Quantum Mechanics, Locality and Realism (an amateur perspective) Marco G. Giammarchi

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Quantum Mechanics, Locality and Realism(an
amateur perspective)Marco G. Giammarchi Infn
Milano
Quantum Mechanics as an abstract theory The
Bohr-Einstein debate about Quantum Mechanics
The EPR statement Bells Inequality Locality
and Realism Classical-Quantum boundary Recent
developments
E. Schroedinger
A. Einstein
J. S. Bell
A. Aspect
W. Heisenberg
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Quantum Mechanics as an abstract theory

• Quantum era was opened in 1900 (Plancks 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

• Quantum Mechanics (end of 20s) 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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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)
Some counter-intuitive characteristics An
intrinsically probabilistic theory The
Heisenberg Uncertainty Principle
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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.
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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)
• Bohrs anwers always included treatment of the
  measuring device

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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
Interference fringes
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.
Einsteins 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.
Pilot waves
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.
Spooky action at a distance
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The general idea of the EPR statement
If one considers the dissociation of a molecule
B
A
Suppose I measure s(xA) ? by momentum
conservation s(xB) is known
Suppose I measure s(yA) ? by momentum
conservation s(yB) is known
Suppose I measure s(zA) ? by momentum
conservation s(zB) is known
s(x,y,zB) (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 !
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Bells 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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Bells Inequality (minimal version)
A set of elements Three dichotomic variables
(trivially true)
Summing up
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)
(symmetric when A ? B)
Initial state vector
Measurement Eigenstates
PA1
PA2

• -
• - -

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Now, express the state vector (in the base of
circular polarizazion state) in the base of
measurement (linear polarization) eigenstates
Using the conversion between linear and circular
polarization eigenstates
v h v h L R
v 1 0 cos2? sin2? 1/2 1/2
h 0 1 sin2? cos2? 1/2 1/2
v cos2? sin2? 1 0 1/2 1/2
h sin2? cos2? 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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So, doing the joint measurement means
The expectation value of the measurement
Since
A fully correlated measurement
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When the polarizers have different angles
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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
Therefore the decomposition of the wave function
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Probabilities for the joint results
Expectation of the a,b correlation
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Quantum mechanical correlation !
The predictions of Quantum Mechanics are based on
the properties of a two-particle 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 Bells Inequality
These quantum correlation violate Bells
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
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What does it mean?
Violation of Bells Inequality has been
demonstrated in thousands of experiments (the
first being the Aspect 1982 experiment)
Between the assumptions of Bells 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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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
The two photons form an EPR-pair
Anticorrelation
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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
and build up the density matrix
A non unitary evolution process that will cancel
off-diagonal (phase dependent) terms
Decoherence
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
Total correlation
The cat "measures" the atomic state
Linear evolution
The system form an EPR pair
Quantum correlations
Atom projected on egtggt
Cat projected on deadgtalivegt
Macroscopic state superposition
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Decoherence
A macroscopic object interacts with its
environment and gets entangled with it
Coherent superposition (deadgt and alivegt)
Decoherence
Statistical mixture (deadgt or alivegt)
Classical correlations in the  natural  basis
No interference between macroscopic states
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Classical Correlations
Quantum Correlations
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The quantum-classical boundary
-Continuous monitoring of the environment -No
entanglement -  Classical behavior
-Entanglement - Schrödinger cat
states -Quantum behavior
Classical world
Quantum world
Continuous parameter to explore the
quantum-classical boundary?
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Microscopic object (S)
Environment (E)
Mesoscopic object (D)
Now let us entangle the quantum system/detector
wavefunction with the environment
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
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A model of Decoherence
Quantum system in interaction with the environment

• Collection of harmonic oscillators ?
• A quantum field ?

Environment ?
A degree of arbitrariness here !
In a popular model a particle with position x and
a potential
One can demonstrate that in the high-T limit, the
evolution of the density matrix is governed by
the Master Equation
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The Master Equation
usual Hamiltonian term
Frictional term (relaxation)
Decoherence term
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
The matrix density features peaks that are on the
x,x diagonal and peaks that are off-diagonal
(which contains the quantum phases information)
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The effect of
Is negligible for the on-diagonal terms while for
the off-diagonal term it will give a decay rate
For a macroscopic object the decoherence time is
many orders of magnitude smaller than the
relaxation time. E.g. for m1 g, T300 K,
separation of 1 cm
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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 wavelength
Particle size
If the particle is a macromolecule
New regime being explored where
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Fullerene experiment
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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