negative probability
John
(except for the article by R. Feynman "Negative probabilities", Quantum
Implication, essays in honour of David Bohm) ???
John Bailey
Fun question. If you had not mentioned the Bohm book, I would have
gone crazy, trying to remember where Feynman's article was. There was
a recent discussion of Imaginary Logic values on alt.math.recreational
http://groups.google.com/groups?hl=en&lr=&safe=off&ic=1&th=d78a03d109ac843&seekd=928905914#928905914
reopening interest in the general topic.
A quick inspection of the Los Alamos National Archive yielded several
papers which mention negative probability in their abstract or title.
This was a quick search, a more determined one might yield a lot more.
http://xyz.lanl.gov/abs/quant-ph/0010091
Local hidden-variable models and negative-probability measures
by Jose L. Cereceda
Elaborating on a previous work by Han et al., we give a general,
basis-independent proof of the necessity of negative probability
measures in order for a class of local hidden-variable (LHV) models to
violate the Bell-CHSH inequality. Moreover, we obtain general
solutions for LHV-induced probability measures that reproduce any
consistent set of probabilities.
http://xyz.lanl.gov/abs/quant-ph/0004109
Hidden Variables or Positive Probabilities?
Authors: Tony Rothman, E. C. G. Sudarshan
Despite claims that Bell's inequalities are based on the Einstein
locality condition, or equivalent, all derivations make an identical
mathematical assumption: that local hidden-variable theories produce a
set of positive-definite probabilities for detecting a particle with a
given spin orientation. The standard argument is that because quantum
mechanics assumes that particles are emitted in a superposition of
states the theory cannot produce such a set of probabilities.
http://xyz.lanl.gov/abs/quant-ph/0008122
Finite resolution measurement of the non-classical polarization
statistics of entangled photon pairs by Holger F. Hofmann
By limiting the resolution of quantum measurements, the measurement
induced changes of the quantum state can be reduced, permitting
subsequent measurements of variables that do not commute with the
initially measured property. It is then possible to experimentally
determine correlations between non-commuting variables. The
application of this method to the polarization statistics of entangled
photon pairs reveals that negative conditional probabilities between
non-orthogonal polarization components are responsible for the
violation of Bell's inequalities. Such negative probabilities can also
be observed in finite resolution measurements of the polarization of a
single photon. The violation of Bell's inequalities therefore
originates from local properties of the quantum statistics of single
photon polarization.
John
John
thanks, i'm checking them out!
The relevance of negative probabilities in QM seems better recognized than the plain mathematics of such
an artifact. I thought mathematicians loved complex, bewildering and imaginative and yet abstract
structures? Well, it seems here is one they have not really studied...
John
I was follwing the discussion you pointed to on Imaginary Logic values on alt.math.recreational.
You may be interested in
quant-ph/0007047 [abs, src, ps, other] :
Title: The Liar-paradox in a Quantum Mechanical Perspective
Authors: Diederik Aerts, Jan Broekaert, Sonja Smets
Comments: 16 pages, 1 figure
Journal-ref: Foundations of Science, 4 (2), 1999, 115-132
John Bailey
first half the paper, I thought it was all doubletalk, but from about
page 6 onward, it looks worth printing and studying.
To return the favor, I commend the article
quant-ph/0001074
An Epistemological Derivation of Quantum Logic
by John Foy
(abstract)
This paper deals with the foundations of quantum mechanics. We
start by outlining the characterisation, due to Birkhoff and Von
Neumann, of the logical structures of the theories of classical
physics and quantum mechanics,as boolean and modular lattices
respectively. Taking all possibilities - in a sense, the set of all
things that may be described by physical theories - gives the lattice
of quantum mechanical propositions. This gives an interpretation of
quantum mechanics as the complete set of such possible descriptions,
the complete physical description of the world.
John
This thread reminds me of the joke about the prisoners who just
swapped the NUMBERS of their jokes. Punchline--some people can tell
them, others can't.
John
> To return the favor, I commend the article
> quant-ph/0001074
> An Epistemological Derivation of Quantum Logic
> by John Foy
Looks interesting. Did you notice that principle IIa looks very much like what Jauch has called
'principle of superposition'. (Jauch, Foundations of QM, Addison & Wesley, 1966, p 106)
Jauch's principle of superposition says:
for any atomic e_1,e_2 distinct, there exists e_3, all distinct such that
e_1 \/ e_2 = e_1 \/ e3 = e_2 \/ e_3
(Jauch writes the set-theoretical union, but he means the same)
Now take c_a=e_1 b_a=e_2, b_a'=e_3
and apply principle IIA of Foy, p12
Remains to be seen if
a= b_a /\ b_a'
John Bailey
>
>
>> To return the favor, I commend the article
>> quant-ph/0001074
>> An Epistemological Derivation of Quantum Logic
>> by John Foy
>
>Looks interesting. Did you notice that principle IIa looks very much like what Jauch has called
>'principle of superposition'. (Jauch, Foundations of QM, Addison & Wesley, 1966, p 106)
> Jauch's principle of superposition says:
My text for this subject has been Hughes, The Structure and
Interpretation of Quantum Mechanics.
An intriguing discussion in Hughes surfaces the idea that logic may be
empirical--just as with geometry's parallel postulate and with the
continuum hypothesis, the distributive law may have variations, one of
which (?) must be selected empirically.
This leads to an amusing question:
Does the distributive law apply to its own validation as a logical
statement?
Lastly, while mulling over the complexities of quantum logic
I ran across this charming quote from Grover:
In a quantum computer, the logic circuitry and time steps are
essentially classical, only the memory bits that hold the variables
are in quantum superpositions - these are called qubits. There is a
set of distinguished computational states in which all the bits are
definite 0s or 1s. In a quantum mechanical algorithm, the quantum
computer consisting of a number of qubits, is prepared in some simple
initial state, and caused to evolve unitarily for some time, and then
is measured. The algorithm is the design of the unitary evolution of
the system. Operations that can be carried out in a controlled way are
unitary operations that act on a small number of qubits in each
step.
A basic operation in quantum computing is the operation M performed on
a single qubit - this is represented by the following matrix:...(here
follows a representation of a 2x2 matrix which will not transcribe to
a newsgroup posting)
(quoted from:
http://xyz.lanl.gov/abs/quant-ph/9712011
Quantum computers can search rapidly by using almost any
transformation by Lov K. Grover (Bell Labs, Murray Hill, NJ)
John
John
> An intriguing discussion in Hughes surfaces the idea that logic may be
> empirical--just as with geometry's parallel postulate and with the
> continuum hypothesis, the distributive law may have variations, one of
> which (?) must be selected empirically.
I read large parts of the book by Hughes when i started doing my PhD on foundations of quantum
mechanics. Actually, the idea that you express above has been brought by many many people from a
quantum probabilistic point of view.
It is easy to show in a simple case, such the probabilities related to thet outcome of an entity
that 'lives' in a two dimensional Hilbertspace, that the set of probabilities itself determines
whether these could have been derived within a Kolmogorvian framework, a real Hilbert space or a
complex Hilbertspace. Names connected with these issues are Mielnik, Pitovski, Accardi, Gudder.
In fact i myself wrote an article "Conditional Probabilities with a Quantal and Kolmogorovian
Limit", Int J. Theor. Phys., vol 35, No 11, november 1996 where i calculate conditional
probabilities related to two possible outcomes on a mechanical and macroscopical sphere model. In
one extreme you find Bayes law, on the other hand the well known square of cosine law (Malus law)
from QM. In between I identify a region that one cannot model by using either of them.
All that to say i am a strong proponent of the idea you express, although i have been looking for it
in a slightly different context, namely quantum probability rather than quantum logic!
Zim Olson
John
-----------------------------------------
John: After reading your post a few more times, I understood the gest
of it better. I am an amateur Mathematician, I have work professionaly
only in Computer Software and Information Science. I am still a little
hazy on what Bell's Inequalities are, but will go over some of your
refrences and links.
Zim Olson