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We study the relation between the possibility of describing quantum correlation with hidden variables and the existence of the Bloch sphere. We derive some proposition concerning a quantum expected value under an assumption about the existence of the Bloch sphere in N spin-1/2 systems. However, the hidden variables theory violates the proposition with a magnitude that grows exponentially with the number of particles. Therefore, we have to give up either the existence of the Bloch sphere or the hidden variables theory. We show that the introduction of curved information and the continuity equation of probability are in agreement with classical quantum mechanics. So we give up the hidden variable theory as local theory and we accept the Bloch sphere as global theory connected with the information space.

The quantum theory (cf. [

As for the foundations of the quantum theory, Leggett-type non-local variables theory [

As for the applications of the quantum theory, the implementation of a quantum algorithm to solve Deutsch’s problem [

We study the relation between a significant specific hidden variables theory and the existence of the Bloch sphere. The results of measurements are either +1 or −1. We derive some proposition concerning a quantum expected value under an assumption about the existence of the Bloch sphere in N spin-1/2 systems. However, the hidden variables theory violates the proposition with a magnitude that exponentially grows with the number of particles. Therefore, we have to give up either the existence of the Bloch sphere or the hidden variables theory. We solve the previous dilemma we introduce the information space and the continuity equation to show how quantum mechanics is consequence of the information locate in all the space so is impossible to represent by local hidden variables. In conclusion, we agree on the existence of the Bloch sphere.

1) Continuity equation

Given the continuity equation of the probability (global conservation of the probability) we have

when the probability density

For

Now in classical mechanics we have for the action S the relation

So

where S is the action and for the wave the phase of the wave.

2) Condition probability from statistical parameters (average value, standard deviation and others)

We assume that to found a particle in a particular state is a probabilistic phenomenon for which we have join probability that the particle can be in a particular state. Now the novelty is to assume that the probability is function of other external elements as parameters. The average of the position for the particle is a parameter the movement of the particle in a particular environment for example inside of a tube or in other boundary condition (see boundary condition in Shrodinger solution) can change the probability for a particular state. Any far or near change of the environment change the probability of the state (Bell theorem and entanglement) In conclusion the join probability of a state of different variables is conditioned by a set of parameters that statically or physically can define the environment where the particle move. We denote all this parameters as the information relate to the environment where the particle is located. The set of external parameters is the information space that can have curvature as in the Berry phase phenomena that show that in the Shrodinger solution any loop can change the original phase. In differential geometry any loop in a space with curvature changes the original phase of the vectors. Now we built the information space which geodesic tensor is the Fisher entropy or Fisher information by which we can compute the covariant derivatives and the curvature.

Given the system of the conditional probabilities

We have the

Given the vector

The derivative is

where

is the Fisher information matrix connected with the Christoffel symbols

Here,

For the Fisher information we have

where E is the average operator so we can write in the first approximation

and

where

and

So

Now for

So we have

where Q is the Bohm quantum potential that is a consequence for the extreme condition of Fisher information (minimum or maximum condition for the Fisher information). We know that the quantum potential as real part and the continuous equation as the imaginary part from the Boltzmann entropic geometry we can generate the Schrödinger equation. We can also use the Schrodinger equation and came back to the Fisher information and to the pure conditional probability interpretation of the quantum mechanics.

Now, we combine the continuity equation of the probability with the covariant derivative in the curved information space we have

where the real part is consequence of the curvature in the information space and the immaginary part is due to the continuous equation for the probability.

Now for

with

In conclusion, we can make a reverse process used by Schrödinger we can generate the Schrödinger equation by the information space and the continuity equation of the probability. In this way, the Hilbert mechanism can be explain only by information, curvature and probability.

Assume that we have a set of N spins

the x-y plane. Let us assume that one source of N uncorrelated spin-carrying particles emits them in a state, which can be described as a multi spin-1/2 pure uncorrelated state. Let us parameterize the settings of the

where r is the hidden result. We assume the value of r is

Also one can introduce a quantum correlation function with the system in such a pure uncorrelated state

where

with

One can write the observable (unit) vector

where

We derive a necessary condition to be satisfied by the quantum correlation function with the system in a pure uncorrelated state given in (27). In more detail, we derive the value of the product of the quantum correlation function,

and

Then, we have

where we use the orthogonality relation

From the convex argument, all quantum separable states must satisfy the inequality (33). Therefore, it is a separability inequality. It is important that the separability inequality (33) is saturated iff

The inequality (35) is saturated iff

when the system is in such a multi spin-1/2 pure uncorrelated state.

A hidden variables correlation function, assuming ergodicity, can be written as a weigthed sum over integer indices. For a function r in a function space

If a formulation with discrete indexing of hidden variables is a sensible way to describe a correlation, then one may select from

Here, the abbreviation

A sign function can then be obtained

and has cardinality m. More explicitly,

The elements of

and

We are very interested in the maximum value of the square of an expected value in a probability interpretation of quantum measurement theory. Therefore we focus on each measurement result providing a probability. And we study the maximum value when we inspect the summation. In short, we can multiply a measurement result by the same measurement result.

Therefore, we wish we would have some sort of Kronecker delta function at our disposal to match proper terms in the sum. In this respect it must be noted that the correlation in (38) can be arbitrary close approximated with the use of

Assuming,

Because, of small differences, we may write in a Taylor like approximation,

The r product in (41) can then be re-written as

If the previous result from (42) is introduced in (41) then the latter can be re-written as

If for

will be true. The form

similar to

for

We study the possibility,

We use the following fact

for properly selected

Clearly, we cannot assign the truth value “1’’ for two propositions (36) (concerning the Bloch sphere) and (46) (concerning the hidden variables theory), simultaneously, when the system is in a multiparticle pure uncorrelated state. Of course we can imagine

To continue we note, each of the theories refers to a spin-1/2 pure state lying in the x-y plane. Therefore, we are in the contradiction when the system is in such a multiparticle pure uncorrelated state. Thus, we cannot accept a general validity of the proposition of a hidden variables theory, if we assign the truth value “1’’ for the proposition (36) (concerning the Bloch sphere).

In conclusion, we have studied the relation between a hidden variables theory and the existence of the Bloch sphere and with a new type of conditional probability and Fisher information as metric for information space we show that the hidden variable are not possible. Now, we have derived some proposition concerning a quantum expected value under an assumption about the existence of the Bloch sphere in N spin-1/2 systems. However, the hidden variables theory has violated the proposition with a magnitude that grows exponential with the number of particles. Therefore, we have had to give up either the existence of the Bloch sphere or the hidden variables theory. The hidden variables theory does not have depictured physical phenomena using the existence of the Bloch sphere with a violation factor that grows exponentially with the number of particles. Now, we point out the problem that when we cannot measure an observable we cannot say nothing on this measure as in the non-commutative case. So we have contradictions. In classical interpretation of quantum mechanics does not exist conditional probability and we cannot measure the probability but with the introduction of the information space and Fisher metric we show that conditional probability is possible but limited to statistical parameters as average value or other parameters. So contradiction is eliminated. Now, entanglement and Bell theorem can be understood in a new type of set theory that includes copula [

Nagata, K., Resconi, G., Nakamura, T. and Geurdes, H. (2016) Information and Conditional Probability to Go beyond Hidden Variables. Journal of Applied Mathematics and Physics, 4, 2203- 2214. http://dx.doi.org/10.4236/jamp.2016.412214