Download Entanglement and Bell theorem

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
[unˆ  (1)  unˆ  (2)  unˆ  (1)  unˆ  (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 nonlocality 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
•Let Aaˆ be the result of a measurement of the spin component
of particle 1 of the pair along the direction â and Bbˆ the result
of a measurement of the spin component of particle 2 of the pair
along the direction bˆ
•We denote a unit spin as h / 2 hence B ˆ , Aaˆ = +1
b
•The expectation value of this observable is:
[ E (aˆ, bˆ)]    1  aˆ 2  bˆ   aˆ  bˆ
•When the analyzers are parallel we have:
[ E (aˆ , aˆ )]  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 
• Let  be the space of these states
• We represent the distribution function for
these states by 


d  1
Bell’s definition:
• A deterministic hidden variable theory is local
if for all â and b̂ and all    we have:
( Aaˆ  Bbˆ )( )  Aaˆ ( )  Bbˆ ( )
• The meaning of this is that once the state  is
specified and the particles have separated
measurements of A can depend on  and â
but not b̂
• The expectation value is taken to be:
E a, b     Aaˆ ( )  Bbˆ ( )d
Proof of Bell’s inequality
[ E (aˆ , aˆ )]  1 Holds if and only if Aaˆ ( )   Baˆ ( )
Hence:


E (aˆ , bˆ)  E (aˆ , cˆ)     Aaˆ ( ) Abˆ ( )  Aaˆ ( ) Acˆ ( ) d
    Aaˆ ( ) Abˆ ( )[1  Abˆ ( ) Acˆ ( )]d
Since A,B=+1
E (aˆ , bˆ)  E (aˆ , cˆ)    [1  Abˆ ( ) Acˆ ( )]d
Proof of Bell’s inequality
Using:


d  1
E a, b     Aaˆ ( )  Bbˆ ( )d
Aaˆ ( )   Baˆ ( )
We have:
E (aˆ , bˆ)  E (aˆ, cˆ)  1  E (bˆ, aˆ )
Violation of Bell inequality
• Taking aˆ, bˆ, cˆ to be coplanar with ĉ making an
angle of 2 / 3 with â , and b̂ making an angle of
 / 3 with both â and ĉ then:
aˆ  bˆ  bˆ  cˆ  1 / 2
aˆ  cˆ  1 / 2
Which gives: E(aˆ, bˆ)  E(aˆ, cˆ)  1 and 1  E (bˆ, aˆ )  1 / 2
What is the meaning of violating the
Bell inequality?
• No deterministic hidden variables theory
satisfying the locality condition and [ E (aˆ, aˆ )]  1
can agree with all of the predictions by quantum
mechanics concerning spins of a pair of spin1/2 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:
1
[ p12 ( )]QM  1 2 f1 g[ 1  2   1 2 F cos( n )]
4
1
[ p1 ]QM  1 f1 1
2
1
[ p1 ]QM   2 f 2 2
2
 i   Mi   mi
 i Effective quantum efficiency of the detector
 Mi  mi max & min transmission of the analyzers
f i Collimator efficiency (probability that appropriate emission enters apparatus
1 or 2
g Conditional probability that if emission 1 enters apparatus 1 then emission 2
enters apparatus 2
F Measure of the initial state purity
n=1 for fermions and n=2 for bosons
Experiment requirements
• Taking the following assumptions:
  1   2 , f1  f 2 ,     1   2
   / 4n
 g  [ 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 (aˆ , aˆ )]  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…
• Ensamble of quantum states 
with probability p
i
i
We can define the density operator
as:
   pi  i  i
i
Crash course in information theory…
• A quantum system whose state  is known
exactly is said to be in pure state, otherwise it
is said to be in mixed state.
• A pure state satisfies      tr( )  1
• A mixed state satisfies tr(  )  1
2
2
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:
pi ..... pn
then the Shannon Entropy associated with it is:
Crash course in information theory…
• Definition of   typical variable X obeys:
Typical sources xi are sources which are highly likely to occur.
Von Neumann entropy
Entanglement distillation and dilution
• Suppose we are supplied not with one copy of
a state  , but with a large number of it.
Entanglement distillation is how many
copies of a pure state  we can
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
• Suppose an entangled state  has a Schmidt
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.