Re: a question on incompatibility of properties in a one particle system

From: bernard.chaverondier (bernard.chaverondier_at_wanadoo.fr)
Date: 10/23/04 

Date: Sat, 23 Oct 2004 17:28:35 +0200

"Bilge" <dubious@radioactivex.lebesque-al.net> a écrit dans le message de
news:slrncnkbck.11k.dubious@radioactivex.lebesque-al.net...

Bilge
> Things like S1US2, is not a word in any language
> nor a standard way of writing anything.

Chaverondier
OK. I used S1 U S2 to mean the system S which is the Union of
subsystem S1 and subsystem S2. My intention was to ascribe a
subscript 1 to the first subsystem denoted S1 and a subscript 2 to the
second one denoted S2. The whole system is denoted S and U stands
for the union symbol often used in classical mechanics when a system
S encompasses two subsystems respectively denoted S1 and S2.

Bilge
>>> A spin singlet is a pure state. It's not decomposable.
> It does encompass all of the information of a pure j = 0 singlet.

Chaverondier
That's what I am stressing. state |psi> = (|+->+|-+>)/2^(1/2)
encompasses all the information about the quantum state
of a singlet like a pair of spin 1/2 particles (as a pair of
electrons in a maximum entangled state for instance).

This quantum state can also be expressed equivalently,
without any loss of information, by a density operator
Rhô which represents perfectly and completely this
pure entangled state of pair of electrons.

Rhô = (1/2) (|+->+|-+>)(<+-|+<-+|)

There is no uncertainty about the pure state Rhô of the singlet.
All what has to be known about the state of the singlet is contained
in the wave vector or equivalently its density operator Rhô.

Now, on the contrary, the _reduced_ density operators
of each electron are the density matrixes

Rhô1 = (|+><+| +|-><-|)/2 for S1
(and the same for Rhô2 the density operator of electron S2)

The singlet state is incompletely described by the mixed state
of each photon modeled separately by there reduced density
operator of each subsystem S1 and S2 of the quantum whole
S because the EPR correlations between S1 and S2 are not
present in this incomplete statistical description.

Now, my point was not about the pure state of a singlet but
about the more general case concerning two EPR correlated
systems like an observed system S1 interacting with a measuring
apparatus S2. I come to that below.

Bilge
> If you want to say something about reduced density matrices,
> write down the reduced density matrices. Then, it will be clear
> what you mean and I won't assume it's meaningless jargon.

Chaverondier
OK.
So, as far as the two systems S1 and S2 don't interact,
S1 being the observed system and S2 the measuring
apparatus in the specific case when we are studying the
measurement problem, they are completely modeled
separately by their two density operators Rhô1 and Rhô2.

Indeed, the density operator Rhô of the whole system S
comprising S1 and S2 is simply the tensor product of the
density operators Rhô1 and Rhô2 modeling the quantum
states of each subsystem.

Now, when S1 and S2 begin to interact, the picture changes.
The density operator Rhô of the whole system S comprising the
observed system S1 and the measuring apparatus S2 accounts
for the EPR correlations between the two parts S1 and S2 (some
time later a third system S3 modeling the environment enters
the picture, but for the sake of simplicity let us first consider
only the observed system S1 and the measuring apparatus S2).

This density operator Rhô is not any more the tensor product
of the reduced density operators Rhô1 and Rhô2 of each part.
The reduced density operators Rhô1 and Rhô2 of each part
of this entangled quantum system S don't provide a complete
description of the quantum whole S (union of S1 and S2).

That's where the uncertainties in the measurement process
come from. When you know only the reduced density
operator Rhô1 of the observed system S1 and not the density
operator Rhô of the quantum whole comprising the observed
system S1 and the measuring apparatus S2, your knowledge
is dincomplete (as reflected by the increase of entropy induced
by the entanglement process caused by a quantum measurement).

Because of your absence of knowledge of the quantum state
of the measuring apparatus S2 (and that of the environment S3),
you can only predict the statistics of quantum measurement
outcomes in agreement with Von Neumann statistics.

Chaverondier
>>>>Photons can be independent and in a pure
>>>>state (ie in a known polarization state).

Bilge
> Then you aren't talking about quantum mechanics.
> A singlet state is _ONE_ wavefunction, |j,j_z> = |0,0>.
> It's not decomposable into two spin 1 particles.

Chaverondier
That's precisely what I am stressing (when EPR correlated
systems are considered). I hope the above details will
clear up any ambiguity on what I am pointing out.

Bilge
>>> ``If \rho is diagonal, the system is said to be an incoherent
>>> superposition of the basis states (provided there is
>>> more than one non-vanishing element.''

Chaverondier
>>And I still completely agree with that last statement.

Bilge
> OK, then you'll have to agree that you don't believe the epr
> correlations exist, since you are saying that the density matrix
> is mixed.

Chaverondier
What density matrix ?
  * If you are evocating the density matrix of an EPR correlated pair
of photons (for instance) before it has interacted, it is a pure state.

Rhô = (1/2) (|+->+|-+>)(<+-|+<-+|)

  * if you are evocating the reduced density matrix of _one_ photon
of the EPR correlated pair, this density matrix is in a mixed state.

Rhô1 = (1/2) |+><+| + (1/2) |-><-|

  * if you are evocating the density matrix of the system comprising
one polarizer and the EPR pair this quantum whole is in a pure state
(as far as the environment has not begun to interact).

  * if you are evocating the reduced density matrix of the EPR pair alone
when it has finished to interact with one polarizer, this is a mixed state.

Rhô = (1/2) |+-><+-| + (1/2) |-+><-+|

A mixed state is a model of the incomplete knowledge of the observer.
Any isolated system S1 which quantum state is perfectly known is in
a pure state and its quantum evolution is deterministic as soon as he
doesn't interact with any system S2 which quantum state is ill known
or too difficult to handle other than approximately (and approximately
modeled as a classical measuring apparatus for instance when a
measurement process is considered).

Chaverondier
> >I was not addressing the question of time ordering of spacelike
> >events here (this question depends on the interpretation of
> >quantum indeterminacy). You can make only one photon of
> >the pair interact. The entanglement between the EPR correlated
> >photons pair and one polarizer occurs as soon as one polarizer
> >interacts with one photon of the pair (the interaction of a second
> >photon with its polarizer is not needed for the entanglement of
> >the EPR pair with one polarizer to take place).

Bilge
> You have completely misunderstood relativity. The points have
> a spacelike separation. ``As soon as'' has no meaning.

Chaverondier
First, even in a classical presentation of relativity, "as soon as"
has a precise meaning when you define what inertial frame you
are considering (when it is not explicitly stated, it means that
there is no ambiguity on the implicitly chosen frame).

Second, There is a possibility to provide a precise meaning to "as soon"
in a manner that doesn't depend on the motion of the observation frame
provided you work in the appropriate framework, ie a geometrical
framework that doesn't assume that all physics phenomena would
satisfy the principle of relativity of motion (without any exception).

You have then to derive the Lorentz transforms in the framework of
Aristotle spacetime (endowed with 3D Euclidean slices of simultaneity
and 1D Euclidean motionless world lines that are the characteristic
foliations of the rank 3 space metric and the rank 1 time metric of
Aristotle space-time A_4 = SE(1)xSE(3)/SO(3) generated by the
Aristotle strict subgroup SE(1)xSE(3) of the restricted Poincaré
group SE(1,3). See http://perso.wanadoo.fr/lebigbang
http://perso.wanadoo.fr/lebigbang/epr.htm and
http://perso.wanadoo.fr/lebigbang/transformation.htm
to get more mathematical and physical details)

Bilge
> The two events that constitute the measurement have no
> intrinsic temporal relationship to each other. ``As soon as''
> implies some reality which contradicts relativity.

Chaverondier
Yes. It contradicts the hypothesis that any phenomenon
would satisfy the principle of relativity of motion. This
hypothesis isn't satisfied if some interaction propagate
Faster Than Light.

Nevertheless, the proof of the impossibility of interactions
propagating Faster Than Light thanks to EPR effect relies
on the hypothesis that quantum indeterminacy would be
fundamental, ie that quantum uncertainties would not be
the consequence of the lack of knowledge of the observer,
hypothesis which I don't believe (to get more details, see
http://perso.wanadoo.fr/lebigbang/no_communication.htm )

Anyhow this additional hypothesis about quantum indeterminacy
nature conflicts (in my opinion) with the unitary, deterministic
and reversible dynamics of quantum evolutions.

Bilge
> You are trying to convolute the meaning of deterministic to suit
> your argument. I have one 22Na nucleus. Give me an equation that
> tells me how long before it decays. Not a half-life or mean lifetime.

Chaverondier
You can't. You don't know the quantum state of the quantum
whole that is involved in the decay's by products observation.
You know only the quantum state of a part of this quantum whole :
the observed 22Na nucleus alone not the EPR correlated quantum
whole comprising the 22Na nucleus, the detectors and the environment
that interact with these detectors.

When this 22Na decays, this quantum system is not anymore
isolated from its surrounding. The by products of this decay have
to be detected. This detection is a position measurement involving
the searched for source of quantum indeterminacy.

This all comes from the lack of knowledge of the quantum state of the
detectors (and of the quantum state of the environment with which these
detectors interact). Here is the source of quantum indeterminacy. In My
opinion, there is no source of quantum indeterminacy that couldn't be
interpreted as a lack of knowledge of the observer.

Bilge
> Chaotic systems are deterministic, Quantum mechanics is not chaotic.

Chaverondier
This has to be proven, more particularly when a measurement process
is involved. My guess is that a deterministically chaotic process
is involved as Paul Budnik seems to argue on his web site.

Chaverondier
>>Quantum dynamics of isolated systems is deterministic.
>>Presently, as far as I know, no known physics observation
>>have provided any proof that an isolated quantum system
>>might, under certain circumstances, exhibit an indeterminist
>>or irreversible behavior.

Bilge
> Then you haven't looked very hard. Stop being ridiculous. The decay
> time for a simple 2p->1s transition is only specified by a probability.

Chaverondier
Indeed, there is a quantum measurement indeterminacy (ie an
observer's loss of information) induced by the detection process
of the by products emerging out of any quantum diffusion process.

Bernard Chaverondier
http://perso.wanadoo.fr/lebigbang/transformation.htm
Derivation of Lorentz transforms and inertial system
of coordinates in the framework of Aristotle space-time.
http://perso.wanadoo.fr/lebigbang/epr.htm
Quantum determinism or Relativist locality ?

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