A Note on Local Unitary Equivalence of Isotropic-like States
aa r X i v : . [ qu a n t - ph ] S e p A Note on Local Unitary Equivalence of Isotropic-like States
Tinggui Zhang , Bobo Hua , Ming Li , Ming-Jing Zhao , and Hong Yang School of Mathematics and Statistics, Hainan Normal University, Haikou 571158, China School of Mathematical Sciences, LMNS, Fudan University, Shanghai 200433, China College of the Science, China University of Petroleum, 266580 Qingdao, China Department of Mathematics, School of Science, Beijing Information Science and Tech-nology University, 100192, Beijing, China College of Physics and Electronic Engineering, Hainan Normal University, Haikou571158, China
Abstract
We consider the local unitary equivalence of a class of quantum states in bipartite case and multipartitecase. The necessary and sufficient condition is presented. As special cases, the local unitary equivalent classes ofisotropic state and werner state are provided. Then we study the local unitary similar equivalence of this class ofquantum states and analyze the necessary and sufficient condition.Pacs numbers: 03.67.-a, 02.20.Hj, 03.65.-wKey words:
Mixed state, Local unitary equivalence, Local unitary similar equivalence
E-mail address: [email protected] (Ming-Jing Zhao)
PACS numbers:
I. INTRODUCTION
Entanglement is one of the most extraordinary features of quantum physics. It plays a vital role in quantuminformation processing, including quantum teleportation, quantum cryptography, quantum computation, etc. [1].One fact is that two entangled states are said to be equivalent in implementing the same quantum information task ifthey can be obtained from each other via local operation and classical communication(LOCC). In particular, all theLOCC equivalent quantum pure states are interconvertible by local unitary operators (LU)[2]. As many properties likequantum correlation, quantum entanglement, quantum discord keep invariant under local unitary transformations, itis significant to classify and characterize quantum states in terms of local unitary transformations.There are a lot of literatures to deal with the LU problem, one approach is to construct invariants of local unitarytransformations [3–10]. Usually the invariants of mixed states are dependent of pure state decomposition. Recently,the invariants of bipartite states independent of the pure states decomposition are studied in [11]. the LU problemfor multipartite pure qubits states has been solved in [12]. By exploiting the high order singular value decompositiontechnique and local symmetries of the states, Ref. [13] presents a practical scheme of classification under local unitarytransformations for general multipartite pure states with arbitrary dimensions, which extends results of n-qubit purestates [12] to that of n-qudit pure states. For mixed states, Ref. [14] solved the LU problem of arbitrary dimensionalbipartite non-degenerated quantum systems by presenting a complete set of invariants, such that two density matricesare locally unitary equivalent if and only if all these invariants have equal values. In [15] the case of multipartitesystems is studied and a complete set of invariants is presented for a special class of mixed states. Recently, we havestudied the local unitary equivalence of multipartite mixed states using the technology of matrix realignment andpartial transpose [16] and solved the LU problem for multi-qubit mixed states with Bloch representation[17].In this paper, we study the LU problem for a special class of quantum states. The necessary and sufficient conditionis provided. Especially, the local unitary equivalence class of isotropic states [18] and Werner states [19] are obtained.Then we study the local unitary similar equivalence of this class of states and give the necessary and sufficientcondition.
II. LOCAL UNITARY EQUIVALENCE
Two multipartite mixed states ρ and ρ ′ in H ⊗ H ⊗ · · · ⊗ H n are said to be equivalent under local unitarytransformations if there exist unitary operators U i on the i -th Hilbert space H i such that ρ ′ = ( U ⊗ U ⊗ · · · ⊗ U n ) ρ ( U ⊗ U ⊗ · · · ⊗ U n ) † . (1)First, we consider the case of bipartite system. Let H be an N − dimensional complex Hilbert space with | i i , i = 1 , , · · · , N an orthonormal basis. A general pure state on H ⊗ H is of the form | φ i = N X i,j =1 a ij | i i ⊗ | j i , a ij ∈ C (2)with the normalization P Ni,j =1 a ij a ∗ ij = 1( x ∗ denotes the complex conjugation of x ). Let A denote the matrix givenby ( A ) ij = a ij , we call A the matrix representation of pure state | φ i . The following quantities are associated with thestate | φ i given by (2). I α = T r ( AA † ) α , α = 1 , , · · · , N, (3)where A † denotes the adjoint of the matrix A . It is well-known that two bipartite pure states | φ i and | φ i in H ⊗ H are local unitary equivalent if and only if their matrix representation give the same values of quantities (3).Here we mainly consider the local unitary equivalence of quantum states ρ = p N I N ⊗ I N + K X i =1 p i | φ i ih φ i | (4)and ρ = p N I N ⊗ I N + K X i =1 p i | ϕ i ih ϕ i | (5)with p i ≥ i = 0 , · · · , K , P Ki =0 p i = 1, and p i = p j for 1 ≤ i < j ≤ K , 1 ≤ K ≤ N . Lemma 1:
Two arbitrary dimensional bipartite non-degenerate density matrices are equivalent under local unitarytransformations if and only if there exist eigenstate decompositions ρ = P i p i | ψ i ih ψ i | such that the following invariantshave the same values for both density matrices: J s = T r ( T r ρ s ) , s = 1 , · · · , N , (6) T r [( A i A † j )( A k A † l ) · · · ( A h A † p )] . (7) Proposition 1:
For two bipartite mixed states in Eq. (4) and Eq. (5), they are local unitary equivalent if and onlyif the corresponding matrix representations of | φ i i and | ϕ i i yield the same values of the invariants (7). Proof: If ρ and ρ are local unitary equivalent, then | φ i i and | ϕ i i are local unitary equivalent under the same localunitary operators. Therefore, | φ i i and | ϕ i i give rise to the same values of the invariants (7).On the other hand, if | φ i i and | ϕ i i give rise to the same values of the invariants (7). By Lemma 1, | φ i i and | ϕ i i are local unitary equivalent under the same local unitary operators, hence ρ and ρ are local unitary equivalent. Remark:
In fact, if the eigenvalues are not all positive in Proposition 1, then the conclusion still holds true. TheProposition 1 can be used to solve the local unitary equivalence of mixed state with only one degenerate eigenvalue.Because if one state has only one degenerate eigenvalues, then it can be transformed to the form like Proposition 1.That is ρ = λ | v ih v | + · · · + λ s | v s ih v s | + P N i = s +1 λ | v i ih v i | , equivalently, ρ = λ I N + ( λ − λ ) | v ih v | + · · · + ( λ s − λ ) | v s ih v s | , where λ i = λ j , i = j, i, j = 0 , , · · · , s .Now we can analyze the LU problem in two-qubit system. First, when quantum state has non-degenerate eigen-values, then Lemma 1 is sufficient to determine the local unitary equivalence. Second, when quantum state haseigenvalues with multiplicity not larger than 2, then one can solve the local unitary equivalence by the method pro-posed in [16]. At last, if there is only one degenerate eigenvalue, then Proposition 1 can be used to deal with the LUproblem of quantum states. Therefore, the LU problem of two-qubit quantum states can be solved in this way.This Proposition can also be used to judge which states are equivalent to isotropic states [18] under local unitarytransformations, which are invariant under transformations of the form ( U ⊗ U ∗ ) ρ ( U ⊗ U ∗ ) † . Isotropic state can bewritten as the mixture of the maximally mixed state and the maximally entangled state | ψ + i = √ d P d − a =0 | aa i ,ρ isot = pd I d ⊗ I d + (1 − p ) | ψ + ih ψ + | , ≤ p ≤
1. Following Proposition 1, the state that is local unitary equivalent to the isotropic states is the form ρ = pd I d ⊗ I d + (1 − p ) | ψ ′ ih ψ ′ | , where | ψ ′ i is a maximally entangled state.Subsequently, we consider the states that are local unitary equivalent to Werner states [19]. We need the techniqueof partial transpose of states. For a density matrix ρ in H ⊗ H with elements ρ mµ,nν = h e m ⊗ f µ | ρ | e n ⊗ f ν i , thepartial transposition of ρ is defined by [20]: ρ T = ( I ⊗ T ) ρ = X mn,µν ρ mν,nµ | e m ⊗ f µ ih e n ⊗ f ν | , where ρ T denotes the transposition of ρ with respect to the second system, | e n i and | f ν i are the bases associated withspaces H and H respectively. The LU problem of the original states can be transformed to that of theirs partialtransposed states [16], since two mixed states ρ and ρ in H ⊗ H are local unitary equivalent if and only if ρ T and ρ T are local unitary equivalent.The arbitrary dimensional Werner states [19] are invariant under the transformations ( U ⊗ U ) ρ ( U ⊗ U ) † for anyunitary U . They can be written as ρ w = 1 d − d [( d − f ) I d ⊗ I d + ( df − X ij | ij ih ji | ] , where − ≤ f ≤
1. The partial transpose of ρ w is ρ T w = d − d ( d − f ) I d ⊗ I d + ( df − d − | ψ + ih ψ + | . Therefore, the statethat is local unitary equivalent to the werner states is of the form ρ = d − d ( d − f ) I d ⊗ I d + ( df − d − ( | ψ ′ ih ψ ′ | ) T , where | ψ ′ i is a maximally entangled state.Now we consider the multipartite case. Before showing the equivalence of multipartite quantum states under localunitary transformations, we give a short review of high order singular value decomposition developed in [21]. For anytensor A with order d × d × · · · × d N , there exists a core tensor Σ such that A = ( U ⊗ U ⊗ · · · ⊗ U N )Σ , (8)where Σ forms the same order tensor with A . Any N − P i n = i obtained by fixing the n-th index to i ,has the following properties h P i n = i , P i n = j i = δ ij σ ( n ) i , with σ ni ≥ σ nj and ∀ i ≤ j for all possible values of n . Here,the singular value σ ni symbolizes the Frobenius norm σ ni = k P i n = i k ≡ q h P i n = i , P i n = i i , where the inner product h A, B i ≡ P i P i · · · P i N b i i ··· i N a ∗ i i ··· i N To calculate the core tensor Σ, one first expresses A in matrix unfolding form A n . Then one derives the singularvalue decomposition of the matrix A n = U n Λ n V n . The core tensor is then given byΣ = ⊗ Nn =1 U † n A . (9) Lemma 2:
Two multipartite pure states are local unitary equivalent if and only if they have the same core tensor upto the local symmetry ⊗ Nn =1 P ( n ) , where P ( n ) is a block-diagonal matrix consisting of unitary blocks with the samepartitions as that of the identical singular values of A n .By Proposition 1 and Lemma 2, one can get the following result easily. Proposition 2:
Two multipartite mixed states of the form ρ = pNM ··· T I N ⊗ I M ⊗ · · · ⊗ I T + (1 − p ) | φ ih φ | and ρ = pNM ··· T I N ⊗ I M ⊗ · · · ⊗ I T + (1 − p ) | ϕ ih ϕ | are local unitary equivalent if and only if | φ i and | ϕ i have the samecore tensor up to the local symmetry ⊗ Nn =1 P ( n ) . Remark:
Two multipartite mixed states ρ = pNM ··· T I N ⊗ I M ⊗ · · · ⊗ I T + (1 − p ) ρ and ρ = pNM ··· T I N ⊗ I M ⊗· · · ⊗ I T + (1 − p ) ρ ′ are local unitary equivalent if and only if the corresponding density matrices ρ and ρ ′ arelocal unitary equivalent. Therefore, the local unitary equivalence of two quantum states does not change under thedisturbance with the white noise. For example, ρ = p I ⊗ I ⊗ I + (1 − p ) ρ and ρ = p I ⊗ I ⊗ I + (1 − p ) ρ ′ with ρ = q ( | i + | i )( h | + h | )+(1 − q ) | ih | and ρ ′ = q ( | i + | i + | i )( h | + h | + h | )+(1 − q ) | ih | ,are not local unitary equivalent because ρ and ρ ′ are not local unitary equivalent [15]. III. LOCAL UNITARY SIMILAR EQUIVALENCE
Definition:
If there exists a unitary matrix U such that ( U ⊗ U ∗ ) ρ ( U ⊗ U ∗ ) † = ρ , we call states ρ and ρ localunitary similar equivalent.In [22] the author studies the unitary invariants and unitary similar equivalence, and the following Specht’s theorem[23] has been presented. Next we use them to deal with the local unitary similar equivalent problem for bipartitemixed states. Lemma 3:
Let A and B be n × n complex matrices. Then A and B are unitary similar, i.e there is a unitary matrix U , such that U AU † = B , if and only if tr ( ω ( A, A † )) = tr ( ω ( B, B † )) holds for every word ω , where ω ( A, A † ) is theresult of taking any monomial ω ( x, y ) in noncommuting variables x and y and replacing x with A and y with A † .The proof of Specht’s theorem can also be applied to two finite sets { A i } ti =1 and { B i } ti =1 of n × n matrices [22, 24]. Lemma 4:
Let { A i } ti =1 and { B i } ti =1 be n × n complex matrices. There is a unitary matrix U such that U † A i U = B i for i = 1 , , · · · , t if and only if for every word ω ( x , y , x , y , · · · , x t , y t ) in the noncommuting variables x i and y i wehave tr ( ω ( A , A † , A , A † , · · · , A t , A † t )) = tr ( ω ( B , B † , B , B † , · · · , B t , B † t )).For pure state | ψ i and φ i with coefficient matrices A and B respectively, if U ⊗ U ∗ | ψ i = | φ i , then U AU † = B .Utilizing this relation, we can get the necessary and sufficient condition for local unitary similar equivalence problem. Proposition 3:
For two bipartite mixed states in Eq. (4) and Eq. (5), they are local unitary similar equiva-lent if and only if tr ( ω ( A , A † , A , A † , · · · , A K , A † K )) = tr ( ω ( B , B † , B , B † , · · · , B K , B † K )) holds true for every word ω ( x , y , x , y , · · · , x t , y t ) in the noncommuting variables x i and y i , with A i and B i the coefficient matrices of | φ i i and | ϕ i i respectively. IV. CONCLUSIONS
In summary, we have studied the LU problem for a special class of states. The necessary and sufficient condition isprovided. Consequently, the local unitary equivalent classes of isotropic state and werner state are obtained. Then wehave investigated the local unitary similar equivalence for this class of state and obtained the necessary and sufficientcondition.
V. ACKNOWLEDGEMENT
We would like to thank Prof. Shao-Ming Fei and Xianqing Li-Jost for useful discussions. This work is supportedby the NSF of China under Grant No. 11401032 and No. 61473325; the NSF of Hainan Province under GrantNo.20151005 and No.20151010; Scientific Research Foundation for the Returned Overseas Chinese Scholars, StateEducation Ministry. [1] R. Horodecki, P. Horodecki, M. Horodecki, K. Horodecki, Rev. Mod. Phys (2009) 865.[2] W. D¨ur, G. Vidal, J.I. Cirac, Phys. Rev. A (2000) 062314.[3] M. Grassl, M. R¨otteler, T. Beth, Phys. Rev. A (1998) 1833.[4] Y. Makhlin, Quant. Info. Proc. (2002) 243.[5] N. Linden, S. Popescu, A. Sudbery, Phys. Rev. Lett (1999) 243.[6] N. Linden, S. Popescu, Phys (1998) 567.[7] S. Albeverio, S.M. Fei, P. Parashar, W.L. Yang, Phys. Rev. A (2003) 010303.[8] S. Albeverio, S.M. Fei, D. Goswami, Phys. Lett. A (2005) 37.[9] B.Z. Sun, S.M. Fei, X. Li-Jost, Z.X. Wang, J. Phys. A (2006) L43.[10] S. Albeverio, L. Cattaneo, S.M. Fei, X.H. Wang, Int. J. Quant. Inform (2005) 603.[11] T.G. Zhang, N. Jing, X. Li-Jost, M.J. Zhao, S.M. Fei, Euro. Phys. J. D (2013) 175.[12] B. Kraus, Phys. Rev. Lett. (2010) 020504; Phys. Rev. A (2010) 032121.[13] B. Liu, J.L. Li, X. Li, C.F. Qiao, Phys. Rev. Lett (2012) 050501.[14] C. Zhou, T. Zhang, S.M. Fei, N. Jing, X. Li-Jost, Phys. Rev. A (2012) 010303.[15] T.G. Zhang, M.J. Zhao, X. Li-Jost, S.M. Fei, Int. J. Theor. Phys (2013) 3020.[16] T.G. Zhang, M.J. Zhao, M. Li, S.M. Fei, X. Li-Jost, Phys. Rev. A (2013) 042304.[17] M. Li, T.G. Zhang, S.M. Fei, X. Li-Jost, N. Jing, Phys. Rev. A (2014) 062325.[18] M. Horodecki, P. Horodecki, Phys. Rev. A (1999) 4206.[19] R.F. Werner, Phys. Rev. A (1989) 4277.[20] M. Horodecki, P. Horodecki, R.Horodecki, Phys. Lett. A (1996) 1.[21] L.D. Lathaumer, B.D. Moor, J. Vandewalle, SIAM. J. Matrix Anal. Appl (2000) 1253.[22] H. Shapiro, Linear Algebra and Appl. (1991) 101.[23] W. Specht, Deutsch. Math.-Verein (1940) 19.[24] N. Wiegmann, J. Austral. Math. Soc2