Π2,opt = (IO,T − PO,T ) ⊗ IE and Π1,opt = 0. n2 +(n+1)2 n2 . This is a standard von Neumann measurement n2 +(n+1)2
Also from the
continuity, we require this structure remain for 0 η1 projecting the input onto the compliment of the totally symmetric subspace with respect to all the odd digits in the program and the data digit.
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Fig. 1. Optimal average success probability, P , vs. the a priori probability, η1 , for n = 2. Dashed line: P1 from Eq. (31), dotted line: P2 from Eq. (32), and solid curve: PPOVM from Eq. (28). The optimal P is given by P2 for η1 < 4/13, by PPOVM for 4/13 η1 9/13 and by P1 for 9/13 < η1 . The lower bound if the optimal average success probability reaches 0.2.
Fig. 2. Optimal average success probability, P , vs. the a priori probability, η1 , for n = 6. The optimal P is given by P2 (dotted line) for η1 < 36/85, by PPOVM (solid curve) for 36/85 η1 49/85 and by P1 (dashed line) for 49/85 < η1 . The lower bound if the optimal average success probability reaches 0.23.
Since the first input is sacrificed completely in this case, we have n n = (1 − η1 ) , P¯2 (n, η1 ) = η2 2(n + 1) 2(n + 1)
Fig. 3. Optimal average success probability, P , vs. the a priori probability, η1 , for n = ∞. In this upper bound limit, the optimal scheme reduces to two full projector probabilities P1 (dashed line) and P2 (dotted line). The minimum value can reach 0.25.
(32)
2
n for 0 η1 n2 +(n+1) 2 . The situation is fully symmetric in the a priori probabilities of inputs. In the intermediate range, where neither of the input states dominates, the POVM does the job of discrimination of them better than von Neumann measurement, while outside the range two full projectors do the work of identifying them at best. The optimal average success probability of the device working in the whole range of a priori probability, as the generalization of the result in [9], is summarized as follows: ⎧ n2 0 η1 < n2 +(n+1) P¯2 (n, η1 ), ⎪ 2, ⎪ ⎪ ⎨ (n+1)2 n2 P¯ opt (n, η1 ) = P¯POVM (n, η1 ), n2 +(n+1) (33) 2 η1 n2 +(n+1)2 , ⎪ ⎪ ⎪ 2 ⎩ ¯ (n+1) P1 (n, η1 ), < η1 1. n2 +(n+1)2
It is seen from the above results that the optimal success probability changes continuously at the boundaries of the respective regions of validity for the three measurements. Some features should be noticed for this device: First, from Eq. (30), the validity range of the POVM measurement will become narrower and narrower with the increase of n, the number of copies used in the program. As n tends to infinity, this range will shrink to a point at η1 = η2 = 0.5. It means that the optimum measurement will reduce to only von Neumann in this extreme. Second, if the preparation probability is fixed, the success probability of the device will increase with the number of copies of the states stored in the program. The larger size of the program yields more information about the unknown states. For example, at η1 = η2 = 0.5, where the identification of the input state is the hardest and the difference between the POVM and two full projectors is the largest, the average success probability, P¯POVM (n) = n/(4n + 2), increases from 1/6 to 1/4 by 50% as n goes from 1 to infinity. We depict these features with Figs. 1–3 for three different n (n = 2, 6, and ∞). Another interesting feature is that the optimal measurement operators and the validity domain depend only on the number of the states we put into the program and their preparation probabilities. Since they are independent of the information encoded in a specific pair of unknown qubits |ψ1 and |ψ2 , the device performs universally. It can be programed with whatever couple of unknown qubits we want to identify and discriminates between the produced registers with the best chance of success and no error. Finally, we compare the obtained results with those of UD for the averages of systems. Suppose the a priori probabilities of the systems prepared as in Eq. (1) are equal. If we take the averages of the density matrices ρ1 = |Ψ1in Ψ1in | and ρ2 = |Ψ2in Ψ2in | over the Bloch spheres of the uniformly and independently distributed |ψ1 , |ψ2 and find the optimal UD solution to these two averages, which are two uniformly distributed known mixed states, we will have the average optimal success probability of 1/3 if the number of the qubit copies tends to infinity [11,12]. This is equal to the optimal IDP average [3–5] for discriminating a pair of known qubits. The corresponding maximum average success probability, 1/4, of our device is obtained through first measuring the systems with the optimal POVM and then taking the averages of the expectation values of Π1 and Π2 , and is lower than this IDP average. However, this device realizes a practical identification for any couple of randomly distributed unknown qubits instead
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of their known average input states. We see from this fact that the discrimination between two unknown systems (with only some information about their symmetry available) and the discrimination between their known averages are not equivalent. So far we have worked out a scheme for a device that unambiguously discriminate between pairs of quantum registers produced with more than one copy of unknown qubits used in the program or reference digits. Its maximum average success probability increases with the number of the copies we store in the program, and also depends on the a priori probability of the states to be distinguished between. Given such an ideal device, we can tune its parameters, c1 and c2 , according to the copy number n and the a priori probability η1 or η2 , so that it reaches the optimal performance for the required measurement. Acknowledgement The authors would like to thank Prof. M. Hillery for valuable discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
C.W. Helstrom, Quantum Detection and Estimation Theory, Academic Press, New York, 1976. A.S. Holevo, Probabilistic and Quantum Aspects of Quantum Theory, North-Holland, Amsterdam, 1982. I.D. Ivanovic, Phys. Lett. A 123 (1987) 257. D. Dieks, Phys. Lett. A 126 (1988) 303. A. Peres, Phys. Lett. A 128 (1988) 19. G. Jaeger, A. Shimony, Phys. Lett. A 197 (1995) 83. A. Chefles, S.M. Barnett, Phys. Lett. A 250 (1998) 223. G.M. D’Ariano, M.F. Sacchi, J. Kahn, Phys. Rev. A 72 (2005) 032310, quant-ph/0504048. J. Bergou, M. Hillery, Phys. Rev. Lett. 94 (2005) 160501, quant-ph/0504201. A. Hayashi, M. Horibe, T. Hashimoto, Phys. Rev. A 72 (2005) 052306, quant-ph/0507237. A. Hayashi, M. Horibe, T. Hashimoto, Phys. Rev. A 73 (2006) 012328, quant-ph/0510015. B. He, J. Bergou, in preparation. M. Dušek, V. Bužek, Phys. Rev. A 66 (2002) 022112.
A generalized programmable unambiguous state discriminator for unknown qubit systems Bing He ∗ , János A. Bergou Department of Physics and Astronomy, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10021, USA Received 25 February 2006; received in revised form 7 April 2006; accepted 2 May 2006 Available online 11 May 2006 Communicated by P.R. Holland
Abstract We discuss a state discriminator that unambiguously distinguishes between two quantum registers prepared with multiple copies of two unknown qubits. This device achieves the optimal performance by von Neumann measurement and general POVM in different ranges of the input preparation probabilities, respectively, and in the limit of very large program registers the optimal measurement reduces to only von Neumann measurements. © 2006 Elsevier B.V. All rights reserved. PACS: 03.65.Bz; 03.67.-a; 03.67.Hk
The discrimination between a couple of linearly independent pure states |ψ1 and |ψ2 is possible only in two situations: We have complete classical knowledge of them, or the states carry some symmetry. Even in these situations, their identification is achieved only with a certain probability if these states are non-orthogonal. There are strategies to achieve the optimal result: First, the strategy that determines the state with minimum probability for the error [1,2] and, second, unambiguous discrimination (UD) that allows the possibility of an inconclusive result (with a minimal probability in the optimal case) [3–7]. In the recent years, the minimax approach to the discrimination of quantum states [8] is also proposed, where the situation with no information about the a priori probabilities of inputs can be dealt with. For practical applications, we need to look for the devices that achieve the optimal discrimination of a couple of quantum states in a versatile way, i.e. first of all, it works universally with any couple of the target states for discrimination and, secondly, it should perform optimally in the whole range of their a priori probabilities, or preparation probabilities, possibly arising in the preparation period. To realize the unknown quantum states (no classical information available) discrimination, we rely on the symmetry from having the available copies of the states we want to identify [9–12]. Here we only consider the UD for the unknown states. In [8] the authors proposed a state discriminator, in which the information about two unknown qubits is supplied in the form of the program in a programmable device [13] using 1 copy of these qubits as the program or reference digits. This problem was generalized to the situation of multiple copies used as the reference [11,12]. In [11], by means of the representation theory of U (n) group, Hayashi et al. obtained the optimal UD solution for the averages of two uniformly distributed unknown qudit systems prepared with n copies of reference plus 1 copy of data, while in [12] we studied by Jordan basis method the corresponding problem for two uniformly distributed unknown qubit systems prepared with n copies of reference plus m copies of data and with arbitrary a priori probabilities. All these results are about the optimal UD for the averages of systems. In this Letter, we construct a device that works on the exact input states without first taking the average of the systems; each time only one prepared input state is fed into the machine and, with a non-zero success probability, the measurement in the machine tells us which one it is.
* Corresponding author.
E-mail address: [email protected] (B. He). 0375-9601/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2006.05.002
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We start with the generalization of the inputs in [9]. Here we store arbitrary n copies of qubits |ψ1 1 and |ψ2 2 in the program part of the quantum registers prepared as follows: in in Ψ = |ψ1 1 |ψ2 2 · · · |ψ1 2n−1 |ψ2 2n |ψ2 2n+1 , Ψ = |ψ1 1 |ψ2 2 · · · |ψ1 2n−1 |ψ2 2n |ψ1 2n+1 , (1) 1 2 where the data, either |ψ1 or |ψ2 , at the tail is appended to the program, a string of tensor product by copies of two unknown qubits. Without loss of generality in the preparation of the program parts, we place |ψ1 in the odd positions and |ψ2 in the even positions of the inputs.1 We need to find an optimal measurement to unambiguously distinguish between these inputs, bearing in mind the data qubits, |ψi = cos(θi /2)|0 + sin(θi /2)eiφi |1,
(2)
where i = 1, 2, are randomly (uniformly or non-uniformly) distributed on Bloch sphere with no available information about θi and φi . For generality, these quantum registers are assumed to be prepared with a priori probabilities η1 and η2 , respectively. The optimal measurement performed by our device is generally achieved by a positive operator value measure (POVM), and we denote its element of unambiguously detecting |Ψ1in as Π1 , that of unambiguously detecting |Ψ2in as Π2 , and that corresponding to failure as Π0 . They satisfy unit decomposition: I = Π1 + Π2 + Π0 .
(3)
The probabilities of successfully identifying these two possible input states are given by in in in in Ψ2 Π2 Ψ2 = p2 , Ψ1 Π1 Ψ1 = p1 ,
(4)
and the condition of no error implies that Π1 Ψ2in = 0. Π2 Ψ1in = 0,
(5)
Without any knowledge about |ψ1 and |ψ2 , what we can make use of in determining the inputs is their symmetry, i.e. |Ψ1in is invariant under the action of symmetric group on the odd positions and the tail position, while |Ψ2in invariant under the corresponding action on the even positions and the tail. To construct Π1 and Π2 that perform the UD of the inputs, we define the following orthonormal basis: 1 |e0 = |0, 0, . . . , 0, Cn0 1 |1, 0, . . . , 0 + |0, 1, . . . , 0 + · · · + |0, 0, . . . , 1 , |e1 = Cn1 .. . 1 |ek = |1, 1, . . . , 0, 0 +|0, 1, 1, . . . , 0 + · · · + |0, 0, . . . , 1, 1 ,
Ck n
k’s 1 in n digits
summation of Cnk terms
.. . 1 |en = |1, 1, . . . , 1, Cnn
(6)
where Cnk is the number of ways to choose k objects from a group of n objects without regard to order. In terms of this basis, the tensor product of n copies of a qubit is expanded as follows: n
⊗n cos(θ/2)|0 + sin(θ/2)eiφ |1 (7) = cosn−k (θ/2) sink (θ/2)eikφ Cnk |ek . k=0
Thus we construct the POVM elements satisfying the unambiguity (no error) in measurement: Π2 = c2 (IO,T − PO,T ) ⊗ IE , Π1 = c1 (IE,T − PE,T ) ⊗ IO , (8) n+1 n+1 where PE,T = k=0 |ek E,T E,T ek | (respectively PO,T = k=0 |ek O,T O,T ek |) is the projection operator onto the totally symmetric subspace with respect to the n even (respectively odd) positions and the tail position, IE,T (respectively IO,T ) the direct 1 The program registers can be set up with the provided 2n qubit copies in any specified order, and then for the POVM elements in Eq. (8) we only need to choose the corresponding projectors onto the totally symmetric spaces of the data and the program registers.
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product of unit matrices in these positions, and c1 , c2 the non-negative parameters arising from the unequal a priori probabilities of |Ψ1in and |Ψ2in . IO and IE are the direct products of the unit matrices in the odd and the even positions of the input states, respectively. Using these Πi , where i = 1, 2, we find their successfully detecting probabilities pi = Ψiin Πi Ψiin = ci − ci Ψiin PR,T ⊗ IS Ψiin , (9) where R = E, O but S = O, E. We use two tricks to calculate the average of the projection operators more conveniently. First, by the expansion k−1 k Cn Cn |e |0 + |ek−1 R |1T , |ek R,T = (10) k R T k k Cn+1 Cn+1 for k = 1, 2, . . . , n, we rewrite the projection operators as PR,T ⊗ IS = |e0 R |0T R e0 |T 0|
+
l−1 l−1 l Cn Cn C n |el R |0T + l |el−1 R |1T R el |T 0| + R el−1 |T 1| l l l Cn+1 Cn+1 Cn+1 Cn+1
n Cnl l=1
+ |en R |1T R en |T 1| ⊗ IS .
(11)
Second, we apply Eq. (7) to expand the input states by two parts: n
in n−l l ilφj Ψ = cos (θ /2) sin (θ /2)e Cnl |el R cos(θi /2)|0T + sin(θi /2)eiφi |1T j j i l=0
×
|Ψi
n
cosn−k (θi /2) sink (θi /2)eikφi Cnk |ek S ,
k=0
|Ψi
(12)
where i = 1, 2 but j = 2, 1. Putting the expectation values from these parts together, we obtain n in in k+1 2 n−k+1 2 Ψi PR,T ⊗ IS Ψi = cos (θi /2) + sin (θi /2) Cnk cos2(n−k) (θj /2) sin2k (θj /2) n+1 n+1 k=0
n 2k + C k cos2(n−k) (θj /2) sin2k (θj /2) cos(θj /2) cos(θi /2) sin(θi /2) cos(φi − φj ). n+1 n
(13)
k=1
The total success probability P to discriminate between the two unknown states, assuming |Ψ1in is produced with a priori probability η1 and |Ψ2in with η2 , is given by P = η 1 p1 + η 2 p2 .
(14)
Since we have no knowledge about the data states, what we use to indicate how well the device performs is the average of this success probability: 2 1 n dφ dθj sin θj η1 Ψ1in Π1 Ψ1in + η2 Ψ2in Π2 Ψ2in = (η1 c1 + η2 c2 ) . j 2 2(n + 1) (4π) 2π
P¯ =
j =1 0
π
(15)
0
We want to maximize this expression subject to the constraint that Π0 is a positive operator. Let H be the Hilbert space of the input states in given in Eq. (1), which, as it is seen from Eq. (10), is spanned by the orthonormal basis {|el O |em E |0T , |el O |em E |1T }, where l, m = 0, 1, . . . , n. To have a nice matrix representation of the inconclusive operator, Π0 = I − Π1 − Π2 = (1 − c1 − c2 )I + c1 PE,T ⊗ IO + c2 PO,T ⊗ IE ,
(16)
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we apply the following orthogonal transformations: Cnm Cnm−1 |e |e |0 + |ηlm = l O m E T m m |el O |em−1 E |1T , Cn+1 Cn+1 Cnm−1 Cnm |χlm = |el O |em E |0T − m m |el O |em−1 E |1T , Cn+1 Cn+1
(17)
where l = 0, 1, . . . , n, and m = 1, 2, . . . , n. The transformed vectors |ηlm and |χlm satisfy ηlm |χl m = 0, ηlm |ηl m = δl,l δm,m , and χlm |χl m = δl,l δm,m . If we put them together with the untransformed |el O |e0 E |0T and |el O |en E |1T , for l = 0, 1, . . . , n, all these unit vectors form an orthonormal basis, {|ηlm , |χlm , |el O |e0 E |0T , |el O |en E |1T }, of H . With this basis, the operator Π0 is represented by the direct sum of two series of block diagonal matrices: (1)
(2)
Π0 = Π0 ⊕ Π0 , where
⎛
⎜ ⎜ (1) Π0 = ⎜ ⎜ ⎝
(18) ⎞
J0
⎟ ⎟ ⎟, ⎟ ⎠
J1 J2 ..
.
⎛ ⎜ ⎜ (2) Π0 = ⎜ ⎜ ⎝
⎞
K0
⎟ ⎟ ⎟. ⎟ ⎠
K1 K2 ..
.
Jn
(19)
Kn
These block diagonal matrices are generated regularly; J0 and K0 are 1 × 1 matrix 1, and J1 is a 3 × 3 matrix: √ ⎞ ⎛ n 1 n c2 c − c 1 − n+1 2 2 3/2 3/2 (n+1) (n+1) ⎟ ⎜ √ ⎟ ⎜ n n3/2 n J1 = ⎜ (n+1)3/2 c2 1 − (n+1)2 c2 c ⎟, 2 2 (n+1) ⎠ ⎝ 3/2 2 n n n − (n+1)3/2 c2 c 1 − c1 − (1+n)2 c2 (n+1)2 2 and then the size of them grows up with the general Jl given as a (2l + 1) × (2l + 1) matrix: √ √ ⎛ l(n+1−l) ln(n+1−l) l 1 − n+1 c2 c − c2 ··· 0 2 3/2 (n+1) (n+1)3/2 ⎜ √ √ ⎜ l(n+1−l) (n+2−2l) n 1 + l−nl−1 c c2 ··· 0 ⎜ (n+1)3/2 c2 (n+1)2 2 (n+1)2 ⎜ √ √ ⎜ 2 Jl = ⎜ − ln(n+1−l) c2 (n+2−2l) n c2 1 − c1 + (l−1)n+1−l−n c2 · · · 0 ⎜ (n+1)3/2 (n+1)2 (n+1)2 ⎜ .. .. .. .. .. ⎜ . . . . . ⎝ 0
0
···
0
1 − c1 +
(20)
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎟ ⎠
(21)
ln−n−n2 c (n+1)2 2
The matrices Kl , for l = 1, 2, . . . , n, differ from the corresponding Jl only with the signs of some off-diagonal elements: √ √ ⎛ ⎞ l(n+1−l) ln(n+1−l) l 1 − n+1 c2 c c2 ··· 0 (n+1)3/2 2 (n+1)3/2 ⎜ √ ⎟ √ ⎜ l(n+1−l) ⎟ (n+2−2l) n 1 + l−nl−1 c − c · · · 0 ⎜ (n+1)3/2 c2 ⎟ 2 2 2 2 (n+1) (n+1) ⎜√ ⎟ √ ⎜ ln(n+1−l) ⎟ 2 (n+2−2l) n (l−1)n+1−l−n Kl = ⎜ ⎟. c2 · · · 0 ⎜ (n+1)3/2 c2 − (n+1)2 c2 1 − c1 + ⎟ (n+1)2 ⎜ ⎟ .. .. .. .. .. ⎜ ⎟ . . . . . ⎝ ⎠ 2 ln−n−n 0 0 0 · · · 1 − c1 + (n+1)2 c2
(22)
All these real symmetric matrices can be diagonalized by finding their eigenvalues. To guarantee the positivity of Π0 , we just need to let all its eigenvalues, which are real due to the symmetry in the above matrices, be non-negative. For any fixed n, we can analytically solve the eigenvalue problem of every Jl or Kl with M ATHEMATICA or M ATHELAB. Through the induction on n, we obtained the eigenvalues of Jl or Kl in the diagonal of the following diagonalized matrices: ! ⎞ ⎛ c12 +c22 (n2 −2n−1)c1 c2 2−c1 −c2 − + 0 · · · 0 2 4 ⎟ ⎜ 2(n+1)2 ⎟ ⎜ ! ⎟ ⎜ 2 2 2 −2n−1)c c c +c (n 2−c −c ⎜ 1 2 1 2 0 + 14 2 + ··· 0⎟ Jl = Kl = ⎜ (23) ⎟. 2 2 2(n+1) ⎟ ⎜ ⎟ ⎜ . . . .. .. .. ⎝ . .. ⎠ 0
0
···
1
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The constant eigenvalue 1 exists for all Jl ’s and Kl ’s, where l = 0, 1, . . . , n. All the rest eigenvalues exist in couples, and the sum of each is 2 − c1 − c2 . After arranging the obtained couples in all Jl ’s and Kl ’s by the members with the minus sign before the square
2 −2n−1 root in an ascending order, we see that the least couple, 1 − 12 c1 − 12 c2 ± 14 c12 + 14 c22 + n2(n+1) 2 c1 c2 , are the common eigenvalues
2 −6n+1 of Jl and Kl for l = 1, 2, . . . , n, and the second least, 1 − 12 c1 − 12 c2 ± 14 c12 + 14 c22 + n2(n+1) 2 c1 c2 , the common eigenvalues of Jl and Kl for l = 2, 3, . . . , n, and so on. Therefore, if we require that the least common eigenvalue of all matrices except for J0 and K0 , the eigenvalue of which is the constant 1, be non-negative, i.e. 1 2 1 2 n2 − 2n − 1 1 1 c1 c2 0, c + c + 1 − c1 − c2 − (24) 2 2 4 1 4 2 2(n + 1)2 the inconclusive operator Π0 will be guaranteed to be positive. Since φ|Πi |φ 1 for ∀|φ ∈ H and all n, we have ci 1 for i = 1, 2 and a non-negative number 1 − 12 c1 − 12 c2 . Then, from the above inequality, we express c2 in terms of c1 : c2
1 − c1 1−
2n+1 c (n+1)2 1
(25)
.
To achieve the maximum success probability, we choose the equal sign, i.e. let the least eigenvalue of Π0 be 0. Inserting the resulting expression into Eq. (15) gives 1 − c1 n ¯ P = η1 c1 + η2 (26) . 2n+1 2(n + 1) 1 − (n+1)2 c1 We can easily find c1,opt where the right-hand side of this expression is maximized and the corresponding c2,opt : ! 1 − η1 η1 (n + 1)2 (n + 1)2 n n 1− 1− c1,opt = , c2,opt = . 2n + 1 n+1 η1 2n + 1 n + 1 1 − η1 Substituting these optimum values into (15) gives the optimum result for POVM: n n + 1 − 2n η1 (1 − η1 ) . P¯POVM (n, η1 ) = 4n + 2 Then we have the following optimum POVM elements: 1 − η1 (n + 1)2 n Π1,opt (n, η1 ) = 1− (IE,T − PE,T ) ⊗ IO , 2n + 1 n+1 η1 ! η1 (n + 1)2 n Π2,opt (n, η1 ) = 1− (IO,T − PO,T ) ⊗ IE . 2n + 1 n + 1 1 − η1
(27)
(28)
(29)
Here we have assumed that a priori probabilities of the inputs satisfy η1 + η2 = 1, i.e. no failure in the preparation period. This optimal POVM, however, holds valid only within the range of η1 or η2 where both 0 c1,opt 1 and 0 c2,opt 1 are satisfied simultaneously. From Eq. (27) we can find the range: (n + 1)2 n2 η , η . 1 2 n2 + (n + 1)2 n2 + (n + 1)2 If η1 =
(n+1)2 n2 +(n+1)2
(η2 =
n2 ), n2 +(n+1)2
(30)
Π1,opt = (IE,T − PE,T ) ⊗ IO and Π2,opt = 0. The continuity to the outside of the POVM’s 2
validity domain requires this structure remain for 1 η1 n2(n+1) . In other words, when the preparation is dominated by the first +(n+1)2 input, the optimal measurement is realized by standard von Neumann measurement, which projects the input onto the compliment of the totally symmetric subspace with respect to all the even digits in the program and the data digit. Then, we get the success probability for each operator: p1,opt = Ψ1in |(IE,T − PE,T ) ⊗ IO |Ψ1in and p2,opt = 0, indicating the second input is sacrificed completely. These results yield the average success probability, n P¯1 (n, η1 ) = η1 (31) , 2(n + 1) for η1
(n+1)2 . n2 +(n+1)2
Conversely, for η1 =
n2 n2 +(n+1)2
(η2 =
(n+1)2 ), Π2,opt = (IO,T − PO,T ) ⊗ IE and Π1,opt = 0. n2 +(n+1)2 n2 . This is a standard von Neumann measurement n2 +(n+1)2
Also from the
continuity, we require this structure remain for 0 η1 projecting the input onto the compliment of the totally symmetric subspace with respect to all the odd digits in the program and the data digit.
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Fig. 1. Optimal average success probability, P , vs. the a priori probability, η1 , for n = 2. Dashed line: P1 from Eq. (31), dotted line: P2 from Eq. (32), and solid curve: PPOVM from Eq. (28). The optimal P is given by P2 for η1 < 4/13, by PPOVM for 4/13 η1 9/13 and by P1 for 9/13 < η1 . The lower bound if the optimal average success probability reaches 0.2.
Fig. 2. Optimal average success probability, P , vs. the a priori probability, η1 , for n = 6. The optimal P is given by P2 (dotted line) for η1 < 36/85, by PPOVM (solid curve) for 36/85 η1 49/85 and by P1 (dashed line) for 49/85 < η1 . The lower bound if the optimal average success probability reaches 0.23.
Since the first input is sacrificed completely in this case, we have n n = (1 − η1 ) , P¯2 (n, η1 ) = η2 2(n + 1) 2(n + 1)
Fig. 3. Optimal average success probability, P , vs. the a priori probability, η1 , for n = ∞. In this upper bound limit, the optimal scheme reduces to two full projector probabilities P1 (dashed line) and P2 (dotted line). The minimum value can reach 0.25.
(32)
2
n for 0 η1 n2 +(n+1) 2 . The situation is fully symmetric in the a priori probabilities of inputs. In the intermediate range, where neither of the input states dominates, the POVM does the job of discrimination of them better than von Neumann measurement, while outside the range two full projectors do the work of identifying them at best. The optimal average success probability of the device working in the whole range of a priori probability, as the generalization of the result in [9], is summarized as follows: ⎧ n2 0 η1 < n2 +(n+1) P¯2 (n, η1 ), ⎪ 2, ⎪ ⎪ ⎨ (n+1)2 n2 P¯ opt (n, η1 ) = P¯POVM (n, η1 ), n2 +(n+1) (33) 2 η1 n2 +(n+1)2 , ⎪ ⎪ ⎪ 2 ⎩ ¯ (n+1) P1 (n, η1 ), < η1 1. n2 +(n+1)2
It is seen from the above results that the optimal success probability changes continuously at the boundaries of the respective regions of validity for the three measurements. Some features should be noticed for this device: First, from Eq. (30), the validity range of the POVM measurement will become narrower and narrower with the increase of n, the number of copies used in the program. As n tends to infinity, this range will shrink to a point at η1 = η2 = 0.5. It means that the optimum measurement will reduce to only von Neumann in this extreme. Second, if the preparation probability is fixed, the success probability of the device will increase with the number of copies of the states stored in the program. The larger size of the program yields more information about the unknown states. For example, at η1 = η2 = 0.5, where the identification of the input state is the hardest and the difference between the POVM and two full projectors is the largest, the average success probability, P¯POVM (n) = n/(4n + 2), increases from 1/6 to 1/4 by 50% as n goes from 1 to infinity. We depict these features with Figs. 1–3 for three different n (n = 2, 6, and ∞). Another interesting feature is that the optimal measurement operators and the validity domain depend only on the number of the states we put into the program and their preparation probabilities. Since they are independent of the information encoded in a specific pair of unknown qubits |ψ1 and |ψ2 , the device performs universally. It can be programed with whatever couple of unknown qubits we want to identify and discriminates between the produced registers with the best chance of success and no error. Finally, we compare the obtained results with those of UD for the averages of systems. Suppose the a priori probabilities of the systems prepared as in Eq. (1) are equal. If we take the averages of the density matrices ρ1 = |Ψ1in Ψ1in | and ρ2 = |Ψ2in Ψ2in | over the Bloch spheres of the uniformly and independently distributed |ψ1 , |ψ2 and find the optimal UD solution to these two averages, which are two uniformly distributed known mixed states, we will have the average optimal success probability of 1/3 if the number of the qubit copies tends to infinity [11,12]. This is equal to the optimal IDP average [3–5] for discriminating a pair of known qubits. The corresponding maximum average success probability, 1/4, of our device is obtained through first measuring the systems with the optimal POVM and then taking the averages of the expectation values of Π1 and Π2 , and is lower than this IDP average. However, this device realizes a practical identification for any couple of randomly distributed unknown qubits instead
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of their known average input states. We see from this fact that the discrimination between two unknown systems (with only some information about their symmetry available) and the discrimination between their known averages are not equivalent. So far we have worked out a scheme for a device that unambiguously discriminate between pairs of quantum registers produced with more than one copy of unknown qubits used in the program or reference digits. Its maximum average success probability increases with the number of the copies we store in the program, and also depends on the a priori probability of the states to be distinguished between. Given such an ideal device, we can tune its parameters, c1 and c2 , according to the copy number n and the a priori probability η1 or η2 , so that it reaches the optimal performance for the required measurement. Acknowledgement The authors would like to thank Prof. M. Hillery for valuable discussions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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