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Entanglement change of mixed states under canonical unitary operations in two qubits
Physics Letters A 337 (2005) 29–36 www.elsevier.com/locate/pla
Entanglement change of mixed states under canonical unitary operations in two qubits Cheng-Zhi Wang ∗ , Chun-Xian Li, Guang-Can Guo Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China Received 26 October 2004; received in revised form 24 January 2005; accepted 24 January 2005 Available online 29 January 2005 Communicated by P.R. Holland
Abstract We investigate entanglement change of mixed states in a class by applying canonical unitary form of two-qubit unitary operations. The class, including Werner states, Bell diagonal states and maximum entangled mixed states, is characterized by three real parameters and geometrically presented by a tetrahedron. We propose a class of mixed states whose entanglement cannot be increased by any canonical unitary operation. 2005 Elsevier B.V. All rights reserved. PACS: 03.65.Ta; 03.67.-a; 03.65.Bz Keywords: Entanglement; Canonical unitaries; Maximally entangled mixed states
1. Introduction Quantum states, as well as quantum operations including unitary operations and measurements, are a useful physical resource and play an essential role in quantum information processing (QIP) [1]. The former are the carriers of information and used to encode quantum information, while the latter as a quantum dynamical resource manipulate the latter. It is of significant importance to study their properties not only for experiment but for pure theory. The properties for the latter can be studied via those of the former, and vice versa. The properties with respect to quantum unitary operations, such as entangling capability or power [2,3], classification and equivalence classes [4,5], have been studied via quantum states. On the other hand, some properties of quantum states under quantum operations was found, the bound entangled states is the case [6]. In particular, Ishizaka and Hiroshima [7] investigated the entanglement of two-qubit states operated by unitary operations, and found maximally entangled * Corresponding author.
E-mail address: [email protected] (C.-Z. Wang). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.01.057
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C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
mixed states (MEMS’s). Those states have an interesting property that their entanglement cannot be increased by any unitary. Further study showed MEMS’s are the only ones with this property [8]. As well known, any two-qubit unitary can be decomposed into local ones and completely nonlocal one [1,9]. The completely nonlocal one is called canonical unitary, which are widely used in QIP [1,2,5,9]. Generally, the process of a unitary operation on a state is equivalent to three steps: first, local unitary operates on a state, and then canonical is applied, finally another local one is applied. However, in many situations one has the freedom to select the canonical unitary, while the local unitaries remain fixed. Indeed, this Letter was motivated by the study of interacting spin chains [10], where the unitary operation is the time evolution under a nearest-neighbour Hamiltonian, and is therefore subject to restrictions of the kind just mentioned above. In this Letter, we study in detail the entanglement change of a class of two-qubit mixed states by canonical unitary operations. The considered class of mixed states contains some important quantum states such as the MEMS’s, Werner states [11] and Bell diagonal states [12], and is characterized by three parameters and geometrically presented by a tetrahedron. By calculating the entanglement changes of the states in the tetrahedron, we find a class of states with the property that their entanglement cannot be increased by any canonical unitary operation. We call the class of states maximally entangled mixed states under canonical unitary operation (MEMSCUO for short). MEMSCUO class contains MEMS class because of our restricted unitary. A possible application of MEMSCUO in QIP and comparison between MEMSCUO and MEMS are also given. The Letter is organized as follows, in Section 2, the decompositions of two-qubit mixed state and unitary operation are reviewed, in Section 3, we investigate the entanglement change in detail, and find the class of MEMSCUO. In addition, the comparison of MEMSCUO with MEMS, and possible application are also given in the section. The Letter ends in Section 4 with a brief conclusion.
2. Two-qubit mixed states and decomposition of unitary operations We start with a brief review on the decompositions of two-qubit quantum state and unitary operation. An arbitrary mixed two-qubit state can be written in terms of Pauli operators as MAB =
3 1 βi,j σi ⊗ σj , I2 ⊗ I2 + u · σ ⊗ I2 + I2 ⊗ v · σ + 4
(1)
i,j =1
σ ⊗ I2 )] and v = TrB [MAB (I2 ⊗ σ )] belong to real where I2 is identity operator in two dimensions, u = TrA [MAB ( vectors, σ = (σ1 , σ2 , σ3 ) with σi being Pauli operator, β is a real matrix with the entry βi,j = TrAB [MAB (σi ⊗ σj )]. According to the equation U(r · σ )U† = (Or ) · σ (U ∈ SU(2) and O ∈ SO(3)), MAB is equivalent to the form, up to local unitary operations ρAB =
3 1 · σ ⊗ I2 + I2 ⊗ n · σ + αi σi ⊗ σi , I2 ⊗ I2 + m 4
(2)
i=1
where m = O1 u, n = O2 v, α = diag(α1 , α2 , α3 ) = O1 βO+ 2 . Since ρAB is locally equivalent to MAB , for simplicity, only the type of the states Eq. (2) is considered in this Letter. As shown in Refs. [1,9], any two-qubit unitary operation can be decomposed into local and completely nonlocal parts U = (A1 ⊗ B1 ) exp −i
3 i=1
θi σi ⊗ σi (A2 ⊗ B2 ),
(3)
C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
where Ai and Bi are local unitary operations on systems A and B, respectively, and completely nonlocal part is defined as the canonical form: Uc (θ1 , θ2 , θ3 ) = exp −i
3
31 π 4
θ1 θ2 |θ3 |. The
θi σi ⊗ σi ,
(4)
i=1
which is equivalent to U up to local unitary operations. For convenience, we call Uc (θ1 , θ2 , θ3 ) the canonical unitary operation (CUO). The canonical unitary operation of any unitary operation is of practical utility in QIP, due to the fact that a general two-qubit Hamiltonian that governs the quantum computing is always locally equivalent to the canonical form: i=1,2,3 ki σi ⊗ σi , with k1 k2 |k3 |. Many important interaction Hamiltonians are of the canonical form, such as Ising interaction H = J σ3 ⊗ σ3 , XYZ model H = J1 σ1 ⊗ σ1 + J2 σ2 ⊗ σ2 + J3 σ3 ⊗ σ3 (J1 = J2 = J3 = 0) and Heisenberg interaction H = J (σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 ). It is noted that the time evolution operations of the canonical form of Hamiltonians exactly give the forms of CUO.
3. Entanglement of the mixed states under CUO In this section, our goal is to find some properties of the states in Eq. (2) by calculating the entanglement change under any CUO. Operated by Uc , the state ρAB changes into = Uc ρAB U†c ρAB 3 1 I2 ⊗ I2 + m = · σ ⊗ I2 + I2 ⊗ n · σ + +ti,j σi ⊗ σj , 4
(5)
i,j =1
where m1 = m1 c2 c3 + n1 s2 s3 ,
n1 = n1 c2 c3 + m1 s2 s3 ,
m2 = m2 c1 c3 + n2 s1 s3 ,
n2 = n2 c1 c3 + m2 s1 s3 ,
m3 = m3 c1 c2 + n3 s1 s2 ,
n3 = n3 c1 c2 + m3 s1 s2 ,
ti,j =
3
εij k (mk ci sj − nk cj si ) + δi,j αi ,
(6)
k=1
with ci ≡ cos(2θi ), si ≡ sin(2θi ), εij k is Levi-Civita symbol. For simplicity, we only consider the following states: ρ=
1 I2 ⊗ I2 + x(I2 ⊗ σ3 + σ3 ⊗ I2 ) + y(σ1 ⊗ σ1 + σ2 ⊗ σ2 ) + zσ3 ⊗ σ3 . 4
(7)
The class of states includes Werner states, Bell diagonal states, MEMS’s, and so on. For a valid physical quantum state in the class, it must be in the tetrahedron with vertices A(0, −1, −1), B(−1, 0, 1), C(1, 0, 1), D(0, 1, −1). According to the partial transposition criterion, a two-qubit state is a disentangled state if and only if its partial 2 transposition is positive [13]. Therefore, a disentangled state in the class satisfies the condition x 2 + y 2 (1+z) 4 , which forms a cone in the parameters space. So the disentangled states are located in the intersection of the tetrahedron and the cone as shown in Fig. 1. The entanglement of two-qubit mixed states is customarily measured by the entanglement of formation (EF ) [14]. The EF for a mixed state ρ is defined as E F (ρ) = minpi ,|ψi i pi E(|ψi ψi |), where the minimum is taken over all possible decompositions of ρ into ρ = i pi |ψi ψi |, and E(|ψi ψi |) is the Neumann entropy of
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Fig. 1. Geometrical representation of the class of states. All the states are in the tetrahedron with vertices A = (0, −1, −1), B = (−1, 0, 1), 2 C = (1, 0, 1), D = (0, 1, −1). The disentangled states are in the intersection of the tetrahedron ABCD and the cone x 2 + y 2 (1+z) 4 .
either of the two subsystems. For two-qubit systems, EF is expressed explicitly as [15]: 1 + 1 − C(ρ)2 EF (ρ) = H , 2
(8)
where H (x) = −x log2 x − (1 − x) log2 (1 − x). The concurrence C(ρ) of the state ρ is defined as: C(ρ) = max{0, λ1 − λ2 − λ3 − λ4 },
(9)
where λ’s are the square roots of eigenvalues of ρ ρ˜ in nonincreasing order, ρ˜ = (σ2 ⊗ σ2 )ρ ∗ (σ2 ⊗ σ2 ), ρ ∗ denotes the complex conjugate of ρ in standard computational basises {|00, |01, |10, |11}. Note that EF is a monotonic function of C(ρ), the concurrence C(ρ) is also a measure of entanglement. According to Eq. (5) and Eq. (7), the final state is given by ρ =
1 I2 ⊗ I2 + x cos θ (σ3 ⊗ I2 + I2 ⊗ σ3 ) + x sin θ (σ1 ⊗ σ2 + σ2 ⊗ σ1 ) 4 + y(σ1 ⊗ σ1 + σ2 ⊗ σ2 ) + zσ3 ⊗ σ3 ,
(10)
with θ = 2(θ2 − θ1 ). The square roots of eigenvalues of ρ ρ˜ are given by 1 − z − 2y , 4 1 − z + 2y λ2 = , 4 1 λ3 = λ4 = (1 + z)2 − 4x 2 , 4 and those of ρ ρ˜ are λ1 =
λ1 = λ1 , λ2 = λ2 ,
1 1 2 2 2 (1 + z) − x cos θ + |x sin θ | , λ3 = 2 4
(11)
C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
λ4
1 = 2
33
1 (1 + z)2 − x 2 cos2 θ − |x sin θ | . 4
(12)
Different cases due to orders of λi and λi are given in calculating concurrence. Case 1. |y| − |x| − z 0 The inequality implies that |y| − |x| − z 0 ⇒ 1 − z + 2|y| 1 + z + 2|x| 1 + z − 2|x| ⇒ 1 − z + 2|y|
(1 + z)2 − 4x 2 .
(13)
So max{λ1 , λ2 } = (1 − z + 2|y|)/4 max{λ3 , λ4 }. As a result, the entanglement of initial state ρ is given by
1 (1 + z)2 − 4x 2 . C(ρ) = max 0, |λ1 − λ2 | − λ3 − λ4 = max 0, |y| − (14) 2 Because max{λ3 , λ4 } = λ3 and the maximum value of λ3 , given by (1 + z + 2|x|)/4 when θ = π2 cannot excess the value of max{λ1 , λ2 }, max{λi } = λ1 for y 0 and max{λi } = λ2 for y 0. The entanglement of the final state ρ is
1 2 2 2 (1 + z) − 4x cos θ . C(ρ ) = max 0, |λ1 − λ2 | − λ3 − λ4 , = max 0, |y| − (15) 2 In order to measure the change of entanglement, we introduce C(ρ, ρ ) ≡ C(ρ ) − C(ρ). Comparing Eqs. (14) and (15) we find that C(ρ, ρ ) 0 always holds for any θ and the equation is taken when cos(4(θ1 − θ2 )) = 1. we conclude that the entanglement of any state in the case can never be increased further by any CUO. Case 2. |y| − |x| − z 0 and 1 − z + 2|y| (1 + z)2 − 4x 2 Based on the latter inequality, the entanglement of ρ is still expressed as
1 2 2 (1 + z) − 4x . C(ρ) = max 0, |y| − (16) 2 To calculate the concurrence of ρ , it need to compare λ1 with λ3 . It is noted that there exists θ = θ meeting λ3 = λ1 , namely, (1 + z)2 − 4x 2 cos2 θ + 2|x sin θ | = 1 − z + 2|y|. (17) If θ satisfies 0 | sin θ | | sin θ |, which means max{λi } = λ1 , the concurrence of ρ is given by
1 C(ρ ) = max{0, λ1 − λ2 − λ3 − λ4 } = max 0, |y| − (1 + z)2 − 4x 2 cos2 θ . 2 While if | sin θ | | sin θ | 1, then max{λi } = λ3 and the concurrence of ρ is
1−z . C(ρ ) = max{0, λ3 − λ1 − λ2 − λ4 } = max 0, |x sin θ | − 2 From Eqs. (18) and (19), it follows that the maximal entanglement of the final state ρ is
1 1−z . (1 + z)2 − 4x 2 , |x| − C(ρ )max = max 0, max |y| − 2 2
(18)
(19)
(20)
The greatest change for entanglement max( C) 0. Different from Case 1, the entanglement of some states in the case can be increased by some CUO.
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Case 3. |y| − |x| − z 0 and 1 − z + 2|y| The concurrence of ρ is simple
(1 + z)2 − 4x 2
C(ρ) = 0. The initial states in the case are disentangled states, while the entanglement for the final states is the form
1−z , C(ρ ) = max{0, λ3 − λ1 − λ2 − λ4 } = max 0, |x sin θ | − 2
(21)
(22)
which archives the maximal value when the canonical unitary is Uc = exp[−i π4 (σ1 ⊗ σ1 + σ2 ⊗ σ2 )]. The greatest change of entanglement max( C) = max{0, |x| − 1−z 2 }. From above results, it is interesting to note that the entanglement of some states in the class can never be increased by any CUO. Those states are called maximally entangled mixed states under CUO (MEMSCUO’s) analogous to maximally entangled mixed states. Next, we compare MEMSCUO’s with MEMS’s. From above discussions, the class of MEMSCUO’s in the tetrahedron is given by ρ = {MEMSCUO | R1 ∪ R2 ∪ R3 },
(23)
where R1 , R2 , R3 are three sets, R1 = {(x, y, z) | z 0}, R2 = {(x, y, z) | z > 0, |x| (1 − z)/2} and R3 = {(x, y, z)| | x| (1 − z)/2, z |y| − |x| ( (1 + z)2 − 4x 2 + z − 1)/2}. In Ref. [7], MEMS’s are those obtained by applying any local unitary transformation to M = p1 Ψ − Ψ − + p2 |00 00| + p3 Ψ + Ψ + + p4 |11 11|, (24) √ where |Ψ ± = (|01 ± |10)/ 2 are Bell states, pi ’s are eigenvalues of M in decreasing order (p1 p2 p3 p4 ), and p1 + p2 + p3 + p4 = 1. MEMS’s may be pure or disentangled state, but their entanglement cannot be increased by any unitary operation. The class of MEMS’s can be rewritten via Eq. (24) as ρ = MEMS | |y| − |x| − z 0, |y| + |x| + z 0, |y| − |x| + z 0 . (25) If z < 0, MEMS’s always exist, and all states are MEMSCUO’s. If z 0, MEMS’s disappear and only some states are MEMSCUO’s. Figs. 2 and 3 give the visual results of Eqs. (23) and (25) for z = −0.2 and 0.2, respectively. Obviously, the class of MEMS’s is a subset of that of MEMSCUO’s, which stems from the fact that Uc is a special unitary operation, that is, a completely nonlocal part of a general unitary operation as shown in Eq. (4). Entanglement for MEMSCUO cannot be increased by any canonical unitary, while some general unitary can. It can be seen from the decomposition of a general unitary. A local unitary changes a state into other, though it cannot change the entanglement. If a canonical unitary operate the changed state, the entanglement may increase, compared with the initial one. A state, which belongs to MEMSCUO class but not to MEMS class, can always be transformed into a MEMS by a unitary operation. In other words, we can always use a given entangling unitary operation assisted by unlimited local unitaries to transform a MEMSCUO into a MEMS. In addition, given the same eigenvalues of MEMS and MEMSCUO, the entanglement of the former is always greater than or equal to that of the latter. Our results can be helpful for experiment. For example, we have a two-qubit state ρAB and a global unitary U = (A1 ⊗ B1 )Uc (A2 ⊗ B2 ). We want to use these resources to create a larger amount of entanglement. First, we check whether the state (A2 ⊗ B2 )ρAB (A2 ⊗ B2 )† is a MEMSCUO state or not. If it is a MEMSCUO state, then we cannot get a sate with a larger amount of entanglement. However if local unitaries are allowed (please note that local unitaries are easily realized in experiment), we first apply local unitary (A ⊗ B)(A2 ⊗ B2 )† to initial state = (A ⊗ B )U (A ⊗ B)ρ ρAB , and then apply the given global unitary U . At last we get the final state ρAB 1 1 c AB (A ⊗ † † † B) Uc (A1 ⊗ B1 ) , which have a larger amount of entanglement. If one choses proper local unitaries A and B, MEMS is obtained [8].
C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
35
Fig. 2. The geometrical representation of the class of mixed states for z = −0.2. Different parts are shown according to the entanglement change, all parts stand for MEMSCUO, the dark (red in the web version of this article) for MEMS and the blank for the states whose entanglement can be increased by a unitary operation. The external of the circle stands for the entangled states, the circle and the inner of the circle for disentanglement states.
Fig. 3. The geometrical representation of the class of mixed states for z = 0.2. The dark (red in the web version of this article) part stands for MEMSCUO, no part stands for MEMS. External of the arc represents the entangled states.
4. Conclusion Based on the canonical decomposition of any unitary operation, we have investigated in detail the entanglement change of states in the considered class under canonical unitary operation. A class of states is found with a property that the entanglement cannot increase under any canonical unitary operation. The relation between MEMSCUO’s and MEMS’s is given. Our results may be used to study entanglement distribution in spin chain serving as quantum information channel and helpful for investigating structure of mixed states. We note that some properties of these states are still largely unknown and require significant exploration. Open questions such as “given a entangling
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C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
unitary operation, what local unitary operations do we need to transform a MEMSCUO to MEMS,” and “do they have advantages over other states in QIP” are the subject of current investigation.
Acknowledgements C.Z.Wang thank Z.W.Zhou for valuable discussion. This work was funded by National Fundamental Research Program (2001CB309300), the Innovation funds from Chinese Academy of Science, National Natural Science Foundation of China under Grant No. 60121503.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
M.A. Nielsen, et al., Phys. Rev. A 67 (2003) 052301. W. Dür, G. Vidal, J.I. Cirac, N. Linden, S. Popescu, Phys. Rev. Lett. 87 (2001) 137901. P. Zanardi, C. Zalka, L. Faoro, Phys. Rev. A 62 (2000) 030301. W. Dür, G. Vidal, J.I. Cirac, Phys. Rev. Lett. 89 (2002) 057901; Y. Makhlin, quant-ph/0002045. J. Zhang, J. Vala, S. Sastry, K.B. Whaley, Phys. Rev. A 67 (2003) 042313. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 80 (1998) 5239. S. Ishizaka, T. Hiroshima, Phys. Rev. A 62 (2000) 022310. F. Verstraete, K. Audenaert, B. De Moor, Phys. Rev. A 64 (2001) 012316. B. Kraus, J.I. Cirac, Phys. Rev. A 63 (2001) 062309. X. Wang, P. Zanardi, Phys. Lett. A 301 (2002) 1. R. Werner, Phys. Rev. A 40 (1989) 4277. O. Rudolph, quant-ph/0202121. M. Horodecki, P. Horodecki, R. Horodecki, Phys. Rev. Lett. 78 (1997) 574. C.H. Bennett, D.P. DiVincenzo, J.A. Smolin, W.K. Wootters, Phys. Rev. A 54 (1996) 3824. W.K. Wootters, Phys. Rev. Lett. 80 (1998) 2245.
Entanglement change of mixed states under canonical unitary operations in two qubits Cheng-Zhi Wang ∗ , Chun-Xian Li, Guang-Can Guo Key Laboratory of Quantum Information, University of Science and Technology of China, Hefei 230026, China Received 26 October 2004; received in revised form 24 January 2005; accepted 24 January 2005 Available online 29 January 2005 Communicated by P.R. Holland
Abstract We investigate entanglement change of mixed states in a class by applying canonical unitary form of two-qubit unitary operations. The class, including Werner states, Bell diagonal states and maximum entangled mixed states, is characterized by three real parameters and geometrically presented by a tetrahedron. We propose a class of mixed states whose entanglement cannot be increased by any canonical unitary operation. 2005 Elsevier B.V. All rights reserved. PACS: 03.65.Ta; 03.67.-a; 03.65.Bz Keywords: Entanglement; Canonical unitaries; Maximally entangled mixed states
1. Introduction Quantum states, as well as quantum operations including unitary operations and measurements, are a useful physical resource and play an essential role in quantum information processing (QIP) [1]. The former are the carriers of information and used to encode quantum information, while the latter as a quantum dynamical resource manipulate the latter. It is of significant importance to study their properties not only for experiment but for pure theory. The properties for the latter can be studied via those of the former, and vice versa. The properties with respect to quantum unitary operations, such as entangling capability or power [2,3], classification and equivalence classes [4,5], have been studied via quantum states. On the other hand, some properties of quantum states under quantum operations was found, the bound entangled states is the case [6]. In particular, Ishizaka and Hiroshima [7] investigated the entanglement of two-qubit states operated by unitary operations, and found maximally entangled * Corresponding author.
E-mail address: [email protected] (C.-Z. Wang). 0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2005.01.057
30
C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
mixed states (MEMS’s). Those states have an interesting property that their entanglement cannot be increased by any unitary. Further study showed MEMS’s are the only ones with this property [8]. As well known, any two-qubit unitary can be decomposed into local ones and completely nonlocal one [1,9]. The completely nonlocal one is called canonical unitary, which are widely used in QIP [1,2,5,9]. Generally, the process of a unitary operation on a state is equivalent to three steps: first, local unitary operates on a state, and then canonical is applied, finally another local one is applied. However, in many situations one has the freedom to select the canonical unitary, while the local unitaries remain fixed. Indeed, this Letter was motivated by the study of interacting spin chains [10], where the unitary operation is the time evolution under a nearest-neighbour Hamiltonian, and is therefore subject to restrictions of the kind just mentioned above. In this Letter, we study in detail the entanglement change of a class of two-qubit mixed states by canonical unitary operations. The considered class of mixed states contains some important quantum states such as the MEMS’s, Werner states [11] and Bell diagonal states [12], and is characterized by three parameters and geometrically presented by a tetrahedron. By calculating the entanglement changes of the states in the tetrahedron, we find a class of states with the property that their entanglement cannot be increased by any canonical unitary operation. We call the class of states maximally entangled mixed states under canonical unitary operation (MEMSCUO for short). MEMSCUO class contains MEMS class because of our restricted unitary. A possible application of MEMSCUO in QIP and comparison between MEMSCUO and MEMS are also given. The Letter is organized as follows, in Section 2, the decompositions of two-qubit mixed state and unitary operation are reviewed, in Section 3, we investigate the entanglement change in detail, and find the class of MEMSCUO. In addition, the comparison of MEMSCUO with MEMS, and possible application are also given in the section. The Letter ends in Section 4 with a brief conclusion.
2. Two-qubit mixed states and decomposition of unitary operations We start with a brief review on the decompositions of two-qubit quantum state and unitary operation. An arbitrary mixed two-qubit state can be written in terms of Pauli operators as MAB =
3 1 βi,j σi ⊗ σj , I2 ⊗ I2 + u · σ ⊗ I2 + I2 ⊗ v · σ + 4
(1)
i,j =1
σ ⊗ I2 )] and v = TrB [MAB (I2 ⊗ σ )] belong to real where I2 is identity operator in two dimensions, u = TrA [MAB ( vectors, σ = (σ1 , σ2 , σ3 ) with σi being Pauli operator, β is a real matrix with the entry βi,j = TrAB [MAB (σi ⊗ σj )]. According to the equation U(r · σ )U† = (Or ) · σ (U ∈ SU(2) and O ∈ SO(3)), MAB is equivalent to the form, up to local unitary operations ρAB =
3 1 · σ ⊗ I2 + I2 ⊗ n · σ + αi σi ⊗ σi , I2 ⊗ I2 + m 4
(2)
i=1
where m = O1 u, n = O2 v, α = diag(α1 , α2 , α3 ) = O1 βO+ 2 . Since ρAB is locally equivalent to MAB , for simplicity, only the type of the states Eq. (2) is considered in this Letter. As shown in Refs. [1,9], any two-qubit unitary operation can be decomposed into local and completely nonlocal parts U = (A1 ⊗ B1 ) exp −i
3 i=1
θi σi ⊗ σi (A2 ⊗ B2 ),
(3)
C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
where Ai and Bi are local unitary operations on systems A and B, respectively, and completely nonlocal part is defined as the canonical form: Uc (θ1 , θ2 , θ3 ) = exp −i
3
31 π 4
θ1 θ2 |θ3 |. The
θi σi ⊗ σi ,
(4)
i=1
which is equivalent to U up to local unitary operations. For convenience, we call Uc (θ1 , θ2 , θ3 ) the canonical unitary operation (CUO). The canonical unitary operation of any unitary operation is of practical utility in QIP, due to the fact that a general two-qubit Hamiltonian that governs the quantum computing is always locally equivalent to the canonical form: i=1,2,3 ki σi ⊗ σi , with k1 k2 |k3 |. Many important interaction Hamiltonians are of the canonical form, such as Ising interaction H = J σ3 ⊗ σ3 , XYZ model H = J1 σ1 ⊗ σ1 + J2 σ2 ⊗ σ2 + J3 σ3 ⊗ σ3 (J1 = J2 = J3 = 0) and Heisenberg interaction H = J (σ1 ⊗ σ1 + σ2 ⊗ σ2 + σ3 ⊗ σ3 ). It is noted that the time evolution operations of the canonical form of Hamiltonians exactly give the forms of CUO.
3. Entanglement of the mixed states under CUO In this section, our goal is to find some properties of the states in Eq. (2) by calculating the entanglement change under any CUO. Operated by Uc , the state ρAB changes into = Uc ρAB U†c ρAB 3 1 I2 ⊗ I2 + m = · σ ⊗ I2 + I2 ⊗ n · σ + +ti,j σi ⊗ σj , 4
(5)
i,j =1
where m1 = m1 c2 c3 + n1 s2 s3 ,
n1 = n1 c2 c3 + m1 s2 s3 ,
m2 = m2 c1 c3 + n2 s1 s3 ,
n2 = n2 c1 c3 + m2 s1 s3 ,
m3 = m3 c1 c2 + n3 s1 s2 ,
n3 = n3 c1 c2 + m3 s1 s2 ,
ti,j =
3
εij k (mk ci sj − nk cj si ) + δi,j αi ,
(6)
k=1
with ci ≡ cos(2θi ), si ≡ sin(2θi ), εij k is Levi-Civita symbol. For simplicity, we only consider the following states: ρ=
1 I2 ⊗ I2 + x(I2 ⊗ σ3 + σ3 ⊗ I2 ) + y(σ1 ⊗ σ1 + σ2 ⊗ σ2 ) + zσ3 ⊗ σ3 . 4
(7)
The class of states includes Werner states, Bell diagonal states, MEMS’s, and so on. For a valid physical quantum state in the class, it must be in the tetrahedron with vertices A(0, −1, −1), B(−1, 0, 1), C(1, 0, 1), D(0, 1, −1). According to the partial transposition criterion, a two-qubit state is a disentangled state if and only if its partial 2 transposition is positive [13]. Therefore, a disentangled state in the class satisfies the condition x 2 + y 2 (1+z) 4 , which forms a cone in the parameters space. So the disentangled states are located in the intersection of the tetrahedron and the cone as shown in Fig. 1. The entanglement of two-qubit mixed states is customarily measured by the entanglement of formation (EF ) [14]. The EF for a mixed state ρ is defined as E F (ρ) = minpi ,|ψi i pi E(|ψi ψi |), where the minimum is taken over all possible decompositions of ρ into ρ = i pi |ψi ψi |, and E(|ψi ψi |) is the Neumann entropy of
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Fig. 1. Geometrical representation of the class of states. All the states are in the tetrahedron with vertices A = (0, −1, −1), B = (−1, 0, 1), 2 C = (1, 0, 1), D = (0, 1, −1). The disentangled states are in the intersection of the tetrahedron ABCD and the cone x 2 + y 2 (1+z) 4 .
either of the two subsystems. For two-qubit systems, EF is expressed explicitly as [15]: 1 + 1 − C(ρ)2 EF (ρ) = H , 2
(8)
where H (x) = −x log2 x − (1 − x) log2 (1 − x). The concurrence C(ρ) of the state ρ is defined as: C(ρ) = max{0, λ1 − λ2 − λ3 − λ4 },
(9)
where λ’s are the square roots of eigenvalues of ρ ρ˜ in nonincreasing order, ρ˜ = (σ2 ⊗ σ2 )ρ ∗ (σ2 ⊗ σ2 ), ρ ∗ denotes the complex conjugate of ρ in standard computational basises {|00, |01, |10, |11}. Note that EF is a monotonic function of C(ρ), the concurrence C(ρ) is also a measure of entanglement. According to Eq. (5) and Eq. (7), the final state is given by ρ =
1 I2 ⊗ I2 + x cos θ (σ3 ⊗ I2 + I2 ⊗ σ3 ) + x sin θ (σ1 ⊗ σ2 + σ2 ⊗ σ1 ) 4 + y(σ1 ⊗ σ1 + σ2 ⊗ σ2 ) + zσ3 ⊗ σ3 ,
(10)
with θ = 2(θ2 − θ1 ). The square roots of eigenvalues of ρ ρ˜ are given by 1 − z − 2y , 4 1 − z + 2y λ2 = , 4 1 λ3 = λ4 = (1 + z)2 − 4x 2 , 4 and those of ρ ρ˜ are λ1 =
λ1 = λ1 , λ2 = λ2 ,
1 1 2 2 2 (1 + z) − x cos θ + |x sin θ | , λ3 = 2 4
(11)
C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
λ4
1 = 2
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1 (1 + z)2 − x 2 cos2 θ − |x sin θ | . 4
(12)
Different cases due to orders of λi and λi are given in calculating concurrence. Case 1. |y| − |x| − z 0 The inequality implies that |y| − |x| − z 0 ⇒ 1 − z + 2|y| 1 + z + 2|x| 1 + z − 2|x| ⇒ 1 − z + 2|y|
(1 + z)2 − 4x 2 .
(13)
So max{λ1 , λ2 } = (1 − z + 2|y|)/4 max{λ3 , λ4 }. As a result, the entanglement of initial state ρ is given by
1 (1 + z)2 − 4x 2 . C(ρ) = max 0, |λ1 − λ2 | − λ3 − λ4 = max 0, |y| − (14) 2 Because max{λ3 , λ4 } = λ3 and the maximum value of λ3 , given by (1 + z + 2|x|)/4 when θ = π2 cannot excess the value of max{λ1 , λ2 }, max{λi } = λ1 for y 0 and max{λi } = λ2 for y 0. The entanglement of the final state ρ is
1 2 2 2 (1 + z) − 4x cos θ . C(ρ ) = max 0, |λ1 − λ2 | − λ3 − λ4 , = max 0, |y| − (15) 2 In order to measure the change of entanglement, we introduce C(ρ, ρ ) ≡ C(ρ ) − C(ρ). Comparing Eqs. (14) and (15) we find that C(ρ, ρ ) 0 always holds for any θ and the equation is taken when cos(4(θ1 − θ2 )) = 1. we conclude that the entanglement of any state in the case can never be increased further by any CUO. Case 2. |y| − |x| − z 0 and 1 − z + 2|y| (1 + z)2 − 4x 2 Based on the latter inequality, the entanglement of ρ is still expressed as
1 2 2 (1 + z) − 4x . C(ρ) = max 0, |y| − (16) 2 To calculate the concurrence of ρ , it need to compare λ1 with λ3 . It is noted that there exists θ = θ meeting λ3 = λ1 , namely, (1 + z)2 − 4x 2 cos2 θ + 2|x sin θ | = 1 − z + 2|y|. (17) If θ satisfies 0 | sin θ | | sin θ |, which means max{λi } = λ1 , the concurrence of ρ is given by
1 C(ρ ) = max{0, λ1 − λ2 − λ3 − λ4 } = max 0, |y| − (1 + z)2 − 4x 2 cos2 θ . 2 While if | sin θ | | sin θ | 1, then max{λi } = λ3 and the concurrence of ρ is
1−z . C(ρ ) = max{0, λ3 − λ1 − λ2 − λ4 } = max 0, |x sin θ | − 2 From Eqs. (18) and (19), it follows that the maximal entanglement of the final state ρ is
1 1−z . (1 + z)2 − 4x 2 , |x| − C(ρ )max = max 0, max |y| − 2 2
(18)
(19)
(20)
The greatest change for entanglement max( C) 0. Different from Case 1, the entanglement of some states in the case can be increased by some CUO.
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Case 3. |y| − |x| − z 0 and 1 − z + 2|y| The concurrence of ρ is simple
(1 + z)2 − 4x 2
C(ρ) = 0. The initial states in the case are disentangled states, while the entanglement for the final states is the form
1−z , C(ρ ) = max{0, λ3 − λ1 − λ2 − λ4 } = max 0, |x sin θ | − 2
(21)
(22)
which archives the maximal value when the canonical unitary is Uc = exp[−i π4 (σ1 ⊗ σ1 + σ2 ⊗ σ2 )]. The greatest change of entanglement max( C) = max{0, |x| − 1−z 2 }. From above results, it is interesting to note that the entanglement of some states in the class can never be increased by any CUO. Those states are called maximally entangled mixed states under CUO (MEMSCUO’s) analogous to maximally entangled mixed states. Next, we compare MEMSCUO’s with MEMS’s. From above discussions, the class of MEMSCUO’s in the tetrahedron is given by ρ = {MEMSCUO | R1 ∪ R2 ∪ R3 },
(23)
where R1 , R2 , R3 are three sets, R1 = {(x, y, z) | z 0}, R2 = {(x, y, z) | z > 0, |x| (1 − z)/2} and R3 = {(x, y, z)| | x| (1 − z)/2, z |y| − |x| ( (1 + z)2 − 4x 2 + z − 1)/2}. In Ref. [7], MEMS’s are those obtained by applying any local unitary transformation to M = p1 Ψ − Ψ − + p2 |00 00| + p3 Ψ + Ψ + + p4 |11 11|, (24) √ where |Ψ ± = (|01 ± |10)/ 2 are Bell states, pi ’s are eigenvalues of M in decreasing order (p1 p2 p3 p4 ), and p1 + p2 + p3 + p4 = 1. MEMS’s may be pure or disentangled state, but their entanglement cannot be increased by any unitary operation. The class of MEMS’s can be rewritten via Eq. (24) as ρ = MEMS | |y| − |x| − z 0, |y| + |x| + z 0, |y| − |x| + z 0 . (25) If z < 0, MEMS’s always exist, and all states are MEMSCUO’s. If z 0, MEMS’s disappear and only some states are MEMSCUO’s. Figs. 2 and 3 give the visual results of Eqs. (23) and (25) for z = −0.2 and 0.2, respectively. Obviously, the class of MEMS’s is a subset of that of MEMSCUO’s, which stems from the fact that Uc is a special unitary operation, that is, a completely nonlocal part of a general unitary operation as shown in Eq. (4). Entanglement for MEMSCUO cannot be increased by any canonical unitary, while some general unitary can. It can be seen from the decomposition of a general unitary. A local unitary changes a state into other, though it cannot change the entanglement. If a canonical unitary operate the changed state, the entanglement may increase, compared with the initial one. A state, which belongs to MEMSCUO class but not to MEMS class, can always be transformed into a MEMS by a unitary operation. In other words, we can always use a given entangling unitary operation assisted by unlimited local unitaries to transform a MEMSCUO into a MEMS. In addition, given the same eigenvalues of MEMS and MEMSCUO, the entanglement of the former is always greater than or equal to that of the latter. Our results can be helpful for experiment. For example, we have a two-qubit state ρAB and a global unitary U = (A1 ⊗ B1 )Uc (A2 ⊗ B2 ). We want to use these resources to create a larger amount of entanglement. First, we check whether the state (A2 ⊗ B2 )ρAB (A2 ⊗ B2 )† is a MEMSCUO state or not. If it is a MEMSCUO state, then we cannot get a sate with a larger amount of entanglement. However if local unitaries are allowed (please note that local unitaries are easily realized in experiment), we first apply local unitary (A ⊗ B)(A2 ⊗ B2 )† to initial state = (A ⊗ B )U (A ⊗ B)ρ ρAB , and then apply the given global unitary U . At last we get the final state ρAB 1 1 c AB (A ⊗ † † † B) Uc (A1 ⊗ B1 ) , which have a larger amount of entanglement. If one choses proper local unitaries A and B, MEMS is obtained [8].
C.-Z. Wang et al. / Physics Letters A 337 (2005) 29–36
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Fig. 2. The geometrical representation of the class of mixed states for z = −0.2. Different parts are shown according to the entanglement change, all parts stand for MEMSCUO, the dark (red in the web version of this article) for MEMS and the blank for the states whose entanglement can be increased by a unitary operation. The external of the circle stands for the entangled states, the circle and the inner of the circle for disentanglement states.
Fig. 3. The geometrical representation of the class of mixed states for z = 0.2. The dark (red in the web version of this article) part stands for MEMSCUO, no part stands for MEMS. External of the arc represents the entangled states.
4. Conclusion Based on the canonical decomposition of any unitary operation, we have investigated in detail the entanglement change of states in the considered class under canonical unitary operation. A class of states is found with a property that the entanglement cannot increase under any canonical unitary operation. The relation between MEMSCUO’s and MEMS’s is given. Our results may be used to study entanglement distribution in spin chain serving as quantum information channel and helpful for investigating structure of mixed states. We note that some properties of these states are still largely unknown and require significant exploration. Open questions such as “given a entangling
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unitary operation, what local unitary operations do we need to transform a MEMSCUO to MEMS,” and “do they have advantages over other states in QIP” are the subject of current investigation.
Acknowledgements C.Z.Wang thank Z.W.Zhou for valuable discussion. This work was funded by National Fundamental Research Program (2001CB309300), the Innovation funds from Chinese Academy of Science, National Natural Science Foundation of China under Grant No. 60121503.
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