Quantum Physics

   

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

Authors: Koji Nagata, Tadao Nakamura

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.

Comments: International Journal of Emerging Engineering Research and Technology, Volume 3, Issue 6 (2015), Page 78--89.

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[v1] 2015-06-26 08:39:20

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