What is quantum information? Information Symmetry and Mechanical Motion

The thesis is: The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections.

Preliminary notes: The conception of quantum information was introduced in the theory of quantum

information studying the phenomena of entanglement in quantum mechanics. The entanglement was

theoretically forecast in the famous papers of Einstein, Podolsky, and Rosen (1935) and independently by Shrödinger (1935) deducing it from Hilbert space, the basic mathematical formalism of quantum mechanics. However, the former three demonstrated the forecast phenomenon as the proof of the alleged “incompleteness of quantum mechanics”. John Bell (1964) deduced a sufficient condition as an experimentally verifiable criterion in order to distinguish classical from quantum correlation (entanglement). Aspect, Grangier, and Roger (1981, 1982) confirmed experimentally the existence of quantum correlations exceeding the upper limit of the possible classical correlations. The theory of quantum information has thrived since the end of the last century in the areas of quantum computer, quantum communication, and quantum cryptography.

The fundament of quantum information is the concept of ‘quantum bit’, “qubit” definable as the normed superposition of any two orthogonal subspaces of complex Hilbert space as follows:

‘Qubit’ is: 𝛼|0⟩+𝛽|1⟩ where 𝛼, 𝛽 are complex numbers such that |𝛼|2+ |𝛽|2=1, and |0⟩, |1⟩ are any two

orthonormal vectors (e.g. the orthonormal bases of any two subspaces) in any vector space (e.g. Hilbert space, Euclidean space, etc.). Thus Hilbert space underlying quantum mechanics is representable as the quantity of quantum information and any wave function, i.e. any state of any quantum system being a point in it can be seen as a value of that quantity. Consequently all physical processes turn out to be quantum-informational, and nature or the universe is a quantum computer processing quantum information.

The qubit is also isomorphic to a ball in Euclidean space, in which two points are chosen: A qubit is
equivalently representable as a unit ball in Euclidean space and two points, the one chosen within the
ball, and the other being the orthogonal projection on its surface, i.e. as a mapping of a unit ball onto its surface (or any other unit sphere).

The main statement: Quantum information is equivalent to the generalization of information from finite to infinite series.

A sketch of the proof:

Indeed information can be interpreted as the number of choices necessary to be reached an ordering of some item from another ordering of the same item or from the absence of ordering. Then the quantity of information is the quantity of choices measured in the units of elementary choice. A bit is that unit of elementary choice: It represents the choice between two equiprobable alternatives. Furthermore, the unit of quantum information, the qubit, can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. Thus it is able to measure the quantity of information as to infinite sets.

The axiom of choice is necessary for quantum information in two ways: (1) in order to guarantee the

choice even if any constructive approach to be chosen an element of the continuum does not exist; (2) to equate the definition in terms of Hilbert space and that as a choice among a continuum of lternatives:

Indeed the theorems about the absence of hidden variables in quantum mechanics (Neumann 1932;
Kochen, Specker 1968) demonstrate that the mathematical formalism of quantum mechanics implies
that no well-ordering of any coherent state might exist before measurement. However, the same
coherent state is transformed into a well-ordered series of results in time after measurement. In order to be equated the state before and after measurement, the well-ordering theorem equivalent to the axiom of choice is necessary. The measurement mediating between them should be interpreted as an

absolutely random choice of an element of the coherent state, for which no constructive way (equivalent to some “hidden variable”) can exist in principle. Thus the quantity of quantum information can describe uniformly the state before and after measurement (equivalent to a choice among an infinite set). Thus, Hilbert space can be understood as the free variable of quantum information. Then any wave function,being a given value of it, “bounds” an unorderable and a well-ordered state as the quantity of qubits (i.e.the “infinite choices”) necessary for the latter to be obtained from the former.The quantity of quantum information is the ordinal corresponding to the infinity series in question. Both definitions of ‘ordinal’ (Cantor 1897; Neumann 1923) are applicable as the ordinals are small. The ordinal defined in Cantor – Russell (Russell, Whitehead any edition) generates a statistical ensemble while that in Neumann, a well-ordering. Both correspond one-to-one to a coherent state as the one and same quantity of quantum information containing in it.

References:

Aspect, A., Grangier, R., Roger, G. (1981) “Experimental Tests of Realistic Local Theories via Bell’s

Theorem”, Physical Review Letters, 47(7): 460-463.

Aspect, A., Grangier, R., Roger, G. (1982) “Experimental Realization of Einstein-Podolsky-Rosen-Bohm Gedanken Experiment: A New Violation of Bell’s Inequalities”, Physical Review Letters, 49(2): 91-94.

Bell, J. (1964) “On the Einstein ‒ Podolsky ‒ Rosen paradox”, Physics (New York), 1(3): 195-200.

Cantor, G. (1897) “Beitrage zur Begrundung der transfiniten Mengenlehre (Zweiter Artikel)”, Mathematische Annalen, 49(2): 207-246.

Einstein, A., Podolsky, B., Rosen, N. (1935) “Can Quantum-Mechanical Description of Physical Reality Be

Considered Complete?” Physical Review, 47(10): 777-780.
Kochen, S., Specker, E. (1968) “The problem of hidden variables in quantum mechanics”, Journal of
Mathematics and Mechanics, 17(1): 59-87.

Neumann, J. von (1923) "Zur Einführung der trasfiniten Zahlen", Acta litterarum ac scientiarum Ragiae

Universitatis Hungaricae Francisco-Josephinae, Sectio scientiarum mathematicarum, 1(4): 199–208.

Neumann, J. von (1932). Mathematical foundation of quantum mechanics [Mathematische Grundlagen der Quantenmechanik, Berlin: Springer, pp. 167-173 (Chapter IV.2).

Russell, B, Whitehead, A. N. (any edition) Principia Mathematica, Vol. 2(*153), Vol. 3(*251).

Schrödinger, E (1935) “Die gegenwärtige situation in der Quantenmechanik”, Die aturwissenschaften,23(48), 807-812; 23(49), 823-828, 23(50), 844-849.

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A related paper: Quantum information as the information of infinite series 
(Abstract: The quantum information introduced by quantum mechanics is equivalent to that generalization of the classical information from finite to infinite series or collections. The quantity of information is the quantity of choices measured in the units of elementary choice. The qubit can be interpreted as that generalization of bit, which is a choice among a continuum of alternatives. The axiom of choice is necessary for quantum information. The coherent state is transformed into a well-ordered series of results in time after measurement. The quantity of quantum information is the ordinal corresponding to the infinity series in question.)