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.
We discuss whether the Stern-Gerlach experiment accepts hidden-variables theories. We discuss that the existence of two spin-1/2 pure states ∣↑⟩ and ∣↓⟩ rules out the existence of probability space of specific quantum measurement. If we detect ∣↑⟩, then measurement outcome is +1. If we detect ∣↓⟩, then measurement outcome is −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 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 and (∣↑⟩+eiθ∣↓⟩)/2. It is 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 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 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.
This article reviews the predictions of quantum mechanics (QM) for one- and two-particle Stern–Gerlach experiments and then frames Bell's results, which rule out hidden-variable alternatives to QM, as attempts by a skeptical Eve to fool Alice and Bob with (first example) classical probability mixtures of non-entangled quantum states and (second example) a classical hidden-variable theory. With hidden variables, Eve can succeed when Alice and Bob limit themselves to two Stern–Gerlach directions, but always fails for some choices of three directions. For three random directions, hidden-variable theories are impossible (we show) exactly 2/3 of the time. All the calculations in this article are available in Mathematica and Python files in the supplementary material, allowing readers to experiment with their own variants.
The Stern-Gerlach experiment (SGE) has been known for decades, but a complete theory that explains all facets of it remains enigmatic. In its classic realization, each spin state of an atom follows a unique path or trajectory in an SGE. The enigmatic part is the final output of SGE. Feynman envisaged the outcome to be an entangled state, a superposition of $|\text{spin}\phantom{\rule{4.pt}{0ex}}\text{state}\ensuremath{\rangle}\ensuremath{\bigotimes}|\text{corresponding}\phantom{\rule{4.pt}{0ex}}\text{path}\ensuremath{\rangle}$. Later works described the final state of SGE as a mixed state, a weighted average of states belonging to separate beams or trajectories. Notably, Schwinger, Englert, and Scully compared the difficulty of obtaining a coherent output (a pure state) to ``putting together Humpty Dumpty'' again. Yet recent experiments using Bose-Einstein condensates indicate the emerging beams are partly coherent, favoring Feynman's description. In this work, we revisit SGE using a recent version of the quantum master equation. We show that both outcomes are probable depending on the time of flight and the timescale of the environmental fluctuations. We predict the output is initially coherent and grows gradually into a mixed state. Moreover, we also estimate how the magnetic-field gradient helps broaden the beam waist. Moreover, our treatment implies that the wave function of an atom passing through a Stern-Gerlach device does not undergo an instantaneous collapse; rather, a gradual decay of coherences results in a mixed state, resulting in an irreversible process.
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.