Qubits

Abstract

The differences between quantum information and classical information are due to the difference of a qubit in a quantum-physical system capable of storing it from a bit in a classical-physical system capable of storing it.¹ This difference arises primarily from the superposition principle of quantum mechanics; despite its being bivalent in the chosen computational basis, a qubit system can be in one of an infinite number of significant states, whereas a bit is capable of being in only one of two significant states.² A qubit system in general also must be considered as at the same time potentially being in one measurable state and/or the other opposite state rather than actually being in just one of the two available states as must necessarily be the case for a bit encoded in a classical physical system. Furthermore, unlike a classical state, a single unknown qubit-system state cannot generally be found by a single measurement. Rather, an ensemble of systems must be measured to discover their unknown shared quantum state.³ It is the nature of quantum potentiality that alternative possibilities for reaching a given quantum state at a given moment superpose, and so are capable of interfering with each other.

Physical bits in traditional digital computers are realized in memory elements, metal-oxide semiconductor field-effect transistors, and electrical wires, all of which carry substantial charge relative to a single electrical quantum [179]. Classical information processors use such elements to store bits of information and perform operations on them, whereas quantum information processors operate on individual quanta. The term “qubit” was coined by Benjamin Schumacher, “...although Holevo’s theorem gives an information-theoretic significance to [quantum entropy]... it does not provide an interpretation of [quantum entropy] in terms of classical information theory. We could not use [it], for example, to interpret the quantum entropy of some macrostate of a thermodynamic system as a measure of resources necessary to represent information about the system’s quantum microstate... [Instead] this is accomplished by replacing the classical idea of a binary digit with a quantum two-state system... These quantum bits, or ‘qubits,’ are the fundamental units of quantum information.” [367].

See Postulate I in Sect. B.1. Paul Dirac noted the unique character of the superposition principle, “the superposition principle that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory, as is shown by the fact that the quantum superposition principle demands indeterminacy in the results of observations in order to be capable of a sensible physical interpretation . . . analogies are likely to be misleading” [Dirac’s emphasis] [136].

Here, the only exceptions to this are situations in which precise information as to the two particular alternative orthogonal states in which a single qubit happens to have been prepared is possessed by the measuring agent and only these potential states are measured. It is precisely this character of individual qubits that provides the possibility of secure quantum key distribution. Here, the term ensemble is meant in the sense of statistical thermodynamics, where it refers to a set of identically prepared systems. See Postulates II and III in Sect. B.1.