Quantum

1
Quantum Information Technology Group in
NUS Singapore
Experimental Section
Theoretical Section
2
QIT Group (Singapore)
Artur Ekert and C.H. Oh

• Janet Anders
• Chia Teck Chee
• Chen Jingling
• Chen Lai Keat
• Choo Keng Wah
• Du Jiangfeng
• Berge Englert
• Feng Xunli
• Ajay Gopinathan
• Darwin Gosal
• Hor Wei Hann
• D. Kaszlikowski
• Christian Kurtsiefer
• L.C. Kwek
• C.H. Lai
• Wayne Lawton
• Lim Jenn Yang
• Antia Lamas Linares

• Alex Ling
• Looi Shiang Yong
• Liu Xiongjun
• Ivan Marcikic
• Neelima Raitha
• Kuldip Singh
• Tey Meng Khoon
• Tong Dianmin
• Wang Zisheng
• Wu Chunfeng
• 5-10 undergraduate students
• www.quantumlah.org

3
First Workshop on Quantum Computation and
Information held in 2001
Speakers

• Prof. A. Ekert (Oxford)
• Prof. C. Bennett (IBM)
• Prof. S. Popescu (Bristol)
• Prof. I. Chuang (MIT)

4
Focus

• Quantum Cryptography
• Quantum Algorithms
• Quantum Games
• Quantum Cloning
• Quantum Channels
• Geometric Phase Computation
• Quantum Entanglement
• Foundation of Quantum Mechanics Bell
  Inequalities, Bures Fidelity

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A New Bell Inequality with Improved Visibility
for 3 qubits

• Chunfeng Wu, Jingling Chen, L. C. Kwek and C. H.
  Oh
• Physics Department
• National University of Singapore
• Presented at the International Conference on
  Recent Progress in Quantum Mechanics and Its
  Applications ,
• Hong Kong, China
• December 13, 2005 to December 16, 2005

6
Outline

• Introduction
• Bell inequalities
• A new Bell inequality for three qubits

7
Introduction

• In a 1935 Einstein, Podolsky, and Rosen (EPR)
  poised the question can quantum mechanical
  description of physical reality be considered
  complete? paper 1
• Element of physical reality If, without in any
  way disturbing a system, we can predict with
  certainty the value of a physical quantity, then
  there exists an element of physical reality
  corresponding to this physical quantity.
• (sufficient, not necessary condition to
  define an element of reality).
• Completeness In a complete theory there is an
  element corresponding to each element of
  reality.
• Locality The real factual situation of the
  system A is independent of what is done with the
  system B, which is spatially separated from the
  former.

1 A. Einstein, B. Podolsky and N. Rosen, Phys.
Rev. 47, 777 (1935).
8
EPR Paradox

• Spooky action the mysterious long-range
  correlations between the two widely separated
  particles.
• Local hidden variables are suggested in order to
  restore locality and completeness to quantum
  mechanics. In a local hidden variable theory,
  measurement is fundamentally deterministic, but
  appears to be probabilistic because some degrees
  of freedom are not precisely known.

9
Entanglement

• Central to EPR paper is an entangled state.
• The notion of entanglement 2 was introduced by
  Schrödinger to describe a situation in which
• Maximal knowledge of a total system does not
  necessarily include total knowledge of all its
  parts, not even when these are fully separated
  from each other and at the moment are not
  influencing each other at all
• 2 E. Schrodinger, The present situation in
  quantum mechanics. In J. Wheeler and W. Zurek,
  editors, it Quantum Theory and Measurement, P
  152, Princeton University Press, 1983.

10
Entanglement

• Understanding of quantum entanglement
• the information in a composite system
  resides more in the correlations than in
  properties of individuals.

11
The Bell Theorem

• In 1964, Bell 3 showed that local realism
  imposes experimentally constraints on the
  statistical measurements of separated systems.
  These constraints, called Bell inequalities, can
  be violated by the predictions of quantum
  mechanics. J. Bells contribution Consider the
  correlations predicted for three spin
  measurements not at right angles but at an
  arbitrary angle ?. He was able to prove that
  correlations predicted by quantum mechanics are
  greater than could be obtained from any local
  hidden variable theory.
• Violation of Bell inequalities is one method to
  identify entanglement.
• 3 J. S. Bell, Physics, 1, 195 (1964).

12
The Bell Theorem

• The original Bell inequalities are not suitable
  for realistic experimental verification. One of
  the most common form of Bell inequalities is
  Clauser-Horne-Shimony-Holt (CHSH) inequality 4
  for two qubit system,

4 J. F. Clauser, M. A. Horne, A. Shimony, and
R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).
13
The Bell Theorem

• The function is the correlation of
  measurements between and for the two
  systems.
• Classically, thus,

14
The Bell Theorem

• Quantum mechanically,

For maximally entangled state
15
The Bell Theorem
Thus, CHSH inequality is violated.
16
Bell Inequalities

• 1982 Aspect experiment two detectors were placed
  13m apart and a container of excited calcium
  atoms midway between them spin states of two
  entangled photons.
• 1997 N Gisin two detectors were placed 11km
  apart
• Rule out local hidden variables

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Bell Inequalities

• Are Bell inequalities violated by all pure
  entangled states?
• Recent developments.
• (1) Gisins theorem 5 every pure bipartite
  entangled state in two dimensions violates the
  CHSH inequality.

5 N. Gisin, Phys. Lett. A 154, 201 (1991) N.
Gisin and A. Peres, Phys. Lett. A 162, 15 (1992).
18
Gisins Theorem (1991)
Phys. Lett. A, 154, 201 (1991)
All entangled pure states violate Bell
inequalities.
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Bell Inequalities

• (2) Mermin-Ardehali-Belinskii-Klyshko
  inequalities for N qubits 6.

6 N. D. Mermin, Phys. Rev. Lett. 65, 1838
(1990) M. Ardehali, Phys. Rev. A 46, 5375
(1992) A. V. Belinskii and D. N. Klyshko, Phys.
Usp. 36, 653 (1993).
20
Bell Inequalities

• Take for example that the
  three-qubit MABK
• inequality is given as

where
If we use Q to describe correlation function, the
above inequality can also be written as
here we take
21
Bell Inequalities

• (3) Scarani and Gisin 7 noticed that there
  exist pure states of N qubits which do not
  violate MABK inequalities.
• For , the states do not
  violate the MABK inequalities.

For example,
7 V. Scarani and N. Gisin, J. Phys. A 34, 6043
(2001).
22
Bell Inequalities
To violate the MABK inequalities, it is required
So for , the generalized GHZ
states do not violate the MABK inequalities.
(4) Zukowski 8 and Werner 9 independently
found the most general Bell inequalities for N
qubits (called Zukowski-Brukner inequalities
here).
8 M. Zukowski and C. Brukner, Phys. Rev. Lett.
88, 210401 (2002). 9 R. F. Werner and M. M.
Wolf, Phys. Rev. A 64, 032112 (2001).
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Bell Inequalities
Zukowski-Brukner(ZB) inequalities for N qubits
The correlation function, in the case of a local
realistic theory, is the average over many runs
of the experiment
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Bell Inequalities
After averaging the expression over the ensemble
of the runs of the experiment, the following set
of Bell inequality is obtained
These are the ZB inequalities.
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Bell Inequalities

• (5) Zukowski et al 10 showed that
• For Neven, the generalized GHZ states
  violate the Zukowski-Brukner inequalities
• For Nodd and , the
  correlations between measurements on qubits in
  the generalized GHZ states satisfy the ZB
  inequalities for correlations.

(6) We constructed Bell inequalities 11 for
three qubits in terms of correlation functions.
These inequalities are violated by all pure
entangled states.
10 M. Zukowski, C. Brukner, W. Laskowski and M.
Wiesniak, Phys. Rev. Lett. 88, 210402
(2002). 11 J. L. Chen, C. F. Wu, L. C. Kwek and
C. H. Oh, Phys. Rev. Lett. 93, 140407 (2004).
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Bell Inequalities

• Bell inequalities are sensitive to the presence
  of noise and above a certain amount of noise, the
  Bell inequalities will cease to be violated by QM.

The strength of violation or visibility (V ) is
considered as the minimal amount V of the given
entangled state that one has to add to pure
noise so that the resulting state still violates
local realism.
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Bell Inequalities

• The Bell inequalities given by us 11 are not
  good enough to the resistance of noise. For the
  GHZ state, threshold visibility is 0.77 (it is
  0.5 for 3-qubit Zukowski-Brukner inequality).
• Our recent research shows that there is one new
  Bell inequality for three qubits with improved
  visibility.
• 11 J. L. Chen, C. F. Wu, L. C. Kwek and C. H.
  Oh, Phys. Rev. Lett. 93, 140407 (2004).

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Bell Inequalities for 3 qubits

1. 3-qubit Zukowski-Brukner inequality 8

(2) Our previous 3-qubit Bell inequality 11
where are three-particle
correlation functions defined as
after many runs of
experiments. Similar definition for two-particle
correlation functions
29
A new Bell Inequality for 3 qubits
(3) Bell inequality with improved visibility
()
30
A new Bell Inequality for 3 qubits

• Quantum mechanically,

• Correlation functions in quantum mechanics

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A new Bell Inequality for 3 qubits
For three qubits, there are only two classes of
genuinely three particle entangled states which
are inequivalent13.
(1) The first class is represented by the GHZ
state 14,
(2) The second by the so-called W state 15.
13 W. Dur, G. Vidal and J. I. Cirac, Phys. Rev.
A 62, 062314 (2000). 14 D.M. Greenberger, M. A.
Horne, and A. Zeilinger, in Bells Theorem,
Quantum Theory, and Conceptions of the Universe,
edited by M. Kafatos (Kluwer, Dordrecht, 1989),
p. 69. 15 A. Zeilinger, M. A. Horne, and D.M.
Greenberger, in Workshop on Squeezed States and
Uncertainty Relations, edited by D. Han et al.,
NASA Conference Publication No. 3135 (NASA,
Washington, DC, 1992), p. 73.
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A new Bell Inequality for 3 qubits
(1) Numerical results for Generalized GHZ states
4.404
For the GHZ state
33
A new Bell Inequality for 3 qubits
(2) Numerical results for Generalized W states
34
A new Bell Inequality for 3 qubits
To show that the new Bell inequality is more
resistant to noise than our previous one. We
rewrite them as follows,
35
A new Bell Inequality for 3 qubits
Usually, the left hand side of Bell inequality
can be described by a quantity , called Bell
quantity. For the two Bell inequalities, we write
separately
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A new Bell Inequality for 3 qubits
Curve A is for our previous inequality and curve
B is for the new inequality.
Quantum violation in the figure is the quantum
prediction for Bell quantity. It is clear that
the new inequality is more resistant to noise
than the previous inequality.
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Generalized Bell Inequalities

• Bell inequalities for M qubits (Mgt3)
• Bell inequalities for M qudits (Mgt3)
• M-qudit M particles in d-dimensional
• Hilbert space.

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Summary

• A new Bell inequality in terms of correlation
  functions with improved visibility for 3 qubits
  is constructed.
• However, the threshold visibility has not reached
  the optimal value 0.5 as exhibited by the maximal
  violation of the ZB inequality by the GHZ state.
  To construct such a Bell inequality for three
  qubits is an open problem.
• Generalization of Gisins theorem for N qubit
  (Ngt3, odd numbers) is still unsolved at this
  stage.

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Thank you!
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Bell Inequalities for 3 qubits

• Our previous inequality is violated by all pure
  entangled states of three qubits.
• The quantum violation strength of the GHZ state
  of the inequality is not as strong as that of ZB
  inequality as seen in (1) and (2) below.

(1)The threshold visibility of the inequality for
the GHZ state is 0.77.
42
Bell Inequalities for 3 qubits
(2)The threshold visibility of the ZB inequality
for the GHZ state is 0.5.
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Generalized Bell Inequalities

• Bell inequalities for M qubits (Mgt3)
• Bell inequalities for M qudits (Mgt3)
• M-qudit M particles in d-dimensional
• Hilbert space.