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Giovanni Andrea Fantasia
Matr. 488159
Corso di Laurea in Scienze dell’ Informazione
E-mail: [email protected]
AN INFORMATION-THEORETICAL APPROACH
TO QUANTUM MECHANICS INTERPRETATIONS
ABSTRACT
This work concern about the fundamental problem of which interpretation we have to give of our
mathematical theories in physical investigation of reality.
We develop to resolve conceptual quantum puzzles originally J. Wheeler ideas that imply
fundamental information-theoretical assumptions on foundations of quantum physics.
The basic idea is that physics could be described using as building bricks the information
achievement that comes from what in traditional quantum mechanics was called wave-collapse.
The work is divided in four parts: in part I we analyse how traditional epistemological categories,
as space-time continuum, are conceptually incompatible with quantum phenomena. In part II we
propose the new information-theoretical approach showing how it resolve conceptual difficulties of
interpretation of quantum mechanics. Part III concern about the ancient philosophical problem of
mind-body dualism and the status of consciousness in physics. In part IV we discuss the role of
informatics in our new interpretation.
This work is to be considered a contribution to the general debate in conceptual foundation of
physics and in the search of links between physics and informatics.
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I would like to thanks all the people who directly or indirectly helped me in these years.
Firstly I would like to thanks Prof. Gianni degli Antoni that believed in the ideas I have developed
in this work and always supported them and my work.
Then my family, for the support they gave me in this years of studying.
Finally all the people who showed interest for my work, so thanks to
Prof. Fornili, Prof. Recami, Dott.ssa Erica, Dott. Ghisi.
And thanks also to Prof. Preparata.
This work is dedicated to his memory.
2
CONTENENTS
PART I
 Space and time: a problematic approach to physics
p.5

The space-temporal continuum as metaphysical assumption
p.12

Discreteness Vs continuum
p.14
PART II
 Introduction
p.16
 It from bit
p.17
 Example: space from bits
p.19
 Other conceptually relevant implications
p.22
 Interference superposition entanglement
p.24
 Mathematical support
p.26
 The Everett’s multiverse approach
p. 30
 Time evolution
p.31
 Appendix I
p.32
PART III
 Introduction
p.34
 The problem of consciousness
p.34
 Consciousness, observation and wave-collapse
p.36
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 “It from bit “ and consciousness
p.38
PART IV

Introduction
p.40

The meaning of information
p.40

The formal equivalence between physics and informatics
p.42
Bibliography
p.38
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PART I
Space and time: a problematic approach to Physics
What is a physical system? This is in effect not a properly physical question. A physical question
may be: how can we describe a physical system? The first answer to this question born with
Galileo' s Kapler’ s Cartesio’ s and others works in first 1600. The answer is to set in a space time
orthonormal vectorial space our " objects " and to describe the physical system evolution as
trajectories in this space e.g.: the system of a body in free fall from a tower.
This was what we call the first era of physics. The geometrical description: body' s parabolas,
planets ellipses, etc.
The question of Isaac Newton was: how do these geometries come? This was the second era of
physics: the era of general laws; e.g. the ellipses comes from the general equation for body’s
gravitational attraction
 G m m
F  0 1 2 r̂
r2
(1.1)
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This era finds his apogee in Maxwell work on electrodynamics and arises just to us with modern
quantum field theories.
In first 1900 Einstein’s work clarifies the non-Euclidean nature of these geometries. Anyway the
space-time frameworks was still present; Einstein's conceptions only overcome the naive vision of
space and time.
This was approximately the situation until 20's quantum revolution; there was two kinds of
dynamics: punctual evolution for particles (viewed as points) and waves propagation for light,
sounds, etc. Some extraordinary experimental results forced the scientists to accept the fact that
light have corpuscular proprieties and particles have ondulatory proprieties. This is emphasised by
Schrödinger equation for matter evolution

 
Ĥψ  i ψ
(1.2)
t


where ψ is a complex function of space and time ψ(x, y, z; t)
This approach unified with special relativity is the modem quantum field theory. So we still have a
space-time framework in which fields evolve (by some wave equation); these fields interact with
our instrument of measure exchanging energy in a discrete way. The vectors for this energy
exchange are the particles of the fields (photons, electrons, mesons...).
We now set our attention: is the wave-particles field description compatible in space-time
framework? I will try to demonstrate that they are conceptually not compatible. Before doing this
we need to explain the classical quantum mechanical interpretation of waves and particles; particles
are interpreted in a very Newtonian way: body points that have mass, momentum, kinetic energy
and their interaction is mediated by energy and momentum exchange; this exchange is transported
by the quanta of a field. E.g. electrons interaction by virtual photons
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These interactions must respect space-time structure: they can' t propagate faster than light. The
wave, that for sake of simplicity we assume for one non relativistic interaction-free particle, is
interpreted following Max Born: The square module of the (complex) wave is the probability of
finding the particle in a infinitesimal region of space at some time: so if dV=dxdydz is our
infinitesimal region of space we have the probability of finding at time t in this region expressed by

|| ψ(x, y, z; t) || 2 dxdydz
Suppose now that we have a wave describing the particle moving on x-axe
We ask: where is the particle now? Can we know its position? No, because we can only have the
probability of finding particle somewhere (if QM is a complete theory i.e. a theory that can say us
all what we can never know about the system we are observing). But we can pose another question:
is, independently from what we can know, the particle anywhere at some moment?
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I think that the answer to this question is a fundamental fact that space-time description is not
compatible with particle wave dualism. We show that thinking the particle somewhere at some time
is wrong. At this purpose we use the two-slit apparatus: we know that when we send a wave
through a two slit window we find the classical interference behaviour
fig 5.1
The wave duplicates from the two slits and interferes in the B zone. If we close one slit the
interference disappears; this description is clearly compatible with space-time structure. Now let' s
describe the whole apparatus in term of particles
fig 6.1
For any possible trajectory that particle uses to go from A to B there are only two possibilities: it
passes through slit 1 or through slit 2. But closing one slit would' t change the statistical behaviour
of particle final positions; in effect we find that closing one slit change the statistical behaviour. So
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the right answer is not " there is a particle in some point at some time" but " the particle is
anywhere at anytime in some space-temporal region
This clearly makes nonsense because when we detect the particle it is always in some point-like
region at some time (with the Heisenberg' s limitation, but this is not relevant for the present
problem of wave collapse). It may be suggested that this problem could be overcome if we attribute
ontological status to the wave-field. So the field is real and has continuos space-temporal
distribution of energy. We have a phase in every space-temporal point that tell us how the field
interacts with other field or with itself. The exchange of energy-momentum between fields is
discrete. But this conceptualisation has some problems: we will show two of these problems.
1. We have one photon field: what happens in field interpretation when a detector absorbs the
photon? When the detector absorbs the photon it means that the energy that was distributed
uniformly in space-time instantaneously is concentrated in a point-like space-temporal region:
how this is possible if nothing can travel faster then light?
2. We set this apparatus: one photon field, two detectors say A and B that cover the whole
extension of the field.
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Let’s imagine that A doesn’t click so B does. After passing A the field collapses in the lower path.
If A doesn’t reveal the photon is because A didn’t interacted with the field. But why the field
collapses in lower path if there was not interaction? Are there two different kind of interaction? Is it
not more logical that field continues evolution over both paths?
A possible solution to these conceptual paradoxes is the so-called many universe interpretation. We
will discuss it later because now we propose the Wheeler "it from bits" interpretation. Before
discussing it let us isolated three fundamental facts that don' t finds an explanation when we mind
at particles: interference, superposition and entanglement. One result of this work is to show how in
Wheeler’s interpretation they are conceptually compatible with particle description; we will see that
in Wheeler' s interpretation we don' t assume the existence of objects, but we find them emerging
from more fundamental entities; the corpuscular and the ondulatory refer to different epistemic
fields.
The interference is well illustrated in two slit experiment that we have discussed before;
superposition and entanglement are better shown by photon polarisation in Chiao' s experiment.
The entanglement is strictly related with the concept of non-locality.
It' s known that the polarisation of a photon determines the probability that the photon will pass
through a polarisation test for example if the photon is vertically polarised we will indicate its state
with |V> the probability that the photon will pass a 45° test it is 0.5. Moreover it's known that two
photons may be put in the so-called entangled state i.e. a state in which the two photons show
strong correlation. A typical example is two photons in this state
1
2
V
V  O O

(1.3)
This state is superposition of the state |V1>|V2> and |O1>|O2> in which all the two photons are
respectively polarised vertically and horizontally. It' s important to note that in that particular state
every photon has 0.5 probability to pass every polarisation test (0°,30°,45°,90°…); moreover, and
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here non locality starts to play its role, once that one of the two photons pass a test (for example a
45° test) even the other photon will remain in the 45° state. The amazing is that the first test forces
the second photon to change his state with respect to the first photon instantaneously and so the
spatial distance between the two photons seems to be irrelevant. This clearly creates a sort of
tension with a fundamental fact of physics that is no signal may propagate faster than light. In
effect non-locality it' s one of the principal problem to been resolved for a coherent interpretation of
QM. Its important to note how non locality is implicated by the same wave function concept and so
in the possibility of superposing or making interference between quantum state. In fact the
possibility of changing the statistical behaviour of a particle not acting on it is what happens when
we stop what we call interference getting of the possibility for the particle to pass through a slit in
classical two slits experiment.
Non-locality, in QM, is not only spatial like but coherently with relativistic symmetry concern time
too. At this purpose we will discuss the so-called delayed choice experiment: the typical apparatus
sets a half-silver mirror (through with a photon has 0.5 probability to pass and 0.5 probability to be
reflected). One photon is shot through a half-silvered mirror SA1 following this scheme:
F’1
R1
S1
SA2
R2
F1
F
SA1
S2
F2
F’2
fig 9.1
From SA1 two identical photons F1,F2 run to S1 and S2 (full reflecting mirrors) and, regulating the
length of one of these paths, we can have different kinds of interference in SA2 (half silvered
mirror) for example that the direction to detector R1 being destructive and to R2 constructive, that
is R1 doesn’t click while R2 does. We suppose now that in S1 and S2 we create the photons F’ 1,F’2
entangled with the hitting photons F1,F2. We know that polarising in P1 F’1 forces F1 to follow new
polarisation verse, so polarising one path destroys interference: at this point R2 starts to click. We
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suppose now to insert another polarisation test p2 that reverse the effect of P1 so that polarised F’1
(so F1) again like F2 so that cancel our gained information and we set all the apparatus to have this
succession of events:
1. The Photon F hits SA1 creating F1 and F2
2. First polarisator P1 acts on F’1 destroying interference.
3. F2 and F1 arrives in SA2 with different polarisation but before reaching R1 and R2...
4. We reverse polarisation on F'1 with P2.
What happens? Against our prevision clicks only R1. Point 4 have temporal feedback on point 3
where there was no interference because we’ve destroyed it in point 2.
Raymond Chiao of Barkeley say that, resolving decoherence problems, point 4 may be done after
revelations of R1 or R2!
We have shown two aspects of modern physics: its space-temporal framework and quantum
phenomenon that don' t find a natural explanation in this context. Trying to resolve this problem we
will develop a new framework for physics in which we abandon space-time continuum for a
discrete theoretical conception of world. The two major results for physics is the self-coherent
interpretation of entanglement, superposition and interference phenomena (part II) and the
conceptual unification of what we call a physical experiment with the concept of quantum
computation (part IV). The philosophical relevance of our work is related to the problem of mindbody dualism and the redefinition of ontological and epistemic fields.
The space-temporal continuum as metaphysical assumption.
The idea and the geometrical intuition of space are effectively appealing; moreover it is natural. As
natural as the sequence 0,1,2,3,4… expression of a fundamental cognitive process that is counting.
Einstein made us know how it is unavoidable to join the time with this geometrical description,
transforming kinematics in geometry where the inertial not accelerated motion is substituted by
geodetics in a 4-dimention Rienmann manifold. Even if we can' t think about an absolute space
(that is a particular reference system that sets the state of motion for all the object in the universe)
and an absolute time (something like universal tic-toc that seizes all the clocks) we need to
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postulate the existence of the 4-dimentional continuum and its absolute curvature (reference system
invariant) caused by energy (momentum-energy tensor). As Heidegger said:" The words are the
house of being " we can say:" space-temporal continuum is the house of energy ". But is it
necessary to postulate space and time in QM? In effect one of the first formulations of QM that is
wave mechanics has got the wave function as fundamental concept. We have already shown its
meaning and the Schrödinger evolution equation (1.1) and it's obvious how this formulation of the
theory needs one well-defined apriori space-temporal structure.
Heisenberg's relation
 X  P 

2
(1.4)
follows from these assumptions: that limits the precision of simultaneous position and momentum
measure. We consider now the empirical problem of building up an orthonormal reference system:
firstly what we need are four things: the origin, a meter, the possibility of degrees measure and the
possibility of tracing locally straight lines. We concentrate on the first problem: the identification of
the origin point; we need at this purpose to identify it with arbitrary precision and to determine the
state of motion with arbitrary precision and all these without mutual limitation. In this contest
arbitrary precision means that the information quantity (bits) necessary to seize without ambiguity
different reference systems may grow up without limit. Now we dispose just one physical resource
to define a point: it must be something that interacts with our instruments (for example a particle)
but this is ruled by Heisenberg's indetermination and so it doesn’t satisfy our demands.
At this point we have two choices:
1) We admit the existence of observable entities that don't obey to Heisenberg’s relations, but there
is no empirical evidence of these entities.
2) We let the apriori space-time hypotheses fall and we try this operation: We postulate the energy
interaction (observations!) and we derive from this the space-temporal structure as an emergent
structure.
It's important underline that we have a similar problem for time, that is in the impossibility of
physically specifying an instant.
In classical physics these problems were solved conceptually by supposing the possibility of
specifying with arbitrary precision the position and the state of motion of one object relatively to
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another object; this procedure is intended as the physical realisation of the abstract concept of
immobile point. The wave particle dualism forbids the possibility of having fixed points: how to
justify the physical and epistemic consistence of a space-time continuum is a proper question.
Discreteness Vs continuum
The problem we face now is if irreducible entities (bricks) that constitute reality exist or if world is
something like a continuum fluid: this is a very old metaphysical question. The today physical
answer is: there are some discrete entities (energy-matters) and some continuous entities (spacetime).
From a mathematical point of view there is a sort of equivalence when we assume that the discrete
quantum is very small with the respect to our measures (macroscopic measurements): we can use
continuous mathematics as if the quantum is infinitesimal; obviously we can even approximate
continuum with discreteness (e.g. numerical routines for infinitesimal calculus).
From a logical point of view the problem is more interesting: we will develop this point to show
that we have almost logical reasons for accepting discreteness versus continuum.
We start to show how irrational numbers were historically introduced in mathematics. We know
that in the framework of geometry we can build up natural and fractional number (e.g. how many
times we can report a small segment in a longer one); this idea is compatible with the fact that there
exist a smallest segment that is a sort of absolute meter. Than let's assume that the postulates of
geometry are compatible with the hypothesis of this absolute meter; via Pitagora's theorem we
arrive to a contradiction: the diagonal of the square is not measurable in this way. So we need to
introduce irrational number; this happens because the irrationality has its seed in some nonconstructive assumption (in the sense of intuitionistic logic) of what is a line and what is a point.
The idea of line conceals an infinitum like process. In fact we know that there are infinite ways of
cutting one segment because there are infinite points in segments. Anyway we use points just like
real entities (e.g. from a point we start line).
The concept of length equivalence also conceals an infinitum like process: in fact A=B is limit of a
process where we do not discover difference. Classical geometry postulate that we can decide when
A<B or A=B or A>B but if we imagine physical praxis we know that we can say A=B within some
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error: the mathematical A=B is the limit of our physical possibilities of discovering difference. So
continuum is a logical consequence of some postulates of ordinary geometry.
From a phenomenological point of view what makes possible perception of reality is the perception
of differences: just a difference that makes possible distinguishing the one in two, the whole in the
particular, the here from the there and the after from the before. A difference as the possibility of
deciding for a yes or a no.
We can see that QM itself forces us in the same direction. The modern quantum field theory tells us
that the world is a space-time continuum where field (continuous entities) interacts between them in
a discrete way (discrete exchange of energy). As Einstein said:" space and time are not entities in
which we live but modalities we think with " we in effect find, in modern quantum field theory,
space and time as parameters more than observables. Our observables imply some energy discrete
exchange (e.g. the revelation of the photons). If these exchanges of energy are epistemically
fundamental we see space and time as construction built up on a sequence of discrete observations.
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PART II
Introduction
In this part we will show the new Wheeler’s conceptual framework for physics. Using it we will try
to explain what is space, what is time, what is a measurement, what is a physical law, what is
entanglement, superposition and interference in QM; We will answer to some why: why
macroscopic reality appears deterministic and why world is not an unknowable chaos (that is why
is possible physics).
The experiments we have discussed in part I force us to consider if our vision of physical reality is
adequate. Non-locality is one of the problems that we have found but is not the only one. The
problem is that we have very old way of thinking (e.g.: 400 years old concept like space-time
continuum) for the news quantum phenomenology. And our old concept fails.
We here conjecture that space and time are not ontologically significant and their epistemic
utilisation is limited and not general. The QM experiments tell us that reality must be thought as a
net of relations between observables where linearity is the way these relations are and evolve.
Physical world is the ensemble of all the interactions that create, fix and destroy these relations: we
call these interactions observations and we give them ontological status.
Experiments tell us that the world is not to be intended localised in space-time but the quantum
relation between observations have statistical regularities that, at some level, let us seeing an
emergent space-time structure.
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It from bit
We start here the explanation of Wheeler interpretation of QM, interpretation that follows our
precedent directives: discreteness, not space-time-based physics, etc. Let hear from Wheeler's
mouth what he thinks about these problems:
" [...] what quantum physics and information theory have to tell us about the age-old question,
"how come existence?" No escape is evident from four conclusions:
1) The world cannot be a giant machine, ruled by any pre-established continuum physical law.
2) There is no such thing at the microscopic level as space or time or space-time continuum.
3) The familiar probability function or functional, and wave equation or functional wave equation,
of standard quantum theory provide mere continuum idealisations and by reason of this
circumstance conceal the information -theoretic source from which they derive.
4) No element in the description of physics shows itself as closer to primordial then the elementary
quantum phenomenon, that is, the elementary device-mediated act of posing a yes-no physical
question and eliciting an answer or, in brief, the elementary act of observer-partecipancy.
Otherwise stated, every physical quantity, every it, derives its ultimate significance from bits,
binary yes-or no indications, a conclusion which we epitomise in the phrase: it from bit."
Let’s start with some examples of yes-no elementary questions: we have a photon polarised at 45°
that travel through a calcite crystal and then stops in one of the two detector A or B (in fact calcite
split a 45° beam in two uniformly one at 0° the other at 90°).
Quantum predictions tell us that we will find the photon in A or B; now we invert this process with
another calcite crystal that reverse the effect of the first one; If we don't destroy coherence in the Zzone, that is we don't observe this zone,  is in the same state of ’ only spatial temporally
translated.
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Well, we call the destruction of such coherence our elementary act of observations; why it is
elementary? Because projects reality in two distinct directions; it makes an irreversible choice: the
photon is 90° or 0°, the photon is in A or B, etc. In the language of Hilbert’s space this observables
are called projectors. Another example of projection that is elementary is the revelation of a photon,
in fact answer to the question: is the photon in this region? Yes there is, no there isn't.
So we can start to consider a numerable set of questions [qi] and the correspondent set of answer
[ai] where ai [0,1]; this two set are clearly in one-o-one correspondence qi ai. When an
elementary act of observation is made a qi becomes an ai. This process builds up reality. With this
point of view when we find somewhere a photon we don't discover it: we construct it; in some
sense when we fail to reveal it somewhere we destroy it that is we take off its possibility of existing
there.
With this point of view we see that, at a deeper level than ordinary, reality consists in an
information-achieving problem. Existence comes only when information is achieved: the universe
will be multiverse until we don't construct it. (We? Who are we? The Observers. What is an
observer? See part III for this). There is not an existence that is not an exclusion of another
existence. We substitute fields in space-time with the ensemble of [ai]; reorganising these answers
forms our idea of world as space-time, particles, trajectories and so on. This is possible because
these answers present regularities, (low algorithmic complexity) that permits a multi-level
recognition of forms and patterns; These regularities have statistical nature but we historically have
first isolated the regularities that looked deterministic (the planets orbits, the body accelerations)
and we have constructed our vision of nature upon this bases: space and time are conceptually
consequences of this regularities.
We need to well understand this information-theoretic approach to physics that the qi ai is not at
any point at some time. This process is apriori, out of space and time that are metaphors of our
mind and don't have ontological status; here and there are not meaningful concept until we don't
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distinguish the here form the there: the qi ai process is behind space and time and is effective,
objective, ontological and constitutive of all. When we find and isolate groups of [ai] that present
the same statistical regularity we start construct the world-structure and have the possibility to
assign a probability to the qi ai. With this point of view physics is a giant puzzle reconstruction:
the [ai] are the piece that nature gives us. The meaning of posing a particular qi and the significance
of the relative ai are possible because we already have a place free for fitting this new ai. The world
is not a chaos just for the reason that the [ai] give us the possibility of being reorganised in
meaningful way so that we have the possibility of "discovering" statistical law for statistical
prediction of the qi ai process.
Examples: space from bits
How does space emerge from the [ai]? The structure of space-time is a scheme that we use to find a
right way to order the [ai] so that this ordered [ai] could be expressed by simple low. Ordering is not
just an [ai] sequence enumeration; we mean it as a relation net construction. The universe presents
regularities at all scales and the deterministic space-time based vision of world has an incredible
philosophical appeal, forcing in these structures every law discovered. Space and time are possible
indeed for these regularities (e.g. the revolution of the sun, the rotation of the earth) but obviously
this conceptualisation may be not extended to all phenomenologies e.g. quantum mechanics.
So first we have to cluster the [ai] than, by these regularities, we generalise our physical laws. Here
we have a first problem: if the qi ai process is elementary it must be the simplest action that is it
can't be subdivided in simpler part; so the qi ai processes have to be all of the same nature,
exactly the same: we shouldn't distinguish them. The problem is how our conscious mind exists and
works with the [ai] outside space and time; this will be widely discussed in part III.
The standard definition of space unity of measure (the meter) is very significant for our discussion:
it's a multiple of the waves length of the light emitted by Ce 133 when a particular energy transition
occurs. It's known that given a light wave (with one photon associated) we have our probabilities of
finding the photon in some region of space.
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This is our epistemic work: we have a theoretical-postulated wave than we empirically find the
photon. Let us invert this epistemic approach: we have a sequence of photon revelations, our [ai],
and we have to reconstruct the wave.
DEFINITION: we define phenomenology a sequence (virtually infinite) of [ai] that comes from
one not variable experimental set.
DEFINITION: we define a cluster a sub-sequence of a phenomenology that repeats itself
conserving its statistical proprieties.
To reconstruct the wave we first notice that we can form the first cluster around couples from [ai],
let's call them a0, a1, a’0, a’1,…: they are always in opposition, that is one is 0 the other is 1.
0 a1 a2 a3 …………..1 a’2 a’3……………..0………………..1……………..0
Fixed the experimental apparatus (external condition) the number and the order of ai between 0 and
1 will be fixed; so we isolate cluster in this way
0
0
………
a1 a’1 ………
a2 a’2 ………
a3 a’3 ……….
.
.
.
.
.
.
then we calculate the frequency of 1's of every line (or equivalently calculate the normalised
probabilities).We have what we call the phenomenological matrix.
So we reorganise the order of lines in the crescent order: in this way we are giving spatial ordering
to our [ai];
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we have to notice that the original 0 and 1 may at some point of the phenomenology change
behaviour, that is the associated probability will be not exactly 1 and 0, but this is not a problem:
the important is the possibility of giving an order (a structure) to our phenomenology. In this
framework space emerges after a number of observation of the same phenomenology.
Given a phenomenology how many observation are necessary to ordinate the structure? This
depend clearly from the number of lines of the phenomenological matrix and from the smallest
interval between the probabilities of the lines, in the best case of m lines with probabilities
uniformly distributed we need n yes-no observation so that n 
m3
(for this calculus see appendix
4
1) in this sense space continuum emerges when m and so n go to infinite and the phenomenological
matrix grows up.
This is not the only way to develop from [ai] a space like structure (a monochromatic wave).
Another way is statistical correlation discover between [ai] coming from different phenomenology;
for example
phen.1 [ai]
phen.2 [bi]
phen.3 [ci]
.....
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Ad infinitum this process constructs the waves with waves length 1, l/2, l/4 ... Anyway wave-like
structure emerges from [ai] with statistical and correlation criterions as obvious more [ai] are
involved more the structure will be dense and our measurement precise. Similar consideration may
be done for time; further correlation between the [ai] of spatial structure and [ai] of temporal
structure may describe other waves that we call particle fields and so on. Generally speaking it
seems reasonable thinking of space-time structure as a constant scaffold built upon some [ai] when
the others [ai] of the phenomenology (describing for example a particle trajectory) changes
statistical behaviour. All this is like a mosaic where periodical geometrical parts encircle individual
motives.
Other conceptually relevant implications
We stress now some important implications of the "it from bit" point of view. A central problem of
QM is the wave-particle dualism: wave tells us what (how much) we don't know about particles
when, thinking to the particle would need complete information on position and momentum. We
have so two fundamental aspects of nature: what we know (epistemic part) and what it is
(ontological part) We could say that wave conceal epistemology when particle the ontology of
nature, but this doesn’t work! We have seen why in part I. In our information-theoretic
interpretation of QM there is no ontology (e.g.: the photon is somewhere in space-time) until we
don't ask for it. What is represented by the vector in Hilbert's space is the full set of possibilities
(different existences) that remains possible; in our interpretation of QM the ontology is step by step
constructed by observation work: at any step we get ontological information.
22
A quantum system is not something that is being: a quantum system represents the residual
possibilities of being. All this is well shown by the following game proposed by Wheeler: The
game is to invite someone to guess the number you have thought only posing you dicotomic
questions, that is questions with only yes-no answers; so the game starts and after some questions
clearly we will guess that number. The fact is that you never have a fixed number in your mind;
you just give him coherent answer to so the number was not a pre-existing reality but was
constructed by coherent answers to coherent questions: this is what happens when we observe
nature.
It seems in quantum effects that nature acts as if every possible different existences exist at the
same time: what we call superposition (e.g.: the photon pass the two slit together). This is really
one of the particular aspects of quantum mechanics; we show now how that finds a natural
explanation in our interpretation. We mind the travel of the photon but the same way of thought is
general of superposition phenomena; if we have the idea that something like a travelling photon
exists superposition is clearly a mystery. But if we assume the "it from bit" ontology we have to
think not a photon but a set of possible question we can do. These questions form the possible way
of existences. When we think at the photon travelling (because in effect we can't see the photon
travelling: this is an (illegal) extrapolation of finding photons in numerous observations-questions)
we are in reality thinking to our possible questions-observations: the wave collapse (e.g.: closing
one slit) is a change from the question to the answer; It closes one possibility and creates the
reality.
Moreover in the classical context we have an idea of what we call a principle of cause end effect;
physical events are determined by some antecedence. We can say that going in the past or in the
future is something like a logical fact ruled by logical connection. In our interpretation we
determine reality by successive answer to questions
(choices to different alternatives). The
causality is only in the correlation we discover in the answers. For example there is a box with a
cat, a poisoned flask, a calcite crystal
23
a 45° polarised photon pass through the calcite crystal:
a) The photon will hit poisoned flask killing the cat
b) The cat is alive
Classically we say that if the cat is dead is because the photon was in the a) path; information
theoretically speaking we say that the system is completely determined (we have answers that make
the experimental set) except for the question: is a) or is b)? This last bit of information resolve the
system. The correlation is between different questions (that maybe the same question) and spacetemporal cause-effect is a conceptual construction on these correlations.
Interference, superposition, entanglement
Interference superposition and entanglement are the most important facts that a good interpretation
of QM must explain. So we test our interpretation on these phenomenologies; for better
understanding we show the standard quantum space-time based on interpretation then our
information theoretical interpretation.
Superposition

standard interpretation
The state of a system is not perfectly defined in all its variables: some of them may have statistical
behaviour, the state of a single system conceals the statistical behaviour of many identical systems.
When we think of the different possibilities that an observation will reveal we say that the system is
in a superposition of all these possibilities, e.g. a 45° polarised photon is a superposition of a
90° polarised photon and a 0° polarised photon. Anyway if we make an observation the state of a
system may change instantaneously to a very different one. This is the so-called quantum jump.

our interpretation
We have yet discussed of our superposition interpretation. We add here that the state of a system
only express statistical behaviour of observation acts in the same experimental condition; the single
system has one defined part and one possible part. After we have fixed the experimental condition,
24
that is we know the answers to some questions, we can call these the experimental setting answers,
it remains different possibilities of existence for our system: these form the superposition state.
Differently to the old interpretation here the set of possibility is equal to the set of all possible
different answers to a question. In the old interpretation we feel like different reality coexist
together. Here only one reality exists: the one sized by the answer.
Interference

standard interpretation
Interference is related to superposition; in superposition we take a state and we say: this state can
be look as the sum of different states; interference is the opposite: we notice that summing different
states gives another state. Both superposition and interference concern the linear structure of QM.

our interpretation
When we get the answer to a question this clearly forces the nature to be some way taking away the
freedom of existence to other possibilities; if we don't get the answer for some question this clearly
leaves other questions related to the former undetermined. But why different questions have mutual
correlation? The answer is always because other way the world would be an unavoidable sprinkling
chaos. When we leave nature different possibility of existence nature is coherent. Interference is
like a calculus that says us: " These different possibilities may give you this result."
Entanglement

standard interpretation
Some observables have correlation; these correlations have non-local nature e.g. polarising a
photon change the polarisation state of another space-time separated photon.

our interpretation
Very simple: the answer to a question affects the statistical behaviour of other answers.
25
Mathematical supports

Hilbert’s spaces
There are many equivalent mathematical foundations of QM. The most popular is perhaps the
Hilbert’s spaces one so we will analyse our concepts in this mathematical framework.
QM can be, in first approximation, axiomatised by saying that:
1
The state of a system is a vector  in a Hilbert’s space 
2
The observable entities we measure have correspondent self-joint operators Ô so that their
eigenvalues are the quantities we measure; if we write a vector  as


ψ  a 
j j
j
where j are eigenvectors of Ô the probability of measuring the eigenvalue j correspondent to
the eigenstate j is ||aj||2
3
After a measure the system will be in a state that is eigenvector of the eigenvalue measured that
is it belong to its autospace.
4
The independent (no observations) evolution of a closed system is linear: if the state at time t0 is
( t0)=a( t0)+b( t0) at time t will be ( t)=a( t)+b( t)
What this mathematics could say about our interpretation of QM? We have already stated that
reality is constructed upon 0-1 blocks. In Hilbert’s space such observables are called projectors. We
can define a projector Ê by
1)Ê 2  Ê
2)Ê   Ê
where Ê+ is the adjoin operator of Ê so Ê is self-adjoin.
It’s known that to a projector we can associate proprieties of a quantum system (e.g.: is the
momentum of a particle in the range [p1,p2] ?) and develop a logical approach to QM (Von
Neumann and others). Here we want only to stress some facts relevant for our ideas.
Let Ô be an observable with eigenvalues o1, o2, o3,… and let be [Êi] the set of projectors associates
with sub-ensemble of [o1, o2, o3,…] (e.g. One of these projectors E may be E=1 if the value of Ô is
26
o2 E=0 otherwise). Anything we can express with observable we can also express it with projector.
In particular position observable can be expressed by infinite numerable sequence of projectors; for
example the position of a particle can be expressed by:
the particle is/isn’t in the R1 region
the particle is/isn’t in the R2 region
the particle is/isn’t in the R3 region
……
if make use of a longer sequence of projectors we obtain more information on the position
Space-time structure is based upon these projectors: mathematical continuous formalism is only a
tool for “ad infinitum” computational process.
We have already said that reality is potential until we don’t click the 0-1 switch. But what represent
the actual Hilbert’s vector ? It is something concerning what is or what it could be? The both
because when we think about it in its quantum discontinuous jump (the projection of the projector)
we realise the decision of two alternatives (the answer to our question); when we think about it as a
fixed vector superposition of others two vectors , belonging to orthogonal complete subspaces
27
A, B of the original Hilbert’s  space AB we see in the vector  the possibility of two
mutual alternatives.
Now we analyse the possible relations between our questions-projectors; considering two
projectors E1 and E2 there are two main general cases:
1)
[E1, E2]=0
E1, E2 commute
2)
[E1, E2]0
E1, E2 don’t commute
In the first case our questions are compatible: their answer can coexist together; moreover the
posing order is not relevant since the last reality we reach is the same: this is one way to say that E1
and E2 leave their respective autospaces unchanged; if we have the answer to E1, E2 will have to be
compatible with the reality build up on E1.
When E1 and E2 don’t commute the posing order make the difference: the two projectors may
change their autospaces the answers change the two reality landscapes, we may say that the two
answers can’t coexist in the same world. In fact if, for example, E2 changes the autospace leaven by
E1 the reality determined by E1 doesn’t exist more. So we can act on the pass? The fact is that if we
are considering a set of non-commuting projectors every time we pose a question Ei this question
must be considered a different question from those we have posed before. For example if [E 1,
E2]0 posing E1(I)E2(II)E1(III)E2(IV) , E1(I) must be interpreted as a completely different question from
E1(III) implying different realities and different correlation with the rest. These considerations may
suggest an arrow of causality ruled by this non-commutative nature. This arrow could be compared
to thermodynamic arrow of time: non-commutativity implies ordering and ordering is the role of
time.
Few words now about non-locality. In our interpretation we have yet noticed that non-locality is
simple the existence of correlation between projectors: in the Chiao’s experiment the entangled
photons was in the state
1
2
V
V  O O
 .We know that there exists no polarisation test that
can separate this entanglement: |V>|V> and |O>|O> must be viewed as single eigenstate of a new
Hilbert’s space (the tensorial product of the two original polarisation space of the single photons):
the original two photons have lost their polarisation individuality. Polarisation test on these photons
must be seen exactly as the same projector independently if we act on photon1 or photon2.
28

Quantum histories
Another mathematical tools very useful to support our ideas are the so-called quantum histories
originally developed by Griffith.
Let E1,E2,E3,…En be a sequence of projectors occurring at an ordered sequences of times
t1,t2,t3,…tn we cal this sequence of projectors a history.
At every history we can associate a probability
p  Tr E n E n 1...........E1ρE1...............E n 1E n 
(2.1)
where Tr is the trace of the operator in the brackets and  is the density operator. This probability
can be justified thinking Ei as a projector associates with the propriety :”the momentum (or
position) of a particle is in some range” so the probability (2.1) is the same that we obtain using
Feynman’s path integral approach.
Given an observable Ak we can build a complete family of disjoints set {Skk} that covers its
spectrum; for every set Skk we have its associate projector Ekk. So for a sequence of observable
{Ak} we have a family of histories depending which projector we associate to every Ak . “All this
histories produce not only one motion picture but a complete family of motion pictures with
different scenarios. One can also think of them as different events as in probability calculus.
Several disjoints histories can be put together to produce another less detailed history.”(Omnes,
1992) Additivity requires that the probability of a larger history is the sum of the probabilities of
the more detailed histories. Mathematical relations may express this consistence condition for
additivity: a sufficient condition for additivity is
 


'
'
'

Tr E n 1 E n  2 ..............E 1 E 1 ............E n 1 E n
n 1
n 2
1
1
n 1
n




0


(2.2)
where the sequence {k}is different from the sequence {’k}. What all these mathematics have to
say to our interpretation? The consistence condition (2.2) gives us a discriminator for what is a
meaningful history and probability (2.1) gives us a tool for quantify its statistical behaviour. So we
can know which possible world we can enter in and how probable are each of them.
29
This is enough for a complete description of Physics and is obviously compatible with our vision of
a world built up on answers to our questions.
The Everett’s multiverse approach
A central problem in QM is the definition of what is a closed system. This is because observer must
be put outside the system and this creates the fracture that brings to wave-collapse “paradox”. We
need to postulate so two distinct physical process :
Process 1: the discontinuous change brought about by the observation of a quantity with eigenstates
1,2,…, in which the state  will be changed to the state j with probability |(,j)|2.
Process 2: the continuous deterministic change of state of an isolated system with time t according
  
to a wave equation Ĥψ  ψ where Ĥ is a linear operator.
t
(Everett ’57)
This approach creates a fracture in the universe between the observer and the system under
observation: an unitary description of the whole observer-observed is lost.
Everett propose an elegant solution that now we describe. He starts his consideration from the
concept of relative state: given two systems S1 and S2 we associate them two hilbert’s space H1 and
H2 (that for our discussion will be the closed system and the apparatus that makes a measure on this
system). When they interact they enter in a new entangled S system that is represented by an
Hilbert’s space H tensor product of H1 and H2 : H= H1 H2. This has the consequence that if the
 
 
sets  Si1 and Sj2 are complete orthonormal basis for H1 and H2 then the general state of H can be
written as a superposition  S   a ij Si1 Sj21 . From this follows that for an entangled system is
i, j
impossible to consider physical propriety of one subsystem independently from the other parts. W
can however for any choice of the state  k of S1 uniquely assign a corresponding relative state in S2
(S 2 ; rel k , S1 )  N k  a kjSj2
j
with Nk normalisation constant.
30
For any choice of basis in S1  i , it is always possible to represent the state of S as a single
superposition of pairs of states, each consisting of a state from the basis  i in S1 and its relative
state in S2. We can write so :
S  
i
1 S1
 i (S 2 ; rel i , S1 ) .
Ni
In this context the measure process is represented by which factorisation we adopt for the state of
entangle system observer –observed. Make a measurement doesn’t make wave collapse is only a
way to factorise the observer-observed system. It can be show that the linear formalism of QM
preserves these proprieties in multiple observation of a system.
These consideration imply that having a valued measure from a quantum system is a just a point of
view. The multiverse  doesn’t collapse when an observation is made. Our individual perception of
a universe (wave-collapse) is one of all the possible equivalent perceptions that come from
factorisation of a vector in superposition of eigenstates.
Our “it from bit” interpretations follow exactly the opposite path from Everett’s interpretation. In
fact to resolve the problem of having the two distinct process of system evolution and system
observation we discard continuum multiverse time evolution hypothesis and we assume the
discontinuous wave-collapse process as ontological. Observer and observed are melted in one
ontological entity that is the information achievement from a set of possibilities.
Time evolution
The apparent flux of time needs to be explained in our information-theoretical approach to physics.
In the framework of Hilbert’s spaces given a closed system we say that time evolution corresponds
to the rotation of the state vector given by the unitary operator e
 it
Ĥ

where t is the time elapsed
from the last preparation of the system and Ĥ is the Hamiltonian operator associate to the system.

So if at time t0 the system is in the state  0 after t seconds we will find it in the state

t  e
 it
Ĥ


0
(3.2)
In this formulation is necessary to postulate the flux of time: time is to be intended as apriori
concept that makes epistemological knowledge possible. This is the so-called Schreodinger
31
representation of time evolution. The observables are kept fixed and what change in time is the
state of the system. Another representation is the Heisenberg one where the vector representing the
state of the system is kept fixed and what change is the observable A according to
A( t )  U 1 A( t 0 ) U( t ) where A(t0) is the observable at time 0 and U( t )  e
 it
Ĥ

.
These two formulations, mathematically equivalent, have very different interpretations: in
Schroedinger representation time is something that every system follows; this imply the clockwork
world interpretation. In Heisenberg representation what change are the observable. So what
changes is the ambient that specify the system under observation. This is compatible with our
assumption that there doesn’t exist a closed system under study but only a set of possibilities that
comes by previous information-achievement.
Once more time is to be considered as an abstraction built up on statistical regularities in the
mosaic drawn by yes-no projectors questions.
All this is very clear if we consider that what makes possible measuring time is the periodical
nature of some phenomena. But periodicity is only a relative concept that comes from relations
between phenomenology: we can say that something has regularities only in relation with
something assumed regular.
Appendix I
Here we make the calculus for having a bound to the number of bits needed in photon detection to
give spatial ordering to a source of m photo-detector or, in other words, how many yes-no
projectors (we detect –we don’t detect the photon) are neede to reconstruct the monochromatic
wave from which they come.
A projector P can be see as a Bernoulli random variable P. Its mean p=[P] can be used as an
n
ordering number for the position of the variable in the space context. The statistical mean
Pi
n
i 1
behave for large n as a Normal distribution of mean p and standard deviation
p(1  p) / n . To
order m Bernoulli variables by their means under the hypothesis they are uniformly distributed
32
between 0 and 1 their standard deviation must be 
1
. So
m
p(1  p)
n

1
that imply
m
n  m 2 p(1  p) . The second member of this relation is maximised for p=1/2 and multiplying form
m3
m that is the number of projectors the n bits needed become n 
.
4
33
PART III
Introduction
In this part we analyse the problem of what is consciousness and in which relation it is with what
we call matter. So we discuss here the very ancient philosophical problem of mind-body dualism.
We firstly analyse the problem from a historical point of view, then we discuss how quantum
mechanics poses urgent problem on which is the role of consciousness in physical investigation of
world, then we propose our interpretation to solve the mind-body dualism.
The problem of consciousness
What is consciousness? Try to put your hand on fire: undoubtedly you fell something. When we
interact with reality we feel: we feel sounds, colours, impressions, emotions, ourselves, our mind
thinking, ultimately we feel. We may say then that consciousness is the general act of feeling.
Consciousness seems a very simple thing but we will show that it doesn’t find a legitimisation in
modern scientific matter-based explanation of world when in our approach to physics
consciousness finds its natural fundamental place.
Consciousness is not explainable in materialistic vision of world; classical scientific approaches try
to explain consciousness as an emergent propriety of brain: matter causes consciousness. We think
that consciousness is not reducible to matter-machinery. Why?
34
Let’s imagine a beam of light travelling towards our eyes: when we see light we have to say
something; we describe this experiment in biophysical context. (fig 1.3)
The light travels in space and reaches our eyes; here hits photoreceptor that start to emit signals.
This signals propagates inside brain then speech brain zone is excited and finally we speak
propagating air-pressure waves in space.
We put this experiment in a time line (fig 2.3) and we colour the hypothetical segment when
consciousness of light appears. But this is a sensible way of thinking?
Bio-chemical processes of brain are not of a different kind of light travel or photoreceptor
stimulation or vocal cords vibration: physics is physics! So, why does consciousness must depend
from particular physical process?
Some neuro-scientist answer that there is not one particular physical process that causes
consciousness but consciousness appears as emergent propriety from the very complex physical
interactions of brain. But the problem is: how complexity makes consciousness emerge?
Consciousness shouldn’t depend from the complexity of a system; if we assume this we have the
problem of deciding how much complexity is needed for consciousness; at what age human phetus
becomes conscious? Which kind of bio-system is conscious? Can a machine be conscious?
35
This problem forced Nobel prized neuro-biologist Sir Eccles to think that God inspires
consciousness in human fetus at three weeks of age .
Cartesio was one of the first thinkers that investigated the problem under the modern scientific
vision of world (post Galileo Galilei). His conclusion is that the mind (consciousness) and body
(matter) are two distinct metaphysical substances. This is the so-called dualistic vision of the
problem. Dualistic philosophy is still present in today thinkers (e.g. Roger Penrose). One problem
of dualism is that is very difficult to justify how matter can act on consciousness if they are
separated metaphysical substances. If matter acts on consciousness we might have physical
possibility of consciousness revelation but clearly we never had such experimental evidence; this is
matter influences matter and vice-versa and matter influences consciousness but not vice-versa.
Others philosophers think that consciousness doesn’t exist! It is illusory. Or that the relations
between matter and consciousness is the same that exists between hardware and software:
consciousness is so a semantic propriety of a formal system (physical world). One problem here is
that a semantic requires a mind that gives interpretation of formal systems: we have a logical loop.
Moreover if consciousness depends from the abstract formal propriety of a system every thing that
have the same formal propriety of human brain must be conscious.
Consciousness, observation and wave collapse
What is an observation act in QM? For example when we detect a photon from a wave-field we do
an observation act. Hilbert’s space formalism mathematically represents observation as projection
of the state vector. This projection needs to be postulated and so the observation process must be
not totally described as physical process.
QM is a linear theory that is quantum systems evolves linearly. If we include in our quantum
description of the system our instruments of measure they linearly entangle with the system. So
more then a collapse of the state quantum theory predicts that the measurement system enter in an
36
entangled state with the system. For example imagine the following experimental apparatus
(fig3.3).
There is a light beam 45° polarised, a calcite crystal that split the beam in two distinct path, two
photo detector that send a signal to an apparatus that show which detector had clicked.
When the beam has passed in the crystal we know the state of the system can be represented as
1
2
( lowerpath  upperpath ) . If we include the two photo-detectors A and B in the quantum
mechanical
1
2
description
when
the
field
reaches
them
their
state
will
be
( A _ clicks B _ doesn ' t _ click  B _ clicks A _ doesn ' t _ click ) . If we include the final
indicator when signals from A and B arrives also it must respect quantum coherence and so it will
be in the usual super positions state. All this can be for every new system that entangles with this
chain also for our brain. But when we observe the process at any moment we find not a
superposition of states but only one of the two alternatives. Some physicist say that observation
causes decoherence of system because implicates the interactions of the quantum system with
classical macroscopic world. So the interaction (entanglements!) of the system with the ambient
causes decoherence of the state. But this way of reasoning doesn’t avoid the fundamental question:
how comes the classical world if QM is the fundamental theory; in other words who is the
responsible of the macroscopic decoherence? In other words if the problem is not why wave
collapse (we postulate it) the problems becomes when and where wave collapse. How can we avoid
the infinite chain of multiple observations? Von Neumann was convinced that to avoid this infinite
chain of superposition a postulate of QM must be that consciousness is the final responsible of
wave collapse, a very extreme and interesting conclusion.
37
“It from bit” and consciousness
Our interpretation of QM gives us a path to follow in the resolution of the problem of wave
collapse and consciousness. In our interpretation they are intimately related. In the previous parts
we have seen that we can substitute to the space-time-matter description of world a description
based on the statistical correlation between answers that nature gives: reality is founded on choice
from a set of possibilities. Reality starts when a decision is taken. This process doesn’t imply nor
requires existence of matter just requires existence of a conscious mind that becomes conscious of
the choice from the different possibilities. So the only postulate we assume in our informationtheoretic approach to reality is the existence of the consciousness. The world is build up on act of
consciousness more than matter: the world evolves through these acts of consciousness more than
in space and time. There is no matter-based brain where consciousness lives but matter exists in
perception of matter. In this context consciousness is not a propriety of some biological systems:
consciousness is ontologically essential; briefly esse est percipi.
We think moreover that that our individual self-consciousness is explainable a part of a general
self-consciousness propriety. We develop some ideas in this direction in part IV.
These are only starting considerations for a more wide theory of consciousness: it would be very
interesting analyse idealistic theory of reality (e.g.: Fichte, Hegel,Schopenauer,etc.) and find out the
possibility of founding these philosophical systems on the quantum phenomenology of wavecollapse.
38
Similar ideas are represented by the J.Wheeler closed loop (fig 4.3) where existence comes out by
the reciprocal process of posing question-distinguish the answers.
Here we start from the assumption that reality is a closed loop: observers ask question (makes
observations) then they distinguish the answer. The process statistics implicate the complex
probability amplitude machinery. Further statistical regularities imply a phase change of the
complex amplitude around a loop (monochromatic waves). Then space-time emerge and fields and
particle. This is physics the permits understanding of what is communication. Communication
permits meaning that is also self-consciousness. Observers have an idea of world that comes from
the questions they posed; they pose other questions and loop is closed.
The critics that realist thinkers pose to idealism is that it implies solipsism that is only our
individual mind exist and how is possible this when we discover objective proprieties in what we
see, proprieties that are the same for all the observers and so the conclusion is that exists a reality
“outside” our mind. But the “it from bit” doesn’t postulate individual consciousness it postulate the
acts of consciousness associated with wave collapse. Individual consciousness emerges from the
closed loop when communications permits the existence of individual communicators: so
individual consciousness is emergent. The only consciousness that needs to be postulated is a
general consciousness that feeds itself as indicated by the closed loop.
39
All this could be epitomised saying that the world is a big idea.
PART IV
Introduction
In this part we discuss the status that (quantum) information have in the new framework we have
set in the previous parts. If the elementary act of reality construction is the yes-no projection that
comes from any observation act the status of information must be fundamental: the Landauer words
“ Information is physical “ must be reversed as “ Physics is information-theoretic”.
The meaning of information
The difference between classical and quantum information is in the different origin of the relative
probabilities associates with bits (classical) and qu-bits (quantum). For classical information we
assume the usual mathematical structure of -algebra: we have a set X called the set of events and
we define on the set of the parts of X (denoted as (X)) a function p:(X)[0,1] so that
p()=0
 E(X) p(E)=1-p(E) whereE is the complement of E
40
for any numerable collection of disjoints set Xi(X) p( X i )   p(X i ) .
i
i
For convention we define p({x})=p(x).
Given the probabilistic structure P=(X, (X),p) we say that the information that comes from
observing xX is -lg2p(x) bits. The average information achievement when we observe en events in
X (called in the information theory the source) is
 p(x)lg
xX
2
1
bits; this is called the entropy of the source.
p(x)
Quantum probabilities and so quantum information have a different origin. To illustrate the main
idea we consider fig1.4 of right-angled triangle made by a vector in a tw0-dimensional space.
a
b
The Pythagoras theorem says that a 2  b 2  c 2 that implies ( ) 2  ( ) 2  1 .
c
c
This is just
cos 2   cos 2   1
(1.4)
where  and  are the angles made by the vector with the x and y axes respectively. However, the
square of the cosine of any angle necessarily lies between 0 and 1, and so the result (1.4) shows that
a model for a probability distribution for an observable A with only two possible values a1 and a2
can be obtained by associating Prob(A=a1;s) and Prob(A=a2;s) with cos 2 
and cos 2 
respectively. Different probability distributions (corresponding, for example, to different states s of
the system, or to a different observable) can be obtained by changing either the vector or the pair of
orthogonal vectors used to define the x and y axes.
This construction can be generalised to any number of dimensions (including, with some care, an
infinite number) and constitutes the essence of the general mathematical framework of quantum
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theory (albeit using vectors whose components are complex, rather than real, numbers). Thus states
are represented by vectors in a vector space, and to each observable there corresponds a set of
vectors that are an appropriate generalisation of the concept of ortho-normal basis in a vector space.
Each one of these special vectors corresponds to a particular value of the observable, and the
probability of obtaining that value is given by the (complex analogue) of the cosine –squared of the
angle between this vector and the state vector.
More important are the conceptual difference of classical and quantum probabilities. Classical
probabilities are to be considered epistemic that is the information associated with these classical
probabilities describes a pre-existing reality that is independently of our knowledge of it.
Quantum probabilities and so quantum information are ontological: when we achieve this
information we construct reality.
The formal equivalence between physics and informatics
A quantum computer is usually described as a quantum system that operates on array of qu-bits. A
quantum computation can be simple viewed as a unitary transformation U in a n-dimensional
Hilbert’s space H space if the input is represented by x in  C n the output of the computation is
x out  Ux in . Moreover it is well know that any n-dimensional unitary U can be composed from
elementary unitary transformation in two-dimensional of Cn. This is usually shown in the context of
parameterisation of the n-dimensional unitary groups. Thereby, a transformation in n-dimensional
spaces is decomposed into transformations in 2-dimensional subspaces. This amounts to a
successive array of U(2) elements, which in their entirety forms an arbitrary time evolution U(n) in
n-dimensional Hilbert's space.
Our “it from bit” interpretation has something very important to say about physics and informatics:
they are mutually dependent. The model of reality that emerges from our interpretation is
something represented by fig 2.4.
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The history of the answers {a1, a2, a3,……,an} of the process of answer-question that builds reality
specify the world of existence that, itself, open the world of the possible existences to come: the
world of possible. In this context the role of Physics is to discover the mathematical structures that
describe the statistical dependence between what is and what it will be.
So given a history of answer {a1, a2, a3,……,an}and given a set of mathematical structure  (e.g.:
Hilbert’s space formalism for quantum mechanics) the role of physics is to discover the functional
dependence d between them:
d: {a1, a2, a3,……,an}
(2.4)
For example given a history {a1, a2, a3,……,an} of answers (observations) that specify experimental
apparatus for polarisation test of a light beam we describe the status of the light beam by a twodimensional Hilbert’s space formalism.
The role of informatics is that of discovering how to compute some functions from NN where N
is the set of positive integers. Quantum computing expand this definition showing that the role of
informatics is to discover how to compute some function
f: N
(2.5)
where  is a mathematical structure that represents statistical dependence of a set of qu-bits that we
use as output. Confronting (2.4) with (2.5) the difference from a physical experiment and a
quantum computation formally disappears. Physics try to understand which functions reality
computes and informatics uses reality as a tool for quantum computation.
The theoretical possibility of a universal (quantum) computer that is a computer that can simulate
all the possible computation and so physically realisable experience shows how reality represent
itself by itself: this possibility of auto-reference can explain the possibility of self-consciousness.
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Self-consciousness can be viewed as the possibility of reality of representing itself by itself. We
think that this is the profound meaning of Wheeler’s conception of universe (fig 3.4).
The self-synthesised universe by a process that evolves step by step taking consciousness of itself.
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