Recently, [{arXiv:0810.3134}] is accepted and published. We present ultimate version of no-hidden-variables theorem. We derive a proposition concerning the quantum theory under the existence of the Bloch sphere in a single spin-1/2 system. The existence of a single classical probability space for measurement outcome within the formalism of von Neumann's projective measurement does not coexist with the proposition concerning the quantum theory. We have to give up the existence of such a classical probability space for measurement outcome in the two-dimensional Hilbert space formalism of the quantum theory. The quantum theory does not accept a hidden-variable interpretation in the two-dimensional space.
Recently, [{arXiv:0810.3134}] is accepted and published. We present ultimate version of no-hidden-variables theorem. We derive a proposition concerning the quantum theory under the existence of the Bloch sphere in a single spin-1/2 system. The existence of a single classical probability space for measurement outcome within the formalism of von Neumann's projective measurement does not coexist with the proposition concerning the quantum theory. We have to give up the existence of such a classical probability space for measurement outcome in the two-dimensional Hilbert space formalism of the quantum theory. The quantum theory does not accept a hidden-variable interpretation in the two-dimensional space.
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.
We derive new type of no-hidden-variables theorem based on the assumptions proposed by Kochen and Specker. We consider N spin-1/2 systems. The hidden results of measurement are either +1 or −1 (in ℏ/2 unit). 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. Also we discuss two-dimensional no-hidden-variables theorem of the KS type. Especially, we systematically describe our assertion based on more mathematical analysis using raw data in a thoughtful experiment.
Recently, a new measurement theory based on the truth values is proposed \cite{NN1}. The results of measurements are either 0 or 1. The measurement theory accepts a hidden variables model for a single Pauli observable. Therefore we can introduce a classical probability space for the measurement theory in this case. On the other hand, we discuss the fact that the projective measurement theory (the results of measurements are either +1 or −1) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.
We discuss the fact that a single spin observable σx in a quantum state does not have a counterpart in physical reality. We consider whether a single spin-1/2 pure state has a counterpart in physical reality. It is an eigenvector of Pauli observable σz or an eigenvector of Pauli observable σx. We assume a state , which can be described as an eigenvector of Pauli observable σz. We assume also a state , which can be described as an eigenvector of Pauli observable σx. The value of transition probability is 1/2. We consider the following physical situation. If we detect , then we assign measurement outcome as +1. If we detect , then we assign measurement outcome as −1. The existence of a single classical probability space for the transition probability within the formalism of the measurement outcome does not coexist with the value of the transition probability = 1/2. We have to give up the existence of such a classical probability space for the state or for the state , as they define the transition probability. It turns out that the single spin-1/2 pure state or the single spin-1/2 pure state does not have counterparts in physical reality, in general. We investigate whether the Stern-Gerlach experiment accepts hidden-variables theories. We discuss that the existence of the two spin-1/2 pure states and rules out the existence of probability space of specific quantum measurement. If we detect , then we assign measurement outcome as +1. If we detect , then we assign measurement outcome as −1. This hidden-variables theory does not accept the transition probability . Therefore we have to give up the hidden-variables theory. This implies the Stern- Gerlach experiment cannot accept the hidden-variables theory. A single spin-1/2 pure state (e.g., is a single one-dimensional projector. In other word, a single one-dimensional projector does not have a counterpart in such physical reality, in general. The one-dimensional projectors are commuting with each other. Our discussion shows that we cannot assign the specific definite values (+1 and −1) to the two commuting operators, simultaneously. We study whether quantum phase factor accepts a hidden-variables theory. We discuss that the existence of two spin-1/2 pure states |0_ = ( + )/√2 and = ( + e iθ )/√2 rules out the existence of probability space of a hidden-variables theory. If we detect , then we assign measurement outcome as +1. If we detect , then we assign measurement outcome as −1. The hidden-variables theory does not accept the transition probability Therefore we have to give up the hidden-variables theory for quantum phase factor. We explore phase factor is indeed a quantum effect, not classical. Our research gives a new insight to the quantum information processing which relies on quantum phase factor, such as Deutsch’s algorithm.
We discuss the fact that a single spin observable σx in a quantum state does not have a counterpart in physical reality. We consider whether a single spin-1/2 pure state has a counterpart in physical reality. It is an eigenvector of Pauli observable σz or an eigenvector of Pauli observable σx. We assume a state ∣+z⟩, which can be described as an eigenvector of Pauli observable σz. We assume also a state ∣+x⟩, which can be described as an eigenvector of Pauli observable σx. The value of transition probability ∣⟨+z∣+x⟩∣2 is 1/2. We consider the following physical situation. If we detect ∣+z⟩, then we assign measurement outcome as +1. If we detect ∣+x⟩, then we assign measurement outcome as −1. The existence of a single classical probability space for the transition probability within the formalism of the measurement outcome does not coexist with the value of the transition probability ∣⟨+z∣+x⟩∣2=1/2. We have to give up the existence of such a classical probability space for the state ∣+z⟩ or for the state ∣+x⟩, as they define the transition probability. It turns out that the single spin-1/2 pure state ∣+z⟩ or the single spin-1/2 pure state ∣+x⟩ does not have counterparts in physical reality, in general. We investigate whether the Stern-Gerlach experiment accepts hidden-variables theories. We discuss that the existence of the two spin-1/2 pure states ∣↑⟩ and ∣↓⟩ rules out the existence of probability space of specific quantum measurement. If we detect ∣↑⟩, then we assign measurement outcome as +1. If we detect ∣↓⟩, then we assign measurement outcome as −1. This hidden-variables theory does not accept the transition probability ∣⟨↑∣↓⟩∣2=0. Therefore we have to give up the hidden-variables theory. This implies the Stern-Gerlach experiment cannot accept the hidden-variables theory. A single spin-1/2 pure state (e.g., ∣↑⟩⟨↑∣) is a single one-dimensional projector. In other word, a single one-dimensional projector does not have a counterpart in such physical reality, in general. The one-dimensional projectors ∣↑⟩⟨↑∣ and ∣↓⟩⟨↓∣ are commuting with each other. Our discussion shows that we cannot assign the specific definite values (+1 and −1) to the two commuting operators, simultaneously. We study whether quantum phase factor accepts a hidden-variables theory. We discuss that the existence of two spin-1/2 pure states ∣0⟩=(∣↑⟩+∣↓⟩)/2 and ∣θ⟩=(∣↑⟩+eiθ∣↓⟩)/2 rules out the existence of probability space of a hidden-variables theory. If we detect ∣0⟩, then we assign measurement outcome as +1. If we detect ∣θ⟩, then we assign measurement outcome as −1. The hidden-variables theory does not accept the transition probability ∣⟨0∣θ⟩∣2=cos2(θ/2). Therefore we have to give up the hidden-variables theory for quantum phase factor. We explore phase factor is indeed a quantum effect, not classical. Our research gives a new insight to the quantum information processing which relies on quantum phase factor, such as Deutsch's algorithm.
We discuss that there is a crucial contradiction within quantum mechanics. We derive a proposition concerning a quantum expectation value under the assumption of the existence of the directions in a spin-1/2 system. The quantum predictions within the formalism of von Neumann's projective measurement cannot coexist with the proposition concerning the existence of the directions. Therefore, we have to give up either the existence of the directions or the formalism of von Neumann's projective measurement. Hence there is a crucial contradiction within the Hilbert space formalism of the quantum theory. This implies that there is no axiomatic system for the quantum theory. This also reveals that we need new physical theories in order to explain the handing of raw experimental data. We discuss that this crucial contradiction makes the quantum-theoretical formulation of Deutsch's algorithm questionable.