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Qubit State Represented by Pendulum Oscillations

Masataka Ohta

Published Oct 1, 2018• 0 citations
OpenAlex

Abstract

Dirac’s misconception that classical and quantum superpositions are different because absolute phase is quantum physically meaningless is, even today, seemingly blindly believed by many physicists. However, there is no such difference because, time translation symmetry means absolute time and, thus, absolute phase are classical physically meaningless, too. As a qubit can be a polarization division multiplexed (PDM) quadrature amplitude modulated (QAM) symbol of light, its state has direct correspondence with polarization state of classical light. Even more intuitively, the state may be represented by two-dimensional mechanical oscillations of a pendulum, which enables intuitive understanding of not-really-quantum superpositioned states of a qubit even by elementary school pupils.

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Dirac’s misconception that classical and quantum superpositions are different because absolute phase is quantum physically meaningless is, even today, seemingly blindly believed by many physicists. However, there is no such difference because, time translation symmetry means absolute time and, thus, absolute phase are classical physically meaningless, too. As a qubit can be a polarization division multiplexed (PDM) quadrature amplitude modulated (QAM) symbol of light, its state has direct correspondence with polarization state of classical light. Even more intuitively, the state may be represented by two-dimensional mechanical oscillations of a pendulum, which enables intuitive understanding of not-really-quantum superpositioned states of a qubit even by elementary school pupils.
With optical communication technology today, it is practical to communicate with polarization division multiplexed (PDM) quadrature amplitude modulated (QAM) symbols, which are quantum superposition of horizontally and vertically polarized photons, which are, so called, qubits. As the number of bits encoded by a PDM QAM symbol is limited, according to Shannon-Hartley theorem, by signal to noise ratio, the degree of parallelism of quantum computers is limited. The result is consistent with quantum threshold theorem. Quantum entanglement between qubits only makes the number of bits encoded by the qubits smaller, because entanglement means correlation between the qubits. Thus, quantum computers are not more powerful than classical ones. Finally, it is shown that purely classical computers can be arbitrarily fast and ideal, that is, noiseless, quantum computers are classical.
Abstract Moving completely into the quantum realm, the key concept of quantum state vector is described in three ways: (1) arrows in state space, (2) Dirac notation symbols that are important for the formal mathematical manipulation of quantum states, and (3) collections of 1s and 0s grouped in columns or rows. A qubit (two-level system) has two basis states. These basis state vectors can be added to give superposition states and multiplied using scalar products to change the state coefficients. These abstract concepts are given a physical meaning in a system of polarized light, a familiar phenomenon because of its use in polarized sunglasses. Light polarization has been widely used to demonstrate many fundamental aspects of quantum computing and quantum information science.
The complementarity between time and energy, as well as between an angle and a component of angular momentum, is described at three different layers of understanding. The phenomena of super-resolution are readily apparent in the quantum phase representation which also reveals that entanglement is not required. We modify Schwinger's harmonic oscillator model of angular momentum to include the case of photons. Therein, the quantum angle measurement is shown to be equivalent to the measurement of the relative phase between the two oscillators. Two reasonable ways of dealing with degeneracy are shown to correspond to a conditional measurement which takes a snapshot in absolute time (corresponding to adding probability amplitudes) and a marginal measurement which takes an average in absolute time (corresponding to adding probabilities). The sense in which distinguishability is a ``matter of how long we look'' is discussed and the meaning of the general theory is illustrated by taking the two oscillators to be circularly polarized photons. It is shown that an odd number of xxx-polarized photons will never have an angle in correspondence with the yyy axis, but an even number of xxx-polarized photons always can! The behavior of an xxx-polarized coherent state is examined and the snapshot angular distributions are seen to evolve into two counter-rotating peaks resulting in considerable correspondence with the yyy axis at the time for which a classical linear polarization vector would shrink to zero length. We also demonstrate how the probability distribution of absolute time (herein a measurable quantity, rather than just a parameter) has an influence on how these snapshot angular distributions evolve into a quantum version of the polarization ellipse.
It is pointed out that, while Suskind and Glogower claim “The phase operator for an oscillator is shown not to exist.”, they actually showed absolute phase operator, which is physically meaningless, does not exist. It is also pointed out that, they showed countably infinitely many trigonometric relative (relative between two oscillators) phase states, which form complete basis and are, in a sense, best possible quantum “phase states”, exist. In addition, continuously infinitely many phase states are constructed at classical, that is, infinitely many quanta, limit. Similarly, relative time states and operator are constructed with two oscillators with different angular velocities.
Time-dependent Schroedinger equation represents the basis of any quantum-theoretical approach. The question concerning its proper content in comparison to the classical physics has not been, however, fully answered until now. It will be shown that there is one-to-one physical correspondence between basic solutions (represented always by one Hamiltonian eigenfunction only) and classical ones, as the non-zero quantum potential has not any physical sense, representing only the "numerical" difference between Hamilton principal function and the phase of corresponding wave function in the case of non-inertial motion. Possible interpretation of superposition solutions will be then discussed in the light of this fact. And also different interpretation alternatives of the quantum-mechanical model will be newly analyzed and new attitude to them will be reasoned.