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We study the relation between hidden variables theories and quantum computation. We discuss an inconsistency between a hidden variables theory and controllability of quantum computation. To derive the inconsistency, we use the maximum value of the square of an expected value. We propose a solution of the problem by using new hidden variables theory. Also we discuss an inconsistency between hidden variables theories and the double-slit experiment as the most basic experiment in quantum mechanics. This experiment can be an easy detector to Pauli observable. We cannot accept hidden variables theories to simulate the double-slit experiment in a specific case. Hidden variables theories may not depicture quantum detector. This is a quantum measurement theoretical profound problem.

Quantum mechanics (cf. [

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

Given the fundamental studies and the application reports, we consider why quantum computer is faster than classical counterpart. It is essential to study the relation between hidden variables theory (classical theory) and quantum mechanics to investigate the quantum computation problem. So we address studying the relation between hidden variables theories and quantum computation.

We study the relation between hidden variables theories and quantum computation. The possible values of the pre-determined result of measurements are

Let us consider controllability of quantum computation. We derive quantum proposition concerning a quan- tum expected value under an assumption about the existence of the orientation of reference frames in N spin-1/2 systems. However, the original hidden variables theory violates the proposition with a magnitude that grows exponentially with the number of particles. To derive the inconsistency, we rely on the maximum value of the square of an hidden variables theoretical expected value. Therefore, we have to give up either the existence of the reference frames or the original hidden variables theory. The original hidden variables theory does not depicture physical phenomena using reference frames with a violation factor that grows exponentially with the number of particles.

The double-slit experiment is an illustration of wave-particle duality. In it, a beam of particles (such as photons) travels through a barrier with two slits removed. If one puts a detector screen on the other side, the pattern of detected particles shows interference fringes characteristic of waves; however, the detector screen responds to particles. The system exhibits the behaviour of both waves (interference patterns) and particles (dots on the screen).

If we modify this experiment so that one slit is closed, no interference pattern is observed. Thus, the state of both slits affects the final results. We can also arrange to have a minimally invasive detector at one of the slits to detect which slit the particle went through. When we do that, the interference pattern disappears. An analysis of a two-atom double-slit experiment based on environment-induced measurements is reported [

We assume an implementation of the double-slit experiment. There is a detector just after each slit. Thus interference figure does not appear, and we do not consider such a pattern. The possible values of the result of measurements are

We consider whether hidden variables theories meet an easy detector model to Pauli observable. We assume an implementation of the double-slit experiment. There is a detector just after each slit. We assume that a source of spin-carrying particles emits them in a state, which can be described as an eigenvector of Pauli observable

Our paper is organized as follows.

In Section 2, we argue a hidden variables theory does not meet the reference frames.

In Section 3, we give a solution of the problem of the hidden variables theory. We find new hidden variables theory meets the reference frames.

In Section 4, we review the Deutsch-Jozsa algorithm using new hidden variables theory.

In Section 5, we discuss the relation between the double-slit experiment and hidden variables theories.

Section 6 concludes this paper.

Assume that we have a set of N spins

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

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 (2). In more detail, we derive the maximum value of the product of the quantum

correlation function,

notations as

and

Then, we have

where we use the orthogonality relation

From the convex argument, all quantum separable states must satisfy the inequality (8). Therefore, it is a separability inequality. It is important that the separability inequality (8) is saturated iff

The inequality (10) is saturated iff

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

On the other hand, a correlation function satisfies the hidden variables theory if it can be written as

where

Assume the quantum correlation function with the system in a pure uncorrelated state given in (2) admits the hidden variables theory. One has the following proposition concerning the hidden variables theory

In what follows, we show that we cannot assign the truth value “1” for the proposition (13) concerning the hidden variables theory. We rely on the maximum value of the square of an expected value. Assume the proposition (13) is true. By changing the hidden variable

An important note here is that the value of the right-hand-side of (13) is equal to the value of the right- hand-side of (14) because we only change the hidden variable.

We abbreviate

We have

Here we use the fact

Since the possible values of

Hence we derive the following proposition if we assign the truth value “1” for a hidden variables theory

Clearly, we cannot assign the truth value “1” for two propositions (11) (concerning the reference frames) and (18) (concerning the hidden variables theory), simultaneously, when the system is in a multiparticle pure uncorrelated state. Of course, each of them is 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 the validity of the proposition (13) (concerning the hidden variables theory) if we assign the truth value “1” for the proposition (11) (concerning the reference frames). In other words, the hidden variables theory does not reveal physical phenomena using reference frames. The reference frames are necessary to control a quantum state. Thus, the hidden variables theory does not reveal physical phenomena controlling a quantum state.

In this section, we solve the contradiction presented in the previous section. We have the maximum possible value of the product as a quantum proposition concerning the reference frames

when the system is in such a multi spin-1/2 pure uncorrelated state. On the other hand, one has the following proposition concerning the hidden variables theory

We cannot assign the truth value “1” for two propositions (19) (concerning the reference frames) and (20) (concerning the hidden variables theory), simultaneously, when the system is in a multiparticle pure un- correlated state. Of course, each of them is 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.

We introduce the following hypothesis:

Hypothesis: We assume the value of

directions are set at

When we accept this hypothesis, the proposition (20) (concerning the hidden variables theory) becomes the following new proposition concerning other hidden variables theory (two-setting model)

We can assign the truth value “1” for both two propositions (19) (concerning the reference frames) and (21) (concerning other hidden variables theory), simultaneously, when the system is in a multiparticle pure un- correlated state. Of course, each of them is a spin-1/2 pure state lying in the x-y plane. Therefore, we are not in the contradiction when the system is in such a multiparticle pure uncorrelated state. Hence, we solve the contradiction presented in the previous section by changing the value of the result of pre-determined mea- surements. Our solution is equivalent to changing Planck’s constant

The earliest quantum algorithm, the Deutsch-Jozsa algorithm, is representative to show that quantum com- putation is faster than classical counterpart with a magnitude that grows exponentially with the number of qubits.

Let us follow the argumentation presented in [

and replies with the result, which is either 0 or 1. Now, Bob has promised to use a function f which is of one of two kinds; either the value of

In the classical case, Alice may only send Bob one value of

times, since she may receive

If Bob and Alice were able to exchange qubits, instead of just classical bits, and if Bob agreed to calculate

Alice has an N qubit register to store her query in, and a single qubit register which she will give to Bob, to store the answer in. She begins by preparing both her query and answer registers in a superposition state. Bob will evaluate

on the query register, and finishes by performing a suitable measurement to determine whether f was constant or balanced.

Let us follow the quantum states through this algorithm. The input state is

Here the query register describes the state of N qubits all prepared in the

state. After the Hadamard transformation on the query register and the Hadamard gate on the answer register we have

The query register is now a superposition of all values, and the answer register is in an evenly weighted superposition of

And

Next, the function f is evaluated (by Bob) using

giving

Here

is the bitwise XOR (exclusive OR) of y and

By checking the cases

Thus

This can be summarized more succinctly in the very useful equation

where

is the bitwise inner product of

Alice now observes the query register. Note that the absolute value of the amplitude for the state

Is

Let’s look at the two possible cases-f constant and f balanced-to discern what happens. In the case where f is constant the absolute value of the amplitude for

is

is of unit length it follows that all the other amplitudes must be zero, and an observation will yield

times for all

If f is balanced then the positive and negative contributions to the absolute value of the amplitude for

cancel, leaving an amplitude of zero, and a measurement must yield a result other than

that is,

on at least one qubit in the query register. Summarizing, if Alice measures all

surement outcome is

We notice that the difference between

question if the Deutsch-Jozsa algorithm in the macroscopic scale is possible or not. This question is open problem.

We see the measurement outcome is predetermined. This is classical situation. We can see the result of the Deutsch-Jozsa algorithm classically. And an input state violates non local realism [

In this section, we consider the relation between the double-slit experiment and the original hidden variables theory. We assume an implementation of the double-slit experiment. There is a detector just after each slit. Thus interference figure does not appear, and we do not consider such a pattern. The possible values of the result of measurements are

Let

The above quantum expected value is zero if we consider only a wave function analysis.

We derive a necessary condition for the quantum expected value for the system in the pure spin-1/2 state

Thus,

We derive the following proposition

On the other hand, a mean value

where

Assume the quantum mean value with the system in an eigenvector

We can assume as follows by Strong Law of Large Numbers,

In what follows, we show that we cannot assign the truth value “1” for the proposition (53) concerning the hidden variables theory. We rely on the maximum value of the square of a mean value.

Assume the proposition (53) is true. By changing the hidden variable

An important note here is that the value of the right-hand-side of (53) is equal to the value of the right- hand-side of (55) because we only change the hidden variable. We have

Here we use the fact

since the possible values of

Hence we derive the following proposition if we assign the truth value “1” for a hidden variables theory

From Strong Law of Large Numbers, we have

Hence we derive the following proposition concerning the hidden variables theory

We do not assign the truth value “1” for two propositions (51) (concerning a wave function analysis) and (61) (concerning the hidden variables theory), simultaneously. We are in the contradiction.

We cannot accept the validity of the proposition (53) (concerning the hidden variables theory) if we assign the truth value “1” for the proposition (51) (concerning a wave function analysis). In other words, we cannot accept the hidden variables theory to simulate the detector model for spin observable

A mean value

where

Assume the quantum mean value with the system in an eigenvector

We can assume as follows by Strong Law of Large Numbers,

In what follows, we show that we cannot assign the truth value “1” for the proposition (63) concerning new hidden variables theory. We rely on the maximum value of the square of a mean value.

Assume the proposition (63) is true. By changing the hidden variable

An important note here is that the value of the right-hand-side of (63) is equal to the value of the right-hand- side of (65) because we only change the hidden variable. We have

Here we use the fact

since the possible values of

Hence we derive the following proposition if we assign the truth value “1” for new hidden variables theory

From Strong Law of Large Numbers, we have

Hence we derive the following proposition concerning new hidden variables theory

We do not assign the truth value “1” for two propositions (51) (concerning a wave function analysis) and (71) (concerning new hidden variables theory), simultaneously. We are in the contradiction.

We cannot accept the validity of the proposition (63) (concerning new hidden variables theory) if we assign the truth value “1” for the proposition (51) (concerning a wave function analysis). In other words, we cannot accept new hidden variables theory to simulate the detector model for spin observable

In conclusion, we have studied the relation between a hidden variables theory and quantum computation. The possible values of the pre-determined result of measurements have been

We have derived some proposition concerning a quantum expected value under an assumption about the existence of the orientation of reference frames in N spin-1/2 systems. However, the hidden variables theory has violated the proposition with a magnitude that grows exponentially with the number of particles. Therefore, we have had to give up either the existence of the reference frames or the hidden variables theory. The hidden variables theory does not have depictured physical phenomena using reference frames with a violation factor that grows exponentially with the number of particles.

We have proposed a solution of the problem. Our solution has been equivalent to changing Planck’s constant

We may have said the Deutsch-Jozsa algorithm is physical. Also we have discussed the fact that both the original hidden variables theory and new hidden variables theory do not meet an easy detector model to a single Pauli observable. Hidden variables theories may not depicture quantum detector. This is a quantum measure- ment theoretical profound problem.

We thank Professor Weinstein for valuable discussions.

Koji Nagata,Tadao Nakamura, (2015) Can Hidden Variables Theories Meet Quantum Computation?. Open Access Library Journal,02,1-12. doi: 10.4236/oalib.1101804