Acta Polytechnica https://doi.org/10.14311/AP.2022.62.0222 Acta Polytechnica 62(1):222–227, 2022 © 2022 The Author(s). Licensed under a CC-BY 4.0 licence Published by the Czech Technical University in Prague A NOTE ON ENTANGLEMENT CLASSIFICATION FOR TRIPARTITE MIXED STATES Hui Zhaoa, Yu-Qiu Liua, Zhi-Xi Wangb, Shao-Ming Feib, ∗ a Beijing University of Technology, Faculty of Science, Beijing 100124, China b Capital Normal University, School of Mathematical Sciences, Beijing 100037, China ∗ corresponding author: feishm@cnu.edu.cn Abstract. We study the classification of entanglement in tripartite systems by using Bell-type inequalities and principal basis. By using Bell functions and the generalized three dimensional Pauli operators, we present a set of Bell inequalities which classifies the entanglement of triqutrit fully separable and bi-separable mixed states. By using the correlation tensors in the principal basis representation of density matrices, we obtain separability criteria for fully separable and bi-separable 2 ⊗ 2 ⊗ 3 quantum mixed states. Detailed example is given to illustrate our criteria in classifying the tripartite entanglement. Keywords: Bell inequalities, separability, principal basis. 1. Introduction One of the most remarkable features that distinguishes quantum mechanics from classical mechanics is the quantum entanglement. Entanglement was first rec- ognized by EPR [1], with significant progress made by Bell [2] toward the resolution of the EPR problem. Since Bell’s work, derivation of new Bell-like inequali- ties has been one of the important and challenging sub- jects. CHSH generalized the original Bell inequalities to a more general case for two observers [3]. In [4] the authors proposed an estimation of quantum entangle- ment by measuring the maximum violation of the Bell inequality without information of the reduced density matrices. In [5] series of Bell inequalities for multi- partite states have been presented with sufficient and necessary conditions to detect certain entanglement. There have been many important generalizations and interesting applications of Bell inequalities [6–8]. By calculating the measures of entanglement and the quantum violation of the Bell-type inequality, a rela- tionship between the entanglement measure and the amount of quantum violation was derived in [9]. How- ever, for high-dimensional multiple quantum systems the results for such relationships between the entan- glement and the nonlocal violation are still far from being satisfied. In [10], an upper bound on fully entan- gled fraction for arbitrary dimensional states has been derived by using the principal basis representation of density matrices. Based on the norms of correlation vectors, the authors in [11] presented an approach to detect entanglement in arbitrary dimensional quan- tum systems. Separability criteria for both bipartite and multipartite quantum states was also derived in terms of the correlation matrices [12]. In this paper by using the Bell function and the generalized three dimensional Pauli operators, we de- rive a quantum upper bound for 3 ⊗ 3 ⊗ 3 quantum systems. We present a classification of entanglement for triqutrit mixed states by a set of Bell inequalities. These inequalities can distinguish fully separable and bi-separable states. Moreover, we propose criteria to detect classification of entanglement for 2 ⊗ 2 ⊗ 3 mixed states with correlation tensor matrices in the principal basis representation of density matrices. 2. Entanglement identification with Bell inequalities We first consider relations between entanglement and non-locality for 3 ⊗ 3 ⊗ 3 quantum systems. Consider three observers who may choose independently be- tween two dichotomic observables denoted by Ai and Bi for the i-th observer, i = 1, 2, 3. Let V̂i denote the measurement operator associated with the variable Vi ∈ {Ai,Bi} of i-th observer. We choose a complete set of orthonormal basis vectors |k⟩ to describe an orthogonal measurement of a given variable Vi. The measurement outcomes are indicated by a set of eigen- values 1,λ,λ2, where λ = exp( i2π3 ) is a primitive third root of unity. Therefore the measurement operator can be represented by V̂i = ∑2 k=0 λ k|k⟩⟨k|. Inspired by the Bell function (the expected value of Bell oper- ator) constructed in [13], we introduce the following Bell operator, B = 2∑ j=1 1 4 (Â1 j ⊗ Â2 j ⊗ Â3 j + λjÂ1 j ⊗ B̂2 j ⊗ B̂3 j + λjB̂1 j ⊗ Â2 j ⊗ B̂3 j + λjB̂1 j ⊗ B̂2 j ⊗ Â3 j ), (1) where Âi j (B̂i j ) denotes the j-th power of Âi (B̂i). Next we construct three Bell operators in terms of Eq. (1). Consider three dimensional Pauli opera- 222 https://doi.org/10.14311/AP.2022.62.0222 https://creativecommons.org/licenses/by/4.0/ https://www.cvut.cz/en vol. 62 no. 1/2022 A Note on Entanglement Classification for Tripartite . . . tors [14] X̂ and Ẑ which satisfy X̂|k⟩ = |k + 1⟩, Ẑ|k⟩ = λk|k⟩, X̂3 = I, Ẑ3 = I, where I denotes the identity operator. Therefore, if we replace Âi and B̂i with the following unitary operators, Â1 = Ẑ, Â2 = λ2X̂Ẑ, Â3 = X̂Ẑ2, B̂1 = Ẑ, B̂2 = X̂Ẑ2 and B̂3 = λ2X̂Ẑ, we obtain B1 = 2∑ j=1 1 4 [Ẑj ⊗ (λ2X̂Ẑ)j ⊗ (X̂Ẑ2)j + λjẐj ⊗ (X̂Ẑ2)j ⊗ (λ2X̂Ẑ)j + λjẐj ⊗ (λ2X̂Ẑ)j ⊗ (λ2X̂Ẑ)j + λjẐj ⊗ (X̂Ẑ2)j ⊗ (X̂Ẑ2)j ]. (2) If we choose unitary operators as follows, Â1 = λ2X̂Ẑ, Â2 = X̂Ẑ2, Â3 = Ẑ, B̂1 = X̂Ẑ2, B̂2 = λ2X̂Ẑ and B̂3 = Ẑ, we have B2 = 2∑ j=1 1 4 [(ω2X̂Ẑ)j ⊗ (X̂Ẑ2)j ⊗ Ẑj + λj (λ2X̂Ẑ)j ⊗ (λ2X̂Ẑ)j ⊗ Ẑj ) + λj (X̂Ẑ2)j ⊗ (X̂Ẑ2)j ⊗ (Ẑ)j + λj (X̂Ẑ2)j ⊗ (λ2X̂Ẑ)j ⊗ (Ẑ)j ]. (3) Taking Â1 = λ2X̂Ẑ, Â2 = Ẑ, Â3 = X̂Ẑ2, B̂1 = X̂Ẑ2, B̂2 = Ẑ and B̂3 = λ2X̂Ẑ, we have B3 = 2∑ j=1 1 4 [(λ2X̂Ẑ)j ⊗ Ẑj ⊗ (X̂Ẑ2)j + λj (λ2X̂Ẑ)j ⊗ (Ẑ)j ⊗ (λ2X̂Ẑ)j ) + λj (X̂Ẑ2)j ⊗ Ẑj ⊗ (λ2X̂Ẑ)j + λj (X̂Ẑ2)j ⊗ Ẑj ⊗ (X̂Ẑ2)j ]. (4) Concerning the bounds on the mean values |⟨Bi⟩| of the operators Bi, i = 1, 2, 3, we have the following conclusions. Theorem 1. For 3 ⊗ 3 ⊗ 3 mixed states, we have the inequality, |⟨Bi⟩| ≤ 54 , i = 1, 2, 3. Proof Due to the linear property of the average values, it is sufficient to consider pure states. Any triqutrit pure state can be written as, |ψ⟩ =c1|000⟩ + c2|011⟩ + c3|012⟩ + c4|021⟩ + c5|022⟩ + c6|101⟩ + c7|102⟩ + c8|110⟩ + c9|111⟩ + c10|120⟩ + c11|122⟩ + c12|201⟩ + c13|202⟩ + c14|210⟩ + c15|212⟩ + c16|220⟩ + c17|221⟩ + c18|222⟩, (5) where c5, c11, c13, c15, c16, c17 and c18 are real and non- negative, |c1| ≥ |ci| for i = 1, 2, . . . , 18, |c9| ≥ |c18| and ∑18 i=1 |ci| 2 = 1. Therefore, |⟨B1⟩| =| 1 4 (−c1c2 + 5c1c5 − c2c5 − c6c10 + 2c7c8 + 2c9c11 + 2c12c15 + 5c12c16 − c13c14 − c13c17 − 4c14c17 + 5c15c16)| ≤ 1 8 × 10 × 18∑ i=1 c2i = 5 4 . (6) Similarly one can prove that |⟨Bi⟩| ≤ 54 for i = 2, 3. □ Theorem 2. If a triqutrit mixed state ρ is fully separable, then |⟨Bi⟩| = 0, i = 1, 2, 3. The proof is straightforward. Due to the linear property of the average values, it is sufficient to consider pure states again. A fully separable pure state can be written as under suitable bases, |ψ⟩ = |0⟩ ⊗ |0⟩ ⊗ |0⟩ ⊗ |0⟩. Therefore |⟨Bi⟩| = |tr(ρBi)| = 0. Theorem 3. For bi-separable states ρi|jk under bi- partition i and jk, i ̸= j ̸= k ∈ {1, 2, 3}, we have |⟨B1⟩| ≤ 34, |⟨B2⟩| = 0, |⟨B3⟩| = 0, |⟨B1⟩| = 0, |⟨B2⟩| ≤ 34, |⟨B3⟩| = 0, |⟨B1⟩| = 0, |⟨B2⟩| = 0, |⟨B3⟩| ≤ 34, for ρ1|23, ρ3|12 and ρ2|13, respectively. Proof It is sufficient to consider pure states only. Every bi-separable pure state ρ1|23 can be written as via a suitable choice of bases [15], |ψ⟩ = |0⟩ ⊗ (c0|00⟩ + c1|11⟩ + c2|22⟩), where |c0| ≥ |c1| ≥ |c2| and ∑2 i=0 |ci| 2 = 1. Therefore, we have by direct calculation, |⟨B1⟩| =| 1 4 (5c2c0 − c0c1 − c1c2)| ≤ 1 8 (5 × (c22 + c 2 0) + (c 2 0 + c 2 1) × (c 2 1 + c 2 2)) ≤ 3 4 . It is straightforward to prove similarly, |⟨B2⟩| = 0 and |⟨B3⟩| = 0. For bi-separable states ρ3|12 and ρ2|13, the results can be proved in a similar way. □ The above relations given in Theorem 1-3 give rise to characterization of quantum entanglement based on the Bell-type violations. If we consider |⟨Bi⟩|, i = 1, 2, 3, to be three coordinates, then all the triqutrit states are confined in a cube with size 54 × 5 4 × 5 4 . The bi-separable states are confined in a cube with size 3 4 × 3 4 × 3 4 , see Figure 1. 223 H. Zhao, Y.-Q. Liu, Z.-X. Wang, S.-M. Fei Acta Polytechnica Figure 1. All states lie in the yellow cube, while in the green cube are bi-separable states. 3. Entanglement classification under principal basis Consider the principal basis on d-dimensional Hilbert space H with computational basis |i⟩, i = 1, 2, ...,d. Let Eij be the d×d unit matrix with the only nonzero entry 1 at the position (i,j). Let ω be a fixed d-th primitive root of unity, the principal basis is given by Aij = ∑ m∈Zd ωimEm,m+j, (7) where ωd = 1, i,j ∈ Zd and Zd is Z modulo d. The set {Aij } spans the principal Cartan subalgebra of gl(d). Under the stand inner product (x|y) = tr(xy) of matrices x and y, the dual basis of the principal basis {Aij } is {(ωij/d)A−i,−j }, which follows also from the algebraic property of the principal matrices, AijAkl = ωjkAi+k,j+l. Namely, A † i,j = ω ijA−i,−j , and thus tr(AijA † kl) = δikδjld [10]. Next we consider the entanglement of 2 ⊗ 2 ⊗ 3 systems. Let {Aij } and {Bij } be the principal bases of 2-dimensional and 3-dimensional Hilbert space, re- spectively. For any quantum state ρ ∈ H21 ⊗ H22 ⊗ H33 , ρ has the principal basis representation: ρ = 1 12 (I2 ⊗ I2 ⊗ I3 + ∑ (i,j) ̸=(0,0) uij Aij ⊗ I2 ⊗ I3 + ∑ (k,l) ̸=(0,0) vklI2 ⊗ Akl ⊗ I3 + ∑ (s,t)̸=(0,0) wstI2 ⊗ I2 ⊗ Bst + ∑ (i,j),(k,l) ̸=(0,0) xij,klAij ⊗ Akl ⊗ I3 + ∑ (i,j),(s,t) ̸=(0,0) yij,stAij ⊗ I2 ⊗ Bst + ∑ (k,l),(s,t) ̸=(0,0) zkl,stI2 ⊗ Akl ⊗ Bst + ∑ (i,j),(k,l),(s,t) ̸=(0,0) rij,kl,stAij ⊗ Akl ⊗ Bst), (8) where I2 (I3) denotes the two (three) dimensional identity matrix, uij = tr(ρA † ij ⊗ I2 ⊗ I3), vkl = tr(ρI2 ⊗ A † kl ⊗ I3), wst = tr(ρI2 ⊗ I2 ⊗ B † st), xij,kl = tr(ρA†ij ⊗A † kl ⊗I3), yij,st = tr(ρA † ij ⊗I2 ⊗B † st), zkl,st = tr(ρI2 ⊗A † kl ⊗B † st) and rij,kl,st = tr(ρA † ij ⊗A † kl ⊗B † st). Denote T 1|231 , T 1|23 2 , T 2|13 1 , T 2|13 2 , T 3|12 1 and T 3|12 2 the matrices with entries given by r01,kl,st, r11,kl,st, rij,01,st, rij,11,st, rij,kl,10 and rij,kl,20 (i,j,k, l ∈ Z2, s,t ∈ Z3), respectively. Let ∥A∥tr = ∑ σi = tr √ AA† be the trace norm of a matrix A ∈ Rm×n, where σi are the singular values of the matrix A. First we note that ∥T 1|231 − T 1|23 2 ∥tr is invariant under local unitary transformations. Denote UAU† by AU . Suppose ρ′ = ρ(I⊗U2⊗U3) with U2 ∈ U(2) and U3 ∈ U(3), AU2ij = ∑ (i′ ,j′ ) ̸=(0,0) mij,i′j′Ai′j′ and BU3ij =∑ (i′ ,j′ ) ̸=(0,0) nij,i′j′Bi′j′ for some coefficients mij,i′j′ and nij,i′j′ . The orthogonality of {AU2ij } and {B U3 ij } re- quires that tr(AU2ij A U2 kl ) = tr(U2AijA † klU † 2 ) =tr(AijA † kl) = 2δikδjl; tr(BU3ij B U3 kl ) = tr(U3BijB † klU † 3 ) =tr(BijB † kl) = 3δikδjl. Hence, we have M = (mij,i′j′ ) ∈ SU(3) and N = (nij,i′j′ ) ∈ SU(8) since any two orthogonal bases are transformed by an unitary matrix. One sees that∑ (i,j),(k,l), (s,t)̸=(0,0) rij,kl,stAij ⊗ AU2kl ⊗ B U3 st = ∑ (i,j),(k,l), (s,t)̸=(0,0) ∑ (k′ ,l′ ),(s′ ,t′ ) ̸=(0,0) rij,kl,stmkl,k′ l′ nst,s′ t′ Aij ⊗ Ak′ l′ ⊗ Bs′ t′ = ∑ (i,j),(k,l), (s,t)̸=(0,0) ( ∑ (k′ ,l′ ),(s′ ,t′ ) ̸=(0,0) mk′ l′ ,klrij,k′ l′ ,s′ t′ ns′ t′ ,st)Aij ⊗ Akl ⊗ Bst. We have T 1|231 (ρ ′) = MtT 1|231 (ρ)N and T 1|23 2 (ρ ′) = MtT 1|23 2 (ρ)N. Therefore, ∥T 1|231 (ρ ′) − T 1|232 (ρ ′)∥tr = ∥T 1|23 1 (ρ) − T 1|23 2 (ρ)∥tr, (9) due to that the singular values of a matrix are the same as those of MtTN when M and N are unitary matrices. Theorem 4. If a mixed state ρ is fully separable, then ∥T 1|231 − T 1|23 2 ∥tr ≤ √ 3. Proof If ρ = |φ⟩⟨φ| is fully separable, we have |φ1|2|3⟩ = |φ1⟩ ⊗ |φ23⟩ ∈ H21 ⊗ H623, where |φ23⟩ = |φ2⟩ ⊗ |φ3⟩ ∈ H22 ⊗ H33 . Then by Schmidt decomposition, |φ1|2|3⟩ = t0|0α⟩ + t1|1β⟩, where t20 + t21 = 1. Taking into account the local unitary equivalence in H22 ⊗ H33 and using (9), we only need to consider that {|α⟩, |β⟩} = {|00⟩, |01⟩}. Then 224 vol. 62 no. 1/2022 A Note on Entanglement Classification for Tripartite . . . |φ1|2|3⟩ = t0|000⟩ + t1|101⟩. T 1|23 1 and T 1|23 2 are given by T 1|23 1 =   0 t0t1 0 0 t0t1 0 0 0 0 0 t0t1 0 0 t0t1ω2 0 0 0 0 0 t0t1 0 0 t0t1ω 0   t , (10) T 1|23 2 =   0 t0t1 0 0 −t0t1 0 0 0 0 0 t0t1 0 0 −t0t1ω2 0 0 0 0 0 t0t1 0 0 −t0t1ω 0   t . (11) with ω3 = 1. Therefore, we have ∥T 1|231 − T 1|23 2 ∥tr =tr √ (T 1|231 − T 1|23 2 )(T 1|23 1 − T 1|23 2 )† = √ 12t20t21 ≤ √ 3. For a fully separable mixed state ρ = ∑ pi|φi⟩⟨φi|, we get ∥T 1|231 (ρ) − T 1|23 2 (ρ)∥tr =∥T 1|231 ( ∑ pi|φi⟩⟨φi|) − T 1|23 2 ( ∑ pi|φi⟩⟨φi|)∥tr ≤ ∑ pi∥T 1|23 1 (|φi⟩⟨φi|) − T 1|23 2 (|φi⟩⟨φi|)∥tr ≤ √ 3, which proves the theorem. □ Theorem 5. For any mixed state ρ =∑ pi|φi⟩⟨φi| ∈ H21 ⊗ H22 ⊗ H33 , ∑ pi = 1, 0 < pi ≤ 1, we have: (1) If ρ is 1|23 separable, then ∥T 1|231 −T 1|23 2 ∥tr ≤ √ 6; (2) If ρ is 2|13 separable, then ∥T 2|131 −T 2|13 2 ∥tr ≤ √ 6; (3) If ρ is 3|12 separable, then ∥T 3|121 −T 3|12 2 ∥tr ≤ √ 3. Proof (1) If ρ = |φ⟩⟨φ| is 1|23 separable, we have |φ1|23⟩ = |φ1⟩ ⊗ |φ23⟩ ∈ H21 ⊗ H623, where H623 = H22 ⊗ H33 . Then by Schmidt decomposition, one has |φ1|23⟩ = t0|0α⟩ + t1|1β⟩, where t20 + t21 = 1. Taking into account the local unitary equivalence in H22 ⊗ H33 and using (9), we only need to consider two cases (i) {|α⟩, |β⟩} = {|00⟩, |01⟩} and (ii) {|00⟩, |11⟩}. For the first case we have ∥T 1|231 − T 1|23 2 ∥tr ≤ √ 3 by Theorem 4. For the second case, we have |φ1|23⟩ = t0|000⟩ + t1|111⟩, where T 1|23 1 and T 1|23 2 are given by T 1|23 1 =   t0t1 0 t0t1 t0t1 0 −t0t1 0 0 0 t0t1 0 t0t1 t0t1ω 2 0 −t0t1ω2 0 0 0 t0t1 0 t0t1 t0t1ω 0 −t0t1ω   t , (12) T 1|23 2 =   t0t1 0 t0t1 −t0t1 0 t0t1 0 0 0 t0t1 0 t0t1 −t0t1ω2 0 t0t1ω2 0 0 0 t0t1 0 t0t1 −t0t1ω 0 t0t1ω   t . (13) Then we have ∥T 1|231 − T 1|23 2 ∥tr =tr √ (T 1|231 − T 1|23 2 )(T 1|23 1 − T 1|23 2 )† = √ 24t20t21 ≤ √ 6. Now consider mixed state ρ = ∑ pi|φi⟩⟨φi|. We obtain ∥T 1|231 (ρ) − T 1|23 2 (ρ)∥tr =∥T 1|231 ( ∑ pi|φi⟩⟨φi|) − T 1|23 2 ( ∑ pi|φi⟩⟨φi|)∥tr ≤ ∑ pi∥T 1|23 1 (|φi⟩⟨φi|) − T 1|23 2 (|φi⟩⟨φi|)∥tr, namely, ∥T 1|231 (ρ) − T 1|23 2 (ρ)∥tr ≤ √ 6. (2) If ρ = |φ⟩⟨φ| is 2|13 separable, we have |φ2|13⟩ = |φ2⟩ ⊗ |φ13⟩ ∈ H22 ⊗ H613, where H613 = H21 ⊗ H33 . Then by Schmidt decomposition, one has |φ2|13⟩ = t0|0α⟩ +t1|1β⟩, where t20 +t21 = 1. Taking into account the local unitary equivalence in H21 ⊗ H33 , we obtain a similar equation of (9). Thus we only need to con- sider again the two cases (i) {|α⟩, |β⟩} = {|00⟩, |01⟩} and (ii) {|00⟩, |11⟩}. In the first case, |φ2|13⟩ = t0|000⟩ + t1|101⟩, and T 2|13 1 and T 2|13 2 are zero matrices. In the second case, |φ2|13⟩ = t0|000⟩ + t1|111⟩, with T 2|13 1 and T 2|13 2 given by T 2|13 1 =   t0t1 0 t0t1 t0t1 0 −t0t1 0 0 0 t0t1 0 t0t1 t0t1ω 2 0 −t0t1ω2 0 0 0 t0t1 0 t0t1 t0t1ω 0 −t0t1ω   t , (14) 225 H. Zhao, Y.-Q. Liu, Z.-X. Wang, S.-M. Fei Acta Polytechnica T 2|13 2 =   t0t1 0 t0t1 −t0t1 0 t0t1 0 0 0 t0t1 0 t0t1 −t0t1ω2 0 t0t1ω2 0 0 0 t0t1 0 t0t1 −t0t1ω 0 t0t1ω   t . (15) Then we have ∥T 2|131 − T 2|13 2 ∥tr =tr √ (T 2|131 − T 2|13 2 )(T 2|13 1 − T 2|13 2 )† = √ 24t20t21 ≤ √ 6. For the mixed state ρ = ∑ pi|φi⟩⟨φi|, we have ∥T 2|131 (ρ) − T 2|13 2 (ρ)∥tr =∥T 2|131 ( ∑ pi|φi⟩⟨φi|) − T 2|13 2 ( ∑ pi|φi⟩⟨φi|)∥tr ≤ ∑ pi∥T 2|13 1 (|φi⟩⟨φi|) − T 2|13 2 (|φi⟩⟨φi|)∥tr ≤ √ 6. (3) If ρ = |φ⟩⟨φ| is 3|12 separable, we have |φ3|12⟩ = |φ3⟩ ⊗ |φ12⟩ ∈ H33 ⊗H412, where H412 = H21 ⊗H22 . Then by Schmidt decomposition, we have |φ3|12⟩ = t0|0α0⟩+ t1|1α1⟩ + t2|2α2⟩, where t20 + t21 + t22 = 1. Taking into account the local unitary equivalence in H21 ⊗ H22 , we obtain similar equation of (9). We only need to consider the case |φ3|12⟩ = t0|000⟩ + t1|101⟩ + t2|210⟩. We have T 3|12 1 =   0 0 00 t20 − ωt21 + ω2t22 0 0 0 0   , T 3|12 2 =   0 0 00 t20 − ω2t21 + ωt22 0 0 0 0   . (16) Using 1 + ω + ω2 = 0, we have ∥T 3|121 − T 3|12 2 ∥tr =tr √ (T 3|121 − T 3|12 2 )(T 3|12 1 − T 3|12 2 )† = √ 3(t20 + t21)2 ≤ √ 3. For the mixed state ρ = ∑ pi|φi⟩⟨φi|, we get ∥T 3|121 (ρ) − T 3|12 2 (ρ)∥tr =∥T 3|121 ( ∑ pi|φi⟩⟨φi|) − T 3|12 2 ( ∑ pi|φi⟩⟨φi|)∥tr ≤ ∑ pi∥T 3|12 1 (|φi⟩⟨φi|) − T 3|12 2 (|φi⟩⟨φi|)∥tr ≤ √ 3. As an example, let us consider the 2 ⊗ 2 ⊗ 3 state, ρ = x|GHZ′⟩⟨GHZ′| + (1 − x)I12, 0 ≤ x ≤ 1, where |GHZ′⟩ = 12 (|000⟩ + |101⟩ + |011⟩ + |112⟩). By Theorem 4, we have that when ∥T 1|231 − T 1|23 2 ∥ = (2 √ 3 2 + 1)x > √ 3, i.e., 0.5021 < x ≤ 1, ρ is not fully separable. By Theorem 5, when ∥T 1|231 − T 1|23 2 ∥ = ∥T 2|131 − T 2|13 2 ∥ = (2 √ 3 2 + 1)x > √ 6, i.e., 0.7101 < x ≤ 1, ρ is not separable under bipartition 1|23 or 2|13. When ∥T 3|121 − T 3|12 2 ∥ = 7 √ 3 4 x > √ 3, i.e., 0.5714 < x ≤ 1, ρ is not separable under bipartition 3|12. 4. Conclusions We have presented quantum upper bounds for triqutrit mixed states by using the generalized Bell functions and the generalized three dimensional Pauli opera- tors, from which the triqutrit entanglement has been identified. Our inequalities distinguish fully separa- ble states and three types of bi-separable states for triqutrit states. Moreover, any triqutrits states are confined in a cube with size 54 × 5 4 × 5 4 and the bi- separable states are in a cube with the size 34 × 3 4 × 3 4 . We have also studied the classification of quantum entanglement for 2 ⊗ 2 ⊗ 3 systems by using the cor- relation tensors in the principal basis representation of density matrices. By considering the upper bounds on some the trace norms, we have obtained the cri- teria which detect fully separable and bi-separable 2 ⊗ 2 ⊗ 3 quantum mixed states. Detailed example has been given to show the classification of tripartite entanglement by using our criteria. Acknowledgements This work is supported by the National Natural Science Foundation of China under grant Nos. 11101017, 11531004, 11726016, 12075159 and 12171044, Simons Foundation un- der grant No. 523868, Beijing Natural Science Foundation (Z190005), Academy for Multidisciplinary Studies, Capi- tal Normal University, Shenzhen Institute for Quantum Science and Engineering, Southern University of Science and Technology (SIQSE202001), and the Academician Innovation Platform of Hainan Province. References [1] A. Einstein, B. Podolsky, N. Rosen. Can quantum-mechanical description of physical reality be considered complete? Physical Review 47(10):777–780, 1935. https://doi.org/10.1103/PhysRev.47.777. [2] J. S. Bell. On the Einstein Podolsky Rosen paradox. Physics 1(3):195–200, 1964. https: //doi.org/10.1103/PhysicsPhysiqueFizika.1.195. [3] J. F. Clauser, M. A. Horne, A. Shimony, R. A. Holt. Proposed experiment to test local hidden-variable theories. Physical Review Letters 23(15):880–884, 1969. https://doi.org/10.1103/PhysRevLett.23.880. [4] P. Y. Chang, S. K. Chu, C. T. Ma. Bell’s inequality and entanglement in qubits. Journal of High Energy Physics volume 2017(9):100, 2017. https://doi.org/10.1007/JHEP09(2017)100. 226 https://doi.org/10.1103/PhysRev.47.777 https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195 https://doi.org/10.1103/PhysRevLett.23.880 https://doi.org/10.1007/JHEP09(2017)100 vol. 62 no. 1/2022 A Note on Entanglement Classification for Tripartite . . . [5] M. Li, S. M. Fei. Bell inequalities for multipartite qubit quantum systems and their maximal violation. Physical Review A 86(5):052119, 2012. https://doi.org/10.1103/PhysRevA.86.052119. [6] D. Collins, N. Gisin, S. Popescu, et al. Bell-type inequalities to detect true n-body nonseparability. Physical Review Letters 88(17):170405, 2002. https://doi.org/10.1103/PhysRevLett.88.170405. [7] S. W. Ji, J. Lee, J. Lim, et al. Multi-setting Bell inequality for qudits. Physical Review A 78(5):052103, 2008. https://doi.org/10.1103/PhysRevA.78.052103. [8] H. Zhao. Entanglement of Bell diagonal mixed states. Physics Letters A 373(43):3924–3930, 2009. https://doi.org/10.1016/j.physleta.2009.08.048. [9] D. Ding, Y. Q. He, F. L. Yan, T. Gao. Entanglement measure and quantum violation of Bell-type inequality. International Journal of Theoretical Physics 55(10):4231–4237, 2016. https://doi.org/10.1007/s10773-016-3048-1. [10] X. F. Huang, N. H. Jing, T. G. Zhang. An upper bound of fully entangled fraction of mixed states. Communications in Theoretical Physics 65(6):701–704, 2016. https://doi.org/10.1088/0253-6102/65/6/701. [11] J. I. de Vicente, M. Huber. Multipartite entanglement detection from correlation tensors. Physical Review A 84(6):242–245, 2011. https://doi.org/10.1103/PhysRevA.84.062306. [12] M. Li, J. Wang, S. M. Fei, X. Li-Jost. Quantum separability criteria for arbitrary dimensional multipartite states. Physical Review A 89(2):767–771, 2014. https://doi.org/10.1103/PhysRevA.89.022325. [13] W. Son, J. Lee, M. S. Kim. Generic Bell inequalities for multipartite arbitrary dimensional systems. Physical Review Letters 96(6):060406, 2006. https://doi.org/10.1103/PhysRevLett.96.060406. [14] D. Gottesman. Fault-tolerant quantum computation with higher-dimensional systems. Chaos, Solitons & Fractals 10(10):1749–1758, 1999. https://doi.org/10.1016/S0960-0779(98)00218-5. [15] H. A. Carteret, A. Higuchi, A. Sudbery. Multipartite generalisation of the Schmidt decomposition. Journal of Mathematical Physics 41(12):7932–7939, 2000. https://doi.org/10.1063/1.1319516. 227 https://doi.org/10.1103/PhysRevA.86.052119 https://doi.org/10.1103/PhysRevLett.88.170405 https://doi.org/10.1103/PhysRevA.78.052103 https://doi.org/10.1016/j.physleta.2009.08.048 https://doi.org/10.1007/s10773-016-3048-1 https://doi.org/10.1088/0253-6102/65/6/701 https://doi.org/10.1103/PhysRevA.84.062306 https://doi.org/10.1103/PhysRevA.89.022325 https://doi.org/10.1103/PhysRevLett.96.060406 https://doi.org/10.1016/S0960-0779(98)00218-5 https://doi.org/10.1063/1.1319516 Acta Polytechnica 62(1):222–227, 2022 1 Introduction 2 Entanglement identification with Bell inequalities 3 Entanglement classification under principal basis 4 Conclusions Acknowledgements References