In quantum information, the spin-based qubit is not represented by a whole spinor in space as illustrated by the z spinor below : <> but by a simplified spinor without spatial extension : <>This simplification is the basis of our first remark about the quantum computer concept. Indeed, the demonstrations explaining the interest of the Deutsch, Glover and Shor algorithms are based on calculations using the factorization of entangled qubits. These factorizations are accurate for spinors without spatial extensions, but only approximate for real spinors with spatial extensions. A more fundamental remark concerns the existence of the qubit as a superposition of quantum states. This reality depends on the quantum mechanics interpretation. In the de Broglie-Bohm interpretation, the wave function (spinor) is not sufficient to completely describe the quantum state of a particle ; it is necessary to add the position of the particle. The spinor spatial extension takes into account the initial position of the particle. However, this position is unknown to the experimenter and is revealed only when measuring. Several particles prepared in the same way have the same wave function (spinor) but different positions. In this interpretation, the superposition of states does not really exist for an individual particle. It is only a linear combination of states resulting from our ignorance of the position of the particle. The superposition of states represents only a statistical collection of particles and the spinor corresponds to a statistical qubit.We present a detailed study of the whole spinor evolution (in space and in time) in the Stern and Gerlach experiment and we explain the decoherence, the individual impacts and the quantization. Our analysis explains very simply, in the de Broglie-Bohm interpretation, the negative results of the NMR technique in attempts to make a large quantum computer (more than 7 qubits). This technique does not use individual quantum objects, but a set of more than 10 8 active molecules diluted in a solvent : the measurement is thus a collective and reliable signal. The results of the NMR data being statistical, then we have a natural explanation of the drop by a factor two of the signal strength.Finally, the parallel quantum computer is feasible in the interpretation of Deutsch and the parallel universes of Hug Everett, but impossible in the de Broglie-Bohm interpretation. In this interpretation, the individual qubit does not exist and we must use two particles to represent it ; the statistical qubit alone exists, such as in Chuang’s computer.
In quantum information, the spin-based qubit is not represented by a whole spinor in space as illustrated by the z spinor below : <> but by a simplified spinor without spatial extension : <>This simplification is the basis of our first remark about the quantum computer concept. Indeed, the demonstrations explaining the interest of the Deutsch, Glover and Shor algorithms are based on calculations using the factorization of entangled qubits. These factorizations are accurate for spinors without spatial extensions, but only approximate for real spinors with spatial extensions. A more fundamental remark concerns the existence of the qubit as a superposition of quantum states. This reality depends on the quantum mechanics interpretation. In the de Broglie-Bohm interpretation, the wave function (spinor) is not sufficient to completely describe the quantum state of a particle ; it is necessary to add the position of the particle. The spinor spatial extension takes into account the initial position of the particle. However, this position is unknown to the experimenter and is revealed only when measuring. Several particles prepared in the same way have the same wave function (spinor) but different positions. In this interpretation, the superposition of states does not really exist for an individual particle. It is only a linear combination of states resulting from our ignorance of the position of the particle. The superposition of states represents only a statistical collection of particles and the spinor corresponds to a statistical qubit.We present a detailed study of the whole spinor evolution (in space and in time) in the Stern and Gerlach experiment and we explain the decoherence, the individual impacts and the quantization. Our analysis explains very simply, in the de Broglie-Bohm interpretation, the negative results of the NMR technique in attempts to make a large quantum computer (more than 7 qubits). This technique does not use individual quantum objects, but a set of more than 10 8 active molecules diluted in a solvent : the measurement is thus a collective and reliable signal. The results of the NMR data being statistical, then we have a natural explanation of the drop by a factor two of the signal strength.Finally, the parallel quantum computer is feasible in the interpretation of Deutsch and the parallel universes of Hug Everett, but impossible in the de Broglie-Bohm interpretation. In this interpretation, the individual qubit does not exist and we must use two particles to represent it ; the statistical qubit alone exists, such as in Chuang’s computer.
We present an analytical solution to the decoherence time for the spin measurement and the diagonalization of the density matrix of spin variables in the Stern-Gerlach experiment.This solution requires the calculation of the Pauli spinor with a spatial extension, which is not found in quantum mechanics textbooks.With this full spinor and the measured position of the particle we demonstrate the three postulates of quantum measurement: quantization, spectral decomposition and wave function reduction.The transition from a quantum superposition to a statistical mixture is well explained in this way, but not the single result that always emerges from a particular experiment.The spinor spatial extension also allows the introduction of the de Broglie-Bohm trajectories which give a very simple explanation of the particles' impact.
We propose a pedagogical presentation of measurement in the de Broglie-Bohm interpretation. In this heterodox interpretation, the position of a quantum particle exists and is piloted by the phase of the wave function. We show how this position explains determinism and realism in the three most important experiments of quantum measurement: double-slit, Stern-Gerlach, and EPR-B. First, we demonstrate the conditions in which the de Broglie-Bohm interpretation can be assumed to be valid through continuity with classical mechanics. Second, we present a numerical simulation of the double-slit experiment performed by Jönsson in 1961 with electrons. It demonstrates the continuity between classical mechanics and quantum mechanics. Third, we present an analytic expression of the wave function in the Stern-Gerlach experiment. This explicit solution requires the calculation of a Pauli spinor with a spatial extension. This solution enables us to demonstrate the decoherence of the wave function and the three postulates of quantum measurement. Finally, we study the Bohm version of the Einstein-Podolsky-Rosen experiment. Its theoretical resolution in space and time shows that a causal interpretation exists where each atom has a position and a spin.
We define the de Broglie-Bohm (dBB) weak interpretation as the dBB interpretation restricted to particles in unbound states whose wave function is defined in the three-dimensional physical space, and the dBB strong interpretation as the usual dBB interpretation applied to all wave functions, in particular to particles in bound states whose wave function is defined in a 3N-dimensional configuration space in which N is the number of particules. We show that the current criticisms of the dBB interpretation do not apply to this weak interpretation and that, furthermore, there are theoritical and experimental reasons to justify the weak dBB interpretation. Theoretically, the main reason concern the continuity existing for such particles between quantum mechanics and classical mechanics: we demonstrate in fact that the density and the phase of the wave function of a single-particle (or a set of identical particles without interaction), when the Planck constant tends to 0, converges to the density and the action of a set of unrecognizable prepared classical particles that satisfy the statistical Hamilton-Jacobi equations. As the Hamilton-Jacobi action pilots the particle in classical mechanics, this continuity naturally concurs with the weak dBB interpretation. Experimentally, we show that the measurement results of the main quantum experiments (Young's slits experiment, Stern and Gerlach, EPR-B) are compatible with the de Broglie-Bohm weak interpretation and everything takes place as if these unbounded particles had trajectories. In addition, we propose two potential solutions to complete the dBB weak interpretation.
The Stern-Gerlach experiment is the fundamental experiment in order to exhibit the quantization of spin and understand the measurement problem in quantum mechanics. However, although the Stern-Gerlach experiment plays an essential role in the teaching of quantum mechanics, no complete analysis of this experiment using Pauli spinors is presented in the pedagogical literature. This paper presents such an analysis and develops implications for the theory of quantum measurement. We first propose an analytic expression of both the wave function and the probability density in the Stern-Gerlach experiment. Our explicit solution is obtained via a complete integration of the Pauli equation over time and space. The probability density evolution describes a slipping of the wave packet into two separate packets due to the measurement device, but it cannot account for impacts. We therefore calculate the de Broglie-Bohm trajectories, which not only explain impacts naturally, but also accounts for the spin quantization following the magnetic field gradient. It is then possible to propose a clear explanation of measurement effects in the Stern-Gerlach experiment.
In the de Broglie Bohm pilot-wave theory and the many-worlds interpretation, unitary development of the quantum state is universally valid. They differ in that de Broglie and Bohm assumed that there are point particles with positions that evolve in time and that our observations are observations of the particles. The many-worlds interpretation is based on the fact that the quantum state can explain our observations. Both interpretations rely on the decoherence mechanism to explain the disappearance of interference effects at a measurement. From this fact, it is argued that for the pilot-wave theory to work, circumstances must be such that the many-worlds interpretation is a viable alternative. However, if this is the case, the de Broglie–Bohm particles become irrelevant to any observer. They are truly hidden. The violation of locality and the corresponding violation of Lorenz invariance are good reasons to believe that dBB particles do not exist.
The dominance of the Copenhagen interpretation is the result of a historical accident (Cushing, 1994). At the Solvay conference in 1927, Louis de Broglie put forward the deterministic pilot wave theory, described later. Never slow to criticize, Pauli jumped up and demolished de Broglie’s proposal, and de Broglie was so discouraged by this that he did not develop his interpretation any further. The Copenhagen interpretation was then developed by Bohr, Heisenberg, Pauli and many others, as described in the previous chapter. It was strongly supported by von Neumann’s proof of the impossibility of hidden variables. Years later, in 1952, Bohm published a deterministic hidden variable theory, showing that there must be something wrong with von Neumann’s proof. The error was found by John Bell in 1966, as described in the last chapter, and he also showed how Pauli’s objections to de Broglie’s original idea can be answered. If this had been done at the conference in 1927, quantum mechanics could have been interpreted deterministically from the beginning. The essential mistake in the Copenhagen interpretation is to treat it as a complete account of the behaviour of each individual system. It is frequently claimed that it enables us to calculate, at least in principle, everything that can be measured. Thus Hooft (1997, p. 11) said that ‘The laws of quantum mechanics have been formulated very accurately. We know exactly how to compute anything we would like to know.’ Also Peierls (1997, p. 25) has said that quantum mechanics ‘had become a complete and consistent scheme capable of giving a unique answer to any questions relating to actual or possible observations’. Even if this were true, it would not imply that it is the final complete theory, for it is always possible that some new phenomenon might be found that cannot be calculated quantum-mechanically. However, it is not true: there are many phenomena such as the time of decay of a radioactive nucleus, or the direction a particle is scattered by a nucleus, that cannot be calculated, even in principle. Quantum mechanics is therefore an incomplete theory. We can only calculate the statistical properties of these systems, such as the half-life of a radioactive decay or the differential scattering cross-section that gives the probabilities that a particle is scattered through various angles. All measurements of quantum systems are of this statistical character. It might be thought that the possibility of making measurements on a single electron provides an exception to this. However, even in such cases the electron is continually bathed in a fluctuating background radiation from nearby atoms. Since this is variable, that electron is a member of an ensemble of electrons subject to different fluctuations. All these examples clearly demonstrate the statistical character of quantum mechanics. ‘I am rather firmly convinced’, Einstein (in Schilpp, 1949, pp. 666, 671) remarked, ‘that the essentially statistical character of contemporary quantum theory is solely to be ascribed to fact that this theory operates with an incomplete description of physical systems’. As a result, ‘the c-function is to be understood as the description not of a single system but of an ensemble of systems’. So, ‘if the statistical quantum theory does not pretend to describe the individual system completely, it appears unavoidable to look elsewhere for a complete description of the individual systems’. Thus ‘the difficulties of theoretical interpretation disappear, if one views the quantum-mechanical description as a description of an ensemble of systems’. If this is achieved, ‘the statistical theory would, within the framework of future physics, take an approximately analogous position to statistical mechanics within the framework of classical mechanics’. ‘Thus, essentially, nothing has changed since Galileo or Newton or Faraday concerning the status of the ‘‘observer’’ or our ‘‘consciousness’’ or of our ‘‘information’’ in physics’ (Popper, 1982, p. 46). Once this is accepted, the quantum paradoxes that plagued the Copenhagen interpretation are easily resolved, as described below. In any textbook of quantum mechanics problems such as the calculation of the energy levels of the hydrogen atom are solved as if that hydrogen atom is alone in the universe. As Feynman (1972) remarked, ‘when we solve a quantum-mechanical problem, what we really do is to divide the universe into two parts – the system in which we are interested and the rest of the universe. We then usually act as if the system in which we are interested comprised the whole universe.’ In fact, however, every hydrogen atom on which measurements are made is surrounded by other atoms and exposed to their radiations. Quantum mechanics somehow takes this into account and gives the average behaviour of an ensemble of hydrogen atoms. A more detailed theory takes these influences into account. One attempt to do this is stochastic electrodynamics, described in a later section.