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

1. What is
quantum information?
Information symmetry and mechanical motion

2. 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

3. Key words:
• axiom of choice
axiom of (transfinite) induction ⦁
• category theory
Hamilton representation of mechanics ⦁
• Hilbert space
information ⦁
• information symmetry
Lagrange representation of mechanics ⦁

4. Key words:
• Minkowski space
Peano arithmetic ⦁
• pseudo-Riemannian space
quantum mechanics ⦁
• quantum information
qubit ⦁
• set theory
special & general relativity ⦁
• Standard model

5. 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 ⦁

6. 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)

7. 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
| ⟩𝟏𝟏
𝜶𝜶, 𝜷𝜷 are two complex numbers:
| ⟩𝟎𝟎 , | ⟩𝟏𝟏 are two orthonormal
vectors or a basis such as two orthogonal
great circles of the unit ball

8. 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:

9. “1”“0”
A bit A qubit
| ⟩𝟎𝟎 | ⟩𝟏𝟏
𝜶𝜶 𝜷𝜷
The axis of symmetry The axis of symmetry
A
x
i
s
o
f
s
y
m
m
e
t
r
y
Lagrange
representation
Hamilton
representation
→
𝑝𝑝
→
𝑥𝑥
→
𝑥𝑥𝑚𝑚𝑚𝑚𝑚

10. 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

11. 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 ⦁

12. 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

13. 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

14. Kategory A
Kategory B
• • • • • • •
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
0
Two viewpoints
to the separable complex Hilbert space
equivalent in the final analysis
Peano arithmetic &
set theory
Kategory theory
Functor &
Cofunctor

15. 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 ⦁

16. 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

17. 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 ⦁

18. [U(1)] X [SU(2)]
The axes of symmetry The axes of symmetry
Lagrange
representation
Hamilton
representation
[SU(3)]
The Standard model by
information symmetry (IS)
Higgs
mechanism
U(1)
𝑆𝑆𝑈𝑈(2)
A privileged
reference frame
Hilbert space
Riemannian space
IS

19. Pseudo-
Riemannian
space
Bird eye’s view
to information
symmetry
• • • • • • • • • • • • • • • • • • •
• • • • • • •
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
Separable
complex
Hilbert
spaceSU(2)U(1)
U(1)X SU(2)Higgs
mechanism
Electro-weak
&HiggsSU(3)
Standard
model

20. It (1) equals asymmetry to symmetry
& as then (2) decomposes asymmetry to both
symmetry and asymmetry newly
Information
symmetry is
the (meta-)symmetry of
symmetry & asymmetry:
Bird eye’s view
to information
symmetry
SU(2)U(1)
U(1)X SU(2)
Higgs
mechanism
Electro-weak
&HiggsSU(3)
Standard
model
It is an iterative scheme
A privileged reference frame &
general relativity
Lagrange
representation
Hamilton
representation
Symmetry Assymmetry
Information
symmetry
Information
symmetry
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

21. 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 ⦁

22. 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

23. 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 ⦁

24. Obrigado de coração
pela sua atenção!

25. The paper:
You can find the entire paper in Internet (after 01.09.16) typing the title
What is quantum information?
Information symmetry and mechanical motion
in any search engine such as Google, Bing, etc.
Thank you so much for your kind attention!
Any questions or comments, please!