Download Quantum Mechanics, Locality and Realism

Quantum Mechanics, Locality and Realism
(an amateur perspective)
Marco G. Giammarchi – Infn Milano
Quantum Mechanics as an abstract theory
E. Schroedinger
The Bohr-Einstein debate about Quantum Mechanics
The EPR statement
Bell’s Inequality: Locality and Realism
A. Einstein
Classical-Quantum boundary
Recent developments
J. S. Bell
A. Aspect
W. Heisenberg
LNGS - 28 June 2012
1
Quantum Mechanics as an abstract theory
• Quantum era was opened in 1900 (Planck’s Law)
• Problems with the classical theory of matter (blackbody radiation, specific heat of solids,
stability of atomic systems)
• Early Quantum Physics (Bohr model) as a quantized classical theory
Still correspondence between the objects of the theory
and the physical concepts
• Quantum Mechanics (end of 20’s) is an abstract theory
• Amazingly succesful (currently, clearly “the best” theory. Actually a meta-theory)
• Relativistic version could be produced (that predicted antimatter, actually
discovered just after the prediction).
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2
Postulates of Quantum Mechanics
Physical System  Hilbert Space with an inner product
Physical States  Vectors in the (separable) Hilbert Space
Hilbert Space of composite systems  Tensor Product of Hilbert Spaces of the subsystems
Physical quantities  Self-adjont operators in the Hilbert Space
Physical Symmetries  (Anti)Unitary operators acting on Hilbert vectors (states)
No trivial correspondence between the objects of the
theory and the physical concepts
Some counter-intuitive characteristics:
An intrinsically probabilistic theory
The Heisenberg Uncertainty Principle
LNGS - 28 June 2012
3
Five Great Problems in Theoretical Physics
(According to Lee Smolin)
The problem of quantum gravity: Combine general relativity and quantum theory
into a single theory that can claim to be the complete theory of nature.
The foundational problems of quantum mechanics: Resolve the problems in the
foundations of quantum mechanics, either by making sense of the theory as it
stands or by inventing a new theory that does make sense.
The unification of particles and forces: Determine whether or not the various
particles and forces can be unified in a theory that explains them all as
manifestations of a single, fundamental entity.
The tuning problem: Explain how the values of the free constants in the standard
model of particle physics are chosen in nature.
The problem of cosmological mysteries: Explain dark matter and dark energy.
Or, if they don't exist, determine how and why gravity is modified on large scales.
More generally, explain why the constants of the standard model of cosmology,
including the dark energy, have the values they do.
LNGS - 28 June 2012
4
The Bohr-Einstein debate about Quantum Mechanics
A double shock to Albert Einstein:
• Introduction of matrix formulation of Quantum Mechanics (W. Heisenberg), with no
spacetime elements
• Introduction of the probabilistic interpretation (M. Born)
A Einstein: “God does not play dice”
This was ok for Niels Bohr who strengthened the role of the (classical) observer:
• Principle of Complementarity: Objects governed by quantum mechanics, when measured,
give results that depend inherently upon the type of measuring device used
Einstein criticism: first phase:
-Gedanken experiments to show that Uncertainty Principles can be violated
(measuring device had an absolute role)
- Bohr’s anwers always included treatment of the measuring device
LNGS - 28 June 2012
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About coherent superpositions
Physical meaning of a
"coherent superposition"?
Any superposition of states
is a possible state
Young's hole experiment
i
  a e b
Interference fringes
P  Pa  Pb
Addition of the probability
amplitudes of each path
Signature of a coherent superposition = interference fringes
Quantum phase of the superposition = phase of the fringes
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Einstein versus the Uncertainty Principle :
We picture the double slit as a coherent
superposition of amplitudes on screen 2.
Any experiment designed to evidence
the corpuscolar part of the process
(detection on b,c of the passing particle)
would destroy the interference pattern
No “welcher weg” information.
Einstein: when the particle goes through S1, it will receive an impulse along x
- Mesure the recoil along x of the S1 screen
- Use momentum conservation
-Then the Vx of the particle is known
The momentum information can be used to know which path the particle has travelled
without disrupting the interferecence pattern!
Uncertainty Principle is violated (positions and velocity can be known,
and the resulting interference pattern comes from a statistical mixture
(since we know, event by event, the path chosen by the particle)
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Bohr response includes the measuring
device as a quantum object:
An extremely precise determination of the
velocity of the screen S1 along x, involves
some uncertainty on the x position of the
screen itself.
The uncertainty on the x position of S1 will change the path
difference between the two paths a-b-d and a-c-d therefore
washing away the interference patter on the screen F.
During the Einstein-Bohr debate, Einstein considered physical quantities (and
their interrelations) as existent without necessarily referring to the measurement
process (REALISTIC approach).
Bohr always considered the result of an experiment, including the role played by
the measuring device (POSITIVISTIC approach)
LNGS - 28 June 2012
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The EPR statement
The the famous EPR (Einstein, Podolsky, Rosen) 1935 paper it is shown that a
consequence of Quantum Mechanics is the existence of long-distance correlations
(Entanglement).
According to Einstein this was the proof that Quantum Mechanics is (probabilistic because)
incomplete. A complete theory would then contain elements that could explain the
entanglement in a causal (deterministic) way.
Einstein’s ideas (and personality) greatly
influenced David Bohm, who built up a
non-local theory based on the concept of
pilot waves (Bohmian Mechanics)
Real path
Bohm, David (1952). "A suggested Interpretation of
the Quantum Theory in Terms of Hidden Variables, I
and II, Physical Review 85.
The particle will go through one single
well defined slit but the (instantaneous,
superluminal) pilot wave will “inform” the
particle of the existence of the second slit.
Pilot waves
“Spooky action at a distance”
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The general idea of the EPR statement:
If one considers the dissociation of a molecule :
A
B
Suppose I measure σ(x;A)  by momentum conservation σ(x;B) is known
Suppose I measure σ(y;A)  by momentum conservation σ(y;B) is known
Suppose I measure σ(z;A)  by momentum conservation σ(z;B) is known
σ(x,y,z;B) (all the components) is an element of reality
(which can be measured without perturbing the system)
But Quantum Mechanics allows to specify only a component (and the modulus
square) of the B spin  Quantum Mechanis is an incomplete theory !
LNGS - 28 June 2012
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Bell’s Inequality: Locality and Realism
After the EPR debate there was still hope that a local realistic theory (based
perhaps on hidden variables) could be the ultimate theory of the micro-world !
A theory with hidden variables perhaps could be local (non-Bohmian, no spooky
action at a distance) and deterministic
A general criteron to
confront Quantum
Mechanics with a local
realistic theory
1964, John Stewart Bell
"On the Einstein Podolsky Rosen paradox"
Bell demonstrated that local realism yields predictions that are in contradictions
with Quantum Mechanics (and measurements)
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Bell’s Inequality (minimal version)
A set of elements
Three dichotomic variables
a, a , b, b , c, c
N (a,b )  N (a,b ,c)  N (a,b ,c )
N (b,c )  N (a,b,c )  N (a ,b,c )
Summing up:
(trivially true)
N (a,b )  N (a,b ,c )
N (b,c )  N (a,b,c )
N (a,b )  N (b,c )  N (a,b ,c )  N (a,b,c )
N (a,b )  N (b,c )  N (a,c )
In the macroworld that seems obvious, but what if a is + polarization of a photon
along axis a and a-bar is the negative polarization along the same axis?
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The study of correlated photons: same-angle polarimetry
(J. Baggot – The meaning of Quantum Theory)
Orientation a
PA1
A
-- h
+ v
Initial state vector:
source
 
Measurement Eigenstates:
Orientation a
B

1
 LA  LB   RA  RB
2
PA1
PA2
h -v +

PA2
(symmetric when A ↔ B)
+
     vA  vB
+
-
     vA  hB
-
+
     hA  vB
-
-
     hA  hB
+
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Now, express the state vector (in the base of circular polarizazion state) in the
base of measurement (linear polarization) eigenstates:
                             
Probabilities of results:
P  (a, a)     
2

1
2
1
      
2
P  (a, a)     
2


1
2
Using the conversion between linear and circular polarization eigenstates:
v
h
v’
h’
L
R
v
1
0
cos2ф
sin2ф
1/2
1/2
h
0
1
sin2ф
cos2ф
1/2
1/2
v’
cos2ф
sin2ф
1
0
1/2
1/2
h’
sin2ф
cos2ф
0
1
1/2
1/2
L
1/2
1/2
1/2
1/2
1
0
R
1/2
1/2
1/2
1/2
0
1
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^
Expectation values
M1 (a) 
^
M1 (a) 
A
v
R 
A
v
A
h
R 
A
v
A
v
A
h
^
M 2 (a)  vB  RvB  vB
^
M 2 (a)  hB  RhB  hB
^
^
M1 (a) M2 (a) 
So, doing the joint measurement means:
^
^
^
1 ^
1
M1 (a) M 2 (a)  
( M1 (a) M 2 (a)     M1 (a) M 2 (a)    ) 
( RvA RvB     RhA RhB    )
2
2
^
^
^
^
^
M1 (a) M 2 (a) 
A
v

B
v
R R 
A
v
B
v
A
v

B
v
^
M1 (a) M 2 (a)  hA  hB  RhA RhB  hA  hB
^
The expectation value of the measurement:
^
^
E (a, a)   M1 (a) M 2 (a)  
Since
Rv 1, Rh   1
^
 M1 (a) M 2 (a) 
1
1
(       ) ( RvA RvB     RhA RhB   )  ( RvA RvB  RhA RhB )
2
2
^
^
E(a, a)   M1 (a) M2 (a)   1
A fully correlated measurement
LNGS - 28 June 2012
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When the polarizers have different angles:
Orientation a
PA1
-- h
+ v
A
source
Orientation b
B
h’ -v’ +
Rotated with respect to PA1
v’
v
h’
b-a
h
v
PA2
Initial state vector:
 
Measurement Eigenstates:

1
 LA  LB   RA  RB
2
PA1
PA2

(symmetric when A ↔ B)
+
+
 '    vA  vB'
+
-
 '    vA  hB'
-
+
 '    hA  vB'
-
-
 '    hA  hB'
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Now, express the state vector (in the base of circular polarizazion state) in the
base of measurement (linear polarization) eigenstates:
   '  '   '  '   '  '   '  ' 
The coefficients are:
 '   

'

1
cos(b  a)
2
1
 
sin( b  a)
2
 '    
 '   
1
cos(b  a)
2
1
sin( b  a)
2
Therefore the decomposition of the wave function:
 

1
 '  cos(b  a)   '  sin( b  a)   '  sin( b  a)   '  cos(b  a)
2
LNGS - 28 June 2012

17
Probabilities for the joint results:
1
P  (a,b)     cos 2 (b  a)
2
1
P  (a,b)     sin 2 (b  a)
2
1
P (a,b)     sin 2 (b  a)
2
1
P (a,b)     cos 2 (b  a)
2
'

'

2
2
2
'

'

2
Expectation of the a,b correlation:
^
^
E (a,b)   M1 (a ) M 2 (b)  
P (a,b) RvA RvB'  P ( a,b) RvA RhB'  P (a,b) RhA RvB'  P ( a,b) RhA RhB' 
 P (a, b)  P (a, b)  P (a, b)  P (a, b)  cos 2 (b  a)  sin 2 (b  a)  cos 2(b  a)
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E (a, b)  cos 2 (b  a)
Quantum mechanical correlation !
The predictions of Quantum Mechanics are based on the properties of a twoparticle state vecotr which, before collapsing into one of the measurement
eigenstates is “delocalized” over the whole experimental arrangement.
The two particles are in effect, always “in contact” prior to measurement and can
therefore exhibit a degree of correlation that is impossible for two Einstein
separable particles
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Quantum correlations and Bell’s Inequality
P  (a, b)  
'

P  (a, b)  
'

1
  cos 2 (b  a)
2
2
1
  sin 2 (b  a)
2
2
P  (a, b)  
P  (a, b)  
'

'

2


2
1
 sin 2 (b  a)
2
1
 cos 2 (b  a)
2
These quantum correlation violate Bell’s Inequality. Let us in fact make the
3 following set of measurements:
Experiment
PA1
orientation
PA2
orientation
Difference
1
a = 00
b = 22.50
b - a = 22.50
2
b = 22.50
c = 450
c – b = 22.50
3
a = 00
c = 450
c – a = 450
N (a,b )  N (b,c )  N (a,c )
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N (a,b )  N (b,c )  N (a,c )
 P (a,b)  
 P (a,b)  
 P (a,b)  
'


2
'


2
'


2
1
1
 sin 2 (c  a)  sin 2 (450 )
2
2
1
1
 sin 2 (c  b)  sin 2 (22.50 )
2
2
1
1
 sin 2 (b  a)  sin 2 (22.50 )
2
2
0.1464  0.2500
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What does it mean?
Violation of Bell’s Inequality has been demonstrated in thousands of
experiments (the first being the Aspect 1982 experiment)
Between the assumptions of Bell’s Inequality there is the idea that physical
quantities in the microwolrd exist before being measured (realism).
This disagrees with the experiments.
Quantum Mechanics (Copenhagen interpretation): we cannot talk about “real”
quantities. We can only talk about quantities being measured.
The observer is part of the physical system and there is no sharp subject/object
separation.
A little epistemological price to pay in order to use the most
powerful physical theory ever invented (actually a meta-theory)
Alternative (still alive): Bohmian Mechanics (with non-local pilot waves)
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22
Since old things are always new:
Copenhagen interpretation (Bohr formulation)
A quantum phenomenon comprises both the “observed” quantum system and
the classical measuring apparata.
It does not make any sense to speak about the quantum system in itself
without specifying the measuring process
(It is senseless to assign simultaneously complimentary attributes – like x,p –
since they cannot be measured at the same time)
The wave function is a representation of the quantum system
An experimental prediction that surpasses the limitations of the theory is not
possible in principle
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Entanglement
Coherent superposition for a bipartite system
"Entangled state" = non factorisable state
No system is in a definite state
Quantum correlations
Violation of Bell's inequalities
Correlations in all the basis
V
V
 V
(2)
(1)
H
S
H
| epr
1

(| H V   | V H )
2
 V
H
The two photons
form an EPR-pair
H
Anticorrelation

Classical-Quantum Boundary
How comes that a quantum system generates at the macroscopic level (on
statistical ensembles) the classical probabilistic additive behaviour ?
Decoherence
A loss of coherence of the phase angles between the components of a
system in quantum superposition
Decoherence has the appearance of a wavefunction collapse
It occurs when a system (irreversibly) interacts with the environment
It is the candidate theory to determine how classical behavior emerges from
a quantum starting point
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Let us start with an entangled state, a
system and a detector
 c    d    d
…and build up the density matrix
   c  c     d  d    *   d  d  
2
c
     d d     d d
2
*
A non unitary evolution process
that will cancel off-diagonal
(phase dependent) terms

r
Decoherence
    d d     d d
2
2
can be interpreted as classical coefficients
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The Schrödinger-cat paradox
Entanglement of a microscopic system with a macroscopic one
A two-level atom and a cat in a box
| e | alive 
| g | dead 
Total correlation
The cat "measures" the atomic state
Linear evolution
The system form an EPR pair
Quantum correlations
Atom projected on |e>+|g>
Cat projected on |dead>+|alive>
Macroscopic state superposition
Decoherence
A macroscopic object interacts with its environment and
gets entangled with it
Coherent superposition (dead> and |alive>)
  alive
cat
e
nucleus
 dead
cat
g
nucleus
Decoherence
  alive
cat
alive  e
nucleus
e  dead
Statistical mixture (|dead> or |alive>)
Classical correlations in the « natural » basis
No interference between macroscopic states
cat
dead  g
nucleus
g
Classical Correlations
 coin 
1
1
H1 H1 T2 T2  T1 T1 H 2 H 2
2
2
Quantum Correlations
1
   A1 B2  B1 A2
2
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
29
The quantum-classical boundary
Microscopic object
-3 parts
-2 time scales
Environment
Tdecoh
Tint
Mesoscopic object
Tint  Tdecoh
-Entanglement
-« Schrödinger cat » states
-Quantum behavior
Quantum world
Tint  Tdecoh
-Continuous monitoring
of the environment
-No entanglement
- ’Classical’ behavior
Classical world
Continuous parameter to explore the quantum-classical boundary?
Microscopic object (S)
Environment (E)
Tdecoh
Tint
Mesoscopic object (D)
Now let us entangle the quantum system/detector wavefunction with the
environment:
 c    d     d      c E0    d  E    d  E  
When the states of the environment corresponding to the different states of the
detector are orthogonal, the density matrix that describes the system-detector
combination is obtained by tracing over the environment degrees of freedom:
 SD  TrE     Ei   Ei
i
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A model of Decoherence
Quantum system in interaction with the environment
• Collection of harmonic oscillators ?
Environment ?
• A quantum field ?
A degree of arbitrariness here !
In a popular model a particle with position x and a potential  ( q,t )
One can demonstrate that in the high-T limit, the
evolution of the density matrix is governed by the
Master Equation:
H int   x
d
dt
d ( x, x ' )
i
  2m k BT
'  
' 2
  H ,   x  x   ' 
x x 
2
dt


 x x 


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

32
The Master Equation
d ( x, x ' )
i
  2m k BT
'  
' 2
  H ,   x  x   ' 
x x 
2
dt


 x x 




usual Hamiltonian
term
Frictional term (relaxation)
R

2
4m
Decoherence term
D
The decoherence term tends to wash away off-diagonal terms responsible for
quantum correlation of spatially separated wavepackets
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How does it work?
Let us start with a 2-gaussian wavepacket
 ( x)    ( x)    ( x)
x     ( x, x ' )  ( x)  * ( x ' )
The matrix density features peaks that are on
the x,x’ diagonal and peaks that are offdiagonal (which contains the quantum phases
information)
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34
The effect of
2m k BT
' 2

x x 
2



Is negligible for the on-diagonal terms while
for the off-diagonal term it will give a decay
rate :
2
2

1  T 
 D  R
  
2
2mkBT x    x 
For a macroscopic object the decoherence
time is many orders of magnitude smaller than
the relaxation time. E.g. for m=1 g, T=300 K,
separation of 1 cm :
D
 10 40
R
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Recent Developments: long distance correlations
Long distance correlations
in quantum criptography
La Palma – Tenerife 144 km
PRL 98 (2007) 010504
Recent Developments: attosecond quantum interference
Interference in the time-energy domain: the role of slits is being played by
windows in time of attosecond duration (F. Lindner et al., PRL 95 (2005) 040401).
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Recent Developments: from elementary particles to big molecules
Interference patterns in double-slit experiments with massive particles (de Broglie
waves interference)
In contrast to classical physics quantum interference can be observed when single
particle arrive at the detector one-by-one
Matter waves interference observed for :
• Electrons, e.g. C. Johnsson, Z. Phys. 161 (1961) 454
• Neutrons, e.g. A. Zeilinger et al., Rev. Mod. Phys. 60 (1988) 1067.
• Atoms, e.g. Phys. Rev. Lett. 61 (1988) 1580, Phys. Rev. Lett. 66 (1991) 2689.
• Molecules, e.g. Science 266 (1994) 1345, Nature 41 (1999) 682, Science 331
(2011) 892.
Nanofabrication and nanoimaging techniques allowed to study
quantum interference patterns with molecules up to ≈ 1000 AMU
e.g. Nature Nanotechnology (2012) doi10.1038/nnano.2012.34
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Another Quantum-Classical boundary: particle size
Particle “size”
In the normal situation for a particle
POINTLIKE
Particle
“wavelength”


2mT


2mT
If the particle is a macromolecule
lP
New regime being explored where
lP  
C60 , C70 , tetraphenylporhyrin
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Fullerene experiment:
lP  
 400  1 nm
C60 , C70 , C60 F48
2.8 pm
1632 AMU
Quantum interferometry experimental study of decoherence
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Conclusion (or beginning?)
Quantum Mechanics : best theory ever in terms of numerical predictions
Copenhagen Interpretation : (among other things) no sharp separation between
observer and quantum system
Entanglement at long distance
Entanglement and classical size
Alternative approach (Bohmian): non local. Still viable.
Classical-Quantum boundary: decoherence as the candidate theory
How can a theory that can account with precision for almost everything we can
measure still be deemed lacking?
The only “failure” of quantum theory is its inability to provide a natural framework
that can accomodate our prejudices about the workings of the universe (W.H.
Zurek). Or the workings of “us and the universe”.
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