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

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)(11

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iimN Nkkkkk

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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 ， ， ，

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;41,34 S 1,34 k

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;32,24 S 2,24 k

44 n 24 L

;31,1,24 S 1,1,24 k

24 n 14 L

;31,1,1,14 S 1,1,1,14 k

04 n 04 L

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.

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NkN

1,,1,1

Nk 11,2 NNk

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.

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2

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A B

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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)

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YXBAABBABABFFBFFF

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1

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2

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2

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1 2

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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.,

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In ━ diagram of the above 3-qubit system, thewhole 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.

1,23 k

33 k

1,1,13 k

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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 polynomialsatisfies the following quadratic Bell-inequalities,

where is the N-th Mermin-Klyshko polynomialand is the entanglement index of this system.”

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NN FF ; NN kS

Nk

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

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1 2

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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

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)2(m

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LnLLnn

LnLnkSkS

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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).

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Thanks! 谢谢 !