1. 1 Classifying Entanglement in Quantum Memory via a Recursive Bell-Inequalities Sixia Yu, Zeng-Bing Chen, Jian-Wei Pan, and Yong-De Zhang Department of Modern Phyiscs University of Science and Technology of China Hefei 230027, China

2. 2 Abstract W e present an entanglement classification for all states of N-qubit (quantum memory) by using a quadratic Bell- Inequalities, consisted of the recursive Mermin-Klyshko polynomials. An integer index can be used to classify the entanglement, where is the total number of entangled qubits in N-qubit, is the number of the groups of entangled qubits. is a sequence of entanglement-partition to N-qubits. The larger the entanglement index of a state, the more entangled is the state in the sense that a larger maximal violation of the Bell-inequalities as follows

3. 3 I ) Entanglement Index [1] As an explanation, we consider an example of N=4. In which there are five different entanglement-types: i) 4 qubits are fully entangled,,, ii) 1 entangled 3-qubit and single separable qubit ， ， ， iii) 2 pairs of entangled 2-qubit ， ， ， iv) 2 entangled 2-qubit and 2 separated qubits ， ， ， v) 4 qubits are fully separable states ， ， ，

4. 4  The entanglement type and M-K polynomials[2 、 3] is invariant under the permutation of qubits. The number of partition in N is also number of irreducible representa- tions of the permutation group of N elements. Therefore we can label different types of entanglement of N-qubit by different partition of N.  For a given N-qubit system, we have i.e., there are (N-2) possible values of the entanglement Index. The largest entanglement index N+1 is possessed only by partition --fully entangled, while the smallest index 3 does not correspond to a single partition. In two cases of and, indexes are equal to 3. Therefore, they can’t be distinguished by the following quadratic Bell-inequalities.

5. 5 2) Mremin-Klyshko polynomials  For a 2-qubit (A+B) system, there are only two types of entanglement, i.e., separated or entangled. If a state is separable (i.e., it is a pure product state or mixture of pure product states), it will satisfy the famous Bell-CHSH inequality, for all observables, where and are Pauli matrices for qubit A and B, and the norms of four real vectors are less than or equal to 1. The maximal violation (upper bound) of this inequality is for maximally entangled states.

6. 6  For A system (N=1), define the 1st M-K polynomial as  For A+B system (N=2), the 2nd M-K polynomial as  For A+B system, using the 2nd M-K polynomial and the upper bound violating the Bell-CHSH inequality, we have[1], for all states (inside circle); while the Bell-CHSH inequality is changed into the following form,, for all separable states (inside square)

7. 7  For A+B+C system (N=3), the entanglement indexes are respectively, Simultaneously, we obtain the 3rd M-K polynomial from the recursive relation between them[3] Therefore, for all three types of entanglement, we have[1] [4]  Indeed, for all cases( ) of fully separated system, their results are same, i.e.,

8. 8  In ━ diagram of the above 3-qubit system, the whole region is divided into three regions as follows: 1, the -type fully separable states are contained inside the region of the smallest square; 2, the -type entangled states are contained inside the region of the small circle; 3, the -type fully entangled states are contained inside the region of the bigger circle. The region outside the bigger circle is non-physical.  It is interesting to note that 3-qubit diagram looks like the shape of ancient-Chinese-coin----Kang-Xi coin( 康熙钱 ) of Qing-Dynasty, while 2-qubit diagram looks like the Wang-mang coin( 王莽钱 )of Han-Dynasty.

9. 9 3) Theorem of General Quadratic Bell- Inequalities[1] Here we present our main result in the form of a theorem. This theorem is stated in terms of the Mermin- Klyshko polynomials, and is expressed as a general quadratic Bell-inequalities with a recursive form. [General Bell-theorem with the entanglement index] “ For a N-qubit system, its Mermin-Klyshko polynomial satisfies the following quadratic Bell-inequalities, where is the N-th Mermin-Klyshko polynomial and is the entanglement index of this system.”

10. 10 Proof: We use induction argument. Firstly, for above case of 3-qubit system (N=3), we have known that the theorem is true. Secondly, we may delete those qubits which are all single separable since they don’t affect this theorem holds: Without lost of generality, we may assume that the qubit added on N-1 qubit system is the N-th qubit and it is on a separable state, i.e.,. Therefore, from the recursive relation of M-K polynomials, we may obtain where we denote for convenience. On the other hand, this don’t change the entanglement index

11. 11  Thirdly, now we assume that the N-th qubit added is entangled with some qubits, and will affect and change the whole entanglement partition configuration, including previous N-1 qubits and the N-th qubit, i.e., Here, without lost of generality, we have assumed that the last qubits are entangled. From the recursive relation of M-K polynomials, we have also[5] Therefore, we obtain On the other hand, we have

12. 12 Therefore, finally, we have References [1] Sixia Yu, Zeng-Bing Chen, Jian-Wei Pan, and Yong- De Zhang, Classifying N-Qubit Entanglement via Bell’s Inequalities, Phys. Rev. Lett. 90, 080401(1-4) (2003) [2] N.D.Mermin, Phys. Rev. Lett. 65, 1838(1990) [3] D.N.Klyshko, Phys.Lett.A172,399(1993); A.V.Belinskii and D.N.Klyshko, Phys. Usp. 36, 653(1993) [4] J.Uffink, Phys. Rev. Lett. 88, 230406(2002) [5] N.Gisin,H.Bechmann-Pasquinucci, Phys. Lett.A246, 1(1998).

13. 13 Thanks! 谢谢 !