L'C' Kwek

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Quantum Entanglement and Bell Inequalities

• L.C. Kwek
• Nanyang Technological University

Presented at the Workshop on Quantum Statistics
and Related Topics, National Institute of
Informatics, Tokyo, 27-28 Jan 2005
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QIT Group (Singapore)
Artur Ekert and C.H. Oh

• Lim Jenn Yang
• Antia Lamas Linares
• Alex Ling
• Looi Shiang Yong
• Ivan Marcikic
• Neelima Raitha
• Kuldip Singh
• Tang Dingyuan
• Tey Meng Khoon
• Tong Dianmin
• Wang Zisheng
• Wu Chunfeng
• about 10 undergraduate students
• www.quantumlah.org

• Janet Anders
• Chia Teck Chee
• Chen Jingling
• Chen Lai Keat
• Chin Mee Koy
• Choo Keng Wah
• Du Jiangfeng
• Berge Englert
• Feng Xinli
• Ajay Gopinathan
• Darwin Gosal
• Hor Wei Hann
• D. Kaszlikowski
• Christian Kurtsiefer
• L.C. Kwek
• C.H. Lai
• Wayne Lawton

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http//www.lasphys.com/workshops/lasphys05/lphys05
.htm
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Scope

• Quantum correlations and Bell inequalities
• Gisins Theorem (1991)
• Overview of development with more particles and
  higher dimensions
• Final remarks

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Hidden Variable Models

• EPR did not doubt that QM is correct
• They claim that QM is an incomplete description
  of physical reality.
• Basic idea the wave function is not the whole
  story some quantity l is needed.
• We call l the hidden variable since we have no
  idea how to calculate or measure it.
• Over the years, a number of hidden variable
  theories have been formulated
• In 1964, J.S. Bell showed that any local hidden
  variable theory is incompatible with QM.

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1
1
-1
1
1
-1
-1
1
-1
-1
E(q,f) p(,) p(-,-)-p(,-)-p(,-)
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• Quantum mechanical correlation function

Classical correlations must satisfy
Quantum mechanics violates the CHSH inequality
for certain settings
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And there is also an inequality in terms of
probabilities Clauser-Horne Inequalities
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• Clauser-Horne (CH) Inequality for classical
  systems (local realism)
• Quantum mechanics gives

where denotes the probability
that detector () at setting i (Alice) and
detector () at setting j (Bob) click
For certain settings,
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Note
It is common to consider the amount of noise can
be admixed to the singlet state so that Bell
inequality is violated by considering the state
The maximum amount of noise for which CHSH and
CH continues to be violated is
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Gisins Theorem (1991)
Phys. Lett. A, 154, 201 (1991)
All entangled pure states violate Bell
inequalities.
Not surprising but probably the first attempt at
quantifying entanglement.
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Higher dimensions
Bell (1964) CHSH (1969)
CH(1974)
MABK (90-93)
Gisins Theorem (1991)
CWKO (2004)
WW-ZB(2002)
KKCMO (2002)
ACGKKOZ (2003-2004)
CGLMP (2002)
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Bell inequalities
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Bell Inequality for 2 Qutrits (KKCZO)

• Consider a maximally entangled state for two
  qutrits

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• The observers measure observables defined by
  6-port (three input and three output ports) beam
  splitter.

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• The observables
• where elements of unitary matrix, U, are

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• Quantum Correlation Functions

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The generalized Bell Inequality for two Qutrits
(KKCZO)
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• Quantum mechanics (for certain settings)

Noise
(qutrit)
(qubit)
Therefore, two maximally entangled qutrits
violate local realism more strongly than two
qubits.
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• More generally (KKCZO),

This inequality has been violated experimentally
(G. Weihs, Private Communication) This
inequality has potential applications in quantum
communication.
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Two-particle systems
Bell inequalities for bipartite quantum systems
of arbitrarily high dimensionality is given as
CGLMP, PRL, 88,040404 (2002)
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Bell Inequality for Three Qutrits

• The inequality for three qutrits given by

(1)
A. Acin et al, PRL, 92, 250404 (2004).
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Remarks

• The maximum amount of noise for maximally
  entangled state for the three-qutrit case is
  F2/5 (0.4), which is less than that for three
  maximally entangled qubits using ZB inequality
  (for which F1/2).
• However, for some non-maximally entangled state
  of 3 qutrits, the maximum amount of noise is F
  0.57.

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Bell Inequalities for Three qubits

• Zukowski-Brukner (Werner-Wolf) inequality with d
  2

The above Bell inequality is not violated by a
family of pure entangled states generalized GHZ
states
for
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Question?

• Can Gisins theorem be generalized to three-qubit
  pure entangled states?
• Can one find a Bell inequality that is violated
  by GHZ state for the whole region?

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Bell Inequalities for 3 qubits

• Consider a Bell-type scenario
• Party Possible measurement
• Alice A1 or A2
• Bob B1 or B2
• Charlie C1 or C2
• Each measurement have 2 possible outcomes
• A1, A2, B1, B2, C1, C20, 1.
• If the observers decide to measure A1, B1, C2,
• the result is (0,1,1) with
• probability P(a10, b11, c21).

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Bell Inequalities for 3 qubits
(2)
where
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Gisins Theorem for 3 Qubits
All generalized GHZ state () for three-qubit
systems violate Bell inequality (2)
All pure 2-entangled states of a three-qubit
system violate inequality (2)
Reference CWKO, PRL, 93, 140407 (2004)
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Numerical results for the family of generalized W
states
Numerical results for the generalized GHZ state
() for inequality (2).
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To show violation for 2-entangled states
Set c00 and c11
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Simplifying
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Eliminate the impossible
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Using (sum of probability 1)
CH inequality for 2 qubits...Apply Gisins Theorem
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Reduction of Three-Qutrit Inequality
For three qubits, inequality (1) can be reduced
to inequality (2), not ZB inequality.

• Inequality (2) is violated by any pure entangled
  states of three qubits
• ZB inequality is maximally violated by GHZ
  state, however, inequality (2) is not maximally
  violated by GHZ state.

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? Richard Cleve et al (quant-ph/0404076)
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Summary

• Brief survey of Bell inequality and Gisins
  theorem
• Presentation of our attempts at higher dimensions
  and larger number of particles.
• Holy Grail?
• Reasons for studying Bell inequalities
  Entanglement witness Quantum Cryptography
  Foundation of QM!

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