Transcript Quantum nonlocality

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Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
1
The parts and the
whole:
a. Collapse Theories
b. Identical Constituents
GianCarlo Ghirardi
Dept. Phys. Univ. of Trieste
The ICTP, Trieste
The INFN, Trieste, Italy
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
2
General Remarks.
The Q. M. view ⇒ The Universe as an Unbroken Whole
The reason being that Q – entanglement, in general, does not consent the
characterization of the constituents of a composite Q-system by forbidding the
attribution of objective properties to the constituents themselves.
Before going on, let me recall that, even for non-composite systems, or for
isolated systems considered as a whole, the non-abelian structure of the set of
their observable quantites allows, in general, to make only nonepistemic
probabilistic predictions concerning the outcomes of prospective measurements.
However, let me also remark that, for physical systems associated to a pure state,
there are ( -many) complete sets of commuting observables such that the theory
attaches probability 1 to precisely one of the collections of the possible outcomes.
It is then quite natural to follow EPR, as we will do, in claiming that an individual
system (a part of the whole) objectively possesses a property (an epr) when the
probability of getting the corresponding outcome in a measurement equals 1.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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•When one takes such a perspective one might state that the lesson that QM
has taught us is that one cannot attribute "too many properties" even to an
isolated system. Some properties are actual and some have the ontological
status of potentialities.
•As we all know, the situation changes radically in the case of composite
systems, just because it may become impossible to attribute any property to its
constituents = parts.
•To briefly discuss this case let me start by considering the case of a bipartite
system S=S1+S2 whose parts are distinguishable.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Within Q.M., the most complete characterization which is in principle possible for
a system is the assignement of its statevector
Note: one can then state that system
S1
possesses the property associated to the
eigenvalue 1 of the projector. E.g., if S1 is
one of two atoms: it is in its ground state
Def.1:
Subsystem S1 of S⇔ associated to the pure density operator
is nonentangled with S2 if
(a projection operator onto a one-dimensional
linear manifold of
) such that
(&)
Theorem 1: Condition (&) implies and is implied by any of the 2
following conditions:
•The reduced statistical operator
is a 1-dim Projector of
•The pure state associated to the system is factorized:
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Let me try to present a synthetic picture of the case of
entangled states.
Theorem:
A necessary and sufficient condition in order that a projection operator
onto
the linear closed manifold
of
satisfies the two following conditions:
(@)
i).
ii).
such that (@) holds for
is that the range of the reduced statistical operator coincides with
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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This approach allows to present a clear picture of the situation making refrence to
the Range of the reduced statistical operator:
a one dimensional linear submanifold of
In this case the statevector is factorized, the systems S1 and S2 are nonentangled and
each of them has a complete set of properties
In this case system S1 is partially entangled with S2 , and it possesses unsharp properties.
Actually, if we consider any operator W(1) such that a subset of its eigenstates (e.g. those
associated to a subinterval [wk , wj] of its spectrum) span
, one can claim that the
variable W(1) has a value lying in the considered interval.
In this case system S1 is totally entangled with S2 : in particular, there is no
variable of this subsystem for which one can claim that its value belongs to a
proper subset of its spectrum. Think, e.g. of the energy, you are not allowed to
claim that the energy of the systems lies, let us say, between 1MeV and 1 GeV
or similar. Moreover, contrary to the case of a single system or of a system as a
whole, this holds for ALL CONCEIVABLE VARIABLES!
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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A simple example: the system e- - e+ , in the state
the reduced statistical operator is then:
If c i ≠ 0, for all i-s and {|fi(1)>} is a c.o.s. then
The situation becomes even more embarrassing in the case of a maximally
entangled state of a system whose Hilbert space is finite (N) dimensional. In such
a case we have :
so that the probability of getting any outcome corresponding to any
nondegenerate eigenvalue of any conceivable onservable has the same
probability (1/N) of being obtained. The subsystem really “has no properties”.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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The paradigmatic case of an embarrassing whole: the macroobjectification or measurement problem of Q.M.
The sketchy ideal von Neumann measurement scheme for
S=smicro+App
1. Eigenvalues and eigenvectors for smicro:
2. Microstates are “measurable”:
3. The macrostates
conscious observer,
correspond to mutually exclusive perceptions of the
factorized
entangled
4. Equation 2 implies:
5. The microsystem and apparatus are entangled ⇒ they have no individual
properties. In particular the apparatus cannot be claimed to possess the
macroproperties which are associated to our definite perceptions.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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One can significantly summarize the macro-objectification problem by making
reference to the quite illuminating sentence by Bell:
Nobody knows what quantum mechanics says exactly
about any situation, for nobody knows where the
boundary really is between wavy quantum systems and
the world of particular events.
J.S. Bell
For a recent quite general and critical analysis of the measurement problem see: A. Bassi
and G.C. Ghirardi: Phys. Lett. A, 275 (2000).
Innumerable proposals have been put forward to overcome
this problem. Each has its pros and cons. I will not discuss
them here, I will simply make reference to “collapse
theories”.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Collapse theories
The central idea is to modify the linear and deterministic evolution implied by
Schrödinger’s equation by adding nonlinear and stochastic terms to it, the aim
being the one of “solving” the measurement problem.
As it is obvious, and as it has been stressed by many scientists (Einstein, Bohm,
Feynman) the situations characterizing macro-objects correspond to perceptually
different locations of (some) of their macroscopic parts (in actual laboratory
experiments, typically the "pointer").
With these premises we can pass to discuss the spontaneous dynamical reduction
(collapse) models, making explicit reference to the so called GRW theory. It is
based on three axioms.
G.C. Ghirardi, A. Rimini and T. Weber, Phys. Rev. D, 36, 3287 (1987).
Leiden, Lorentz Center, march 2010
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1. States. A Hilbert space
is associated to any physical system and the state
of the system is represented by a (normalized) vector
in
2. Dynamics. The evolution of the system obeys Schrödinger’s
equation.
Moreover, at random times, with a Poissonian distribution with mean frequency l,
each particle of any system is subjected to a spontaneous localization process of
the form :
the probability density for a collapse at x beeing
3. Ontology. Let
be the wavefunction in configuration space.
Then
Note: localizations occur with
higher probability where there
would be an higher probability of
finding the particle in a standard
measurement process
is assumed to describe the density of mass distribution of the system in threedimensional space as a function of time.
G.C. Ghirardi, R, Grassi, F. Benatti, Found. Phys 25, 5 (1995).
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Localization of a
microscopic system
The fundamental trigger process in the
case of a macroscopic almost rigid body
SIMPLIFIED
VISUALIZATION OF
THE LOCALIZATION
PROCESSES FOR
MICRO AND MACRO
OBJECTS
For simplicity I will deal with the pointer as
if it would be a point like object.
Accordingly, I wil identify its position with its
c.o.m. position and I will disregard its actual
spatial extension.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Note:
1.A universal dynamical equation,
2.No mention of measurements, observers and so on,
3.Macrosystems become extremely well localized (for 1g, c.o.m spread sq≃10-12
cm)
The basic ideas (an oversimplified version):
1.The standard Q-dynamics leads to definite “different positions” of the pointer
(different mass densities) according to the specific eigenstates triggering the
apparatus,
2.The experiment must be calibrated (establishing the correspondence),
3.Our perceptions correspond to definite positions (definite mass density
distribution).
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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The dynamical emergence of the
properties of the parts
of the
Unbroken Universe
Suppose a spontaneous localization
occurs at this point
Then
Alternatively
Thus, we end up, with the correct
quantum probabilities, with a state :
which is “practically” an extremely
well localized and non entangled
(system-apparatus) state: the pointer
has a precise objective location.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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We have used the expression “practically” for two main reasons:
1.The wavefunction of the c.o.m has “tails” extending to infinity,
2. Multiplying the wavefunction corresponding to an outcome times the
localization function centered at a point corresponding to a different outcome
does not suppresses completely the associated state.
However:
a.Difficulty 1 is not characteristic of the GRW theory, it also affects the standard
theory, just because no wavefunction can have compact support in space. The
tail prolem is not the measurement problem.
b. The specific dynamics of the theory, in the case of a macroscopic object leads
immediately to an incredibly well localized state (10-12cm). Correspondingly,
the norm of the other terms which “survive” and yield entaglement turns out to
be absolutely negligible.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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c. The so called “tail problems” has seen a lively debate because it gives rise to
the counting anomaly. It seems that, at the end, an agreement concerning the
fact that it is not a problem emerged. I will limit myself to stress that, for a pointlike macro-pointer of 1 gr and for a separation d of the nothches of the ruler of a
small fraction of a centimeter, the total mass outside the interval [ar-(d/4),
ar+(d/4)] (ar being the “position on the scale” corresponding to the outcome)
amounts to
P:Lewis, Br. J. Phil. Sci. 48, 313 (1997)
R. Clifton and B. Monton, Br.J.Phil.Sci. 50, 697 (1999), 51, 155 (1999)
G.C. Ghirardi and A. Bassi, Br.J.Phil.Sci. 50, 49 (1999), ibid. 719 (1999), 52,125
(2001)
P. Lewis, Br. J. Phil. Sci. 54, 165 (2003)
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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I have shortly outlined how the Dynamical Reduction
Program “solves” the measurement problem of quantum
mechanics on the basis of a universal dynamical principle
and of absolutely natural assumptions concerning the
functioning of our experimental apparatuses and our
perceptions of the location of macroscopic systems.
What is important from the perspective of this meeting is that
the new dynamics forbids the superpositions of states of
macroscopic systems corresponding to different locations in
space. Correspondingly, macro objects are non-entagled with
each other and they emerge clearly as actual parts of the
Unbroken Quantum World, just because they must always be
in extremely well defined positions.
This is the relevant part of the story for what concerns us here.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Before concluding this part there is something more to say.
We have used, in our formulation only the universal dynamical principle, the
calibration of the experiment and the assumed correspondence of our
perceptions to the definite positions of the pointers. But much more is implied.
We have proved that, by taking into account all our assumptions and the
implications of the formalism and by resorting to the Riesz representation
theorem the probabilities concerning the various possible outcomes implied by
the formalism can be expressed as the average values over the initial state of
the effects associated to a Positive Operator Valued Measure (POVM) on the Hilbert
space of the measured system.
Moreover if we require reproducibility of the experiments (i.e. that repeating a
measurement one gets the same outcome he has just obtained) then the POVM
reduces to a Projection Valued Measure (PVM).
Concluding: our general physical approach leads to a natural deduction of the
quantum rules in their most general and axiomatic form
A. Bassi, G.C. Ghirardi, D.G.M. Salvetti, J. Phys. A: Math. Theor., 40, 13755 (2007).
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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The parts in the case of
composite systems
involving identical
constituents
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Preliminary remarks.
•The so called “ principle of individuality” of physical systems has a long history in
philosophy,
•Leibniz: “there are never in nature two exactly similar entities in which one cannot find
an internal difference” We all know that two electrons exhibit no internal
differences.
•In classical physics one can try to “individuate” absolutely identical objects by
considering their locations in space and time.
•This is not possible in quantum physics since in it trajectories are
meaningless.This has led some philosophers to claim that quantum particles
cannot be considered as individuals in any traditional meaning of such a term.
We will not enter into this debate. We will never be interested in questions like
“presently, is the electron which we have labeled as 1 initially at a given position or is its
spin up in some direction?”
We believe that the correct question must sound like: on the basis of our
knowledge of the state of a composite system can we consider legitimate to claim
that “there is an electron (we do not care which one) in a certain region and it has its spin
pointing up along a given direction”?
Some of the many refrences: P. Teller, Phil. Sci., 50, 309 (1983), M. Redhead and P. Teller, BR. J. Phil.
Sci., 43,201 (1992), M.L. Dalla Chiara and G. Toraldo di Francia, Bridging the gap, Kluwer (1993), N.
Huggett, The Monist, 80, 118 (1997), etc.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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It is precisely due to the just mentioned situation that the problem of identifying
whether two subsystems of a composite system are entangled or not has been
a serious source of difficulties in the case in which the constituents are
identical.
All problems derive from not having taken properly into account the real
physical meaning and the subtle implications of entanglement itself and by a
naive transposition to the case of identical constituents of some formal aspects
of the case of distinguishable constituents.
Two paradigmatic examples:
I would like to remind you that there is a
universal correlation of the EPR type which
we do not have to cleverly set up …, it is
simply the total antisymmetrization of a
many fermion state, which does correlate the
electrons of my body with those of any
inhabitant of the Andromeda Galaxy.
J.M.
One may not draw conclusions about
entanglement in configuration space by
looking at the state in Fock space.
Lévy-
D.M. Greenberger,
M.A. Horne,
A. Zeilinger.
Leblond
The argument which follows is based, essentially, on the following paper,
1.
G.C. Ghirardi, L. Marinatto and T. Weber, J. Stat. Phys., 108, 49 (2002),
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Two identical constituents
• If one takes factorizability as a criterion for non-entanglement one is
mistakenly lead to claim that (exception made for the case of two bosons in
the same state) non-entagled states cannot exist
•To tackle the problem in the correct way one has to stick to the idea that the
physically most interesting and fundamental feature of non-entangled states,
in the case of distinguishable particles, is that both constituents possess
objectively a complete set of properties.
•We have taken precisely this attitude also with reference to the case of
systems with identical constituents. Starting with the appropriate definitions
we have derived two theorems. Let us discuss them.
Leiden, Lorentz Center, march 2010
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Definition 1: In the case of a composite quantum system of two identical
constituents, they are non-entangled when both constituents possess a complete
set of properties.
Definition 2: One constituent of a system of two identical particles in the state
|Y(1,2)> possesses a complete set of properties iff there exists a one-dimensional
single particle projection operator P(i), i=1,2, such that :
Y(1, 2) EP (1, 2) Y(1, 2)  1
EP (1, 2)  P (1)  [I (2)  P (2) ][I (1)  P (1) ]  P (2)  P (1)  P (2)
This guarantees that at least one of the particles possesses the complete set of
properties associated to P.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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This physically meaningful criterion implies the two following
theorems:
Theorem 1: The identical fermions S1 and S2 of a composite
system S=S1+S2 described by the pure normalized state
|Y(1,2)> are non-entangled iff |Y(1,2)> is obtained by
antisymmetrizing a factorized state.
Theorem 2: The identical bosons S1 and S2 of a composite
system S=S1+S2 described by the pure normalized state
|Y(1,2)> are non-entangled iff either |Y(1,2)> is obtained by
symmetrizing a factorized product of two orthogonal states
or it is the product of the same state for the two particles.
Leiden, Lorentz Center, march 2010
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Summarizing:
Given that:
, we have
Non-entangled fermions:
Y(1, 2) 
1
 (1)   (2)   (1)   (2)

2

1
 (1)   (2)   (1)   (2)

2

Non-entangled bosons:

Either
Y(1, 2) 
Or

Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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Non-entanglement, correlations and all that.
The fact that our criteria
are appropriate to characterize
nonentangled
states is strengthened and
clarified by the consideration of the physical
implications of the form of
the state-vector.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
Actually, in the case of
the non-factorized states
we have just identified
as non-entangled, it is
not possible to take
advantage of the form of
the
statevector
to
perform
teleportation
processes or to violate
Bell’s inequality.
27
Two elementary examples
Consider the state (obtained by antisymmetrizing a factorized state):
Y(1, 2) 
1
z 1 R 1  z 2 L 2  z 1 L 1  z 2 R

2
2

|R> and |L> having compact disjoint spatial supports. For it one can state that
“there is one particle at R with spin up and one at L with spin down”.
The spin correlation function is:
E(a,b )  z s  a z  z s  b z 
and it does not imply nonlocal correlations of the Bell’s type.
It is useful
 to compare this case with a true case of the EPR-Bohm type:
1
(1, 2) 
z  1 z  2  z  1 z  2   R 1 L 2  L 1 R

2
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
2
.
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The case of many (N) indistinguishable particles.
The problem of entanglement becomes much more complicated in this situation,
but it deserves at least a quick discussion.
Definition: Given a quantum system of N identical particles described by the
properly (anti)symmetrized pure state
we wil state that it contains two
nonentangled subgroups of particles of cardinality M and K=N-M, when both
subgroups possess a complete set of properties
It would take too much time to go through the complicated mathematics which is
necessary to deal with this problem. I will try to make clear some crucial points
which will be relevant in what follows.
To fix our ideas let us consider two states of systems with identical constituents:
belonging to the Hilbert spaces
,
which are appropriate for
the systems of M, K, identical bosons (fermions), respectively.
Leiden, Lorentz Center, march 2010
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We now introduce an important mathematical concept
We will state that two states as those just considered are “one particle orthogonal”
if:
Remarks:
•The variable which is saturated is absolutely irrelevant (due to the (anti)symmetry)
•The condition amounts to requiring that there exist a single particle basis
such that if one writes the Fourier decomposition of the two states:
the sets of indices for which
and
are disjoint.
•Note that, in the case in which the first M particles are strictly confined in a region A
and the remaning ones in a region B, and such regions have an empty intersection,
the two states are automatically one-particle orthogonal.
Leiden, Lorentz Center, march 2010
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Having made clear this point we can now formulate the basic theorems:
Theorem A: A necessary and sufficient condition in order that a state
of
the Hilbert space of N identical fermions allows the identification of two subsets
of particles which possess a complete set of properties is that
be
obtained by antisymmetrizing and normalizing the direct product of two oneparticle-orthogonal antisymmetrized states
Theorem B: A necessary and sufficient condition in order that a state
of
the Hilbert space of N identical bosons allows the identification of two subsets of
particles which possess a complete set of properties is that
be obtained
by antisymmetrizing and normalizing the direct product of two one-particleorthogonal symmetrized states
Leiden, Lorentz Center, march 2010
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Obviously, in the case of an even number of identical bosons one can also
consider the case in which the system can be split in two subsystems in the
same state:
Besides the basic reason that, when the conditions of the previous theorems are
satisfied, one can claim that "the group of M particles" and the group of the
remaining K particles have precise properties, i.e., those associated to the states
and
respectively, there are other, physically more meaningful reasons to state that,
in the considered case, the two subsystems are really "parts" of a "whole".
To discuss this point we need some further formal analysis. Let us discuss it.
Leiden, Lorentz Center, march 2010
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Let us consider a complete orthonormal set of single particle states
•Let d and d* be two disjoint subsets of the index set {j}.
•Let V(M) d , V(M)d* be the linear manifolds spanned by the normalized, antisymmetrized states
•Let V(K) d , V(K)d* be the corresponding linear manifolds for the system of K fermions
Consider an orthonormal basis
and
of V(M) d and a vector
spanning V(K)d* , two arbitrary vectors
of V(K)d* . Then
an equation which shows clearly that one can do the physics of the M particles
within their pertinent manifold by completely ignoring the other identical particles.
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A precise, physically meaningful example.
Suppose one has a Helium atom here (at his origin) and a Lithium atom there (at
a macroscopic distance D) and let us, for simplicity, concentrate our attention to
the electrons which enter into the game. We have then a state:
Now we are in trouble: the two states appearing in this equation are not oneparticle orthogonal. However, they are not so because of the overlap integrals
which involve the tails of the electronic wavefunctions. For a distance of the
order of 1 cm the relevant integrals turn out to be of the order of
Considering the two states as one-particle ortogonal implies an error of the same
order of, e.g., disregarding the Helium in evaluating the Lithium energy levels
Lithium
D
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Accordingly, our claim: “there is a Helium atom near the origin" is, strictly speaking,
not fully correct; but it has an extremely high degree of validity.
To fully allow to appreciate the relevance of the above remarks let us consider
also a state like:
which can be produced and would not make legitimate, in any way whatsoever,
to make claims about what is here being a Helium rather a Lithium atom.
Let me conclude by visually summarizing all what I have said.
Leiden, Lorentz Center, march 2010
GianCarlo Ghirardi
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A particle, with
either spin up or
spin down along
the z-axis goes
through a SternGerlach apparatus
and it triggers the
firing of the engine
of a shuttle.
Andromeda
Andromeda
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1. Each one of the two
superposed final states
is
(essentially)
a
nonentangled state (in
spite of Lévy-Leblond’s
opinion).
2. The final linear
superposition exhibits a
puzzling entanglement
concerning mine and
hers location
3. Any spontaneous
localization involving one
of the microconstituents
of mine/hers shuttle or
body, leads (essentially)
to a non entangled state
in which the parts (me
and she) regain their
individuality, and are
taken
out
of
the
“Unbroken Whole”.
Leiden, Lorentz Center, march 2010
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Greenberger, Horne and Zeilinger:
The state:
is manifestly non-entagled, while its position representation
(obviously the same state) :
is, in their opinion, manifestly entangled.
This shows that our approach is the corret one. For us the
second state is obviously nonentangled. Being entangled
or not is a property of the state, not of the way you write it.
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It is obvious that also in the case of a single
particle in an harmonic potential in a state:
there is no proper subset of the energy
spectrum to which you can claim that the
energy belongs.
But in this case there are for sure selfadjoint operators such that
and one can claim that the property G=gs is
objectively possessed by the particle.
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The so called measurement problem does not arise from the fact that one has a
function (e.g. of the c.o.m. position) which is not of compact support (everybody
knows that such functions cannot persist for more than one instant)
It consists in the fact that the theory implies, if it assumed to govern all natural
processes, that superpositions of differently located macroscopic systems are
possible, and actually unavoidable.
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