Lecture Note 1 Quantum Information Processing

Lecture Note 1Quantum Information Processing

Jian-Wei Pan

Physikalisches Institut der Universität Heidelberg Philosophenweg 12, 69120 Heidelberg, Germany

Outlines

• A quick review of quantum mechanics• Quantum superposition and noncloning theorem• Quantum Zeno effect• Quantum entanglement • Quantum nonlocality

Basic Principles of Quantum Mechanics

• The state of a quantum mechanical system is completely specified by wave function , which has an important property that

is the probability that the particle lies in the volume element at time , satisfying the normalized condition

),( tx3* ),(),( dxtxtx

3dx t

1),(),( 3*

dxtxtx

• To every observable in classical mechanics there corresponds a linear, Hermitian operator in quantum mechanics (indicates superposition principle ).

Basic Principles of Quantum Mechanics

• In any measurement of the observable associated with operator , the only values that will ever be observed are the eigenvalues , which satisfy the eigenvalue equation

a

aA

3* ˆˆ dxAA

If a system is in a state described by a normalized wave function , then the average

value of the observable corresponding to

A

Basic Principles of Quantum Mechanics

• The wave function or state function of a system evolves in time according to the time-dependent Schrödinger equation

where is the Hamiltonian.

H

ti

H

• The total wave function must be antisymmetric (or symmetric ) with respect to the interchange of all coordinates of one fermion (boson) with those of another.

Quantum Superposition Principle

.||,||

,

,||

222

211

21

212

pp

p

Quantum Superposition Principle

which slit?

| | or

| | +

Classical Physics:

“bit”

Quantum Physics: “qubit”

Quantum foundations: Bell’s inequality, quantum nonlocality…Quantum information processing: quantum communication, quantum computation, quantum simulation, and high precision measurement etc …

Quantum Superposition Principle

Qubits: Polarization of Single Photons

One bit of information per photon(encoded in polarization)

"1|"|

"0|"|

V

H

Qubit: VH |||

Non-cloning theorem:

An unknown quantum state can not be copied precisely!

1|||| 22 2||

2||

Non-Cloning Theorem

,1110

,0000

W. Wootters & W. Zurek, Nature 299, 802 (1982) .

)10)(10(11000)10(

According to linear superposition principle

Zeno Paradox

Origin of Zeno effect

Can the rabbit overtake the turtle?

Quantum Zeno Effect

Quantum Zeno Effect

)1)1(0)1((2

10 ii eeHH

Quantum Zeno Effect

Interaction-free measurement !

Quantum Zeno Effect

Considering neutron spin evolving in magnetic field, the probability to find it still in spin up state after time T is

where is the Larmor frequency .

)2

(cos2 Tp

up down

Quantum Zeno Effect

T=0 T=π/2 T=π

T=0 T=π/2 T=π

G (cake is good)=G0×

If we cut the bad part of the cake at time T=π/2 , then at T=π we have G=1/4×G0

)1(,2

cos1

T

Experiment

P. Kwiat et. al., Phys. Rev. Lett. 74 4763 (1995)

N

NP )]

2([cos2

In the limit of large N

)(4

1 22

NON

P

Bell states – maximally entangled states:

212112

212112

||||2

1|

||||2

1|

HVVH

VVHH

Polarization Entangled Photon Pair

)|||(|2

1

)|||(|2

1|

2121

212112

HVVH

HVVH

)|(|2

1|

)|(|2

1|

VHV

VHH

Singlet:

where

45-degree polarization

Polarization Entangled Photons

GHZ states – three-photon maximally entangled states:

)(2

1

)(2

1

)(2

1

)(2

1

321321123

321321123

321321123

321321123

HHVVVH

VHVHVH

HVVVHH

VVVHHH

Manipulation of Entanglement

0000 01011110 1011

Manipulation of Entanglement

000)10()1000()1100(

10000)10(00

Manipulation of Entanglement

00000)10(0)1000(0)1100(

110000111000110000

0)1100(0)1000(00)10(000

Einstein-Poldosky-Rosen Elements of Reality

. . .

Einstein-Podolsky-Rosen Elements of Reality

In quantum mechanics in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other. Then either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) these two quantities cannot have simultaneous reality.Quantum-mechanical description of physical reality cannot be considered complete.

non-commuting operators, i.e., momentum P and position X of a particle

12 1 2 1 2

1 2 1 2

1 2 1 2

1| (| | | | )

21

(| | | | )2

1(| | | | )

2

H V V H

H V V H

R L L R

zVH ,1

0,

0

1

x

VHV

VHH,

)(2

1

)(2

1

y

iVHL

iVHR,

)(2

1

)(2

1

Polarization Entangled Photon Pairs shared by Alice and Bob

Bohm converted the original thought experiment into something closer to being experimentally testable.

Bohm’s Argument

Bohm’s Argument

According to the EPR argument, there exist threeelements of reality corresponding to and

!yx ,

zHowever, quantum mechanically, and are three noncommuting operators !

yx ,

z

Bell’s Inequality and Violation of Local Realism

The limitation of the EPR !

Both a local realistic (LR) picture and quantum mechanics (QM) can explain the perfect correlations observed.

Non-testable!

[J. S. Bell, Physics 1, 195 (1964)]

Bell’s inequality states that certain statistical correlations predicted by QM for measurements on two-particle ensembles cannot be understood within a realistic picture based on local properties of each individual particle.

Sakurai's Bell Inequlaity

Correlation measurements between Alice’s and Bob’s detection events for different choices for the bases ( indicted by a and b for the orientation of the PBS).

)(2

12121 HVVH

Pick three arbitrary directions a, b, and c

Sakurai's Bell Inequlaity

43),( PPbaP

42),( PPcaP

73),( PPbcP

724343 PPPPPP

It is easy to see

Sakurai's Bell Inequlaity

The quantum-mechanical prediction is

For example , the inequality would require

),(),(),( bcPcaPbaP

2)2

(sin2

1),(

babaP

。。。 ，， 04590 bca

1464.02500.0

5.22sin2

15.22sin

2

145sin

2

1 222

。。。

CHSH Inequality

Considering the imperfections in experiments, generalized Bell equality — CHSH equality,

Quantum Mechanical prediction:

22MAXS

Local Reality prediction: 2MAXS

J. Clauser et al., Phys. Rev. Lett. 23, 880 (1969)

A. Aspect et al., Phys. Rev. Lett. 49, 1804 (1982)

Experimental Test of Bell Inequality

Drawbacks: 1. locality loop hole 2. detection loop

hole

020.0101.0exp S violates a generalized inequality 0Sby 5 standard deviations.

G. Weihs et al., Phys. Rev. Lett. 81, 5039 (1998)

02.073.2exp S

Experimental Test of Bell Inequality

violates CHSH inequality

2S

by 30 standard deviations.

Drawback: detection loop hole