The concept of matter in physics can be considered as a generalized form of information, that of quantum information involved by quantum mechanics Even more, quantum information is a generalization of classical information: So, information either classical or quantum is the universal foundation of all in the world In particular, the ideal or abstract objects also share information (the classical one) in their common base

1. Quantum Information
As the Substance of the World

2. Vasil Penchev
vasildinev@gmail.com, vaspench@abv.bg
http://www.scribd.com/vasil7penchev
http://www.wprdpress.com/vasil7penchev
CV: http://old-philosophy.issk-bas.org/CV/cv-
pdf/V.Penchev-CV-eng.pdf
___________
5 Sep 14 17:00
Calouste Gulbenkian Foundation and School of
Human and Social Sciences of Nova University in
Lisbon, on 5 and 6 September 2014

3. The thesis
• The concept of matter in physics can be
considered as a generalized form of
information, that of quantum information
involved by quantum mechanics
• Even more, quantum information is a
generalization of classical information: So,
information either classical or quantum is the
universal fundament of all in the world
• In particular, the ideal or abstract objects also
share information (the classical one) in their
common base

4. The substance of all

6. Measurement and interpretation
Language as ontology
Measurement Interpretation
UnderstandingReality

7. The unity of mass and energy
• Contemporary physics introduces the notion
of matter and quantity of mass as a form of
energy according to Einstein’s famous
equation (𝐸 = 𝑚𝑐2
)
• Another possible step would be the equivalent
of that quantity of mass-energy to
information, and more especially to quantum
information
• The equivalent mass-energy of classical
information would be exactly zero

8. “Q” means quantum information
“f” means some unknown function

9. The substance of the physical world

10. The substance of all

11. Information
and thermodynamic entropy
• The dimensionless physical quantity of
thermodynamic entropy shares the same or similar
mathematical formula as information, and more
especially that of classical information
• Indeed any statistical ensemble consists of many
physical items, which are always a finite number
though perhaps very large:
• Classical information (or the physical quantity of
entropy) reflects order (respectively disorder) of
that finite set of items whatever be: Its quantity is a
number unambiguously assigned to that set

12. .
.
.
.
. .
.
.
.
.
..
.
.
A statistical ensemble:
A huge but finite number
of material items

14. The “river” between infinity and finiteness
Here is the state of finiteness
And here is the state of infinity
Inhabitants:
Quantum information,
material items having nonzero mass-energy
Inhabitants:
Classical information,
ideal or abstract items having zero mass-energy

15. The quantity of information
• Information can be considered as a quantity
describing the degree of ordering (or
disordering, or complexity) of any collection
• If the collection is a finite set, the corresponding
information should be classical
• Accordingly, if the collection is infinite, the
corresponding information should be quantum
• Even more, if the collection is infinite, it can be
as a set as anything else: Nevertheless its
information can be defined as quantum in both
cases

16. The “river” between infinity and finiteness
Here is the state of finiteness
And here is the state of infinity
Quantum information:
Well-ordering of infinite sets:
i.e. transfinite ordinals
Classical information:
Well-ordering of finite sets:
i.e. finite ordinals
Thus information
is well-ordering
in both cases,
on both sides of
the boundary
Thus information
is well-ordering
in both cases,
on both sides of
the boundary

17. Physical being as quantum information

18. Any physical item
Some corresponding nonzero quantity
of quantum information
A corresponding transfinite ordinal number
Some corresponding mathematical item
The “river” between physics and mathematics

19. Quantum information
as generalized information
• Quantum information can be seen as that
generalization of information, which is relevant
to infinite collections
• The classically defined information can refer
only to finite ones
• Thus quantum information corresponds to that
generalization of information, which passes
from finite to infinite sets
• Indeed the information of any infinite set can
be represented as the ordinal number of this
set

20. 1 n ω ω+1 ω+n ωω
“Little” transfinite ordinals
“Bits” “Qubits”
1 n
The most
entangled
qubit
Ω

22. ω ω+1
1
That qubit is absolutely
extraordinary and even
unique

23. Matter as the information of infinite
collections
• Consequently, the physical concept of mass, energy
and matter can be interpreted as the notion of
information in the case of quantum information, i.e.
of an infinite collection
• Indeed: A transfinite ordinal number corresponds to
the infinite set in question. Further, a value of
quantum information corresponds to that ordinal
number; a wave function, to that value of quantum
information; a value of mass-energy, to that wave
function
• That is: A transfinite ordinal number corresponds to
any given value of mass-energy in final analysis

24. The information of
an infinite set
A transfinite ordinal
A value of
quantum information
A wave function
A value of
mass-energy

25. Infinity and matter

26. Infinity as a whole
(“actual infinity”)
Infinity as a process
(“potential infinity”)
A variety of
different infinities
A well-ordering
of different transfinite
ordinals: ω+1; ω+2; ...
Different
cardinal
numbers
Different
cardinal
numbers
A well-ordering
of different infinite
cardinals:
A definite and FINITE value of mass-energy

27. Choice and information
• Further, the notion of choice grounds that of
information
• Information can be thought as a few equated
relations:
• Between two orders in the present, or as the
steps (choices) to the latter order starting from
the former one
• Between two well-orderings in the past, or as the
corresponding steps (choices) as above
• Between two possible states in the future, or
corresponding steps (choices)

28. Future
Past
The leap of
the present
An order
Another
order
A well-ordering Another well-ordering
A possibility Another possibility

30. Future
Past
The leap of
the present
An order
Another
order
A well-ordering Another well-ordering
A possibility Another possibilityAn order
A possibility
A well-ordering

31. The unit of information
as an elementary choice
• The quantity of information can be seen as the
quantity of elementary choices or that of units of
choice, which are also units of information
• Indeed the elementary choices can be equated to
the number of minimal discrete steps necessary to
achieve a given order, well-ordering, or future state
from another
• For example, a bit (i.e. a Binary digIT) of information
is the minimal discrete step between the two binary
digits, such as “0” and “1”, or the choice between
them

32. 0
1
One elementary choice
One elementary step

33. Quantum information
and the axiom of choice

34. 1 n ω ω+1 ω+n ωω
“Bits” “Qubits”
1 n Ω
The sense of the axiom of choice:
The choice is uniform
from the finiteness to the infinity

35. The notion of infinite choice
• One can demonstrate that quantum mechanics
involves and even develops implicitly the concept of
choice as to infinity
• The same makes also set theory (the so-called
paradox of Skolem based on the axiom of choice)
• One can define the infinite generalization of a bit
(i.e. an elementary choice between two
equiprobable alternatives) as the elementary choice
between infinitely many alternatives
• Further, one can demonstrate that this is equivalent
to a quantum bit (qubit) involved by quantum
mechanics and information

36. Quantum
mechanics
Physics Mathematics
The foundation of
the physical world
The foundation
of mathematics
Set theory

38. Subject Object
The dualism of classical philosophy
Two fundamental approaches in philosophy:

39. The concept of quantum information
• The concept of quantum information can be
introduced in different ways. Some of them
(and the most important ones in the present
context) are:
• By Hilbert space
• By wave functions
• By operators in Hilbert space
• By a quantum Turing machine (tape)
• By quantum computations

40. Quantum information in terms of
Hilbert space Wave functions
Turing machines Quantum computations

41. Quantum information by Hilbert space

42. Hilbert space
as the free variable
of quantum information
Any point in it
as the value of
that free variable
A wave function
A state of
a quantum system
=
=

43. A quantum bit (qubit)
• The notion of quantum bit (or ‘qubit’) underlies
quantum information
• A quantum bit is usually defined as
the normed superposition of any two subspaces of
Hilbert space orthogonal to each other:
• That is: 𝛼|0 + 𝛽|1 where 𝛼, 𝛽 are two 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.)

44. Quantum “Turing tape”
• Furthermore, Hilbert space can be represented as a
“tape” of qubits: Given any point in (complex)
Hilbert space as a vector 𝐶1, 𝐶2, … 𝐶 𝑛, 𝐶 𝑛+1, … ,
one can replace any successive couple of its
components such as ( 𝐶1, 𝐶2 , 𝐶2, 𝐶3 , …
𝐶 𝑛−1, 𝐶 𝑛 … ) with a single corresponding qubit
{𝑄1, 𝑄2, … , 𝑄 𝑛, 𝑄 𝑛+1, … } such that:
𝛼 𝑛 =
𝐶 𝑛
(+) 𝐶 𝑛
2+ 𝐶 𝑛+1
2
; 𝛽 𝑛 =
𝐶 𝑛+1
(+) 𝐶 𝑛
2+ 𝐶 𝑛+1
2
if 𝐶 𝑛, 𝐶 𝑛+1 are not both 0. However if both are 0 one
needs to add conventionally the center (𝛼 𝑛 = 0, 𝛽 𝑛 =
0) to conserve the mapping of Hilbert space and an
infinite qubit tape to be one-to-one

45. Components
“Axes”Hilbert space
Quantum Turing tape
1 ... n n+1 ...
The/No
last
cell
... ...
......
..
𝒆𝒊.𝟏.𝝎
𝒆𝒊.𝒏.𝝎
𝒆𝒊.(𝒏+𝟏).𝝎 𝒆𝒊.∞.𝝎
𝑪 𝟏
𝑪 𝒏 𝑪 𝒏+𝟏
𝑪"∞"

46. Bit and qubit
• Then if any bit is an elementary binary choice
between two disjunctive options usually
designated by “0” and “1”, any qubit is a
choice between a continuum of disjunctive
options as many (or “much”) as the points of
the surface of the unit ball:
• Indeed a qubit is equivalently representable as
a unit ball in Euclidean space and two points,
the one chosen within the ball, and the other
on its surface, i.e. as a mapping of a unit ball
onto its surface (or any other unit sphere)

47. 𝜶| 𝟎 defines a point of the unit ball
𝜶| 𝟎 and 𝜷| 𝟏 define a point of the unit sphere
𝜶 𝟐
+ 𝜷 𝟐
= 𝟏| 𝟎
| 𝟏
𝜶, 𝜷 are two complex numbers:
| 𝟎 , | 𝟏 are two orthonormal
vectors or a basis such as two orthogonal
great circles of the unit ball

48. Choice, information, and computation
• Thus the concept of choice is the core of
computation and information
• The choice unifies them for:
• Information and computation mean one and
the same from two different viewpoints:
• Information is a relation between an order,
well-ordering, or possible state and another
• Computation is the same relation represented
as a process, i.e. step by step
• That is: The computation can be considered as
the constructive analog of the information

49. Infinity as a whole
“actual infinity”
Infinity as a process
“potential infinity”
A relation between
two infinities
(as possible states
as orderings)
Information
A process from one
to another infinity
(only as well-orderings)
Quantum computation

51. Finiteness Infinity Mathematics
Informatics
Quantum
information
Reality Physics
Classical
information
Ideality

52. Information
as the number of primary choices
• The concept of information can be interpreted as the
quantity of the number of primary choices
• It can be visualized as the minimal length of
the Turing tape, either finite or transfinite (quantum),
necessary for this information to be written
• For example, the quantity of classical information is
equivalent to that number written in binary notation on
the Turing tape (e.g. as the ultimate result of processing
by a Turing machine)

53. 1 n ω ω+1 ω+n ωω
“Bits” “Qubits”
1 n Ω
Information as the number correspondingly
Of finite choices
(such as bits)
Of infinite or transfinite
choices (such as qubits)

54. The concept of Turing machine as a bridge
between computation and information

55. A “classical” Turing machine
A quantum Turing machine
1 ... n n+1 ...
The
last
cell
A classical Turing
tape of bits:
A quantum Turing
tape of qubits:
1 ... n n+1 ...
The
/No
last
cell
The list of all
operations on a cell:
1. Write!
2. Read!
3. Next!
4. Stop!

56. Information “bit by bit”
• This equivalence of information and computation
allows of discussing the information “step by step”
or “one by one”:
• The quantity of information can be thought as the
sum of the change bit by bit or qubit by qubit, i.e.
as the change of number written by two or infinitely
many digits
• The only difference between the two cases is the
number of different digits: two (or equivalently any
finite number) versus infinite ones
• Information is classical in the former case and
quantum in the latter

57. 1 n ω+1 ω+n
“Bits” “Qubits”
1 n
A classical Turing
machine
A quantum Turing
machine

59. 0
1
0
1
One bit (a finite choice)
One qubit (an infinite choice)
∞
Choice Well-ordering

60. Information within a sell
• Thus, the distinction between the classical and
quantum case can be limited within any cell of an
algorithm or (qu)bit of information:
• Any bit or qubit is a digit of the written information:
• If the system of notation contents of a finite
number of “digits” (or the number of alternatives of
an elementary choice), the corresponding written
information is classical
• If not (i.e. the “digits” are infinitely many or much),
the information is quantum as in the physical world

61. ... ...
......
..
An example of a binary number (i.e. a well-ordered
series of bits and binary values):
An example of a decimal number (i.e. a well-ordered
series of digits and decimal values):
An example of a number in a counting system of
“infinite base” (i.e. a well-ordered series of qubits
and infinite values):

62. Components
“Axes”Hilbert space
Quantum Turing tape
1 ... n n+1 ...
The/No
last
cell
... ...
......
..
𝒆𝒊.𝟏.𝝎
𝒆𝒊.𝒏.𝝎
𝒆𝒊.(𝒏+𝟏).𝝎 𝒆𝒊.∞.𝝎
𝑪 𝟏
𝑪 𝒏 𝑪 𝒏+𝟏
𝑪"∞"

63. Physical processes as quantum
computations

64. Different ways to represent a physical process
as different, but equivalent changes
Of wave functions:
Of values of quantum information:
Of transfinite ordinal numbers:
Thus as a quantum computation

65. Physical states and quantities
in terms of Hilbert space
• Quantum mechanics states that any physical state
or its change is a self-adjoint operator in Hilbert
space as any physical system can be considered as
a quantum one
• Consequently the states can be interpreted as the
quantities to the state of absolute rest
• Then the boundary between ’quantum states’
and ‘quantum quantities’ is relative:
• Indeed any point in Hilbert space can be
interpreted as the relevant operator transforming
the zero point into the point in question

66. The temporal interpretation of the definition
of physical quantity (A) in quantum mechanics
Ψ*(x) Ψ∗(x)

68. The physical
processes
in space-time
Computations
of a quantum
computer

69. Physical processes as changes of
information
• Consequently all physical processes are
informational in fact:
• The boundary between the physical and the
informational can be represented temporally:
• The physical is all till now: This is the current
computational result as well all previous results
constituting a well-ordering
• The informational includes the physical: However it
includes the future as a single coherent state as well
as its gradual transformation in the present and
past of the physical:

70. Time and entanglement
A coherent
state
A few entangled
states
A few well-ordered series in time

71. Quantum mechanics as a universal
doctrine

72. All is quantum information because:
Quantum mechanics offers
an universal scientific theory:
Any physical item and process
are quantum in final analysis
Thus all should be quantum
information and/ or its processing

73. Quantum processes in terms of
quantum information
• Any quantum process is informational in terms of a
generalized kind of information: quantum
information
• The course of time implies this:
• The absolutely coherent future containing all
possibilities of development is being transformed
into the well-ordered series of events in the past by
mediation of the present, within which the choice
necessary for that transformation is being
accomplished

74. The principles of least action and most probability:
1st stage:
2nd stage:
Physical motion: A leap from past
into future
A choice of a smooth
trajectory from future
to past, “from the end
to the beginning”

77. Quantum information as the base
• Quantum information is the real fundament of the
world
• If the time is what creates the world, the quantum
information is what builds the world
• Indeed the world in all variety needs freedom and
choice to be able to be created:
• It is impossible both in the absolutely rigorously
ordered and thus lifeless past and in the quite
chaotic and elemental future:
• The world is possible only in the thin strip, the
precinct or the phase transition between them: the
present

79. All wave functions
as values of quantum information

81. The quantum quantities as the
changes of quantum information
• All physical quantities in the world are
a certain kind of changes of wave functions
and thus of quantum information
• Thus they can be as interpreted as a
“distraction” of transfinite ordinal numbers
• If that difference is a finite, the information is
classical
• If that difference is transfinite, the information
is quantum: There is some physical change

82. 1 ω+n ω
“Bits” “Qubits”
n 0
n
Quantum information =
a physical change
Classical information =
an ideal change

84. StaffEnergy
Matter
Information
Ideality
Reality
Time
Being
A local viewpointA global viewpoint
The philosophical map of concepts

85. Quantum information and matter
• Quantum information is the substance of the physical
world, its “matter”
• However quantum information being a generalization
of classical information and therefore including it as
the particular case as to finite sets underlies also the
abstract or ideal world
• Thus quantum information underlies both the
material and ideal world, i.e. anything existing
• Indeed anything, which is being now, requires or make
some choices, the corresponding quantity of which is
that of information

86. StaffEnergy
Matter
Quantum information
Ideality
Reality
The choices of time
Another map of concepts
Classical information

87. Information and the unity of physics
and mathematics

88. StaffEnergy
Matter
Quantum information
Mathematics Physics
The choices of time
The union of physics and mathematics
Classical information

89. Quantum information and the unity of
the material and ideal world
• As information is a dimensionless quantity equally
well referring both to a physical entity or to a
mathematical class, it can serve as a “bridge”:
• Between physics and mathematics
• And thus a bridge between the material and ideal
world
• A bridge between the concrete and abstract objects
• This bridge connects the areas of infinity and
finiteness over the gap of ... complementarity

90. The bridge over the gap
Information and
quantum information
Mathematics Physics
The material worldThe ideal world
Abstract objects Concrete objects
... of complementarity
Finiteness Infinity

92. Finite sets Infinite sets
Classical
information
Quantum
information

93. Conclusions:
• What is is being now
• Thus all which is is in time
• The fundamental structure of all is defined
between the relation between the future and
the past in the present by means of choices
• Choice is the only way for the unorderable
future to be transformed into the well-
ordered past
• Choice can happen only at the boundary of
the future and the past, i.e. the present

94. More conclusions:

95. Further conclusions:
• The unit of classical information, i.e. a bit,
means the elementary choice between two
equiprobable alternatives
• Analogically, the unit of quantum information,
i.e a quantum bit (a qubit) means the
elementary choice between an infinite set of
alternatives
• Quantum information is equivalent to Hilbert
space. Any point in it, i.e. a wave function, is a
value of quantum information

97. References I:
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99. Muito
obrigado
pela sua
atenção!

100. Congratulamo-nos
com suas
perguntas e
comentários!