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Does a Single Spin-1/2 Pure Quantum State Have a Counterpart in Physical Reality? (Accepted Version)

Koji Nagata, Tadao Nakamura

Published Jun 1, 2015• 0 citations
OpenAlex

Abstract

We discuss the fact that a single spin observable $\sigma_x$ 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 $\sigma_z$ or an eigenvector of Pauli observable $\sigma_x$. We assume a state $|+_z\rangle$, which can be described as an eigenvector of Pauli observable $\sigma_z$. We assume also a state $|+_x\rangle$, which can be described as an eigenvector of Pauli observable $\sigma_x$. The value of transition probability $|\langle +_z|+_x\rangle|^2$ is 1/2. We consider the following physical situation. If we detect $|+_z\rangle$, then we assign measurement outcome as $+1$. If we detect $|+_x\rangle$, 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 $|\langle +_z|+_x\rangle|^2=1/2$. We have to give up the existence of such a classical probability space for the state $|+_z\rangle$ or for the state $|+_x\rangle$, as they define the transition probability. It turns out that the single spin-1/2 pure state $|+_z\rangle$ or the single spin-1/2 pure state $|+_x\rangle$ 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 $|\uparrow\rangle$ and $|\downarrow\rangle$ rules out the existence of probability space of specific quantum measurement. If we detect $|\uparrow\rangle$, then we assign measurement outcome as $+1$. If we detect $|\downarrow\rangle$, then we assign measurement outcome as $-1$. This hidden-variables theory does not accept the transition probability $|\langle\uparrow|\downarrow\rangle|^2=0$. 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., $|\uparrow \rangle\langle \uparrow|$) 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 $|\uparrow\rangle\langle\uparrow|$ and $|\downarrow\rangle\langle \downarrow|$ 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 $|0\rangle=(|\uparrow\rangle+|\downarrow\rangle)/\sqrt{2}$ and $|\theta\rangle=(|\uparrow\rangle+e^{i \theta}|\downarrow\rangle)/\sqrt{2}$ rules out the existence of probability space of a hidden-variables theory. If we detect $|0\rangle$, then we assign measurement outcome as $+1$. If we detect $|\theta\rangle$, then we assign measurement outcome as $-1$. The hidden-variables theory does not accept the transition probability $|\langle 0|\theta\rangle|^2=\cos^2(\theta/2)$. 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.

Related papers

Does a Single Spin-1/2 Pure Quantum State Have a Counterpart in Physical Reality?

Koji Nagata, Tadao Nakamura

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.

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Stern-Gerlach Experiment, Quantum Phase Factor, and Hidden-Variables Theories

Koji Nagata, Tadao Nakamura

We discuss whether the Stern-Gerlach experiment accepts hidden-variables theories. We discuss that the existence of two spin-1/2 pure states $|\uparrow\rangle$ and $|\downarrow\rangle$ rules out the existence of probability space of specific quantum measurement. If we detect $|\uparrow\rangle$, then measurement outcome is $+1$. If we detect $|\downarrow\rangle$, then measurement outcome is $-1$. This hidden-variables theory does not accept the transition probability $|\langle\uparrow|\downarrow\rangle|^2=0$. Therefore we have to give up the hidden-variables theory. This implies the Stern-Gerlach experiment cannot accept the specific hidden-variables theory. And we study whether quantum phase factor accepts hidden-variables theories. We use the transition probability for two spin-1/2 pure states $(|\uparrow\rangle+|\downarrow\rangle)/\sqrt{2}$ and $(|\uparrow\rangle+e^{i \theta}|\downarrow\rangle)/\sqrt{2}$. It is $\cos^2(\theta/2)$. We discuss that the phase factor does not accept another specific hidden-variables theory. We explore the 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.

•Mar 1, 2014•0 citations

Kochen-Specker Theorem and the Two Quantum Measurement Theories

Koji Nagata, Tadao Nakamura, Han J.F. Geurdes +3 more

We consider the two quantum measurement theories for measuring a single Pauli observable. We assume also the existence of a classical probability space for the two measurement theories. We cannot avoid the Kochen-Specker (KS) contradiction when we measure the Pauli observable by using the projective measurement theory if we introduce a classical probability space. The results of measurement are either $+1$ or $-1$ (in $\hbar/2$ unit) when we consider a spin-1/2 system. The projective measurement theory does not accept a classical probability space when we measure the Pauli observable. We propose a new measurement theory based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either $+1$ or 0 (in $\hbar/2$ unit). We avoid the KS contradiction when we measure the Pauli observable by using the new measurement theory if we introduce a classical probability space. The new measurement theory accepts a classical probability space when we measure the Pauli observable.

Studies in Applied Mathematics•Jan 6, 2018•1 citation

Does a classical probability space for two-dimensional quantum measurement theory exist?

Koji Nagata, Tadao Nakamura

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.

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A Quantum Mechanical Measurement Leading to Simultaneous Spin-Up and Spin-Down State of a Single Electron

Vesselin Noninski

In is shown that when two observers carry out a simultaneous measurement on a pair of spin-? particles in a "singlet" state a possibility exists for an outcome that lacks physical meaning. More specifically, despite the fact that the two commuting operators formally possess simultaneous eigenvectors it is not possible for physical reasons these eigenvectors to exist simultaneously. It is pointed out that the possibility for such non-physical outcome is observed only in the case of "singlet" state which puts into question the physical meaning of such state, and following from it exotic notions such as "non-locality". This hints that probably QM in its present form should be reconsidered with regard to its limitations.

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Bell, Spinors, and the Impossibility of a Classical Spin-Vector Model

G. A. Koroteev

We revisit the Bell--CHSH scenario for two spin-\(\tfrac{1}{2}\) particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-\(\tfrac{1}{2}\) is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-\(\tfrac{1}{2}\) particles, with the singlet state yielding the usual correlation \(E(a,b) = -\,a\cdot b\) and Tsirelson's bound \(2\sqrt{2}\). On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as \(\{\pm1\}\)-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra \(\mathcal{C}(Λ)\). We show that there is no representation of the spinor algebra of spin-\(\tfrac{1}{2}\) (with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the \(\{\pm1\}\) spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C\(^*\)-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.

quant-pharXiv:2512.11476•Dec 15, 2025•0 citations
We discuss the fact that a single spin observable σx\sigma_xσx​ 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\sigma_zσz​ or an eigenvector of Pauli observable σx\sigma_xσx​. We assume a state ∣+z⟩|+_z\rangle∣+z​⟩, which can be described as an eigenvector of Pauli observable σz\sigma_zσz​. We assume also a state ∣+x⟩|+_x\rangle∣+x​⟩, which can be described as an eigenvector of Pauli observable σx\sigma_xσx​. The value of transition probability ∣⟨+z∣+x⟩∣2|\langle +_z|+_x\rangle|^2∣⟨+z​∣+x​⟩∣2 is 1/2. We consider the following physical situation. If we detect ∣+z⟩|+_z\rangle∣+z​⟩, then we assign measurement outcome as +1+1+1. If we detect ∣+x⟩|+_x\rangle∣+x​⟩, then we assign measurement outcome as −1-1−1. 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|\langle +_z|+_x\rangle|^2=1/2∣⟨+z​∣+x​⟩∣2=1/2. We have to give up the existence of such a classical probability space for the state ∣+z⟩|+_z\rangle∣+z​⟩ or for the state ∣+x⟩|+_x\rangle∣+x​⟩, as they define the transition probability. It turns out that the single spin-1/2 pure state ∣+z⟩|+_z\rangle∣+z​⟩ or the single spin-1/2 pure state ∣+x⟩|+_x\rangle∣+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 ∣↑⟩|\uparrow\rangle∣↑⟩ and ∣↓⟩|\downarrow\rangle∣↓⟩ rules out the existence of probability space of specific quantum measurement. If we detect ∣↑⟩|\uparrow\rangle∣↑⟩, then we assign measurement outcome as +1+1+1. If we detect ∣↓⟩|\downarrow\rangle∣↓⟩, then we assign measurement outcome as −1-1−1. This hidden-variables theory does not accept the transition probability ∣⟨↑∣↓⟩∣2=0|\langle\uparrow|\downarrow\rangle|^2=0∣⟨↑∣↓⟩∣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., ∣↑⟩⟨↑∣|\uparrow \rangle\langle \uparrow|∣↑⟩⟨↑∣) 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 ∣↑⟩⟨↑∣|\uparrow\rangle\langle\uparrow|∣↑⟩⟨↑∣ and ∣↓⟩⟨↓∣|\downarrow\rangle\langle \downarrow|∣↓⟩⟨↓∣ are commuting with each other. Our discussion shows that we cannot assign the specific definite values (+1+1+1 and −1-1−1) 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|0\rangle=(|\uparrow\rangle+|\downarrow\rangle)/\sqrt{2}∣0⟩=(∣↑⟩+∣↓⟩)/2​ and ∣θ⟩=(∣↑⟩+eiθ∣↓⟩)/2|\theta\rangle=(|\uparrow\rangle+e^{i \theta}|\downarrow\rangle)/\sqrt{2}∣θ⟩=(∣↑⟩+eiθ∣↓⟩)/2​ rules out the existence of probability space of a hidden-variables theory. If we detect ∣0⟩|0\rangle∣0⟩, then we assign measurement outcome as +1+1+1. If we detect ∣θ⟩|\theta\rangle∣θ⟩, then we assign measurement outcome as −1-1−1. The hidden-variables theory does not accept the transition probability ∣⟨0∣θ⟩∣2=cos⁡2(θ/2)|\langle 0|\theta\rangle|^2=\cos^2(\theta/2)∣⟨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 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 whether the Stern-Gerlach experiment accepts hidden-variables theories. We discuss that the existence of two spin-1/2 pure states ∣↑⟩|\uparrow\rangle∣↑⟩ and ∣↓⟩|\downarrow\rangle∣↓⟩ rules out the existence of probability space of specific quantum measurement. If we detect ∣↑⟩|\uparrow\rangle∣↑⟩, then measurement outcome is +1+1+1. If we detect ∣↓⟩|\downarrow\rangle∣↓⟩, then measurement outcome is −1-1−1. This hidden-variables theory does not accept the transition probability ∣⟨↑∣↓⟩∣2=0|\langle\uparrow|\downarrow\rangle|^2=0∣⟨↑∣↓⟩∣2=0. Therefore we have to give up the hidden-variables theory. This implies the Stern-Gerlach experiment cannot accept the specific hidden-variables theory. And we study whether quantum phase factor accepts hidden-variables theories. We use the transition probability for two spin-1/2 pure states (∣↑⟩+∣↓⟩)/2(|\uparrow\rangle+|\downarrow\rangle)/\sqrt{2}(∣↑⟩+∣↓⟩)/2​ and (∣↑⟩+eiθ∣↓⟩)/2(|\uparrow\rangle+e^{i \theta}|\downarrow\rangle)/\sqrt{2}(∣↑⟩+eiθ∣↓⟩)/2​. It is cos⁡2(θ/2)\cos^2(\theta/2)cos2(θ/2). We discuss that the phase factor does not accept another specific hidden-variables theory. We explore the 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 consider the two quantum measurement theories for measuring a single Pauli observable. We assume also the existence of a classical probability space for the two measurement theories. We cannot avoid the Kochen-Specker (KS) contradiction when we measure the Pauli observable by using the projective measurement theory if we introduce a classical probability space. The results of measurement are either +1+1+1 or −1-1−1 (in ℏ/2\hbar/2ℏ/2 unit) when we consider a spin-1/2 system. The projective measurement theory does not accept a classical probability space when we measure the Pauli observable. We propose a new measurement theory based on the truth values, i.e., the truth T (1) for true and the falsity F (0) for false. The results of measurement are either +1+1+1 or 0 (in ℏ/2\hbar/2ℏ/2 unit). We avoid the KS contradiction when we measure the Pauli observable by using the new measurement theory if we introduce a classical probability space. The new measurement theory accepts a classical probability space when we measure the Pauli observable.
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+1+1 or −1-1−1) 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.
In is shown that when two observers carry out a simultaneous measurement on a pair of spin-? particles in a "singlet" state a possibility exists for an outcome that lacks physical meaning. More specifically, despite the fact that the two commuting operators formally possess simultaneous eigenvectors it is not possible for physical reasons these eigenvectors to exist simultaneously. It is pointed out that the possibility for such non-physical outcome is observed only in the case of "singlet" state which puts into question the physical meaning of such state, and following from it exotic notions such as "non-locality". This hints that probably QM in its present form should be reconsidered with regard to its limitations.
We revisit the Bell--CHSH scenario for two spin-12\tfrac{1}{2}21​ particles and isolate the precise algebraic origin of the Bell contradiction. On the quantum side, spin-12\tfrac{1}{2}21​ is described by a noncommutative spinor (Clifford) algebra acting on the Hilbert space of two spin-12\tfrac{1}{2}21​ particles, with the singlet state yielding the usual correlation E(a,b)=− a⋅bE(a,b) = -\,a\cdot bE(a,b)=−a⋅b and Tsirelson's bound 222\sqrt{2}22​. On the classical side, the standard Bell assumptions amount to describing all measurement outcomes as {±1}\{\pm1\}{±1}-valued random variables on a single Kolmogorov probability space, i.e.\ elements of a commutative algebra C(Λ)\mathcal{C}(Λ)C(Λ). We show that there is no representation of the spinor algebra of spin-12\tfrac{1}{2}21​ (with its singlet state and locality structure) into any such commutative Kolmogorov algebra that preserves the {±1}\{\pm1\}{±1} spectra of local spin components and the singlet correlations entering the CHSH expression, under the standard Bell assumptions of locality (factorization) and measurement independence. In this sense, the Bell--CHSH contradiction is exhibited as an algebraic mismatch between a noncommutative spinor/Clifford description of spin and the classical assumption of a single global Kolmogorov space supporting all outcomes. In the language of quantum probability, this is a C∗^*∗-algebraic reformulation of the known fact that the singlet correlations admit no local hidden-variable model with jointly distributed outcomes on one probability space. We also give an explicit realization of the same spinor structure within the author's Quantum Index Algebra (QIA) framework, where locality appears as disjoint index slots and the singlet state as a simple index cocycle.