I presume that it must be more than just be the logical notion of orthogonality; that is, that hilbert inner product =0, which would reduce it to probabilistic addivitity F(A or B)= F(A) +F(B) when A ,B disjoint on given bases

definitely not Cauchy's equation F(A +B)=F(A) +F(B) for a unique-ne probability functionn spin 1/2 systems, as there nothing to add up except normalization giving you F(1/2)=1/2 and maybe the degenerate F(0)=0 and F(1)=1;

which applies to arbitrary events whether atomic, or disjunctive, and where can A=B can be repeated often arbitrarily many times o

given its recent application to spin 1/2 systems. 

Is it a norm, plus a the logical notion, or a geometric and logical notion (so that it could be called geometrically orthogonal, or complementary /non commutting (in the logical sense) additivity)

Which do not have enough additive structure given probabilism. It looks to me that it sounds closer to a geometric notion (or Strong completementary or non commuting addivitity closer in spirit to cauchy's equation( l)

linearity/functional additivity (1)F(x+y)=F(x)+F(y)

then probabilism/additivity (2)F(x or y)=F(x)+F(y) where x and y are disjoint.

I can only presume its

(2) (hilbert inner product=0) plus a significant portion of

(1) where is construed as some kind of norm, or weird addivity rule relationship, between  events on distinct,geometrically orthogonal  bases, or between events oncomplementary events / non commutting bases . So as to compare, to compare sums, metrics and differences.

I presume  that it cannot be purely logical, and whilst it is often said that it is not purely geometric (otherwise in quantum mechanics it would appear to denote 'complementary additivity). I presume it must be the standard logical notion which is not functional additivity so much as probabilistic addivitity combined with something else, which amounts to adding up events, or comparing differences of events on non-commutting spacse

I cannot help but feel,  that this must me closer to it, given its use in spin 1/2 systems for the born rule of late

So it cant be purely logically orthogonalitity, this would reduce orthogonal additivity to probabilism

ie P(A or B)=P(A) +P(B) where A and B are mutually exclusive alternative on the same probability space.

In the logical approach, where orthogonality denotes , the Hilbert inner product=0,isn't this only the case when the bases completely commute; and in spin half systems, the only basis/probability space that antwo outcome probability/space/basis/ortho-normal basis/ etc commutes with is itself, So it would amount to not much much more that on each basis the outcome must sum one, P(spin up)+PR(spin down)=1, where even if one has a total order between and within bases, all you will get is F(0.5)=0.5 Pr(impossible event, in the pure state)=0 and PR(certain event)=1 whilst every other event will bigger than 0.5, and some events ranked higher again, but how much higher is another story (likewise with those less likely than not).

So I presume that without some kind of metric, such as in the james-ean Gudder version, nothing would come out of it, even if absolute continuity presumed, and the function presumed to be strictly increasing function of the amplitudes squared, events with bigger born rule probabiltiies might get assigned bigger frame functions values, and so it will represent the ordering of the born rule, but without being able to assess difference, 0.55=F(0.75)>F(0.7)>F(0.65>0.5; and the sums may not come out correctly ie if x+y>m+G there is not guarantee that

F(x) +F(y)>F(m) +F(G), if one of the inequalities is reversed, ie x>>G>m>y, x>>m, x>y, yF(m)=0.49,> F(y)=0.2

x=0.95,>   G=0.51>m=0.31>y=0.3,

yet x+y1.3>>m+G=0.82 ie 1.3>> 0.82

Yet the degree of difference or numerical metric is not

F(x)+F(y)=0.8G>m>y, F(x)>F(G)>F(m)>F(y)

the function would still represents the born amplitudes where x>y iff F(x)>G(y), x=y, iff F(x)=F(y), x

There is a whole jumble if different stuff mixed up here, but Ill try to clear up ideas by parts.

In usual

 probability theory independent events A and B means

that P(A) = P(A/B), the last is probability of A given B. ie. that A is not influenced by B. Look it up in probability books.

Equivalently P of A and B is the product of both.

If you start to tread into Hilbert space, state vectors etc. this is math related to quantum theory, nothing to do with classical probability. It has its own different probabilities.

The wave function for a complete system is the sum of all

the wave functions for the particle to be in separate states i, weighed by coeficients.

The wave function of the different states are all orthogonal to each other, if the index is different. These states are eigenstates of some hermitian hamiltonian.

The coeficients, the square modulus of them, are the probabilites to find the particle in that particular state. The amplitudes or coeficients are the inner product of the full wave function with that of a particular state i.

Thus the state of spin 1/2 is ortogonal that of -1/2

Simultaneous measurement in the quantum theory, without one

measurement  influencing anoth is obtained from commutivity of

 operators representing that dynamical variable which is measured.

This is sort of analogous to independent events of classical probability theory.

But obviously the calculation of probablility in the quantum is specific to physics and has nothing to do with throwing dice.

@juan, I am wondering about one main theme. Basically distinct conditions are under which two events are considered are disjoint, complementary or orthogonal. And which hilbert space metric denotes it..

And given that to which notion does orthogonal complement or mutually orthogonal events, or triples pertain (if its vectors) it seems like (2); as these are events on distinct basis (they may commute perhaps I do not know the exact terminology), but they are not events on the very same basis which are mutually exhaustive and exclusive.

, - disjoint-ness (in the sense of  being  that they sit on the same basis and mutually exclusive - in the kolmorov like sense ) so if one were to presume just  a probablistic representation of the amplitude moduli squared, one would be able to add the frame function probabilities up. Ie mere probabilistic additivity ,

- Complementary, where one cannot explicitely assume  this even given (1) , where the events might be on distinct bases or at right angles to an another. 

As it would be akin to cauchy additivity, (ie it would presume uniqueness essentially) . It was derivable by Gleason for N\geq 3 from (1) and the quantum nature of spin 1 system due to equalities in the rank roughly (as a result of quantum logic and non -contextuality).As far as I can see which would allow you to find common terms in different bases which must be assigned the same values (this will never happen in two atomic probability quantum or not 1/3 sits alongside 2/3 is not connected to every other event ie 1/3 1/3 1/3 1/3 1/2 1/6 (any two events which add up to 2/3)

I presume that they could assume (1) for a spin half system but a nothing much would result from it (ie a unique representation)

I presume that GLeason derived the orth-gonal triple property, and odd-ness for the frame function, and did not assume it unless, he just meant probabilism (1) and he derived a form of (2) from it

. As otherwise it would be close to presuming (2) cauchy ie unique-ness: So I get confused because there is sometimes some conflation as to which kind of additivity is meant

'Cauchy additivity' or 'probabilistic additivity in the kolmorov like sense', the former being much stronger uniqueness property,

the latter , kolmogorv/quantum analog of probabilistic representability-(as opposed to a unique probabilistic representation) not that one cant sometimes from the probabilistic additivity/under disjoint union/normalization

PR(A)+PR(B)+PR(C)=1 where A, B C disjoint mutually exclusive ie the frame functions sum to 1 in the horizontal on each basis (which is where the amplitudes square sum to one, and denote mutually exclusive and exhaustive events

for instance (A) F(1/3)+F(1/3)+F(1/3)=1   on the very same basis x,y,z of mutually exluive and exhaustive events or F(1/2)+ F(1/4)+F(1/4) if those events are on the same basis

Any arbitary atomic event /or disjunctive? events whose amplitude mod square, sums to one must have frame function values which sum to one, whether the coordinate are the same or not? Or a restricted version of this ;

ie for any events full stop if x+y+z=1 or ||x||^2+||y||^2+||z||^2 F(z)+F(y)+F(z)=1

spin up at direction/basis with amplitude moduli=33

spin up at 2\neq at direction/basis with amplitude moduli=1/6

spin up at 3\neq at ,  \neg direction/basis with amplitude moduli=1/6

where the bases may be something like 11/3 1/3 1/3

                                                               2  1/2 1/3  1/6  

                                                                2 1/6  1/2       1/3

Where one is not add the frame function values to one along the horizontals (the same basis but also along, the columns , and the diagonals as well whenever the amplitude mod squared or chancessum to one in said columns/ diaonglas as well.

Ie in this first colum and the diagonal

Any arbitary three events in the system disjoint or not, on different vectors/basis,  atomic, or disjunctive or not? if the amplitudes mod squared sum, to one, likewise the frame function probabilities

(the gleason property) a strong step towards cauchy attivitiy

- orcauchy addivity in certain systems.

, if the means unrestrictedly events three events or at least events on distinct basis whose amplitude mod squared sum to one, must have frame function values which sum to one (that is a unique-ness property you can have probabilistic function as a function of the amplitude squared, at least on the face of it, which represent or preserve the order of the amplitudes squared without that property as n=2 shows.

@juan i am more concerned about what counts as orthogonal, I presumed it was whenn the sense of evenst being on the same basis were disjoint ; but now i presume its when the vectors are orthogonal and that the hilbert inner product is zero, where this is not always the case

if a state is spin up 1/2 and another is spin down  -1/2, i presume that they will only be orthogonal if their vectors are distinct

u=

v= so that u1*v1+u2*v2=-1/2 + 1/2\times 1/2=0,

I presume that orthogonality meant disjointedness which was precisely whenever the events or vectors were not at right angles but were one and the same,on the same angle (not perpendicular).

Or is there an inner product for events within the same vector, as how does one take inner product of two events, i thought it was a relation between two basis, is if an event is spin up 1/2, that is just the first component of the vector,  does one just consider the second component to be zero  so that for

v  on the same basis, is ' the event inner product  of v1 with v2 zero, the basis inner product v^2= (-1/2_^2+1/2^2 =1/2> 0= (and perhaps normalized to one, so that it equals 1  which is still >0 ) with itself would be one, entailing that is the same angle/basis not geometrically orthogonal, events on Spin 1/2 system events are generally logically orthogonal/disjoint, can add (sit on the same basis) 

whenever they re not geometrically orthogonal, ie the two notions are opposed

When events are non-commuting or geometrically orthogonal i presume that the euclidean inner product of both bases is 0, but their event inner product is not 0 or rather logical Hilbert space inner product is not zero, so they cannot add?)

Which would be diametrically opposed to when the geometric inner product is zero (at right angles ) maximally non commuting  in a spin 1/2 system which to do with orthogonality of vectors or coordinates.

Is there a distinct kind of inner product for this, euclidean?, within the context of the same spin system but when considering distinct bases?

The very same vectors would appear to have an inner product with itself of 1 (but I though that this was precisely when the events are considered orthogonal,

An event in QM would be a measurement of one of the posible eigenvalues.

I understand complementarity as more a concept of set theory,

that is everything in the universal set not contained in the set whoes complement is sought. It depends on what problem one is dealing with. (ie what the set elements are)

Orthogonal is when the inner product is zero (a common math term). for vectors , or functions as a generalization of vectors in functional analysis. In QM such vectors are

Complex in general, so in the inner product one takes the complex conyugate of one of them in the inner product.( integral over all space of their spatial arrgument.)

Orthogonality means physically that one wave does not interfere with the other., so the different states have orthgonal eigenvectors or state vectors.. Hermitian operators have orthogonal eigenstates.

The operators may be comutating or not. Different dynamical variables are assigned different operatos.. I already explained the meaning of commutation.

The eigenstate are the effective base vectors in the infinite dimenional hilbert

space, an arbitrary vector or wave function is epressible as a linear combination

of these,in exact analogy of R3 vectors. Look into functional análisis for systems of ortogonal functions. (or special functions)

The result of a measurement is one of the eigenvalues of the corresponding operator with the probability already explained

I presume they mean orthogonal in hilbert space norm and just geometrically, so I presume that events can be orthogonal whilst being on distinct bases of the a spin system. At least if its a spin 1 system. I presumed that in spin 1/2 system all of the bases do not commute and thus the only event that is othogonal to spin up x, direction is spin down in the x direction, as an orthogonal projection (or where orthogonality of the vectosr is given when the  the kronecker delta=1) , which when its very same basis.

Events are mutually exclusive.

If you measure spin 1/2 you dont measure spin -1/2

The corrsponding eigenstates are ortogonal and normalized

say (1 0) for spin up and (0 1) for spin down

usually vertical brackets

The general spin state would be (a b)

 a normalized bracket to 1

that just looks like probabilism within each basis; by spin 1/2 you mean the eigen value designated to up and down. I am taking more about orthogonality of vectors

where vector 1,

1/sqrt2 spin 1/2 + 1/sqrt 2 spin -1/2

-1/sqrt2 spin 1/2 + 1/sqrt 2 spin -1/2

=0 =1, =1, unit vectors which also denote that the event on the same basis count as orthogonal,  but also that u,v =0, as i\neq j, so that the vectors u. v are orthogonal does that mean one can add the probabilities of events on orthogonal vectors. not just orthogonal projections of the same vector/basis

I presumed that orthogonal additivity was stronger in and of itself, then probabilism. It seems odd that the issue with the born rule was really unique-ness not probabilism. 

A

nd that assuming probabilism a second time, is not going to make it any more unique for n=2. I thought orthogonal addivitiy was closer to cauchy additivity and thus stronger then probabilistic additivity of mutually exclusive events.  Not that cauchy additivity  and the gleason global property of normalization cannot be derived from local within basis normalization given the structure of quantum logic and non contextuality.

I presume that Gleason meant something stronger then three outcome normalization by his orthogonal triangle property. I presume in my language is closer to (1) then (2). I presume he meant three orthogonal vectors (which could have nine events, three on each basis) and if any three of those 9 events sum to one in amplitude mod squared, so  did the frame function; which is much closer to this

where what I mean by probabilism is horizontally additive, F(x)+F(y)+F(y)=1, x+y+z=1,

F(x1)+F(y1)+F(y1)=1, x1+y1+z1=1,

Fx2)+F(y2)+F(y2)=1, x2+y2+z2=1,

 where what I mean by something more global is not just the above, but if is if any of those nine events sum to one. x1+ y2+ z3=1,then F(x1)+F(y2)+F(z3)

or x+x1+x2=1 then F(x1)+F(x1)+F(x2)=1

ie adding up the events vertically or along the diagonals. which is far stronger.

a and likewise for the other two. Note also

i(1)e forall x,y,z in [0,1] if , x+y+z= 1  then F(x)+F(y)+F(z)=1,

which is significantly stronger then

(1.a)merely A, B , C disjoint and mutually exclusive F(A)+F(B)+F(C)=1,

given F(0)=0 alone (1) collapse into cauchy additivity function over the unit triangle with F(1)=1,  as it entails if, x+y =1 thenF(x)+F(y)=1, which is the any given two analogue. given non negativity or codomain is [0,1] continuity, strict monotone increasing ness F(x)=x is immediate if the domain is [0,1] and

Although what counts as mutually exclusive , in quantum systems spin (1) in quantum mechanics might make it look (2),  as you can derive cauchy additivity

F(x+y)=F(x)+F(y) for all x,y, in [0,1], 0

There are rules of what happens when you Exchange the places

of otherwise two identical states according to Fermi or bose

statistics

Both the states you wrote down indicate equal likelihood of having

spin up or spin down

But they can be distinguished according to the statistics

In this case spin 1/2 are Fermions, so the sign changes by -1

You are only left  with the second one.

Psi(1,2) = -psi(2,1)  for fermions

psi(1,2) = psi(2,1) for bosons

Dear julian thanks for this, But am I correct in presuming that because the vector inner product of those states, are zero, that the corresponding found  for any two events in distinct state of that form,  such as spin 1/2 or spin up in state 1  with amplitude -1/sqrt(2) and thus probability 1/2 and spin up  in state two ie spin 1/2 with probability  0.5 amplitude positive 1/srtqr(2) are allowed, to add to be added in a probabilistic representation. Or rather count as disjoint or orthogonal in the probabilistic sense of the world.

Bosons are are spin particles that pertain to spin systems with a odd number of atomic elements dont they  ie spin is a whole number such at 1 (three outcomes) 2, (5 outcomes); fermions are those with an even number of atomic elements or projections on each basis or for precisely spin is fractional; spin 1/2.(2 outcomes , spin 3/2 (4),  outcomes

psi(1,2) = psi(2,1) I presume you mean psi(1)psi(2)=-psi(2)psi(1) roughly where when |moduli of spin up| = |moduli spin down| on the same state, the spins must be distinct ; 

Or I presume that whenver the norm of the amplitudes on each basis are the same, for each outcome, as the eigenvalues swap so do the amplitudes, where I presume one take the absolute value of the eigen values (which are always the same in  a spin system) and the probability amplitudes of spin up and spin down, which must be opposed in sign whenever the their absolute values are the same.

so that the first vector is ruled out. Ie a sign change in the spin eigenvalue corresponds to the amplitudes having an opposite sign.

or whenver  |moduli spin up| =|moduli spin up|

for any vector in the system, as the eigenvalues are the same, the amplitudes must have a distinct sign

between vectors where as these have the same eigenvalues, the amplitudes must have a distinct sign

so spin down  onthe second vector would not be possible  as a possible basis, if the first basis is possible, with the first unless I changed it with 2 with (3) as the amplitude of spin down on both have the same sign, the same moduli,  the eigen values are the same.  and they both denote spin down, and thus the amplitudes themselves have to swap.

(3)1/sqrt2 spin 1/2 + -1/sqrt 2-spin -1/2

the eigenvalues(1)*amplitudes(spin upS 1)= -eigenvalues (2)

f each given state.

So I presume that  one of these states is simply not possible at all, (or clearly are not c-measurable)..

1/sqrt2 spin 1/2 + 1/sqrt 2 spin -1/2

when you say  this do you mean that the equiprobable state orthogonal quantum states, are those pairs of states where the amplitude (roughly)  of spin up in basis 1= amplitude of spin down in basis 2 and conversely;

S1=1/sqrt2 spin 1/2 + 1/sqrt 2 spin -1/2 is impossible

I know that by the pauli exclusion principle that the same quantum system if fermionic (not an integer spin ) cannot be in the same state at the same time, by this I presume it doest

means the or norm of the two (possible) super-positions vectors S(1), S2) expressed as states in a hilbert space cant be the same state, roughly have the same absolute value and thus if i presume

, in the same spin 1/2 system prepared in the same eigen state, yet to be prepared.

||S1||=1, ||S2||=1 given by the inner product over vector space, as these are presumably unit vectors,

And I presume it does not mean meerely |phi(x)|=phi(y) which presumably gives the moduli of the amplitude a positive number a for a state phi to transition into outcome x=spin up . where but rather some kind of norm of over Phi_1 or phi_2

I think I am confusing the orthogonality/disjointness of the very same singular measurement,  of outcomes within a measurement,.basis,

spin up =x1 @ superposition 1 versus  spin down=x2 @=super-position 1

= ⟨, which I presume is expressed rather as \phi y\phiA(x)=-\phi(y)\phi x

disjoint-ness,

with orthogonality of vectors,  (Ψ1, Ψ2) = 0

Φ_2=1/sqrt2 spin 1/2 + 1/sqrt 2 spin -1/

Φ_1=-1/sqrt2 spin 1/2 + 1/sqrt 2 spin -1/

and orthogonal vectors which is given by the kronecker delta function where if these are the same superposition it is 1, else zero; where orthogonal projections may be the events themselves on said vectors, whose Hilbert inner product is 1 or 0.

I presume that there must be some terminology that distinguishes those Events that are orthogonal in the sense that they lie on the same vectors, whose inner product with itself is given by 1, and those which are 0, which I presume are events orthogonally complements x,y {^\perp}

and x\perp y, i am not sure which of the two is being used in orthogonal additivity, ie events which sit on orthogonal vectors, or events which are themselves are orthogonal in the context of probabilistic representations.

Both of which I presume are distinct from geometrically orthogonal vectors which are (maximally non commutting) and are at right angles in the cartesian euclidean norm.

And orthogonal projections which might be an encompassing same for events that orthogonal but restricted those which sit on distinct vectors (orthogonal vectors) in inner product (ie =0).  

or which are events that lie on distinct vectors whose inner product with one another is 0, sum to 90 degrees( in some sense

I presume orthogonal complement or perpendicular complement refers to the case where its zero, where these

and where the norm may not be positive ||x||, (unless its a unit vector for instance ); which I presumed were just those consisted of three outcomes whose moduli amplitude squared summed one

and orthogonal projections which come somewhere in between.

unlike the moduli | | of an outcome |spin up|

 which may again be distinct from the geometric orthogonality relation which

or bases or superposition of distinct possible measurement (all of which I thought did not commute within the one system ) i; with non commutatibility which may be different again.

I thought one expressed orthogonality, as the a disjoint-ness of outcomes (spin up and spin down on S1) such as spin up or spin down, for the same preparation and measurement on a spin system, within the same vector, or componennts or projections of the same vector/basis/superpositions

whilst complementarity or non-commuttation;  was with orthogonality with respect to the the vectors, in hilbert, space, itself. where generally bold face is used, and a distinction is made between ||S1|| and the moduli |si(x_1)| where S1 is a vector whilst s_i expressed the amplitude moduli of an outcome or component of a that vector S1, ie a projection such as spin up x occurs on basis S1 with amplitude =||si(x_1)| 

S1, S2, super-positions themselves, orthogonality of distinct measurements of the same spin system prepared in the same way, ie distinct basis/super-positions 

but measured at distinct angles(geometrical orthogonality) or dis-jointness of measurements, and thus where such events that are components of such systems are conditionally orthogonally or complementary (orthogonal complements)

e that geometrically orthogonal superposition's were often non commutting measurements; and could not add, (in probability)unlike disjoint outcomes of the same measurement and superposition

above being the same, or the moduli (of the amplitudes of each of the individual outcomes spin up and spin down in s1 and spin up and spin down being the the ) within

I apologize as I sometimes confuse notation I note that two identical particles, cannot in the same quantum system cannot be in the same quantum state simultaneously I presume this means the the vectors or the wavefunction itself must suffer a change in signe F(x,y)=F(x)F(y)=-F(y)F(x

|ψ(x) |r Ψ.

Does this mean identity of superposition identified with having the same norm ||x|| such as superposition vector 1 and vector 2, or the observables spin up and spin down having the same moduli within each superposition

I presume that when this means in a spin system prepared to be a in a certain state, subject to a certain measurement,

or for fermions both entries have their sign change by - 1 in one case; i presume you can cant have :

state; 1 1/sqrt2 spin 1/2 + -1/sqrt 2 spin -1/2

as such a state denoing the same measurement would have spin up and spin down as complementary?

state -1/sqrt (2) spin 1/2 + -1/sqrt(2) spin -1/2

I think what I amtrying to get out as the distinction between a sum over states which is cauchy additive and the sum of probabilities within each state,/vectors /superposition,. And the huge distinction between cauchy additive and probabilism additivity

the former being closer to the standard notion of mutually exclusiveness; or an orthogonal frames and probabilities as sum overs states rather then just considering each superposition/vector, to be its own finitely additivity probability function/space ordered tripe with a continuum order over these vectors. Justified for whatever reason. so long that N>=3 then from local probabilism of both entities one gets Cauchy additivity (and otherwise ones does not ). And this holds whether its quantum or not generally. That is whatever occurs locally, can be induced globally, putting a metric on the order, despite the fact that these events are not disjoint in the sense that they sit on  disinct events. Which essentially ensure unique-ness. 

I get the impression that some people think that cauchy additivity is a trivial presumption and that there is no distinction between it and probabilism. THere is all the difference in the world! Once cauchy addivitity is shown, generally, the unique-ness of the probability is guaranteed, there is nothing more to prove.

 So what is the common agreed upon properites of quantal probabiliism that both functions can be presumed, if compatible with it, to satisfy. Orthogonal adidivity>

with * events in F for a spin 1 system that is continuum de.

As in or a given quantum state, one could think of a spin vector, where one has not additivity within a spin vector, but over them, a probability as a sum over the distinct states, on each distinct vectors.

where they form a ladder like standard sequence, which allows one to you  to insert solved value into other spaces, due to the equalities in the rank, from quantum logic and non contextuality

I think its even called probability as a sum over states.

And that Gleason property could be an orthogonal frame property. I presume that is was derived from within spin vector orthogonality of events plus equalities in the rank. And so I am asking whether the quantum space is closed under unions within each vector not over the vectors. as such

I am wondering if this was presumed. or derived as result of probability only p

In the quantum you dont add probabilities. You add states

Adding probabilities (under suitable conditions) is a concept of classicl probabilityy

yes, i was about to note that I found an article on spin vectors as connected continuum dense sets of connected finite probability spaces 3 atomic probability space, I came across that. However, in the context of unique-ness i presume that is somewhat related.

For isntance,, when working towards a unique representation; for instance if one has a conception of quantum objective chance or moduli squared, what are the common agreed upon rules that both entities satisfy,

So one can have for arguments sake that that domain function is linear if that is how it is defined, already; as that is not what one trying to represent as such. I presume that same goes for the born rule or amplitudes squared; they are what they are.

One is trying to show that given the per-theoretical agreed upon rules of quantum probabilism, if there be such, that both function agree on; Given some ranking methods, and then show  that both functions are necessarily the same. BY deriving the richness contraints/structural constraints.

If i were to look at the axioms of a frame function I would consider, non-negativity , separability (or some part of it) part of quantal probabilism, and thus necessary conditions for respresentation not for unique-ness,. Along with quantal probabilism (if there be such)

and I would classify odd-ness and the triple function and (over and above) properties, F(1)=1, or richness properties that were derived, from the rank order, and quantal probabilism that lead to uniqueness.

Butt if the frame functions conditions just are the axioms of quantum probabilism. Then quantum probability representations are unique by constructions, that is they are More additive, then classical functions. As they are essentially cauchy additive, and I  presume that there is a distinction in quantum mechanics between probabilistic additivity and cauchy additivity.; there is a huge difference. When they say that there is a huge difference between additivity and linearity they are not saying there is a huge difference between cauchy addivitity and linearity (there isnt), they are saying there is a huge difference between cauchy additivity and probabilistic addivity. Continuity is automatic generally given cauchy additivity,

Otherwise the distinction between representation and unique representation would disappear.

Then that is saying, or at least nearly, that quantum probability functions (not measures) are more additive), they are globally and unrestrictedly additive, whether the events are disjoint or not or completely unrelted,

Some of the language you use drives me wild.

Instead of Quantum probability function use wave function, for it has nothing to do

with your Clasical distribution function (whih carries probability density)

As I partly mentioned before the full state of the Physical system before measurement

is described by the full wave function, which is a linear superposition of different eigenstates the particle may be in (The one particle case to make it simple).

Solutions to Schrödinger are subjet to superposition principle.

Just after measurement you get as a result one of the eigenvalues that you

previously calculated, and the full wave function has collapsed to the corresponding

eigenstate.

Of course this collpse sound strange, and bother many people. The very concept of

measurement is  problematic and very debated.

Quantum mechanics consists in  bit of math, plus interpretational rules, which are rather important. It is not enough to know just the math.

Dear Julian I apologize for this.

. When I speak of  the infinitely many possible albeit distinct but connected unit triples/states/bases, within a  singular prepared spin system , is the correct terminology, for these entities or probability vectors, a spin vectors

 triads or doubles in the case of spin half system  spin vector?,

I dont see why you would use three ortogonal unit vectors as in R3. in these cases.,

much less many of them?

. There is an underlying spatial and time frame for Schrödinger but that is all.

There are it is true three components of angular momenrum operators, or

a spin operator which do not commute among themselvs, are you talking about this?

For spin 1/2  you can represent as Pauli matrices.

You are left with just two you can use simultaneously L(z) and the total angular momentum operator. LL

look up spin and angular momentum algebra in your books.

For electrons S(z) has eigenvalues 1/2 or -1/2, that is what is talked about.

ie

the quantum analogue of the triples in the unit triangle.

And the frame function not being so much a classical probability measure but akin o a triangle function; of course that hearkens to the notion of a distribution function.I do not mean that however, I just use a simplex model to build a notion of a lot of ontically connected hypothetical and logically propensity spaces, that are all modally- connected

It has nothing to with classical statistics.

(propensity is my analysis if the chance concept; as single case objective in-deterministic ). Not

I too, do not like classical notions of probability. When I use the word function.

It may be conflated a probability function or probability measure in the classical sense.

When I say a function, I mean function in the usual sense,  that just so happens to based on a map between two kinds of probabilities.

Which is really nothing much like a classical probability function or measure of distributiion function in the usual sense, at all. Which is generally a function between events in a sigma algebra and a numerical range. or between some quantity, and a probability value, which specifies the distribution of density of it. I mean this more in the unique-ness theorem representation sense, where the domain represents ones analysis of chance propensity etc(translate amplitude mod squared), and the range or the function itself, represents the true objective credal measure (translate, frame function values, true quantum rule). ANd one wishes to show that F(x), the creal measure or true quantum rule is equal to the domain, the rule specified by amplitude mod squared or the chances or propensities

There is no density function as such at all. And its not an enterprise in statistsics.

A

So generally one gives some reason as to why the function values must be ordered by the domain elements, this can be done for n=2. and then one proceeds to show that one must not only respect the chances, but ones' credences must be numerically identical (which generally cannot be done, without further splitting mixtures etc, or other contrivances, unless n>=3 and its entangled, the worlds are, in quantum like way, which gives rise to enough additive structure despite the fact that there are only three disjoint events at any one time., its reducible, the equalities in the rank allow one to put the frame values assigned to elements in one triple onto others.

Or to put atomic events onto disjunctive events. It can be doine with out closure under union as well. If one has degenerate spacse and so without the any given two rule rule x+y=1 iff F(x)+F(y)=1 in the vertical; if one assignes F(0)=0, and one has spaces like this which allows you to get the same result through horizontal normalization which always gives rise to vertical and global normalization (and thus global any given two, ;

The distinction being when I say vertical I mean adding up events on distinct triples

Strong sense (global vertical sense); its essentially jensens equation

(1)x+y+z=1 iff F(x)+F(y)+F(z)=1 ie cauchy normaliztion, this is a uniqueness property

 As opposed to this weak sense (probabilistic) or merely horizontal sense

where, (2) A B, C, A,\cap B, A\cap B, B\cap C=emptyset, Acup B\cup=T, are mutually exclusive exhaustive, sit on the world/triple, and A, B C are atomic events

(1) F(A)+F(B)+F(C)=1,

The same goes for the distinction between x+y=1 iff F(x)+F(y)=1 which is also global (often called symmetry) and normalization for two outcomes or complement additivity in the weak sense :

Versus strong sense (symmetry)

(1) x+y=1 iff F(x)+F(y)=1, everywhere, same triple vector no/double or not,

or oddness.

x+-x iff F(-x)=-F(x), which oddness around 0. for [-1,-1

=3 quantum probability function must be a quantum probability frame function(ie unique linear) cauchy additive. I presume he had to first show that a quantum probability function for n>=3 must be a frame function , because that is over and above being a quantum probability function, and if its isnt, it seems to almost imply uniqueness from the outset, as its essentially cauchy additive?

nd whilst what I talk about my look like a cumulutative distribution function which gives areas under the curve for different parameters. Its nothing of the sort. Its rather a function F(x) from a one type of probability values, x to another kind F(x)  where aims to show that one entity the domain uniquely gives the values of the other other kind of probability

its a uniqueness theorem. for instance, to show why if. the if the propensity or objective chance in A is 0.5, your credence in A should be 0.5 its a multidimentionsal representation, global propensity to credence function

Or the numerical Principle principal. Sometimes the born rule is called the quantum principle Principal. So I am asking about when one is trying to show that given two probabilistic like entities, propensity and credence in my case, or in the quantum case, the onitic  quantum chances/amplitude mod squared and the true objective frame function. Where one already has the prescribed rules for the ontic chances. What are you allowed to presume about the frame function or credence function in quantum cases- in order for a uniqueness theorem to be non circular

? That is apart that its satisfies the basic axioms of quantum probability jif there be such, for the frame function, one can use the basic axiomatic structure of the as the amplitudes mod squared do, of quantal probabailism but not cauchy addivitity; as there would be no point to the exercise. That the two entities are one and the same in value would given to you on a plate

that they must match each other (ie a uniquetheorem). what are you allowed to presume about the

. That why I am interesting in quantum ideas.

which doesnt denote the probability value or the area under which that probability value 0.5 or less would be picked ou,

F(0.5)=0.5, for example means in my case,

t it means the objective credence that an agent should have in any arbitary propensity 0.5 event in the system is 0.5 regardless of which hypothetical, triple they are on. or which 'event ' has that value, where that they do or measure or beleive has implications for what would have been had they been somwhere else in the simplex. or had they believed differently

It gives general triangle /like unique-ness function and so in that sense is not like classical uniqueness theorems at all. For instance that is why i ahve the n=2/n=3 issue, each space is an ontic finite set propensity space, where additivity is presumed only to be local within each space. (the credence function would sum to more than one otherwise, in anycase, as the credence function as a function of the propensity values, sum to one, on each vector individually). 

So when I speak of vertical additivity I mean presuming or deriving the ability to add, credence function values  of geometrically distinct yet connected events/ on distinct vectors.ie to derive that as that the credence one should have in 0.25 event occurings, had one been  at world 1 and the credence one should have in the propensity 0.25, events at world 2   were one there ust somehow some add up to the credence one has in two propensity

wolrd 3 events, already resolved at F(0.5,A0=0.5, F(0.5\neg A)=0.5, due to finite normalization

that s the two 0.25 propensity events they are equally probable for whatever reason,  as each other,

then you must form a  weird inter world disjuinction to say that they are logically equivalent to either of the two 0.5 propensity events, at world 3 and thus their credence sum must be assigned the same credence values   as as each other and must be equal to the credence the agent has in the propensity 0.5 events, which is resolved say at credence =0.5; to resolve that that the objective credence value of the two 0,25 events are each individually 0.25, and somehow sum up to this weird event on another world. Thats what I mean by vertical additivity. I presume that its not a rule of quantum or classical probability  in the context of single case ontic chance to objective credence uniqueness theorems. Its something aims at deriving

, already resolved at 0.5

that they your credence in the 0.25 event must add up

to

Propensity space in my case. In any case its not a distribution over propensities

propensity

on distinct probability triples

(that has nothing to with classical probabilism) ie trhat any give any 7 events iun the plane whose propensities sum to 5, must have credence values which sum to 5

but where the agent has hypothetical views about what would have been were they on a distinct vector in the simplex

n that I cannot just the entirety of the events probability values up (it would be infinite). Nor can, as they are not disjoint, they are present, in that sense they are non commuting, 

,. The idea is to show that the propensity function should be the agents credence function, that their credence should be given by the chance values

(ie the events of which there are uncountably which ahve a density function over them) y

But of course i do not mean that. in the sense that the density is one over the cartesian plane). As that need not be the case , and that would be a density over parameters. or sets of probablity of events. I still mean the probability of event, where there are m

The three outcomes might sum to one on each space, but I mean something closer to an ontic projective space, where these are all of the possible worlds on may be in. NOt

( as if there a probability function over, which desribed the probability of ociking out three parameters,riple).

Sometimes when the word basis is used (it is meant to refer to the degerate , out of which each simplex/spin system is contructed.

each basis or state state, which looks at first sight to be an individual finite qm amplitude mod squared 'space' , where the function however, is global (over the entire infinitude of these distinct albeit connected vectors in the sphere/spin system)  In that sense its not a classical probability function in the usual sense

all of the un-countably many triples of events, each triple being located, as it were, at a different spatial position in the sphere. in a projective or hypothetical kind of states space, each triple oif three distinct events,  being itself a state.

_(x,y,z);

where  this a probability triple, roughly, or a unit vector , at angle (x,y,z) or rather perhaps I should use polar coodintes, and the 'probabilities; are given by the moduli amplitude squared, a, b,c have amplitudes, x,y,z, respective, roughtly. These triples, in say spin 1 systems, I was going to identify as a connected finitely (sum  of squares, mod amplitude) qm space, but the function is really over the uncountable infinitude of these entities in the sphereical quantum simplex. I presume that these entities are called spin vectors, (i may have accidentally called them possible superpositions, or projective,frames, or some such ;

i think the latter sounds better, they are akin to distinct hypothetical sets of amplitudes that the same events, in some sense, could have had); that are all modally, or logically entangled in some sense in spin 1 systems (again might not be the correct terminology, but it relates to the quantum logic connections that inspired gleason, ie, what obtains on one basis or vector has a logical connection with what obtains on the other) which was the first half of the quantum 'ranking methods.

of the very same prepared spin system)

space, (I was going to use quantum probability function ) but I t may not be appropriate

these  i not that in quantum mechanic there  in spin systems there is also a distinct between the number and type of distinct eigenvalues events can can have

However, I deliberately use the word function because essentially all I mean is a function in the classical sense (which happens to have some basis in the concept of probability, whether classical or quantum)

. For instance a classical probability measure (density or ) or function, is not a function in the usual sense o

f F;[0,1] \to [0,1] that is bijective,  ie the events have to sum to, or have density of one (in the classical case) of 1, roughly.

In my case, my model, as it were, its not one large probability space non atomic kolmogorian probability space. Its a kind of lattice, of infinitely many finite space, where only the disjoint and mutually exclusive atomic events (projections)

on each space or triple or vector are presumed to sum to one ( (tthe same number of events on each triple/wolrd) may only be three, may or may not be closed under finite union) . Its then a function over the distinct triples, of which there are uncoutnably many, where there is a global rank between the events  in the structure/system, whether they are on the same triple or not. Additivity, of credence or the triangle credence function is only presumed to hold within each vector, Although I know that global or 'orthogonal;' or cauchy additivity can be derived, due to ability to insert values of events on other bases/spaces/triples/vectors, into other vectors. There also consists the three degenerate spaces , , , where 1 here is distinct from \Omega

I am not sure if I am using this terminology correctly but I presume; please correct me if wrong (which I presumably am). That if, on a spin system. When it comes to a single case 'probability rule'  unique-ness theorem,

I understand, that some the terminology I use may be incorrect. I am just having a little difficulty at the moment ensuring that I dont confuse certain distinction- orthogonality of spin vectors, vectors, geometric orthogonality, orthogonality of vectors and states.  I presume then that orthogonal addivity as a functional equation as used by Gudder, and De Zelah, in spin 1/2 system is is meant to apply only to events or projections which whose spin vectors,  have hilbert inner product zero. That is either events on the same ' spin vector'  or those for which the vector inner product is zero,.

I presume there is a notion of orthogonality of projections (or events) in quantum mechanics as well; disjoint outcomes of the very same measurement spin vector

Given the strong duality between states, superpositions, and outcomes, measurements. I presume that the same notation is sued

where if restricted only to cases where the hilbert inner product denotes orthogonality=0 (rather than an additional convexity condition)