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
32 NNNN LnkS
NnNL
NS
)(11
21
m
iimN Nkkkkk
NN kSNN FF
222
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 , , ,
32 NNNN LnkS
44 k ;544 S44 n 14 L
;41,34 S 1,34 k
34 n 14 L
;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.
13 NkS NN
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.
NN FF ,
2
BABABAAB
BBAA bBbBaAaA ;;;
A B
bbaa ,,,
22
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)
YXBAABBABABFFBFFF
YXBAABBABABFFBFFF
2
1
2
1
2
1
2
12
1
2
1
2
1
2
1
11112
11112
22
2
2
2
FF
AA aAFaAF 11 ,
1, 22
FFMax
22
1 2
2F
2F01
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.,
YCCXCFFCFFF
CYXCCFFCFFF
22223
22223
2
1
2
12
1
2
1
31,31,2,43 3333 SSS
333 1,2, CBAFFMax
3;16
1,2;8
2
3
2
3
2
3
2
3
ABCFF
CABFF
NNk 1
NNNN AAAFFMax 1,2, 21
2
22
4
3F
3F
3N
8
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
3F 3F
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.”
NN kSNN FF
222
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
1,1 NN kk
1111
22
1 2
11, NNNNNNNNNN kkAAkk
22
NNN FFn
11111 321, NNNNNN kSLnkS
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
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mmmNmmmNN FFFFFFF 4
1
mmmNmNmmNN kkkk
8
1,
)2(m
6262
3232
NNmmNmmN
mmmNmNmmmNmN
LnLLnn
LnLnkSkS
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).
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mNmNmmmmNN kkkk
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1,
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Thanks! 谢谢 !