^{1}

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^{2}

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We discuss the fact that there is a crucial contradiction within Von Neumann’s theory. We derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in N spin-1/2 systems (1 ≤ N < +∞). This assumption intuitively depictures our physical world. However, the quantum predictions within the formalism of Von Neumann’s projective measurement violate the proposition with a magnitude that grows exponentially with the number of particles. We have to give up either the existence of the directions or the formalism of Von Neumann’s projective measurement. Therefore, Von Neumann’s theory cannot depicture our physical world with a violation factor that grows exponentially with the number of particles. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm relies on Von Neumann’s theory. We investigate whether Von Neumann’s theory meets the Deutsch-Jozsa algorithm. We discuss the fact that the crucial contradiction makes the quantum-theoretical formulation of Deutsch-Jozsa algorithm questionable. Further, we discuss the fact that projective measurement theory does not meet an easy detector model for a single Pauli observable. Especially, we systematically describe our assertion based on more mathematical analysis using raw data. We propose a solution of the problem. Our solution is equivalent to changing Planck’s constant to a new constant . It may be said that a new type of the quantum theory early approaches Newton’s theory in the macroscopic scale than the old quantum theory does. We discuss how our solution is used in an implementation of Deutsch’s algorithm.

Von Neumann introduces the Hilbert space and he tries to present axiomatic system for quantum mechanics [

A quantum computer is a device for computation that makes direct use of quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Quantum computers are different from digital computers based on transistor gates. Whereas digital computers require data to be encoded into binary digits (bits), quantum computation utilizes quantum properties to represent data and perform operations on these data [

As a famous physical theory, the quantum theory (cf. [

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

To date, the quantum theory seems to be a successful physical theory and it looks to have no problems in order to use it experimentally. Several researches address [

We discuss the fact that there is a crucial contradiction within Von Neumann’s theory [

We know that a theory means a set of propositions. Unfortunately, we have to abandon that the quantum theory satisfies consistency, which is necessary in order to have axiomatic system. This implies that there is no axiomatic system for the quantum theory. A theory K may be said to be consistent if any proposition,

We propose the solution of the problem. Our solution is equivalent to changing Planck’s constant

On the other hand, 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 [

We try to implement 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 ±1 (in

It is discussed [

problem. Let us follow the argumentations. Assume a pure spin 1/2 state. We have

jective measurement theory is true. Hence the expected values of two spin observables

not be measured by using projective measurement theory. But, we have

quantum measurement theory is true. The different point is that the values of the result of quantum measurements are

It is also discussed [

rection. We have

vables

We consider whether projective measurement theory meets an easy detector model for Pauli observable. We try to implement 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. We assume that a source of spin-carrying particles emits them in a state, which can be described as an eigenvector of Pauli observable

At this stage we are in the following situation.

1) We cannot measure an expected value of a single spin observable by using projective measurement theory.

2) New measurement theory covers the problem mentioned above.

3) We can use new measurement theory for an implementation of Deutsch’s algorithm.

Our discussion is very important. The reason is that our discussion reveals that we need new physical theories in order to explain our physical world informationally, to create new information science, and to predict new unknown physical phenomena efficiently. What are new physical theories? We cannot answer it at this stage. However, we expect that our discussion in this paper could contribute to creating new physical theories in order to explain our physical world, to create new information science, and to predict new unknown physical phenomena efficiently.

Throughout this paper, we confine ourselves to the two-level (e.g., electron spin, photon polarizations, and so on) and the discrete eigenvalue case.

Our paper is organized as follows.

In Section 2, we provide the notations and preparation to show a contradiction within Von Neumann’s theory.

In Section 3, we discuss the fact that there is a problem within the mathematical formulation of Von Neumann’s theory.

In Section 4, we review Deutsch’s algorithm along with Ref. [

In Section 5, we discuss a problem of Deutsch’s algorithm.

In Section 6, we show that Von Neumann’s theory does not meet our physical world.

In Section 7, we modify Von Neumann’s projective measurement theory.

In Section 8, we propose a new type of the Deutsch-Jozsa algorithm along with our modification of Von Neumann’s measurement theory.

In Section 9, we consider the relation between double-slit experiment and projective measurement theory. We cannot measure a single spin observable by using the projective measurement theory.

In Section 10, we consider many double-slit experiments. And we propose a solution of the problem concerning projective measurement theory.

In Section 11, we discuss how our solution is used in an implementation of Deutsch’s algorithm.

Section 12 concludes this paper.

We consider a two-dimensional space H. Let N denote a set of the numbers

that contains the countably infinite. Let S be_{1} denote a set of the numbers

that contains the countably infinite. Here we introduce_{2} denote a set of the numbers

that contains the countably infinite. Here we introduce_{3} denote a set of the numbers

that contains the countably infinite. Here we introduce_{4} denote a set of the numbers

that contains the countably infinite. Here we introduce

the vector of Pauli operators. The measurements (observables) of ^{3}. One measures an observable

We consider the following propositions:

Proposition: M (measurement outcome),

Proposition: E (quantum expected value),

Lemma: T

if

then

and

In this section, we investigate if Von Neumann’s theory can be almighty.

We assume a pure spin-1/2 state ^{3}.

We have a quantum expected value

We have

Here, the angle

We derive a necessary condition for the quantum expected value for the system in a pure spin-1/2 state lying in the x-y plane given in (3.1). We derive the possible values of the scalar product

We use the decomposition (3.2). We introduce simplified notations as

and

Then, we have

where we use the orthogonality relation

From a proposition of the quantum theory, the Bloch sphere (the orientation of reference frames) with the value of

is bounded as

The reason of the condition (3.8) is the Bloch sphere

Thus we derive a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames (in a spin-1/2 system). The proposition is

This inequality is saturated and iff

Hence, we derive the following proposition concerning the existence of the orientation of reference frames when the system is in a pure state lying in the x-y plane

We assign the truth value “1” for Proposition M and Proposition E. Let A_{k} be

From Proposition E and Lemma T, the quantum expected value in (3.1)

From Proposition M, the possible values of the actually measured result

From Lemma T, the same quantum expected value is given by

From Proposition M, the possible values of the actually measured result

From Proposition E and Lemma T, the quantum expected value in (3.1)

From Proposition M, the possible values of the actually measured result

From Lemma T, the same quantum expected value is given by

From Proposition M, the possible values of the actually measured result

We derive a necessary condition for the two quantum expected values for the system in a pure spin-1/2 state lying in the x-y plane given in (3.16) and (3.19). We derive the possible values of the scalar product

From Proposition M, we have

The above inequality (3.22) is saturated when

This implies

The above condition (3.25) can be possible since, as we have said,

and

Thus we derive a proposition concerning the two quantum expected values under an assumption that we assign the truth value “1” for Proposition M and Proposition E, (in a spin-1/2 system). The proposition is

We cannot assign the truth value “1” for two propositions (3.15) (concerning the existence of the orientation of reference frames) and (3.28) (concerning Proposition M and Proposition E), simultaneously, when the system is in a pure state lying in the x-y plane. We do not assign the truth value “1” for three propositions:

1) Proposition M;

2) Proposition E;

3) The existence of the orientation of reference frames simultaneously. In other words, we do not assign the truth value “1” for two propositions:

a) The existence of measurement outcome;

b) The existence of the orientation of reference frames simultaneously.

In this section, we review Deutsch’s algorithm along with Ref. [

Quantum parallelism is a fundamental feature of many quantum algorithms. It allows quantum computers to evaluate the values of a function

is a function with a one-bit domain and range. A convenient way of computing this function on a quantum com- puter is to consider a two-qubit quantum computer which starts in the state

With an appropriate sequence of logic gates it is possible to transform this state into

where

a name,

Deutsch’s algorithm combines quantum parallelism with a property of quantum mechanics known as interference. Let us use the Hadamard gate to prepare the first qubit

as the superposition

but let us prepare the second qubit as the superposition

using the Hadamard gate applied to the state

The Hadamard gate is as

Let us follow the states along to see what happens in this circuit. The input state

is sent through two Hadamard gates to give

A little thought shows that if we apply

then we obtain the state

Applying

The final Hadamard gate on the first qubit thus gives us

Realizing that

so by measuring the first qubit we may determine

In this section, we suggest a problem of Deutsch’s algorithm. We see that the implementation of Deutsch’s algorithm is not possible if we give up either observability of a quantum state or controllability of a quantum state.

We introduce the following quantum proposition concerning controllability:

We may consider the following non-quantum-theoretical proposition:

The proposition (5.2) implies the validity of Proposition M and Proposition E (observability and the existence of measurement outcome). The proposition (5.2) implies

Thus,

However, the validity of Proposition M and Proposition E does not imply the proposition (5.2). We see that the proposition (5.1) is not equivalent to Proposition M and Proposition E (observability and the existence of measurement outcome). From Truth Value

On the other hand, the proposition (5.1) implies that

when the system is in a pure state lying in the x-y plane. The reason is as follows: Assume a pure state lying in the x-y plane as

where

and

Then we have

and

Therefore, we see

We thus see the proposition (5.1) implies the existence of the orientation of reference frames in the Hilbert space formalism of the quantum theory.

From the discussion presented in Section 3, we see that the quantum proposition (5.1) concerning controllability (the existence of the orientation of reference frames) cannot coexist with the validity of Proposition M and Proposition E (observability and the existence of measurement outcome), which states

when the system is in a pure state lying in the x-y plane.

A | B | A ∨ B |
---|---|---|

1 | 1 | 1 |

1 | 0 | 1 |

0 | 1 | 1 |

0 | 0 | 0 |

Deutsch’s algorithm shows the importance of the ability of the Hadamard gate (controllability and the existence of the orientation of reference frames) for quantum computation. The ability of the Hadamard gate is valid only when we assign the truth value “1” for the proposition (5.1) (the existence of the orientation of reference frames). We see that the quantum state

is a pure state lying in the x-y plane. We can assign the truth value “1” for the ability of the Hadamard gate (controllability and the existence of the orientation of reference frames)

only when we assign the truth value “1” for the proposition (5.1) concerning controllability (the existence of the orientation of reference frames) and we give up the validity of Proposition M and Proposition E (observability and the existence of measurement outcome). The validity of the proposition (5.1) implies that

Thus applying H twice to a quantum state does nothing to it if we assign the truth value “1” for the proposition (5.1). When we assign the truth value “1” for the proposition (5.1), we have

We conclude that the step in which transforms the state

Assume that we have a set of N spins

sume 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 jth observer with a unit vector

where r is the projective result. We assume the value of r is ±1 (in

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

where ^{2},

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 (6.2). 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 (6.8). Therefore, it is a separability inequality. It is important that the separability inequality (6.8) is saturated iff

The inequality (6.10) 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 projective measurement theory if it can be written as

where l denotes a label and r is the result of Von Neumann’s projective measurement of the dichotomic observables parameterized by the directions of

Assume the quantum correlation function with the system in a pure uncorrelated state given in (6.2) admits projective measurement theory. One has the following proposition concerning projective measurement theory

In what follows, we show that we cannot assign the truth value “1” for the proposition (6.13) concerning projective measurement theory.

Assume the proposition (80) is true. By changing the label l into

An important note here is that the value of the right-hand-side of (6.13) is equal to the value of the right-hand- side of (6.14) because we only change labels.

We abbreviate

We have

We use the following fact

The inequality (6.15) is saturated since we have

Hence one has the following proposition concerning projective measurement theory

Clearly, we cannot assign the truth value “1” for two propositions (6.11) (concerning our physical world) and (6.18) (concerning projective measurement 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 (6.13) (concerning projective measurement theory) if we assign the truth value “1” for the proposition (6.11) (concerning our physical world). In other words, such projective measurement theory does not reveal our physical world.

In this section, we solve the contradiction presented in the previous section. We have the maximal possible value of the scalar product as a quantum proposition concerning our physical world

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

We cannot assign the truth value “1” for two propositions (7.1) (concerning our physical world) and (7.3) (concerning projective measurement 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.

We introduce the following hypothesis:

Hypothesis: We assume the value of r is

directions are set at

When we accept this hypothesis, the proposition (7.2) (concerning projective measurement theory) becomes the following new proposition concerning a quantum measurement theory (two-setting model)

We can assign the truth value “1” for both two propositions (7.1) (concerning our physical world) and (7.3) (concerning the quantum measurement 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 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 quantum measurements. Our solution is equivalent to changing Planck’s constant

The earliest quantum algorithm, the Deutsch-Jozsa algorithm, is representative to show that quantum computation 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 x in each letter. At worst, Alice will need to query Bob at least

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 x and z, modulo 2. Using this equation and (8.10) we can now evaluate

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 +1. Because

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

times for all N qubits in the query register. Thus, global measurement outcome is

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

We notice that the difference between

In this section, we consider the relation between double-slit experiment and projective measurement theory. We try to implement 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 ±1 (in

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

Hence we have

Thus,

On the other hand, a mean value E admits projective measurement theory if it can be written as

where l denotes a label and r is the result of projective measurement of the Pauli observable

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 (9.6) concerning projective measurement theory.

Assume the proposition (9.6) is true. By changing the label l into

An important note here is that the value of the right-hand-side of (9.6) is equal to the value of the right-hand-side of (9.8) because we only change labels. We have

We use the following fact

The inequality (9.9) is saturated since we have

Thus we derive a proposition concerning the quantum mean value under an assumption that projective measurement theory is true (in a spin-1/2 system), that is

From Strong Law of Large Numbers, we have

Hence we derive the following proposition concerning projective measurement theory

We do not assign the truth value “1” for two propositions (9.4) (concerning a wave function analysis) and (9.14) (concerning projective measurement theory), simultaneously. We are in the contradiction. This implies that we cannot perform the following Deutsch’s algorithm.

・ The control of quantum states relies on the wave functional analysis.

・ The observation of quantum states relies on projective measurement theory.

We cannot accept the validity of the proposition (9.6) (concerning projective measurement theory) if we assign the truth value “1” for the proposition (9.4) (concerning a wave function analysis). In other words, such projective measurement theory does not meet the detector model for spin observable

In this section, we consider many double-slit experiments. In a macroscopic system, we solve the contradiction presented in the previous section.

We consider an implementation of N double-slit experiments. We assume that N sources of spin-carrying particles emit them in a state, which can be described as an eigenvector of Pauli observable

Each of them can be described as an eigenvector of Pauli observable

then we have the following quantum expected value from a wave function analysis

Thus we have the following proposition concerning a wave function analysis

Hence we have

Thus,

On the other hand, a mean value E admits a quantum measurement theory if it can be written as

where l denotes a label and r is the result of quantum measurement of the Pauli observable

Assume the quantum mean value with the system in an eigenvector

In what follows, we show that we can assign the truth value “1” for the proposition (10.8) concerning the quantum measurement theory in the macroscopic system

Assume the proposition (10.8) is true. By changing the label l into

An important note here is that the value of the right-hand-side of (10.8) is equal to the value of the right-hand- side of (10.9) because we only change labels. We have

We use the following fact

The inequality (10.10) is saturated since we have

Thus we derive a proposition concerning the quantum mean value under an assumption that such a quantum measurement is true (in a spin-1/2 system), that is,

From Strong Law of Large Numbers, we have

Therefore we have

Thus,

We can assign the truth value “1” for both two propositions (10.6) (concerning a wave function analysis) and (10.16) (concerning the quantum measurement theory), simultaneously. Hence, we solve the contradiction presented in the previous section by changing the value of the result of quantum measurements and by considering an implementation of double-slit experiments macroscopically. This implies that we can perform the following Deutsch’s algorithm.

・ The control of quantum states relies on the wave functional analysis.

・ The observation of quantum states relies on the measurement theory.

In other words, such a measurement theory meets the detector model for spin observable

In this section, we discuss how our solution is used in an implementation of Deutsch’s algorithm. Now, we can measure Pauli observable

We can consider

Therefore if we can measure an expected value of

We see one measurement is enough to determine which state is realized. We can omit the final Hadamard gate on the first qubit.

In conclusion, we have discussed the fact that there is a crucial contradiction within Von Neumann’s theory. We have derived a proposition concerning a quantum expected value under an assumption of the existence of the orientation of reference frames in N spin-1/2 systems (1 ≤ N < +∞). This assumption intuitively has depictured our physical world. However, the quantum predictions within the formalism of Von Neumann’s projective measurement have violated the proposition with a magnitude that grows exponentially with the number of particles. We have had to give up either the existence of the directions or the formalism of Von Neumann’s projective measurement. Therefore, Von Neumann’s theory cannot have depictured our physical world with a violation factor that grows exponentially with the number of particles. The theoretical formalism of the implementation of the Deutsch-Jozsa algorithm has relied on Von Neumann’s theory. We have investigated whether Von Neumann’s theory meets the Deutsch-Jozsa algorithm. We have discussed the fact that the crucial contradiction makes the quantum-theoretical formulation of Deutsch-Jozsa algorithm questionable. Further, we have discussed the fact that projective measurement theory does not meet easy detector model for a single Pauli observable. Especially, we have systematically described our assertion based on more mathematical analysis using raw data. We have proposed a solution of the problem. Our solution has been equivalent to changing Planck’s constant

What are new physical theories? We cannot answer it at this stage. However, we expect that our discussion in this paper could contribute to creating new physical theories in order to explain our physical world, to create new information science, and to predict new unknown physical phenomena efficiently.

We would like to thank Professor Niizeki and Dr. Ren for valuable comments.

KojiNagata,TadaoNakamura, (2015) Von Neumann’s Theory, Projective Measurement, and Quantum Computation. Journal of Applied Mathematics and Physics,03,874-897. doi: 10.4236/jamp.2015.37108