What is quantum information? Information symmetry and mechanical motion

What is quantum information?

Information symmetry and mechanical motion

Vasil Penchev Bulgarian Academy of Sciences: Institute for the Study of Societies and Knowledge (Institute for Philosophical Research): Dept. of Logical Systems and models vasildinev@gmail.com

2nd PORTUGUESE CONGRESS OF PHILOSOPHY Porto, Portugal: 8-9 September 2016 (14: 30, 8 Sep, room 201) University of Porto - Faculdade de Letras

Key words: • axiom of choice • category theory • Hilbert space

axiom of (transfinite) induction ⦁

Hamilton representation of mechanics ⦁

information ⦁

• information symmetry Lagrange representation of mechanics ⦁

Key words: • Minkowski space • pseudo-Riemannian space • quantum information • set theory • Standard model

Peano arithmetic ⦁

quantum mechanics ⦁ qubit ⦁

special & general relativity ⦁

Quantum information and qubits • The unit of quantum information, a qubit is introduced as both: Standardly: normed superposition of two orthogonal subspaces of the separable complex Hilbert space, and ⦁ Newly: invariance of Hamilton and Lagrange representation of any mechanical system ⦁ • The base for that mechanical interpretation of information is the isomorphism of The standard introduction, and ⦁ The geometric representation of a qubit to a 3D unit ball, in which two points are chosen ⦁

The qubit as a 3D ball • ‘Qubit’ is: đ?›źđ?›ź|0â&#x;Š + đ?›˝đ?›˝|1â&#x;Š where đ?›źđ?›ź, đ?›˝đ?›˝ are two complex numbers such that đ?›źđ?›ź 2 + đ?›˝đ?›˝ 2 = 1, and |0â&#x;Š, |1â&#x;Š 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.) • 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 • A qubit (being a unit of quantum information) is at the same time that unit which unifies the discrete quantum leaps (representable by the separable complex Hilbert space) and smooth motion (representable as a continuum of 3D unit balls)

|đ?&#x;Žđ?&#x;Žâ&#x;Š

|đ?&#x;?đ?&#x;?â&#x;Š

đ?œśđ?œś, đ?œˇđ?œˇ are two complex numbers: đ?œśđ?œś đ?&#x;?đ?&#x;? + đ?œˇđ?œˇ

đ?&#x;?đ?&#x;?

= đ?&#x;?đ?&#x;?

|đ?&#x;Žđ?&#x;Žâ&#x;Š, |đ?&#x;?đ?&#x;?â&#x;Š are two orthonormal vectors or a basis such as two orthogonal great circles of the unit ball

đ?œśđ?œś|đ?&#x;Žđ?&#x;Žâ&#x;Š defines a point of the unit ball

đ?œśđ?œś|đ?&#x;Žđ?&#x;Žâ&#x;Š and đ?œˇđ?œˇ|đ?&#x;?đ?&#x;?â&#x;Š define a point of the unit sphere

The qubit as an unit of transfinite counting • A qubit means the equivalence of the discrete (such as quantum leaps) and the continuous (such as the continuum of smooth motion) Thus and furthermore, it can be interpreted as the unit unifying the standard and the nonstandard interpretation in the sense of Robinson’s analysis, or the proper and non-proper interpretation in the sense of Skolem’s “relativity of ‘set’” • Thus and furthermore, it can be also thought (as by us now) as the unit unifuing the Lagrange and Hamilton representation of mechanics (including quantum mechanics) Here (in the next slide) is visualized how:

The axis of symmetry

The axis of symmetry

Lagrange representation đ?‘šđ?‘šđ?‘šđ?‘šđ?‘š

“1�

“0�

A bit

đ?œśđ?œś

|đ?&#x;Žđ?&#x;Žâ&#x;Š

đ?œˇđ?œˇ

A qubit

|đ?&#x;?đ?&#x;?â&#x;Š

→ đ?‘?đ?‘?

→ ��

A x i s o f

→ ��

Hamilton representation

s y m m e t r y

Hilbert space as quantum information • The separable complex Hilbert space is considered as the free variable of quantum information Any point in it is its value as the bound variable then ⦁ • That value of quantum information is interpreted in quantum mechanics as a wave function describing a state of a quantum system what any physical entity is Consequently, the metaphysical conclusion is: the substance of the physical world is (quantum) information ⦁ • A generalization and possible hypothesis would be: all (i.e. not only physical world, but even the mental one) is information, among which the physical world is properly quantum information

The mental as mathematical, and the material as physical • That hypothesis about the omnipresence of quantum information would be exemplified by the interpretation of the mental as mathematical Then, the entire description of all being as quantum information would mean both ⦁ • The unity of the mental and material as quantum information The unity of our knowledge of them (the knowledge is mental) as the common base of mathematics and physics in quantum information ⦁

Hilbert space and Peano arithmetic • A qubit is equivalent to the generalization of ‘bit’ from the set of two equally probable alternatives to an infinite set of alternatives Then, that Hilbert space can be considered as a generalization of Peano arithmetic where: • Any unit is substituted by a qubit, and thus The set of natural numbers is mappable within any qubit as the complex internal structure of the unit ⦁ • That complex internal structure (neglected in Peano arithmetic) of any unit is a different state of all Peano arithmetic as whole

Hilbert space and category theory • Any mathematical structure being reducible to set theory is representable as: A set of wave functions, and thus: • As a subspace of the separable complex Hilbert space Then, it can be identified as the category of all categories for any functor represents an operator transforming a set (or subspace) of the separable complex Hilbert space into another ⦁ • Thus, category theory turns out to be isomorphic to the Hilbert-space representation of set theory & Peano arithmetic as in the previous slide

Peano arithmetic & set theory

0 1 Kategory B 0 1 0 1 Functor & 0 1 Cofunctor 0 1 Kategory A

0 1 0 1

Kategory theory Two viewpoints to the separable complex Hilbert space equivalent in the final analysis

0 1 0 1 0 1

• • • • • • •

Both physical and mathematical interpretations of quantum information • What that consideration implies: Given any value of quantum information (i.e. a point in the separable complex Hilbert space), it always admits two equally acceptable interpretations: • The one is physical, and The other is mathematical ⦁ • The former is a wave function as the exhausted description of a certain state of a certain quantum system The latter chooses a certain mathematical structure among a certain category ⦁

The indistinguishability of ‘mathematical structure’ and ‘physical state’ • Thus there is no way to be distinguished a mathematical structure from a physical state for both are described exhaustedly as a value of quantum information.

This statement in turn can be utilized to be defined quantum information by the identity of any mathematical structure to a physical state, and also vice versa ⦁ • Further, that definition is equivalent to both: Standard definition as the normed superposition, and: • Invariance of Hamilton and Lagrange interpretation of mechanical motion

Information symmetry • Then, the concept of information symmetry can be involved as the symmetry between three elements or two pairs of elements: The self-identical pair of Lagrange representation, and: • Each counterpart of the pair of Hamilton representation The sense and meaning of information symmetry may be visualized by a single (quantum) bit and its interpretation as: • Both (privileged) reference frame, and: The symmetries U(1), SU(2), and SU(3) of the Standard model ⦁

The axes of symmetry

The axes of symmetry

U(1) Higgs mechanism

Lagrange representation Riemannian space A privileged reference frame

[SU(3)]

đ?‘†đ?‘†đ?‘ˆđ?‘ˆ(2)

[U(1)] X [SU(2)] The Standard model by information symmetry (IS)

IS

Hilbert space Hamilton representation

• • • • • • • • • • • • • • • • • • •

Bird eye’s view to information symmetry

1 0 1 0 1 0 1

Standard model Electro-weak SU(3) &Higgs Higgs X SU(2) U(1) mechanism U(1)

0 1 0 1

0 1 PseudoRiemannian 0 1 space Separable complex 0 1 Hilbert SU(2) space 0 1 • • • • • • •

Information Information symmetry symmetry Symmetry Assymmetry • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

Hamilton representation

Bird eye’s view to information symmetry

Lagrange representation Standard A privileged reference frame & Information general relativity model symmetry is the (meta-)symmetry of Electro-weak SU(3) &Higgs symmetry & asymmetry: It is an iterative scheme Higgs U(1) X SU(2) mechanism It (1) equals asymmetry to symmetry & as then (2) decomposes asymmetry to both U(1) SU(2) symmetry and asymmetry newly

The structure of the paper instead of conclusions • Section 1, Introduction is a general outline of the paper Section 2, Quantum mechanics in terms of quantum information considers how quantum mechanics can be reinterpreted as an information theory ⦁ • Section 3, Hamilton and Lagrange interpretations unified in quantum information explains how the concept of (quantum) bit unifies both ways for mechanic to be interpreted Section 4, Information as the quantity of choice(s) discusses why information is the quantity of choices in the final analysis ⦁

The structure of the paper instead of conclusions • Section 5, HS as a generalization of Peano arithmetic deduces how the separable complex Hilbert space can be seen as a generalization of Peano arithmetic and the conclusion about the foundation of mathematics Section 6, Models of set and category theory in HS elucidates how HS can unify set theory and category theory ⦁ • Section 7, Identifying physics and mathematics as interpretations of quantum information reveals why a state in quantum mechanics and a mathematical structure in mathematics are isomorphic to each other as two equally admissible interpretations of quantum information

The structure of the paper instead of conclusions • Section 8, Information symmetry introduces the concept of information symmetry on the base of the equivalence of Hamilton and Lagrange interpretations Section 9, Information symmetry visualised by impressing examples exemplifies it by the symmetries of three qubits and their interpretation as a privileged reference frame ⦁ • Section 10, Metaphysical and philosophical interpretations discusses (quantum) information as the general substance of all mental and material phenomena Section 11, Summary addressing future work offers a few main directions for future work ⦁

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Information symmetry and mechanical motion

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