Transcript Bell inequality & entanglement
Bell inequality & entanglement
The EPR argument (1935) based on three premises: 1. Some QM predictions concerning observations on a certain type of system, consisting of two spatially separated particles, are correct.
2. A very reasonable criterion of the existence of ‘an element of physical reality’ is proposed: ’if, without any way disturbing a system, we can predict with certainty (i.e. with probability equal to unity) the value of a physical quantity, then there exists an element of physical reality corresponding to this physical quantity’ 3. There is no action-at-a-distance in nature.
EPR paradox
EPR paradox
• Before making the measurement on spin 1 (in z direction) the state vector of the system is: • After measurement on particle 1, (for argument’s sake say we measured spin down), the state of particle 2 is:
EPR paradox
• Since there is no longer an interaction between particle 1 and 2, and since we haven’t measured anything of particle 2, we can say that it’s state before the measurement is the same as after:
EPR paradox
• We could apply the same argument if we have measured the spin in the x direction and receive: In other words: it is possible to assign two different state vectors to the same reality!
Bell’s theorem
• If premise 1 is taken to assert that all quantum mechanical predictions are correct, then Bell’s theorem has shown it to be inconsistent with premises 2 & 3.
• • • • Deterministic local hidden variables and Bell’s theorem Bohm’s theorem: spatially separated spin ½ particles produced in singlet state: 1 2 [
u
( 1 )
u n
( 2 )
u
( 1 )
u n
( 2 )] All components of spin of each particle are definite, which of course is not so in QM description => HV theory seems to be required.
The question asked by Bell is whether the peculiar non locality exhibited by HV models is a generic characteristic of HV theories that agree with the statistical predictions by QM.
He proved the answer was: YES.
LHV and Bell’s theorem
•
A
ˆ
a
ˆ
b
ˆ of a measurement of the spin component of particle 2 of the pair along the direction
b
ˆ •
h A a
• The expectation value of this observable is: [
E
(
a
ˆ ,
b
ˆ )] 1 2
b
ˆ
b
ˆ • When the analyzers are parallel we have: [
E
(
a
ˆ ,
a
ˆ )] 1 • The EPR premise 2 assures us that if we measure A we know B
Local Hidden Variables defined.
• Since QM state does not determine the result of an individual measurement, this fact suggests that there exists a more complete specification of the state in which this determinism is manifest. We denote this state by • • We represent the distribution function for these states by
d
1
Bell’s definition:
• A deterministic hidden variable theory is local (
A
B
)( )
A
( )
B
( ) • specified and the particles have separated • but not The expectation value is taken to be:
E
A
( )
B b
ˆ ( )
d
Proof of Bell’s inequality
[
E
( ˆ ,
a
ˆ )] 1 Holds if and only if
A
( )
B
( )
E
(
a
ˆ ,
b
ˆ )
E
(
a
ˆ ,
c
ˆ )
A
A a
ˆ Hence: ( )
A b
ˆ ( ) ( )
A b
ˆ ( )[ 1
A b
ˆ ( )
A c
ˆ ( )]
d
A
( )
A c
ˆ ( )
d
Since A,B=+1
E
( ˆ ,
b
ˆ )
E
(
a
ˆ ,
c
ˆ ) [ 1
A b
ˆ ( )
A
( )]
d
Proof of Bell’s inequality
E
Using:
d
1
A
( )
B
( )
d
A
( )
B
( ) We have:
E
( ˆ ,
b
ˆ )
E
(
a
ˆ ,
c
ˆ ) 1
E
(
b
ˆ ,
a
ˆ )
Violation of Bell inequality
• ˆ ,
b
ˆ ,
c
ˆ 2 / 3
a
ˆ
b
ˆ
b
ˆ
c
ˆ 1 / 2
a
ˆ
c
ˆ 1 / 2 Which gives: and 1
E
(
b
ˆ , ) 1 / 2
What is the meaning of violating the Bell inequality?
• No deterministic hidden variables theory satisfying the locality condition and [
E
(
a
ˆ , )] 1 can agree with all of the predictions by quantum mechanics concerning spins of a pair of spin 1/2 particles in the singlet case.
In other words: once Bell’s inequality is violated we must abandon either locality or reality!
Requirements for a general experiment test • Let us consider the following apparatus:
Experiment requirements
• The QM predictions take the following form:
i M
i i m f i g F
[
p
12 ( [
p
1 ]
QM
)]
QM
1 4 1 2
f
1
g
[ 1 2 1 2
F
cos(
n
)] 1 2 1
f
1 1 [
p
1 ]
QM
1 2 2
f
2 2
i
i M
i m
Effective quantum efficiency of the detector max & min transmission of the analyzers Collimator efficiency (probability that appropriate emission enters apparatus 1 or 2 Conditional probability that if emission 1 enters apparatus 1 then emission 2 enters apparatus 2 Measure of the initial state purity n=1 for fermions and n=2 for bosons
Experiment requirements
• Taking the following assumptions: 1 / 4
n
2 ,
f
1
g
[
f
2 , 1 2 2 ( / ) 2
F
1 ] 2 If the experimental values are within the domain of the above inequality, then we can distinguish between QM prediction and inequalities.
• • • • • • Summery: for direct test of inequality the requirements are: A source must emit pairs of discrete-state systems, which can be detected with high efficiency.
QM must predict strong correlations of the relevant observables of each pair, and the pairs must have high QM purity.
Analyzers must have extremely high fidelity to allow transmittance of desired states and rejections of undesired.
The collimators must have high transmittance and not depolarize the emissions.
A source must produce the systems via 2-body decay, or else g becomes g<<1.
For locality’s sake: Analyzer parameters must be changed while particles are in flight. (no information exchange between detectors.
CHSH
• • Since no idealized system exists, one can abandon the requirement: [
E
( ˆ ,
a
ˆ )] 1 CHSH arrived at the following inequality:
E
(
a
,
b
)
E
(
a
' ,
b
)
E
(
a
,
b
' )
E
(
a
' ,
b
' ) 2 Which was violated by Alain’s experiment => Proof of non-local correlations occur on a time scale faster than the speed of light.
experiments
experiments
• • The third Alain Aspect experiment: faster than light correlation.
Using the following setup:
Crash course in information theory… • with probability
p i
We can define the density operator as:
i p i
i
i
Crash course in information theory… • • • exactly is said to be in
pure state
, otherwise it is said to be in
mixed state
.
A pure state satisfies
tr
( 2 ) 1 A mixed state satisfies
tr
( 2 ) 1
Schmidt de-composition
Crash course in information theory… • • Shannon Entropy: quantifies how much information we gain, on average, on a random variable X, or the amount of uncertainty before measuring the value of X. If we know the probability distribution of X:
p i
.....
p n
then the Shannon Entropy associated with it is:
• Crash course in information theory…
x
Von Neumann entropy
Entanglement distillation and dilution • Suppose we are supplied not with one copy of
Entanglement distillation
is how many convert into entangled Bell state.
Entanglement dilution
is the reverse process.
Entanglement distillation and dilution • • Defining a specific bell state as a ‘standard unit’ of entanglement, we can quantify entanglement.
Defining an integer n which represents the number of Bell states, and an integer m representing the number of pure states that can be produced, then the limiting ratio n/m is the
entanglement of formation of the state
Setting the limits
• decomposition
Setting the limits
• An m-fold tensor product can be defined
Alice and Bob live by the limits
=> We have an upper limit for entanglement formation!
In a similar manner it was shown that there is a lower limit for entanglement distillation which is also
bibliography
• • • • • Quantum optics- an introduction/ M. Fox p.304-323 Quantum optics/ M.Scully & M.Tsubery p.528-550.
Bell’s theorem: experimental tests and implications, J. Clauser & A. Shimony, Rep. Prog. Phys, Vol.41, 1978.
Experimetal tests of realistic local theories via Bell’s theorem, PRL vol.47, nu.7, 1981 A.Aspect et al.
Quantum computation and quantum information, M.Nielasen and I.Chuang, p.137,580, 607.