Dichotomic Observables; GP(1992) vs Psudospin observable(2002). - - PowerPoint PPT Presentation

Dichotomic Observables; GP(1992) vs Psudospin observable(2002).

• Gisin-Peres observable for Bell test of N-dim.

state A(θ) = Γx sin θ + Γz cos θ + E – E is a matrix whose only non-vanishing ele- ment is EN,N = 1 when N is odd and E is zero when N is even. – It also can be written as ˆ U(θ)

N−1

• n=0

(|2n2n| − |2n + 12n + 1|) ˆ U(θ)† where ˆ U(θ) is tensor sum of SU(2) unitary

• perator

– With the observable, any pure entangled state (N-dim.) will violate CHSH Inequ.

• Pseudospin operator s = (sx, sy, sz) for a nonlo-

cality test of continuous variables sz =

∞

• n=0
• |2n + 12n + 1| − |2n2n|
• ,

a · s = sz cos θ + sin θ(eiϕs− + e−iϕs+), where s− = ∞

n=0 |2n2n + 1| = (s+)† and a is

a unit vector. – Even and Odd parity measuremant – SU(2) rotational symmetry – Pseudospin operator corresponds to the GP

• bservable in the limit of N → ∞.
• REMARK: Any bi-partite pure Continuous Vari-

able Entangled state are nonlocal.

Dichotomic Observables; Wiger(parity,CHSH) vs Q (present,CH)

Banaszek and Wodkiewicz, PRL(1998)

• Parity measurement and displacement operation

Π(α) = Π+(α) − Π−(α) = D(α)

∞

• n=0
• |2n2n| − |2n + 12n + 1|
• D†(α)

– D(α) is the displacement operator D(α) = exp[αˆ a† − α∗ˆ a] for bosonic operators ˆ a,ˆ a†. – The Wigner fuction from the operator W(α) = TrΠ(α)ˆ ρ1 Wab(α; β) = TrΠ(α) ⊗ Π(β)ˆ ρab – CHSH inequality can be constructed |BCHSH| = π2 4 |W(α, β) + W(α, β′) + W(α′, β) − W(α′, β′)| ≤ 2,

• Photon presence measurement with dispalcement
• peration

ˆ Q(α) = ˆ D(α)|00| ˆ D(α)† ˆ P(α) = ˆ D(α)

∞

• n=1

|nn| ˆ D(α)† – It satisfies the completeness relation ˆ Q(α) + ˆ P(α) = 1 – Assign 1 to no counts and 0 to triggering – The Q-Function Qab(α, β) = 1 π2Tr ˆ Q(α) ⊗ ˆ Q(β)ρab – CH inequality can be constructed −1 ≤ π2[Qab(α, β) + Qab(α, β′) + Qab(α′, β) −Qab(α′, β′)] − π[Qa(α) + Qb(β)] ≤ 0

Continuous Variable State; EPR State

• EPR state ; Two mode momentum eigenstate

with total momentum p1 + p2 = 0 and relative position x1 − x2 = x0 – EPR state corresponds to a two-mode squeezed state for infinity squeezing limit |TMSS =

∞

• n=0

(tanh r)n cosh r |n, n =

• dxC(x, r)|x, x

– Gaussian state – It can be generated by beam splitter or para- metric down conversion

Kim and Sanders, PRA(1996);Hong and Mandel,PRL(1984)

• Optimized violation of CHSH inequality

– Psudospin operator (dotted) – Parity op. (BW solid, Our dashed)

Continuous Variable State; EPR State

• Optimized violation of CH inequality

– Psudospin operator (CHSH, dotted) – Photon presence measurement. (BW solid, Our dashed)

• Violation of CHSH inequality goes maximum as

r → ∞ while CH inequality goes 0.

• For the case of EPR state, maximum violation of

CHSH is |BBW|max → 8/39/8 ≃ 2.32.

• EPR state does not maximally violate CHSH in

the generalized BW observable.

• Parity meaurement gives higher violaiton for

TMSS altough it is experimentally difficult.

Unitary Operators

• Rotation of pseudospin operator a · s

a · s|2n + 1 = cos θ|2n + 1 + sin θ|2n a · s|2n = − sin θ|2n + 1 + cos θ|2n

• The expectation value of the parity operator for a

number state P(n, |α|) = n|Π(α)|n = e−|α|2|α|2n n!

∞

• k=0

(2k)! |α|4k

• L(n−2k)

2k

(|α|2) 2 −(2k + 1)! |α|4k+2

• L(n−2k−1)

2k+2

(|α|2) 2 , where L(p)

q (x) is an associated Laguerre polyno-

mial.

Bell’s inequality with finite outcome measurement (W.Son et al., will be appeared)

• Collins version of Bell’s inequality (2002)

– Bell’s inequality for N-dim. system using finite

• utcome measurement.

– Agree with the results that noise robustness

• f violations of local realism. (D. Kaszlikowski et

al.,2000)

• Bell’s inequality test for TMSS using finite out-
• come. measurement

– Always violate the Bell’s inequality. – Infinite squeezing does not gives maximum vi-

• lation.

– Higher squeezing ; The more outcome the more violation. – Small squeezing ; The more outcome the less violation.

Summary and Future work

• Summary

– We discussed for the link between the nolocal- ity of finite and infinte dimensional systems by showing origin of psudospin operator. – We found the maximal bound of violation of Bell’s inequality for BW formalism. – Nonlocality test for different CV state with different dichotomic observables – We discussed the relation between the nonlo- cality and unitary operators of measurement process. – Nonlocality test with more than two outcome.

• Future work

– Noise effect for the nonlocality tests. – Find the implication of the state and the mea- surements in quantum information precessing. (cf Quantum cryptography)