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Local discrimination of maximally entangled states in canonical form
Physics Letters A 333 (2004) 232–234 www.elsevier.com/locate/pla
Local discrimination of maximally entangled states in canonical form ✩ Hongen Cao ∗ , Mingsheng Ying State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China Received 8 July 2004; received in revised form 22 October 2004; accepted 25 October 2004 Available online 30 October 2004 Communicated by P.R. Holland
Abstract It is shown that two copies are enough to distinguish a complete basis of maximally entangled states in canonical form by constructing an explicit protocol. In particular, in such a protocol, no auxiliary system is needed and the symmetrical roles of certain indexes of the canonical form are embodied properly. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 03.67.Mn; 03.65.Ud Keywords: Local operation and classical communication; Local discrimination; Maximally entangled states
In quantum information theory, orthogonal states can always be perfectly distinguished from one another under global operation. However, many global operations cannot be performed using only local operations and classical communication (LOCC). A natural question arises what is and what is not locally possible in state discrimination. In [1], Bennett et al. showed that there exist sets of orthogonal pure product ✩ This work was partly supported by the National Foundation of Natural Sciences of China (Grant No. 60321002) and the Key Grant Project of Chinese Ministry of Education (Grant No. 10403). * Corresponding author. E-mail address: [email protected] (H. Cao).
0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.10.057
states that cannot be distinguished by LOCC, which is a rather surprising result counter to our intuition. But Walgate et al. [2] proved that any two orthogonal pure states can be distinguished perfectly under LOCC. Recently, considerable attention has been drawn to the study of some special cases of local discrimination of multipartite pure states. In [3], Ghosh et al. considered the distinguishability of Bell states and found that it is impossible to discriminate (deterministically or probabilistically) among the four Bell states locally if only one copy is provided. They also proved that discriminating among any three Bell states is not possible under LOCC. Walgate and Hardy [4] derived a necessary and sufficient condition for perfect dis-
H. Cao, M. Ying / Physics Letters A 333 (2004) 232–234
crimination among orthogonal pure states of an arbitrary (2 × n)-dimensional quantum system. Horodecki et al. [5] showed that any full basis of a bipartite system is not distinguishable by LOCC if at least one of the vectors is entangled. In [6], Chen and Li derived a Schmidt number criterion for the distinguishability of orthogonal pure states of an arbitrary (m × n)dimensional quantum system. Fan [7] demonstrated that the d 2 pairwise orthogonal maximally entangled states in canonical form can never be distinguished deterministically or probabilistically under LOCC provided there is only one copy. Some other authors considered the conclusive and inconclusive local discrimination of multipartite pure states. In [8], Virmani et al. showed that for inconclusive discrimination of any two pure states, the optimal probability of error can be attained under LOCC only. Chen and Yang [9] proved that the same is true in the case of conclusive discrimination provided the two states are prepared with equal prior probability. More recently, Ji et al. [10] extended their result to the general case that the two states are prepared with arbitrary prior probability. In [11], Chefles derived a necessary and sufficient condition for unambiguous state discrimination using only LOCC. This Letter is highly motivated by the result of Walgate et al. [2]. It was shown in [2] that any m orthogonal pure states can be perfectly discriminated using only LOCC if (m − 1) copies are provided. However, there do exist m orthogonal pure states that can be discriminated locally with less than (m − 1) copies. One such example is the four Bell states ± Ψ
AB
1 = √ |00AB ± |11AB 2
(1)
1 = √ |01AB ± |10AB 2
(2)
and ± Φ
AB
which can be discriminated by LOCC perfectly if two copies are provided. In this case, the two parties Alice and Bob need only perform von Neumann measurement locally on their own system in the computational basis {|0A , |1A } and {|0B , |1B }, respectively. Through classical communication, they can exchange information about the measurement result and identify which of the four Bell states they share.
233
In [2], only the four Bell states were considered which compose a maximally entangled basis set of a 2 × 2 quantum system. In this Letter, we consider the distinguishability of maximally entangled basis sets in canonical form in a general d × d quantum system. Here, by ‘canonical form’, we mean any set of maximally entangled states that can be written as d−1 1 jn ω |j A ⊗ (j + m) mod d B , |ψmn AB = √ d j =0
m, n = 0, . . . , d − 1,
(3) 2π i d
as defined in [12], where ω = e and {|iA }, {|iB } are standard orthonormal bases of d-dimensional Hilbert spaces HA and HB , respectively. Since the states in Eq. (3) generalize Bell states in a d × d quantum system, it is natural to ask how many copies are needed to distinguish locally the d 2 states in Eq. (3) above. We have the following theorem which is the main result of this Letter. Theorem. Two copies are sufficient to distinguish locally the d 2 orthogonal maximally entangled states in Eq. (3). Proof. We prove this by constructing an explicit protocol as follows. Suppose the two particles of each copy are shared between Alice and Bob. (1) For the first copy d−1 1 jn |ψmn AB = √ ω |j A ⊗ (j + m) mod d B . d j =0
(4) Alice and Bob both perform von Neumann measurement locally on their own system in the computational basis {|iA } and {|iB }, respectively. Through classical communication, they can exchange information about the measurement result and identify m. (2) For the second copy, first, Alice performs adjoint Fourier transformation F † and Bob performs Fourier transformation F locally on their own system, respectively. Then the state turns into d−1 1 j m |ψ˜ mn AB = √ ω (j + n) mod d A ⊗ |j B . d j =0
Repeat step (1), Alice and Bob will identify n.
(5)
234
H. Cao, M. Ying / Physics Letters A 333 (2004) 232–234
Obviously, m and n uniquely determine the state |ψmn AB . This completes the proof. 2 After this work is done, we notice that Ghosh et al. [12] have already independently found this result using a different method. In their proof, they use the teleportation protocol. Alice first teleports the states |0 and √1 (|0 + |1 + · · · + |d − 1) through the two d copies of the shared channel |ψmn AB to Bob. After the teleportation protocol is over, the two states of Bob’s system to the two copies are |m d−1 corresponding j n |(j + m) mod d , respectively. Oband √1 ω B j =0 d viously, Bob himself can identify m and n locally. Compared to the method of Ghosh et al., our proof is much simpler in that we need no auxiliary system. Moreover, our proof embodies the symmetrical roles that m and n play in the state |ψmn AB . Finally, there may be a close connection between our proof and that of Ghosh et al. since Fourier transformation plays an important role in the teleportation protocol [13,14]. We would like to point out that our discrimination protocol only applies to maximally entangled basis set in the form shown in Eq. (3). For general cases, our method does not work since the most general form of any maximally entangled basis set of a d × d quantum system is not known [15]. To summarize, we independently find that two copies are enough to distinguish locally a complete basis of maximally entangled states in canonical form. We also make some comparison between our proof and that of Ghosh et al.
Acknowledgements The authors thank Z.F. Ji, R.Y. Duan and Y. Feng for useful discussion in this field. We also thank anonymous referee for helpful comments and informing us of a valuable reference.
References [1] C.H. Bennett, D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, W.K. Wootters, Phys. Rev. A 59 (1999) 1070. [2] J. Walgate, A.J. Short, L. Hardy, V. Vedral, Phys. Rev. Lett. 85 (2000) 4972. [3] S. Ghosh, G. Kar, A. Roy, A. Sen(De), U. Sen, Phys. Rev. Lett. 87 (2001) 277902. [4] J. Walgate, L. Hardy, Phys. Rev. Lett. 89 (2002) 147901. [5] M. Horodecki, A. Sen(De), U. Sen, K. Horodecki, Phys. Rev. Lett. 90 (2003) 047902. [6] P.X. Chen, C.Z. Li, Phys. Rev. A 68 (2003) 062107. [7] H. Fan, Phys. Rev. Lett. 92 (2004) 177905. [8] S. Virmani, M.F. Sacchi, M.B. Plenio, D. Markham, Phys. Lett. A 288 (2001) 62. [9] Y.X. Chen, D. Yang, Phys. Rev. A 65 (2002) 022320. [10] Z.F. Ji, H.E. Cao, M.S. Ying, quant-ph/0407120. [11] A. Chefles, Phys. Rev. A 69 (2004) 050307. [12] S. Ghosh, G. Kar, A. Roy, D. Sarkar, quant-ph/0205105. [13] C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [14] D.N. Mermin, Phys. Rev. A 65 (2001) 012320. [15] R.F. Werner, J. Phys. A: Math. Gen. 34 (2001) 7081.
Local discrimination of maximally entangled states in canonical form ✩ Hongen Cao ∗ , Mingsheng Ying State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing 100084, China Received 8 July 2004; received in revised form 22 October 2004; accepted 25 October 2004 Available online 30 October 2004 Communicated by P.R. Holland
Abstract It is shown that two copies are enough to distinguish a complete basis of maximally entangled states in canonical form by constructing an explicit protocol. In particular, in such a protocol, no auxiliary system is needed and the symmetrical roles of certain indexes of the canonical form are embodied properly. 2004 Elsevier B.V. All rights reserved. PACS: 03.67.-a; 03.67.Mn; 03.65.Ud Keywords: Local operation and classical communication; Local discrimination; Maximally entangled states
In quantum information theory, orthogonal states can always be perfectly distinguished from one another under global operation. However, many global operations cannot be performed using only local operations and classical communication (LOCC). A natural question arises what is and what is not locally possible in state discrimination. In [1], Bennett et al. showed that there exist sets of orthogonal pure product ✩ This work was partly supported by the National Foundation of Natural Sciences of China (Grant No. 60321002) and the Key Grant Project of Chinese Ministry of Education (Grant No. 10403). * Corresponding author. E-mail address: [email protected] (H. Cao).
0375-9601/$ – see front matter 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2004.10.057
states that cannot be distinguished by LOCC, which is a rather surprising result counter to our intuition. But Walgate et al. [2] proved that any two orthogonal pure states can be distinguished perfectly under LOCC. Recently, considerable attention has been drawn to the study of some special cases of local discrimination of multipartite pure states. In [3], Ghosh et al. considered the distinguishability of Bell states and found that it is impossible to discriminate (deterministically or probabilistically) among the four Bell states locally if only one copy is provided. They also proved that discriminating among any three Bell states is not possible under LOCC. Walgate and Hardy [4] derived a necessary and sufficient condition for perfect dis-
H. Cao, M. Ying / Physics Letters A 333 (2004) 232–234
crimination among orthogonal pure states of an arbitrary (2 × n)-dimensional quantum system. Horodecki et al. [5] showed that any full basis of a bipartite system is not distinguishable by LOCC if at least one of the vectors is entangled. In [6], Chen and Li derived a Schmidt number criterion for the distinguishability of orthogonal pure states of an arbitrary (m × n)dimensional quantum system. Fan [7] demonstrated that the d 2 pairwise orthogonal maximally entangled states in canonical form can never be distinguished deterministically or probabilistically under LOCC provided there is only one copy. Some other authors considered the conclusive and inconclusive local discrimination of multipartite pure states. In [8], Virmani et al. showed that for inconclusive discrimination of any two pure states, the optimal probability of error can be attained under LOCC only. Chen and Yang [9] proved that the same is true in the case of conclusive discrimination provided the two states are prepared with equal prior probability. More recently, Ji et al. [10] extended their result to the general case that the two states are prepared with arbitrary prior probability. In [11], Chefles derived a necessary and sufficient condition for unambiguous state discrimination using only LOCC. This Letter is highly motivated by the result of Walgate et al. [2]. It was shown in [2] that any m orthogonal pure states can be perfectly discriminated using only LOCC if (m − 1) copies are provided. However, there do exist m orthogonal pure states that can be discriminated locally with less than (m − 1) copies. One such example is the four Bell states ± Ψ
AB
1 = √ |00AB ± |11AB 2
(1)
1 = √ |01AB ± |10AB 2
(2)
and ± Φ
AB
which can be discriminated by LOCC perfectly if two copies are provided. In this case, the two parties Alice and Bob need only perform von Neumann measurement locally on their own system in the computational basis {|0A , |1A } and {|0B , |1B }, respectively. Through classical communication, they can exchange information about the measurement result and identify which of the four Bell states they share.
233
In [2], only the four Bell states were considered which compose a maximally entangled basis set of a 2 × 2 quantum system. In this Letter, we consider the distinguishability of maximally entangled basis sets in canonical form in a general d × d quantum system. Here, by ‘canonical form’, we mean any set of maximally entangled states that can be written as d−1 1 jn ω |j A ⊗ (j + m) mod d B , |ψmn AB = √ d j =0
m, n = 0, . . . , d − 1,
(3) 2π i d
as defined in [12], where ω = e and {|iA }, {|iB } are standard orthonormal bases of d-dimensional Hilbert spaces HA and HB , respectively. Since the states in Eq. (3) generalize Bell states in a d × d quantum system, it is natural to ask how many copies are needed to distinguish locally the d 2 states in Eq. (3) above. We have the following theorem which is the main result of this Letter. Theorem. Two copies are sufficient to distinguish locally the d 2 orthogonal maximally entangled states in Eq. (3). Proof. We prove this by constructing an explicit protocol as follows. Suppose the two particles of each copy are shared between Alice and Bob. (1) For the first copy d−1 1 jn |ψmn AB = √ ω |j A ⊗ (j + m) mod d B . d j =0
(4) Alice and Bob both perform von Neumann measurement locally on their own system in the computational basis {|iA } and {|iB }, respectively. Through classical communication, they can exchange information about the measurement result and identify m. (2) For the second copy, first, Alice performs adjoint Fourier transformation F † and Bob performs Fourier transformation F locally on their own system, respectively. Then the state turns into d−1 1 j m |ψ˜ mn AB = √ ω (j + n) mod d A ⊗ |j B . d j =0
Repeat step (1), Alice and Bob will identify n.
(5)
234
H. Cao, M. Ying / Physics Letters A 333 (2004) 232–234
Obviously, m and n uniquely determine the state |ψmn AB . This completes the proof. 2 After this work is done, we notice that Ghosh et al. [12] have already independently found this result using a different method. In their proof, they use the teleportation protocol. Alice first teleports the states |0 and √1 (|0 + |1 + · · · + |d − 1) through the two d copies of the shared channel |ψmn AB to Bob. After the teleportation protocol is over, the two states of Bob’s system to the two copies are |m d−1 corresponding j n |(j + m) mod d , respectively. Oband √1 ω B j =0 d viously, Bob himself can identify m and n locally. Compared to the method of Ghosh et al., our proof is much simpler in that we need no auxiliary system. Moreover, our proof embodies the symmetrical roles that m and n play in the state |ψmn AB . Finally, there may be a close connection between our proof and that of Ghosh et al. since Fourier transformation plays an important role in the teleportation protocol [13,14]. We would like to point out that our discrimination protocol only applies to maximally entangled basis set in the form shown in Eq. (3). For general cases, our method does not work since the most general form of any maximally entangled basis set of a d × d quantum system is not known [15]. To summarize, we independently find that two copies are enough to distinguish locally a complete basis of maximally entangled states in canonical form. We also make some comparison between our proof and that of Ghosh et al.
Acknowledgements The authors thank Z.F. Ji, R.Y. Duan and Y. Feng for useful discussion in this field. We also thank anonymous referee for helpful comments and informing us of a valuable reference.
References [1] C.H. Bennett, D.P. DiVincenzo, C.A. Fuchs, T. Mor, E. Rains, P.W. Shor, J.A. Smolin, W.K. Wootters, Phys. Rev. A 59 (1999) 1070. [2] J. Walgate, A.J. Short, L. Hardy, V. Vedral, Phys. Rev. Lett. 85 (2000) 4972. [3] S. Ghosh, G. Kar, A. Roy, A. Sen(De), U. Sen, Phys. Rev. Lett. 87 (2001) 277902. [4] J. Walgate, L. Hardy, Phys. Rev. Lett. 89 (2002) 147901. [5] M. Horodecki, A. Sen(De), U. Sen, K. Horodecki, Phys. Rev. Lett. 90 (2003) 047902. [6] P.X. Chen, C.Z. Li, Phys. Rev. A 68 (2003) 062107. [7] H. Fan, Phys. Rev. Lett. 92 (2004) 177905. [8] S. Virmani, M.F. Sacchi, M.B. Plenio, D. Markham, Phys. Lett. A 288 (2001) 62. [9] Y.X. Chen, D. Yang, Phys. Rev. A 65 (2002) 022320. [10] Z.F. Ji, H.E. Cao, M.S. Ying, quant-ph/0407120. [11] A. Chefles, Phys. Rev. A 69 (2004) 050307. [12] S. Ghosh, G. Kar, A. Roy, D. Sarkar, quant-ph/0205105. [13] C.H. Bennett, G. Brassard, C. Crépeau, R. Jozsa, A. Peres, W.K. Wootters, Phys. Rev. Lett. 70 (1993) 1895. [14] D.N. Mermin, Phys. Rev. A 65 (2001) 012320. [15] R.F. Werner, J. Phys. A: Math. Gen. 34 (2001) 7081.