Quantum theory is the result of the confrontation with some enthralling questions: “Is nature discrete or not?” “Are the ultimate elements of nature particles or fields?” “Do the phenomena of life and light belong to the same rank?” “How should we interpret the probabilistic results of the augmentation of entropy in thermodynamics?” The last question points out also to the unification of quantum theory and relativity, because of the significance of entropy in cosmology.
The development of computer science motivated scientists to explore the problem of simulating physics with computers. The relation between entropy and information was presented by Feynman with the mental experiment of a gas, treated with slow, isothermal compression. Since information is considered as the opposite of disorder, entropy can be used to decipher the information content of a probe. Based on the reversibility of the laws of microphysics, quantum theorists like Feynman stipulated a backward capacity of a quantum computer, constructed with consideration of diminished heat dissipation and decreased entropy, by avoiding loss of information.
All conventional logic gates, with the only exception of NOT, lose information irretrievably; for instance, in the conventional logic gate of implication, only when the output is F can we know with certainty that the input is T → F, thus, the likelihood to avoid loss of energy and information by knowing the input from the output is 0,25.
However, the reversible nature of physics permits that whenever we erase information in a computer, the previous information may be practically lost, but it becomes entropy, randomized information, taking the form of heat (Frank, 2017). The project of reversible computing is compatible with the conception of Simondon (2006; 2009) that the tools and the machines correspond to information and energy resources. Quantum reversible computing promises, therefore, an energy saving computing.
A simulation of quantum physics with computers should be based on structural empiricism, which argues that empirical evidence in microphysics comes from finite quanta, observable by their line spectra. This approach would comply with the quantum-philosophical principle of the restriction of theoretical description to observable facts and the mathematical introduction of the matrix mechanics (Born, 1956, Heisenberg, 1925).
Paul Benioff and Richard Feynman showed that information processing corresponds to a simulation of the values of the spin of a particle to one of the outputs of a logic gate (Bennett and Landauer, 1985). In the non-relativistic context, we represent the spin states as being up and down. In the relativistic context, we consider the electron as being polarised, just like a photon. Polarisations for 1/2 particles are usually called helicities. “To get spin 1 from spin 1/2 and spin 1/2, the electron and positron have to be polarised in the same direction” as Schwartz (2014, p. 86) explains.
Experiments, however, had shown that it was impossible to simultaneously assign exact values to the x and y components of the spin of a particle. Particles with spin 1/2 under a 2π rotation transform as ψ → – ψ, because they cannot be smoothly deformed to a point. This experimental data shows that quantum field theory cannot conform to classical logic.
Quantum mechanics differ from classical, as their “dynamical variables do not obey the commutative law of multiplication,” since position and momentum are conjugate variables. Take two particles with opposite momenta -p and p and opposite positions x and –x. We cannot measure the position and the momentum of the same particle because quantum mechanics says that this is impossible because position and momentum are conjugate variables. These dynamic variables are not ordinary c-numbers but what Dirac calls “q-numbers”, representable “by matrices whose elements are c-numbers (functions of a time parameter)” (Dirac, 1927, p. 621). The q-numbers are the foregoers of the qubits. Quantum information is reducible to qubits, to one- and two-qubit gate operations. Qubits are the quantum counterparts of the binary bits. Their physical counterparts are polarised photons.
A qubit can have a value that is either 0, 1 or a quantum superposition of 0 and 1. A two-dimensional column vector of real or complex numbers represents a possible quantum state held by a qubit that is a qubit state with the complex numbers α and β satisfying
|α|^2+ |β|^2= 1.
If photons pass through polarization filters, they cannot amount to irrational or rational numbers, but only to natural numbers. For this reason, their quantity is substituted by probability (Laidler, 1998). We distinguish between circular polarisation (called left and right helicity) and linear polarization. “Linearly polarised electrons are like linearly polarised light, and the polarisations must be transverse to the direction of motion. So the electron moving in the z direction can either be polarised in the y direction or in the x direction” (Schwartz, 2014, p. 86).
Orthogonality and reversibility are the critical functions of quantum computing. The polarization of a photon has basically two distinguishable states: horizontal and vertical. And you can distinguish them by using a pair of Polaroid sunglasses. The horizontal ones go through, and the vertical ones do not, or vice versa, depending on which way you rotate your glasses. The spin of 1/2 spin particle corresponds to the quantum states up and down. These are the basic states of the quantum state space, whose geometrical representation is given by the Bloch sphere: a qubit is a two-dimensional complex vector space, rendering the unit vector to a two-dimensional convex vector space, which is equivalent to a three-dimensional real vector space.
A quantum state space of a system is a vector space. In quantum mechanics, everything has complex numbers in it, thus we regard a complex vector space. A pure quantum state corresponds to a ray in a Hilbert space over the complex numbers, while mixed states are represented by density matrices. Individual qubits composing a multi-qubit system, do not always have individual (pure) states of their own. Hence, we give a statistical description of an individual qubit, or a group of qubits, in terms of a density matrix or mixed state (Mermin, 2007).
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Note: This commentary is a review of author’s presentation at the Congrès 2021 de la Société de Philosophie des Sciences – UMONS (Belgique) – 08-10 septembre 2021.
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