Classifying Entanglement in Quantum Memory via a Recursive Bell-Inequalities

1. Classifying Entanglement in Quantum Memoryvia a Recursive Bell-Inequalities Sixia Yu, Zeng-Bing Chen, Jian-Wei Pan, and Yong-De Zhang Department of Modern PhyiscsUniversity of Science and Technology of ChinaHefei 230027, China

2. Abstract We 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Thanks! 谢谢!