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纠缠态
Entangled states:
Hilbert space,
unit vector = pure
s t1 a te,n
1 n
state,
j isj a product
outhve:r w ise, enutavnw gl edv，wwuhere
pj uj uj
pj 0,pj 1
n 1
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从期望到方差
• 方差
ˆn 2 ( n 1 ) /2G H G Z H G Z H G Z HZ
0n, ˆn2n1
1 n ,U * ˆn U 2 n 1
UU1Un
1 n , U * ˆ n U 2 n 1
Z.Chen, PRL 93, 110403(2019)
Copier
Alice
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Eve
Bob
2 bits Alice
1 qubit
2 bits Bob
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Entanglement enhanced
1 qubit Alice
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2 bit
1 qubit Bob
Entanglement enhanced
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N kN 1 N j
j 1
kj
G(H ) Z co 0n ssi n 1 n
• 3-qubit概率不等式(没有严格证明): J.L.Chen,etal, PRL93,140407(2019)
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Wigner-Yanase信息(WYI)
• Wigner&Yanase, A49,910(1963):
Entanglement enhanced
1 qubit
2 bit
1 qubit
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2-qubit纠缠态
• Bell不等式:对乘积态-Gisin定理， Phys.Lett.A 154,201(1991)
a 1 1 1 2 ( a 2 2 a 2 2 ) a 1 1 1 2 ( a 2 2 a 2 2 ) 1
I 0 0 1 3 0 1 3 2 3 1 ( 2 2 ) 2
I(000 11) 13 , 1 213 2
Z.Chen,PRA71, 052302(2019)
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MKபைடு நூலகம்Bell算子
• Mermin-Klyshko不等式
,
1a11
1 a1 1
n n 1 1 2 ( a n n a n n ) n 1 1 2 ( a n n a n n ) n n 1 1 2 ( a n n a n n ) n 1 1 2 ( a n n a n n )
j
j
Mixed states:
uj
is separable, if each 2020/4o/29therwise, entangled.
is product,
量子位(qubit)
C2
qubit system,0 10, 1 10
0 x 1
1
1
0 z 0
01
0 i
y i 0
x, y, z
a a 1 x a 2 y a z z
aR3
n-qubit: C2 n i jkij k i, j,k0,1
0n 00 1n 11
EPR (Einstein, Podolsky & Rosen) pairs = Bell states
1 0011 2
1 0110
2
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I(t,A)I(,A)
iddttA,t,0
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Luo-Heisenberg不等式
• 骆顺龙（中科院应用数学所）, PRL91,180403(2019)
,A t r A 2 t r A 2
I(,A)(,A)
I ,A ,A A 2 A 2
1tr[A,B]2I,AI,B
maximum problem ma0xnU21nU2
or max0n U , where the maximum is taken
over all local unitary transformations U on n qubits, such that both 0n U~ and 1n U~ are real
I , 1 x 2 x 2 4 pq
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非局域Wigner-Yanase信息
• N-qubit非局域信息
I ( ) sI u ,a 1 1 p a n n
I 1 nn
I 212 n 12 n12 n n2 GHGZH 1Z 2 nI
I( ,A ) 1 t r 1 /2 ,A 2 t r A 2 t r 1 /2 A 1 /2 A 2 I ( 1 2 ,A ) I ( 1 ,A ) I ( 2 ,A ) I ( 1 2 , A 1 1 1 A 2 ) I ( 1 , A 1 ) I ( 2 , A 2 )
• PPT判别: Peres,PRL77,1413(2019) • 结论完整
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N-qubit纠缠态
• 广义GHZ态（N>2）对某些参数不满足 任何标准Bell不等式-
Żukowski,etal,PRL88,210402(2019)
2 1 s s A (n ) 1 Ns1 , , sn 1 ,1k 1 , , kN 1 ,21 k 1 1
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目录
• 纠缠态 • 量子位(qubit) • 物理意义 • N-qubit纠缠态 • Wigner-Yanase信息(WYI) • Bell-WYI不等式 • 非局域Wigner-Yanase信息 • MK Bell算子 • 从期望到方差 • Chen-Xu的判别法 • 广义GHZ态的判别 • 讨论
Can Quantum Information be translated into Classical Information?
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“No-Cloning Theorem“ Copier:
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Detected eavesdropping on quantum information
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Chen-Xu的判别法
• 结论(许全华教授,法国佛朗什-孔泰大学)
A pure state of n qubits is entangled if and
only if ,U ~ * ˆn U ~ 2 n 1 for a local unitary
transformation U~ resolving the following
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numbers. (/quant-ph/0505166)
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广义GHZ态的判别
• 广义GHZ态
G(H ) Z co 0n ssi n 1 n
U ~I2 I2
G ( ) H ˆn , 2 Z n 1 c2 o 2 s
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讨论
• EPR局域实在性 • Bell不等式 • 纠缠态与量子非局域性 • 问题：N-qubit混合纠缠态的判别
4
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Bell-WYI不等式
•
乘积态
I 1 n , a 1 1 a n n n
• 纠缠态
I ,a 1 1 a n n n 2
• 最大违背：GHZ态
1
GH Z2
0n
1n
• 2-qubit纠缠： p 0 0 q 1,p 1 ,q 0 ,p 2 q 2 1