Some questions on decoherence and QM.
Frank Hellmann
Hi, I'm a 2nd Year student from Germany who is gripping with the
concepts underlying Quantum Mechanics beyond the simple interpretations
and phenomena that were the subject of the first lectures on QM.
First "simple" question:
How does decoherence lead to the collapse of superposition? As I
understand it decoherence makes the of diagonal terms of the density
matrix disappear, and that means the diagonal terms can no longer
interact with each other.
U*rho*U° (with ° denoting the selfadjoint) becomes (for a simple spin model)
U[1,1]*rho[1,1]*U[1,1]° (with [] being the matrix indices)
U[2,2]*rho[2,2]*U[2,2]°
All other terms being zero. I understand that this statistical operator
then cannot be reduced to a state in the Hilbertspace.
A meassurement of the quantum system described by rho in generally still
has a propability for both classically exclusive states though, so we
still have a superposition of classically exclusive states.
In particularly in quantph/0202169 it is claimed that decoherence
resolves the Schrödingers cat experiment, as the cat becomes classical
after a very short period of time. However the dissapearance of the
offdiagonal elements makes certain that no quantum interference occurs
between the states but not at all that only classically allowed states
are taken.
I assume that I'm somehow quite fundamentally misguided on the whole
thing and have made some in hindsight obvious mistake. Any suggestions
what to read to get a better view of the whole decoherence thing (as
well as a better grasp of the density matrix/statistical operator
formalism which didn't appear at all in my university lectures by now.)?
Second Question:
I'm currently reading the beginings of Landau/Lifschitz 2: Classical
Field Theorie, and I'm slightly unsatisfied with the
mathematical/geometric background given on tensor calculus, could
anybody recommend some good supplementary reading on the subject?
Third question:
One of the epistemologicaly and onthologicaly most remarkable features
of classical mechanics, which enables it to describe reality over many
different scales coherently, is that it reproduces it's conceptual
assumptions within itself in justifying that one views a group of point
particles as a new point particle at the center of mass.
Does anything similar exist for QM? A theorem describing under which
circumstances a group of wave functions can be described by a single
wavefunction? Or alternatively a theroem describing how the center of
mass of several wave functions develops?
Perhaps in connection with the Ehrenfest Theorems?
Thanks,
Frank Hellmann
Greg Egan
wrote:
>Any suggestions
> what to read to get a better view of the whole decoherence thing (as
> well as a better grasp of the density matrix/statistical operator
> formalism which didn't appear at all in my university lectures by now.)?
_Decoherence and the Appearance of a Classical World in Quantum Theory_
by Giulini et al, Springer, 1996. ISBN 3-540-61394-3
Alfred Einstead
> First "simple" question:
> How does decoherence lead to the collapse of superposition?
A "superposition" occurs when two quantum systems are on "different
bases" in Hilbert space; in particular when two or more sets of
states are (virtually) indistinguishable, e.g., by their energy level
or other observations.
What an interaction does is to tie the two systems undergoing
interaction onto a common basis.
It might help to develop a little perturbation theory in
coordinate-free form first, before seeing this.
Suppose you have a system described by 2 parts, each being
described by its own interaction (a 'free' Hamiltonian H0
equal to H1 for part 1, H2 for part 2). Without an interaction,
the total state |psi> factors into |psi_1>|psi_2> which
evolves independently. So, there's no correlation between
them and, as a result, someone in system 1 would see someone
in system 2 in a "superposition".
To actually observe system 2 by system 1 means to have an
interaction between them. This interaction breaks the
symmetry of the state space and forces a natural Hilbert
space basis on the overall system, which destroys most
degeneracies and their corresponding superposition.
So, suppose the free Hamiltonian is amended by an interaction
H = HO + a V, with a being a small parameter. The original
state space decomposes into a set of eigen states:
|m1>|m2>,
with
H0|m1>|m2> = (E1(m1) + E2(m2))|m1>|m2>.
Label these and their energies by a single value |n>, and
the eigenvalues too E(n). Then in the basis |n>, H reduces
to
H0 = E = sum E(n) |n><n|,
with
<m||n> = delta_{mn}, sum |n><n| = I.
The matrix E is diagonal.
In this spectra there is supposedly a large degree of
degeneracy, otherwise the different states are all
distinguishable by their energy levels and you won't
have a coherent superposition, once the energy is observed.
The interaction destroys this symmetry, breaking the
degeneracy, and also as a by-product imposes a natural
basis on each sector (i.e., at each energy level).
Suppose the interaction is small enough (that is: a is
small enough) that the effect is to warp this nice
symmetric state space into:
Eigenstates: |n:a>, Energies E(n:a),
with
(H0 + aV) |n:a> = E(n:a) |n:a>,
with
<m:a||n:a> = delta_{mn}, sum |n:a><n:a| = I.
Continuity is assumed:
E(n:a) -> E(n), as a -> 0.
Then, define the operator
U(a) = sum |n:a><n|.
Its adjoint is
U(a)+ = sum |n><n:a|.
The last set of identities for the deformed state space
then reduce to:
U(a)+ U(a) = sum |m><m:a||n:a><n|
= sum |m> delta_{mn} <n| = sum |n><n| = I,
U(a) U(a)+ = sum |m:a><m||n><n:a|
= sum |m:a> delta_{mn} <n:a| = sum |n:a><n:a| = I.
So, U(a) is a unitary operator -- the "deformation operator".
>>From it, a common basis will emerge in the total state space.
Define the diagonal matrix E(a) by:
E(a) = sum E(n:a) |n><n|.
Then
U(a) E(a) = sum |m:a><m| E(n:a) |n><n|
= sum E(n:a) |m:a><m||n><n|
= sum E(n:a) |m:a> delta_{mn} <n|
= sum E(n:a) |n:a> <n|
= sum H |n:a><n|
= H U(a),
so that
(E + aV) U(a) = U(a) E(a),
with
E(a) -> E, as a -> 0,
U(a) U(a)+ = I = U(a)+ U(a).
These are the basic equations, in coordinate-free form, that
form the background to Perturbation Theory.
The last relations imply that U = U(0) must be unitary.
So assume the expansion,
U(a) = (I + a D1 + a^2 D2 + ...) U
E(a) = E + a E1 + a^2 E2 + ...
Then up to the 2nd order:
(E + aV) (I + a D1 + a^2 D2) U = (I + a D1 + a^2 D2)U(E + a E1 + a^2 E2)
(I + a D1 + a^2 D2) U U+ (I + a D1+ + a^2 D2+) = I
U+ (I + a D1+ + a^2 D2+) (I + a D1 + a^2 D2) U = I.
The last two relations imply that
(I + a D1 + a^2 D2), (I + a D1+ + a^2 D2+)
are inverses (up to the 2nd order):
So, you get:
EU = UE
(E D1 + V) U = D1 UE + U E1
(E D2 + V D1) U = D2 U E + D1 U E1 + U E2
D1 + D1+ = 0
D2 + D1 D1+ + D2+ = 0,
so that
[E, U] = 0
[E, D1] + V = U E1 U+
[E, D2] + V D1 = D1 (U E1 U+) + U E2 U+
= D1 ([E, D1] + V) + U E2 U+,
or
[E, D2] + [V, D1] = D1 [E, D1] + U E2 U+.
The relations between the D's imply,
D1+ = -D1,
D2+ = D1^2 - D2.
Now, with the perturbation theory preliminaries ot of the way,
we've come at last to the point. Without the deformation, the
U matrix freely mixes the states that reside within each
energy sector. So, if you fix the energy of the system,
the state will be in a superposition of the states in
that sector.
Switching on the interaction causes the deformation.
This deformation destroys the symmetry of each sector
and imposes a fixed basis on the state space in that
sector. The first condition is:
E1 = U+ ([E, D1] + V) U is diagonal,
= [E, U+ D1 U] + U+ V U is diagonal, with D1+ = -D1.
The requirement that E1 be diagonal constrains U in part.
This fixes part of the basis for each sector. The
equation also determines the values in E1 -- with a result
that is familiar in Perturbation Theory. In particular,
you get:
(Em - Ep) <m|D1|p> + <m|V|p> = 0, when Em != Ep,
or
D1 = sum D1_{mp} |m><p|, taken over states with Em = Ep
+ sum |m><m| V/(Ep-Em) |p><p|, taken over states with Em != Ep
or, define the projector
P(e) = sum |m><m|: taken over Em = e,
which projects onto sector e, and
D1(e) = P(e) D1 P(e),
which is the restriction of D1 to sector e.
Then you get
D1 = sum D1(e): over all energies e
+ sum (P(e) V P(f))/(f - e), over all energies e != f,
with the adjoint relation:
D1(e)+ = -D1(e).
But you also get the constraint for each energy level e:
E1(e) must be diagonal
with
E1(e) = P(e) E1 P(e)
= P(e) U+ V U P(e)
= U+ P(e) V P(e) U
= U+ V(e) U ... diagonal.
where
V(e) = P(e) V P(e)
is the interaction restricted to sector e.
So, the state space in sector e is diagonalized by the
interaction V. The interaction imposes a natural basis
on the state space. Each sector-restricted interaction
V(e) must be diagonalized.
This puts both systems, partly, on a common basis. There
may be further degeneracy and this may get further
reduced by the 2nd set of relations:
E2 must be diagonal, with
E2 = U+ ([E, D2] + [V, D1] - D1 [E, D1]) U
and
D2+ = D1^2 - D2.
These won't be worked out in detail, the results
correspond to those familiar from 2nd-order perturbation
theory. The net effect is that further constraints may be
imposed by the requirement that E2 be diagonal (while allowing
you find an expression for D2, and in part for the residual
terms D1(e) which were left undefined above).
Urs Schreiber
> Any suggestions
> what to read to get a better view of the whole decoherence thing (as
> well as a better grasp of the density matrix/statistical operator
> formalism which didn't appear at all in my university lectures by now.)?
Apart from the standard textbook by Giulini, which was already
mentioned, for a quick introduction also see the nice recent
article in "Physik Journal":
W. Strunz, G. Alber, F. Haake, Dekohaerenz in offenen
Quantensystemen - Von den Grundlagen der Quantenmechanik zur
Quantentechnologie, Physik Journal, November, p. 47-52 (2002)
W. Strunz was a postdoc here in Essen (has now moved to
Freiburg). I just come from a talk he gave on his work on
robust state dynamics (states that are in some sense stable
under decoherence and can become classically observable). He
has a beautiful pedagogic style. On his website
http://tqd1.physik.uni-freiburg.de/~walter/forschung/paperle.html
you'll find more pointers to his work including further recent
online-available articles.
> All other terms being zero. I understand that this statistical operator
> then cannot be reduced to a state in the Hilbertspace.
> A measurement of the quantum system described by rho in generally still
> has a propability for both classically exclusive states though, so we
> still have a superposition of classically exclusive states.
The last phrase must read: "a *mixture* of classical states".
Using the density operator one is bound to talk about
statistics only. Decoherence cannot and does not explain "how"
a system chooses from the possible outcomes a specific one
when we measure it. Decoherence only explains how the "quantum
probability" becomes a "classical probability", very roughly
speaking, but it still only gives probabilities.
> In particularly in quantph/0202169 it is claimed that decoherence
> resolves the Schrödingers cat experiment, as the cat becomes classical
> after a very short period of time. However the dissapearance of the
> offdiagonal elements makes certain that no quantum interference occurs
> between the states but not at all that only classically allowed states
> are taken.
This is very good point that is often not emphasized in the
older texts on decoherence. The mere fact alone that the
density operator becomes diagonal does not yet yield the
classical world we observe. I believe among the first this has
been pointed out by Penrose in some of his lectures, where he
essentially remarks that a matrix is diagonal in more than one
basis. For instance
|0><0| + |1><1| = ( (|0> + |1>)(<0| + <1|) + (|0> - |1>)(<0|
- <1|) )/2 .
Hence it is not enough to argue that we observe a system in
either of the diagonal entries of the density operator: We
need to know in which basis!
This is resolved by understanding "robust states", which is
conceptually very simple. The point is that in addition to
"tracing out" the environment one needs to look at the
temporal evolution of the remaining reduced density operator.
This is governed by the precise nature of the coupling of the
system to its environment. Schematically the coupling term I
generically looks like
I = S (x) B ,
where S is an operator of the system and B an operator of the
bath. For instance consider a system consisting of a single
harmonic oscillator coupled to a bath of other harmonic
oscillators. Then the coupling is usually of the form
I = a (x) a+ + h.c.,
where the left "a" is the annihilation operator of the system
oscillator and the right a+ the creation operator of a bath
oscillator (details suppressed). This means the system and the
bath interact by exchanging quanta of energy - no surprise.
Now consider an initial superposition of the system:
(|0> + |1>) (x) |bath initially> .
Obviously when this state evolves it will generically lead to
something like
|0(t)> (x) |bath finally having interacted with 0>
+|1(t)> (x) |bath finally having interacted with 1>
When you now take the trace over the bath, the modification of
the system's density matrix is the larger the lesser the
"overlap" between the final bath states. But the final bath
states will only be appreciably different when the interaction
with the system's states was, too. This means that a measure
for the decoherence of the initial superposition is the
difference between the eigenvalues alpha of the interaction
operator a on the states |0> and |1>. The decay rate of the
coherence is proportional to
D ~ |alpha_0 - alpha_1|.
For general interaction operators s with eigenvalues sigma
this of course generalizes to
D ~ |sigma_0 - sigma_1|.
This law holds quite universally for arbitrary setups of
system and bath (e.g. quant-ph/0204129).
When you look at a classical object you do not see it
decohering. Hence classical states are states which have D=0.
These are also called robust states. They are obviously
characterized as being eigenstates of the system's interaction
term!
Hence for the oscillator-oscibath setup the robust states are
the eigenstates of the annihilator which are just the coherent
states. Note that these states are only "robust" as opposed to
being stable. They evolve in time as
|alpha(t)> = |alpha e^(-i omega - gamma)t>
i.e. they narrow in on the classical ground state |0>, but
while doing so they do not decohere.
This effect has been experimentally confirmed in 1996, or so,
in a famous experiment by some group in Paris (I forget the
details), which used as the system a single mode of an EM
field in a resonator.
So the upshot is that the trace over the environment explains
how a superposition becomes a mixture, while the coupling to
the environment determines which states we actually observe
when measuring this mixture, namely the robust states.
This has a nice and simple consequence for how to build a
quantum computer: In order for its RAM not to decohere, it
must store its qbits in terms of states which are all
eigenstates to (approximately) the same eigenvalue of the
interaction operator s. That is, a good quantum computer will
couple to its environment via an operator with large
eigenspaces and will perform its calculation within one such
eigenspace.
> Second Question:
> I'm currently reading the beginings of Landau/Lifschitz 2: Classical
> Field Theorie, and I'm slightly unsatisfied with the
> mathematical/geometric background given on tensor calculus, could
> anybody recommend some good supplementary reading on the subject?
I have frequently recommended Frankel, The Geometry of
Physics, Cambridge Univ. Press. People here have pointed out
some deficiencies of this book, but when you understand these
you won't need this book anymore, anyway. If nothing else, it
is not as old as Landau/Lifschitz. There has been progress
since then in concepts and notation. READ IT.
> Third question:
> One of the epistemologicaly and onthologicaly most remarkable features
> of classical mechanics, which enables it to describe reality over many
> different scales coherently, is that it reproduces it's conceptual
> assumptions within itself in justifying that one views a group of point
> particles as a new point particle at the center of mass.
> Does anything similar exist for QM?
Yes. Simply use the classical Hamiltonian function in a form
where the center-of-mass motion is explicit and quantize that.
Look at some simple two-particle system in your textbooks, for
instance at the hydrogen atom, and see how this works out as
expected.
> A theorem describing under which
> circumstances a group of wave functions can be described by a single
> wavefunction?
Group of wave functions? Shortly after deBroglie wave
functions ceased to be waves in space associated with a single
particle. A state of your system, irrespective of how many
particles it contains, is always described by a single wave
function, which is a function over the system's configuration
space. What you are really asking for is, if you can
parameterize the configuration space in such a way, that one
subset of parameters describes the center of mass motion. Yes,
this is just the same reparametrization as in classical
mechanics.
> Or alternatively a theroem describing how the center of
> mass of several wave functions develops?
> Perhaps in connection with the Ehrenfest Theorems?
Yes, the expectation value of the center of mass follows the
classical equation of motion. As I said, have a look a any
2-particle system and you will quickly be de-confused about
this point.
Beste Gruesse,
Urs
Alfred Einstead
Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
> > In particularly in quant ph/0202169 it is claimed that decoherence
> > after a very short period of time. However the dissapearance of the
> > offdiagonal elements makes certain that no quantum interference occurs
> > between the states but not at all that only classically allowed states
> > are taken.
> This is very good point that is often not emphasized in the
> older texts on decoherence. The mere fact alone that the
> density operator becomes diagonal does not yet yield the
> classical world we observe.
The extra missing piece to the puzzle is precisely what I related
in my last response, where it was shown how an interaction will
deform a state space and select out a natural basis.
A classical world, therefore, will emerge in which everyone
is on the "same basis" if there exists an interaction with
the properties that it be:
(1) universal
(2) long-range
and
(3) unshieldable.
So, for instance, in the case of an actual Schroedinger cat,
because of (3) you'll still be able to distinguish between
the two states. Then because of (2) the distinction between
the two states will entangle everyone else far away into a
common basis. And because of (1), the entanglement will be
universal, so everyone will be on a common basis.
Fortunately, the Universe has provided us with an instance
of one (1) force satisfying the 3 properties above.
Mike Mowbray
> Frank Hellmann wrote:
>> Any suggestions what to read to get a better view of the
>> whole decoherence thing (as well as a better grasp of the
>> density matrix/statistical operator formalism [...]) ?
> _Decoherence and the Appearance of a Classical World in Quantum Theory_
> by Giulini et al, Springer, 1996. ISBN 3-540-61394-3
I'm considering whether to buy this book sight-unseen.
Most of the (lanl) papers on decoherence I've read so far tend to
waffle rather too much, without rigorously deriving the extra terms
that generate non-unitary evolution for the off-diagonal elements in
the density matrix. Does this book give the detailed derivations?
TIA.
- MikeM.
Alfred Einstead
Frank Hellmann <C....@gmx.net> wrote:
> How does decoherence lead to the collapse of superposition? As I
> understand it decoherence makes the of diagonal terms of the density
> matrix disappear, and that means the diagonal terms can no longer
> interact with each other.
Actually, that's the description of "superselection". States that
reside in different sectors are orthogonal and so have no cross-terms.
There is no coherent superposition between states that reside in
different sectors. This is called superselection.
A sector is defined by a set of quantum numbers corresponding to
conserved quantitities, like total energy. The cross-terms can
only occur in states that have the same quantum numbers. So,
looking for these, you have to look for places where there's
sufficient symmetry to allow two or more states to have the
same quantum numbers.
The most prominent example of that is an electron in a
Hydrogen atom. In isolation, the electron in each energy
level has 2 or more states. So, these can exist in
coherent superposition.
But once you turn on an interaction, such as a magnetic
field, this warps the nice symmetric state space (in the
way I demonstrated with my little exercise in perturbation
theory), creating what amount to "principle directions" in
each sector. The eigenvectors of the interaction, itself,
when restricted to the sector bring about the natural "basis".
In the case of a magnetic field, the warping of the symmetry
amounts to the occurrence of a fine structure splitting.
With the splitting, the states that used to be at the same
level and in the same sector are now in different levels of
energy and in different sectors, which means you now have
superselection between them -- and thus: no coherent
superposition or off-diagonal terms.
And so we refer to the imposition of the magnetic field on
the atom as a "measurement", particularly: "a measurement
of the electron's spin".
All measurements are, in essence, of this nature.
So, the ultimate question that comes about is: why do you
even need the rather advanced (and contrived) explanations
about the excessive degrees of freedom and thermodynamic
noise of macroscopic systems and the like when the
explanation for decoherence was already at hand, in a
far simpler fashion, as nothing more than a simple
consequence of basic perturbation theory.
> In particularly in quantph/0202169 it is claimed that decoherence
> resolves the Schrödingers cat experiment,
This extra element, as described in my second follow-up article
on the subject is actually an independent element the necessary
conditions of whose resolution provides a far more interesting
clue to something whose existence is little noted, deemed little
relevant and rarely spoken of in the context of quantum physics.
But it might be a useful exercise to approach the same conclusion
from a slightly different direction. In order to have
decoherence, you need the superselection that arises from the
destruction of symmetry in the state space brought about by
an interaction.
In order for it to take place through the box which supposedly
shields the cat from the outside world, the interaction, itself,
has to be of such a nature that no box and no device known to
us can shield it.
In order for it to take place at macroscopic distances from
the box, the interaction has to be long-range.
Finally, in order for it to bring the entire world within the
same fold as the cat, the interaction has to be of such a
nature that all participate in it: it has to be universal.
Answer the 3 clues provided by this riddle and you'll have in
place the final piece to your puzzle of decoherence.
Calvin Ritchie
>How does decoherence lead to the collapse of superposition? As I
>understand it decoherence makes the of diagonal terms of the density
>matrix disappear, and that means the diagonal terms can no longer
>interact with each other.
I see that you already have some very good responses on this, so I'll
just add that there's a very well written lecture, "Lectures on
Decoherence", available, free, on the 'net:
www.nottingham.ac.uk/~ppxada/lec1d.pdf
Submitted afternoon 11 Jan 03
Don Ritchie
DonRit...@csWebmail.com
Urs Schreiber
> A sector is defined by a set of quantum numbers corresponding to
> conserved quantities, like total energy. The cross-terms can
> only occur in states that have the same quantum numbers. So,
> looking for these, you have to look for places where there's
> sufficient symmetry to allow two or more states to have the
> same quantum numbers.
I don't get what you mean. Take the 1-dimensional harmonic oscillator
as an example. There is essentially one conserved quantity, namely
energy, and states are labeled uniquely by this "quantum number". Now
what exactly is your claim in this situation?
> In the case of a magnetic field, the warping of the symmetry
> amounts to the occurrence of a fine structure splitting.
> With the splitting, the states that used to be at the same
> level and in the same sector are now in different levels of
> energy and in different sectors, which means you now have
> superselection between them -- and thus: no coherent
> superposition or off-diagonal terms.
States of different energy are not generally in different
superselection sectors. Also, one can certainly have coherent
superposition of orthogonal states. Otherwise the concept would be
empty. I don't understand what you are getting at.
[...]
> Answer the 3 clues provided by this riddle and you'll have in
> place the final piece to your puzzle of decoherence.
While in principle gravity may be an important clue to questions that
concern the quantum nature of the universe in total, I am wondering
how much effect it really has in laboratory-accessible experiments on
decoherence. I am not an expert myself, but I have heard experts talk
about decoherence in lab experiments and they have so far never
mentioned the effect of gravity. Do, or will, quantum computers suffer
noticeably from gravitationally induced decoherence?
Mike Mowbray
Urs Schreiber wrote:
> While in principle gravity may be an important clue to questions that
> concern the quantum nature of the universe in total, I am wondering
> how much effect it really has in laboratory-accessible experiments on
> decoherence. I am not an expert myself, but I have heard experts talk
> about decoherence in lab experiments and they have so far never
> mentioned the effect of gravity. Do, or will, quantum computers suffer
> noticeably from gravitationally induced decoherence?
Some interesting papers by Ford & O'Connell recently appeared on LANL:
quant-ph/0301054, quant-ph/0301057. They show how decoherence occurs
in a prototypical Schrodinger cat state (actually a pair of Gaussian
wave packets) merely by giving the packets a (thermal) distribution of
velocities. The onset of decoherence is shown to be *extremely* fast.
They point out that it doesn't matter how the distribution of velocities
arose, - just as in the equation of state for classical thermodynamic
equilibrium it doesn't explicitly mention the collisions that drove the
system towards equilibrium.
What do others think of these papers? If a small distribution of
velocities is indeed sufficient to cause such rapid decoherence, I'm
wondering whether gravitational interactions with the environment are
strong enough to produce sufficiently wide distributions.
- MikeM.
Frank Hellmann
Alfred Einstead wrote:
> Frank Hellmann <C....@gmx.net> wrote:
>
Hmmm... actually I do not understand at all what you are doing there.
The fixing of the Eigenbasis of the Hamiltonian of the Hilbert Space
through interaction does fixes the "pointer basis" if every single state
becomes distinguishable by the meassurement, e.g. if we measure the
energy, but that is not what decoherence does, is it?
After all we are after vanishing offdiagonal elements in the density
Matrix not in the Hamiltonian, or would there be a connection between
the two? I think not, a diagonal Hamiltonian acts quite coherently upon
a non decoherent (and therefor nondiagonal) density matrix, simply as
upon an arbitrary (and thus quite possibly superposed) state.
Essentially a fixed basis says nothing about the superpositions occuring
within it. A fixed basis in regard to the enviromental interaction term,
together with the (thermodynamic) decoherence process induced by this
interaction term however truely eliminates coherent superpositions I think.
Or am I completely off-track here?
Thanks,
Frank.
P.S. I really liked the coordinate free pertubation theory. Thanks a lot
for the lesson.
Frank Hellmann
struggling with all this, won't fit in my head that easily yet.
I have now started to read up on the robust states, this post here
really touches upon what I was puzzled with most in the begining.
Alain Stalder wrote:
> In article <3E1C9025...@uni-essen.de>,
> Urs Schreiber <Urs.Sc...@uni-essen.de> wrote:
>>Frank Hellmann wrote:
>>>A measurement of the quantum system described by rho in generally still
>>>has a propability for both classically exclusive states though, so we
>>>still have a superposition of classically exclusive states.
>>The last phrase must read: "a *mixture* of classical states".
I would object to this terminology.
A classical statistic I think is either given if you have many systems
or incomplete knowledge on the initial state of the system.
Roughly speaking it thus only becomes a classical statistic if you
accept either a Many World interpretation (it is then a statistic over
the different possible worlds) or a hidden variable version (Bohmian
mechanics is the only one I'm aware of and that is again a many world
theory in essence).
A single particle can not classically exist as a mixture, it's
description as mixture is due to incomplete knowledge and is no longer
reasonable or neccesary for the time evolution after a measurement.
OTOH a quantum state can decohere into a mixture (even if it's only
approximately), and after a measurement might well develop into a
mixture again.
Thus a decohered superposition and a classical mixture still have IMO
vastly different interpretations, even though their mathematical
appearance is deceivingly similar.
I also wonder whether the superselection rules really require classical
mixtures or decohered superpositions.
The problem in all this being of course that the two states are
experimentally virtually identical.
> It is worthwhile to explain what exactly "classical" means in
> this context. This is maybe most easily seen if Schroedinger's
> Gedankenexperiment is combined with the experiment for testing
> Bell's Inequality:
[snip]
Are there experiments on this interesting combination verifying the
Gedankenexperiment?
> In conclusion, decoherence is a big step towards understanding
> measurement in quantum mechanics, but does not go all the way,
> at least not yet.
>
> Alain Stalder
Thanks a lot, that was more or less exactly what I wanted to know there.
Frank Hellmann
> Frank Hellmann wrote:
>>Any suggestions
>>what to read to get a better view of the whole decoherence thing (as
>>well as a better grasp of the density matrix/statistical operator
>>formalism which didn't appear at all in my university lectures by now.)?
> W. Strunz was a postdoc here in Essen (has now moved to
> Freiburg). I just come from a talk he gave on his work on
> robust state dynamics (states that are in some sense stable
> under decoherence and can become classically observable). He
> has a beautiful pedagogic style. On his website
> http://tqd1.physik.uni-freiburg.de/~walter/forschung/paperle.html
> you'll find more pointers to his work including further recent
> online-available articles.
Thanks, this robust states dynamics sound fascinating and important. I'll
read up more there.
>>Second Question:
>>I'm currently reading the beginings of Landau/Lifschitz 2: Classical
>>Field Theorie, and I'm slightly unsatisfied with the
>>mathematical/geometric background given on tensor calculus, could
>>anybody recommend some good supplementary reading on the subject?
> I have frequently recommended Frankel, The Geometry of
> Physics, Cambridge Univ. Press. People here have pointed out
> some deficiencies of this book, but when you understand these
> you won't need this book anymore, anyway. If nothing else, it
> is not as old as Landau/Lifschitz. There has been progress
> since then in concepts and notation. READ IT.
Thanks a lot, this sounds like exactly what I've been looking for.
(Guess Guilini's book will have to wait then though...)
>>Third question:
>>One of the epistemologicaly and onthologicaly most remarkable features
>>of classical mechanics, which enables it to describe reality over many
>>different scales coherently, is that it reproduces it's conceptual
>>assumptions within itself in justifying that one views a group of point
>>particles as a new point particle at the center of mass.
>>Does anything similar exist for QM?
> Yes. Simply use the classical Hamiltonian function in a form
> where the center-of-mass motion is explicit and quantize that.
Yes but what I would like would be to make this step without going the
way through classical mechanics.
>>A theorem describing under which
>>circumstances a group of wave functions can be described by a single
>>wavefunction?
> What you are really asking for is, if you can
> parameterize the configuration space in such a way, that one
> subset of parameters describes the center of mass motion. Yes,
> this is just the same reparametrization as in classical
> mechanics.
>
> Yes, the expectation value of the center of mass follows the
> classical equation of motion. As I said, have a look a any
> 2-particle system and you will quickly be de-confused about
> this point.
Hmmm... ok, the textbook example indeed does just that classical
reparametrization (surprise ;).
Question is: can one prove that such a reparametrization (which splits
the Hamiltonian) is always possible for closed systems, and that in the
approximation of constant exterior field over the area of the wave
function the center of mass parameters behave accordingly?
This is absolutely straightforward in classical mechanics, and looking
at the calculations it seems that nothing is done with the variables
which could not be done with the according operators...
So that's off my list for one :)
Thanks a lot,
Frank
David Winsemius
A850F5.131...@news.bluewin.ch:
> Can we conclude that whether the cat is dead or alive is already
> determined, that an experimentator who looks inside to discover
> either a dead or a living cat will only note what was already
> determined before ?
>
> No, because Bell's Inequality.......snipped
And what if the cat "reads the meter"and lives or dies thereupon? Why does
it have to be human consciousness that reaches the verdict? Or any
consciousness?
Ah, reading on, I see that Wigner was way ahead of me (no surprise there.)
I suspect that this layering of "someone has to look" on top of the
measurement question has been very misleading. The Sun continues to shine,
trees fall in the woods, all without experimentors. I know that common
sense is not always the ruler in QM, but the orderly progression of the
universe (both macro and micro) ought to but some constraints on the
axioms.
David Winsemius
John Baez
In article <astalder-3C452F...@news.bluewin.ch>,
Alain Stalder <asta...@exactphilosophy.net> wrote:
>My understanding of decoherence
>may not be deep enough, but if decoherence is just applied quantum
>mechanics, a system prepared in a state |S> = a|A> + b|B>, with
>|A> and |B> linearly independent, cannot be brought into a state
>proportional to either |A> or |B> by any linear operator,
Eh? You can certainly find a linear operator that maps
any state |S> = a|A> + b|B> to either |A>, or to |B>.
You can even find a unitary one that does the job.
This isn't too relevant to decoherence, though.
Frank Hellmann
Alain Stalder wrote:
> In article <3e26a...@news.arcor-ip.de>,
> Frank Hellmann <C....@gmx.net> wrote:
>
>
>>>It is worthwhile to explain what exactly "classical" means in
>>>this context. This is maybe most easily seen if Schroedinger's
>>>Gedankenexperiment is combined with the experiment for testing
>>>Bell's Inequality:
>>
>>[snip]
>>
>>Are there experiments on this interesting combination verifying the
>>Gedankenexperiment?
>
>
> that I know of is the one by the Zeilinger group of 1998, where
> the distance was about 400 meters, see quant-ph/9810080 and
> http://www.quantum.univie.ac.at/research/bellexp
>
What I thought of was to get eh entangled states to decohere as a whole.
In Bells equality we meassure the state
rho=(0.5 |1>|-1> + 0.5 |1>|-1>)(0.5 <1|<-1| + 0.5 <1|<-1|)
correct? And the probabilities extracted there are decidedly non classical.
If we let this state decohere we get
rho=0.5 |1>|-1><1|<-1| + 0.5 |1>|-1><1|<-1|
Do the same Bell propabilities hold for this evolution?
This would be a definite non classical behavior of a decohered state then?
Frank Hellmann