What is quantum information?

Information symmetry and mechanical motion

Vasil PenchevBulgarian 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 PHILOSOPHYPorto, Portugal: 8-9 September 2016 (14: 30, 8 Sep, room 201)

University of Porto - Faculdade de Letras

Key words:

• axiom of choiceaxiom of (transfinite) induction ⦁

• category theoryHamilton representation of mechanics ⦁

• Hilbert spaceinformation ⦁

• information symmetryLagrange representation of mechanics ⦁

Key words:• Minkowski space

Peano arithmetic ⦁• pseudo-Riemannian space

quantum mechanics ⦁• quantum information

qubit ⦁• set theory

special & general relativity ⦁• Standard model

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 ofThe 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: 𝛼𝛼| ⟩0 + 𝛽𝛽| ⟩1 where 𝛼𝛼,𝛽𝛽 are two complex numbers such that 𝛼𝛼 2+ 𝛽𝛽 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.)

• 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)

𝜶𝜶| ⟩𝟎𝟎 defines a point of the unit ball

𝜶𝜶| ⟩𝟎𝟎 and 𝜷𝜷| ⟩𝟏𝟏 define a point of the unit sphere

𝜶𝜶 𝟐𝟐+ 𝜷𝜷 𝟐𝟐 = 𝟏𝟏| ⟩𝟎𝟎

| ⟩𝟏𝟏𝜶𝜶,𝜷𝜷 are two complex numbers:

| ⟩𝟎𝟎 , | ⟩𝟏𝟏 are two orthonormalvectors or a basis such as two orthogonal great circles of the unit ball

The qubit as an unit of transfinite counting

• A qubit means the equivalence of the discrete (such as quantumleaps) 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, orthe 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:

“1”“0”

A bit A qubit

| ⟩𝟎𝟎 | ⟩𝟏𝟏

𝜶𝜶 𝜷𝜷

The axis of symmetry The axis of symmetry

Axis

of

symmetry

Lagrangerepresentation

Hamiltonrepresentation

→𝑝𝑝

→𝑥𝑥

→𝑥𝑥𝑚𝑚𝑥𝑥𝑚

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 isConsequently, 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 Peanoarithmetic 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 spaceThen, 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 & Peanoarithmetic as in the previous slide

Kategory A

Kategory B

• • • • • • •

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0

0

0

0

0

0

0

11

11

1

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1

1

0

Two viewpoints to the separable complex Hilbert spaceequivalent in the final analysis

Peano arithmetic &set theory

Kategory theory

Functor &Cofunctor

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, andThe 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 ⦁

[U(1)] X [SU(2)]

The axes of symmetry The axes of symmetry Lagrangerepresentation

Hamiltonrepresentation

[SU(3)]

The Standard model byinformation symmetry (IS)

Higgsmechanism

U(1)

𝑆𝑆𝑈𝑈(2)

A privilegedreference frame

Hilbert space

Riemannian space

IS

Pseudo-Riemannian

space

Bird eye’s view to information

symmetry• • • • • • • • • • • • • • • • • • •

• • • • • • •

00

0

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Separablecomplex

HilbertspaceSU(2)U(1)

U(1)X SU(2)Higgsmechanism

Electro-weak &HiggsSU(3)

Standard model

It (1) equals asymmetry to symmetry& as then (2) decomposes asymmetry to both symmetry and asymmetry newly

Informationsymmetry isthe (meta-)symmetry ofsymmetry & asymmetry:

Bird eye’s view to information

symmetry

SU(2)U(1)

U(1)X SU(2) Higgsmechanism

Electro-weak &HiggsSU(3)

Standard model

It is an iterative scheme

A privileged reference frame &general relativity

Lagrangerepresentation

Hamiltonrepresentation

Symmetry Assymmetry

Informationsymmetry

Informationsymmetry

• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

The structure of the paperinstead of conclusions

• Section 1, Introduction is a general outline of the paperSection 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 paperinstead 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 paperinstead 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 examplesexemplifies 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 ⦁

Obrigado de coraçãopela sua atenção!

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