The parts and the whole: a. Collapse Theories b. Identical Constituents

Leiden, Lorentz Center, march 2010 1GianCarlo Ghirardi

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The parts and the whole:

a. Collapse Theoriesb. Identical Constituents

2GianCarlo GhirardiLeiden, Lorentz Center, march 2010

GianCarlo GhirardiDept. Phys. Univ. of Trieste

The ICTP, TriesteThe INFN, Trieste, Italy


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.

•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.


Within Q.M., the most complete characterization which is in principle possible for a system is the assignement of its statevector

•The pure state associated to the system is factorized:

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

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


Let me try to present a synthetic picture of the case of entangled states.

Theorem:

(@)

such that (@) holds for

i).

ii).

is that the range of the reduced statistical operator coincides with

A necessary and sufficient condition in order that a projection operator onto the linear closed manifold of satisfies the two following conditions:


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 such that a subset of its eigenstates (e.g. those associated to a subinterval [k , j] of its spectrum) span , one can claim that the variable 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!


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”.

A simple example: the system e- - e+ , in the state

the reduced statistical operator is 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 :

If c i ≠ 0, for all i-s and {|i(1)>} is a c.o.s. then


The paradigmatic case of an embarrassing whole: the macro-objectification or measurement problem of Q.M.

The sketchy ideal von Neumann measurement scheme for S=smicro+App

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.

1. Eigenvalues and eigenvectors for smicro:

2. Microstates are “measurable”:

3. The macrostates correspond to mutually exclusive perceptions of the conscious observer,

4. Equation 2 implies:

factorized entangled


One can significantly summarize the macro-objectification problem by making reference to the quite illuminating sentence by Bell:

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”.

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).


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).


1. States. A Hilbert space is associated to any physical system and the state of the system is represented by a (normalized) vector in

Note: localizations occur with higher probability where there would be an higher probability of finding the particle in a standard measurement process

2. Dynamics. The evolution of the system obeys Schrödinger’s equation. Moreover, at random times, with a Poissonian distribution with mean frequency , 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

is assumed to describe the density of mass distribution of the system in three-dimensional space as a function of time.

G.C. Ghirardi, R, Grassi, F. Benatti, Found. Phys 25, 5 (1995).


Localization of a microscopic system

SIMPLIFIED VISUALIZATION OF THE LOCALIZATION PROCESSES FOR

MICRO AND MACRO OBJECTS

The fundamental trigger process in the case of a macroscopic almost rigid body

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.


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 q 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).


Suppose a spontaneous localization occurs at this point

Then

Alternatively

The dynamical emergence of the properties of the parts of the Unbroken Universe

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.


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.


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 point-like 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)


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.

This is the relevant part of the story for what concerns us here.

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.


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).

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.

A. Bassi, G.C. Ghirardi, D.G.M. Salvetti, J. Phys. A: Math. Theor., 40, 13755 (2007).

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.

Concluding: our general physical approach leads to a natural deduction of the quantum rules in their most general and axiomatic form


The parts in the case of composite systems involving identical

constituents


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 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”?

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?”

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.


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. Lévy-Leblond

One may not draw conclusions about entanglement in configuration space by looking at the state in Fock space.

D.M. Greenberger,

M.A. Horne,

A. Zeilinger.

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),


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.


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 |(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 :

€

(1,2) EP (1,2) Ψ(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.


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 |(1,2)> are non-entangled iff |(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 |(1,2)> are non-entangled iff either |(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.


Summarizing:

Given that: , we have

Non-entangled fermions:

€

(1,2) =12

ϕ (1) ⊗ χ (2) − χ (1) ⊗ ϕ (2)[ ]

Non-entangled bosons:

€

(1,2) =12

ϕ (1) ⊗ χ (2) + χ (1) ⊗ ϕ (2)[ ]Either

Or


Non-entanglement, correlations and all that.

The fact that our criteria are appropriate to cha-racterize nonentangled states is strengthened and clarified by the considera-tion of the physical implications of the form of the state-vector.

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.


Two elementary examples

Consider the state (obtained by antisymmetrizing a factorized state):

€

(1,2) =12

z↑1

R1⊗ z↓

2L

2− z↓

1L

1⊗ z↑

2R

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(r a ,

r b ) = z↑

r σ ⋅

r a z↑ z↓

r σ ⋅

r 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:

€

Φ(1,2) =12

z↑1

z↓2

− z↓1

z↑2[ ]⊗ R

1L

2+ L

1R

2[ ].


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.

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.

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

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.


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.


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 one- particle-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-particle-orthogonal symmetrized states


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

respectively, there are other, physically more meaningful reasons to state that, in the considered case, the two subsystems are really "parts" of a "whole".

and

To discuss this point we need some further formal analysis. Let us discuss it.


Let us consider a complete orthonormal set of single particle states

•Let and * be two disjoint subsets of the index set {j}.

•Let V(K) , V(K)* be the corresponding linear manifolds for the system of K

fermions

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.

Consider an orthonormal basis spanning V(K)* , two arbitrary vectors

and of V(M) and a vector of V(K)* . Then

•Let V(M) , V(M)* be the linear manifolds spanned by the normalized, antisym-

metrized states

Leiden, Lorentz Center, march 2010 GianCarlo Ghirardi 34

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:

Lithium

D

Now we are in trouble: the two states appearing in this equation are not one-particle 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


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.


A particle, with either spin up or spin down along the z-axis goes through a Stern-Gerlach apparatus and it triggers the firing of the engine of a shuttle.

Andromeda

Andromeda


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 GianCarlo Ghirardi 38


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.


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.

and one can claim that the property =s is objectively possessed by the particle.

But in this case there are for sure self-adjoint operators such that


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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The parts and the whole: a. Collapse Theories b. Identical Constituents