Exploring Hardy's Axioms and Their Impact on Quantum Theory

[MathematicalPhysicist]

what are they ?
i know they are related to quantum theory.

 

[PRodQuanta]

Why explain shortly and possibly misinterpret when YOU can read?
Here you go: http://arxiv.org/PS_cache/quant-ph/pdf/0101/0101012.pdf

Enjoy
Paden Roder

 

[marcus]

Originally posted by PRodQuanta
Why explain shortly and possibly misinterpret when YOU can read?
Here you go: http://arxiv.org/PS_cache/quant-ph/pdf/0101/0101012.pdf

Enjoy
Paden Roder

I wouldn't always want to start downloading a PDF file from arxiv without first
looking at the abstract. Some articles have hundreds of pages.
And the title and brief summary can sometimes tell you enough. Here is the abstract for what Paden recommends reading. If you like the short summary in the abstract then click on "PDF" button right below it.

http://arxiv.org/quant-ph/0101012

 

[marcus]

http://arxiv.org/quant-ph/0101012

I'm impressed. Thanks for the link. It is Hardy's original article, only 34 pages, and gives the 5 axioms

Here is an exerpt from Hardy's article, "Quantum Theory from Five Reasonable Axioms"
This quote gives a taste of what it's like:

------------
[[[Definition:

The state associated with a particular preparation
is defined to be (that thing represented by) any
mathematical object that can be used to deter-
mine the probability associated with the out-
comes of any measurement that may be per-
formed on a system prepared by the given prepa-
ration.
Hence, a list of all probabilities pertaining to all pos-
sible measurements that could be made would cer-
tainly represent the state. However, this would most
likely over determine the state. Since most physical
theories have some structure, a smaller set of prob-
abilities pertaining to a set of carefully chosen mea-
surements may be sufficient to determine the state.
This is the case in classical probability theory and
quantum theory.

Central to the axioms are two inte-
gers K and N which characterize the type of system
being considered.

* The number of degrees of freedom, K, is defined
as the minimum number of probability measure-
ments needed to determine the state, or, more
roughly, as the number of real parameters re-
quired to specify the state.

* The dimension, N, is defined as the maximum
number of states that can be reliably distinguished from one another in a single shot measurement.
We will only consider the case where the number
of distinguishable states is finite or countably infinite. As will be shown below, classical probability theory has K = N and quantum probability theory has K = N² (note we do not assume that states are normalized).

The five axioms for quantum theory (to be stated again, in context, later) are

Axiom 1 Probabilities. Relative frequencies (mea-
sured by taking the proportion of times a par-
ticular outcome is observed) tend to the same
value (which we call the probability) for any case
where a given measurement is performed on a
ensemble of n systems prepared by some given
preparation in the limit as n becomes infinite.
of N (i.e. K = K(N)) where N = 1; 2; : : : and
where, for each given N, K takes the minimum
value consistent with the axioms.

Axiom 2 Simplicity. K is determined by a function
of N (i.e. K = K(N)) where N = 1; 2; : : : and
where, for each given N, K takes the minimum
value consistent with the axioms.

Axiom 3 Subspaces. A system whose state is con-
strained to belong to an M dimensional subspace
(i.e. have support on only M of a set of N possi-
ble distinguishable states) behaves like a system
of dimension M.

Axiom 4 Composite systems. A composite system
consisting of subsystems A and B satisfies N =
N_AN_B and K = K_AK_B

Axiom 5 Continuity. There exists a continuous re-
versible transformation on a system between any
two pure states of that system.

The first four axioms are consistent with classical
probability theory but the fifth is not (unless the
word "continuous" is dropped). If the last axiom is
dropped then, because of the simplicity axiom, we
obtain classical probability theory (with K = N) in-
stead of quantum theory (with K = N² ). It is very
striking that we have here a set of axioms for quan-
tum theory which have the property that if a single
word is removed (namely the word "continuous" in
Axiom 5) then we obtain classical probability theory
instead.]]]

 

[marcus]

what strikes me is the
two core definitions and the fact that in a quantum
system the "degrees of freedom", as Hardy tells it, is equal
to the SQUARE of the dimension
(while in mere probability theory it is simply equal to the dimension itself) so I want to consider what he means by these two key numbers

Central to the axioms are two inte-
gers K and N which characterize the type of system
being considered.

* The number of degrees of freedom, K, is defined
as the minimum number of probability measure-
ments needed to determine the state, or, more
roughly, as the number of real parameters re-
quired to specify the state.

* The dimension, N, is defined as the maximum
number of states that can be reliably distinguished from one another in a single shot measurement.
We will only consider the case where the number
of distinguishable states is finite or countably infinite. As will be shown below, classical probability theory has K = N and quantum probability theory has K = N² (note we do not assume that states are normalized).

In the quantum case, with K = N² , we have that

"the minimum number of probability measurements needed to determine the state" is equal to the square of
"the maximum
number of states that can be reliably distinguished from one another in a single shot measurement"

Paden Roda, any comment about where the N-squared comes from?

 

[PRodQuanta]

I am nowhere near the expert in this area, but if I were to make a guess, here it goes...

In the statement, axiom 5 states that there exists a continuous reversible transformation on a system between any two pure states of that system.

I think the term "reversible" means that it adds a whole new set of probabilities to the system. Thus, it would give the N it's square.

This may not make sense for 2 reasons:
1)I'm speaking in a language that I can understand, but maybe not descriptive enough for others.
2)Lack of knowledge on the subject.

There are my thoughts. Make of them what you will.
Paden Roder

 

[MathematicalPhysicist]

Originally posted by marcus
I wouldn't always want to start downloading a PDF file from arxiv without first
looking at the abstract. Some articles have hundreds of pages.
And the title and brief summary can sometimes tell you enough. Here is the abstract for what Paden recommends reading. If you like the short summary in the abstract then click on "PDF" button right below it.

http://arxiv.org/quant-ph/0101012

i agree with you.