11institutetext: Giacomo Mauro D’Ariano 22institutetext: Dipartimento di
Fisica, University of Pavia, via Bassi 6, 27100 Pavia, 22email:
<EMAIL_ADDRESS>33institutetext: Federico Faggin44institutetext: Federico and
Elvia Faggin Foundation

# Hard Problem and Free Will:
an information-theoretical approach

Giacomo Mauro D’Ariano and Federico Faggin

###### Abstract

We explore definite theoretical assertions about consciousness, starting from
a non-reductive psycho-informational solution of David Chalmers’s hard
problem, based on the hypothesis that a fundamental property of ”information”
is its experience by the supporting ”system”. The kind of information involved
in consciousness needs to be quantum for multiple reasons, including its
intrinsic privacy and its power of building up thoughts by entangling qualia
states. As a result we reach a quantum-information-based panpsychism, with
classical physics supervening on quantum physics, quantum physics supervening
on quantum information, and quantum information supervening on consciousness.
We then argue that the internally experienced quantum state, since it
corresponds to a definite experience–not to a random choice–must be pure, and
we call it ontic. This should be distinguished from the state predictable from
the outside (i.e. the state describing the knowledge of the experience from
the point of view of an external observer) which we call epistemic and is
generally mixed. Purity of the ontic state requires an evolution that is
purity preserving, namely a so-called atomic quantum operation. The latter is
generally probabilistic, and its particular outcome is interpreted as the free
will, which is unpredictable even in principle since quantum probability
cannot be interpreted as lack of knowledge. We also see how the same purity of
state and evolution allow solving the well-known combination problem of
panpsychism.
Quantum state evolution accounts for a short-term buffer of experience and
contains itself quantum-to-classical and classical-to-quantum information
transfers. Long term memory, on the other hand, is classical, and needs
memorization and recall processes that are quantum-to-classical and classical-
to-quantum, respectively. Such processes can take advantage of multiple copies
of the experienced state re-prepared with ”attention”, and therefore allowing
for a better quality of classical storing.
Finally, we explore the possibility of experimental tests of our theory in
cognitive sciences, including the evaluation of the number of qubits involved,
the existence of complementary observables, and violations of local-realism
bounds.
In the appendices we succinctly illustrate the operational probabilistic
theory (OPT) framework for possible post-quantum theories of consciousness,
assessing the convenient black-box approach of the OPT, along with its
methodological robustness in separating objective from theoretical elements,
guaranteeing experimental control and falsifiability. We finally synthetically
compare the mathematical postulates and theorems of the most relevant
instances of OPTs–i.e. classical and quantum theories–for convenience of the
reader for better understanding our theory of consciousness. The mathematical
notation is provided in a handy table in the appendices.

> We are such stuff as dreams are made on, and our little life is rounded with
> a sleep.  William Shakespeare

##  1 A quantum-informational panpsychism

In his book The Character of Consciousness Chalmers2010  David Chalmers states
what he calls the hard problem of consciousness, namely the issue of
explaining our experience–sensorial, bodily, mental, and emotional, including
any stream of thoughts. Chalmers contrasts the hard problem with the easy
problems which, as it happens in all sciences, can be tackled in terms of a
mechanistic approach that is useless for the problem of experience. Indeed, in
all sciences we always seek explanations in terms of functioning, a concept
that is entirely independent from the notion of experience. Chalmers writes:

> Why is the performance of these functions accompanied by experience? … Why
> doesn’t all of this information processing go on “in the dark” free of any
> inner feel?… There is an explanation gap between the function and the
> experience.

An effective paradigm for comprehending the conceptual gap between
“experience” and “functioning” is that of zombie, which is behaviourally
indistinguishable from a conscious being, nevertheless has no inner
experience.

There are currently two main lines of response to the hard problem: 1) the
Physicalist view–with consciousness “emergent from a functioning”, such as
some biological property of life dennett2017consciousness ; 2) the Panpsychist
view–with consciousness as a fundamental feature of the world that all
entities have. What is proposed here is:

  * Panpsychism with consciousness as a fundamental feature of “information”, and physics supervening on information.

The idea that physics is a manifestation of pure information processing has
been strongly advocated by John Wheeler wheeler89  and Richard Feynman
feynman1982simulating ; hey1998feynman , along with several other authors,
among which David Finkelstein finkelstein1996quantum , who was particularly
fond of this idea PWP . Only quite recently, however, the new informational
paradigm for physics has been concretely established. This program achieved:
1) the derivation of quantum theory as an information theory QUIT-Arxiv ;
purification ; CUPDCP , and 2) the derivation of free quantum field theory as
emergent from the nontrivial quantum algorithm with denumerable systems with
minimal algorithmic complexity PhysRevA.90.062106 ; bisio2016 .111The
literature on the informational derivation of free quantum field theory is
extensive, and, although not up to date, we suggest the review PWP  written by
one of the authors in memoriam of David Finkelstein. The algorithmic paradigm
has opened for the first time the possibility of avoiding physical primitives
in the axioms of the physical theory, allowing a re-foundation of the whole
physics over logically solid grounds DArianoHilbert . In addition to such
methodological value, the new information-theoretic derivation of quantum
field theory is particularly promising for establishing a theoretical
framework for quantum gravity as emergent from the quantum information
processing, as also suggested by the role played by information in the
holographic principle Susskind:1995aa ; Bousso:2003aa . In synthesis: the
physical world emerges from an underlying algorithm, and the kind of
information that is processed beneath is quantum.

The idea that quantum theory (QT) could be regarded as an information theory
is a relatively recent one fuchs2002quantum , and originated within the field
of quantum information Nielsen:1997p655 . Meanwhile what we name “information
theory” has largely evolved, from its origins as a communication theory
Schannon , toward a general theory of “processing” of information, which
previously had been the sole domain of computer science.

What do we mean by “information theory”?

Recently, both in physics and in computer science (which in the meantime
connected with quantum information), the theoretical framework for all
information theories emerged in the physics literature in terms of the notion
of Operational Probabilistic Theory (OPT) purification ; CUPDCP ;
Chiribella2016 . an isomorphic framework that emerged within computer science
in terms of Category Theory CoeckePV ; TullPV ; Coecke2012 ; Coecke2016 ;
abramsky_heunen_2016 ; Coecke2018 . Indeed, the mathematical framework of an
information theory is precisely that of the OPT, whichever information theory
we consider–either classical, quantum, or ”post-quantum”. The main structure
of OPT is reviewed in the Appendix.222 The reader who is not familiar with the
notion of OPT can regard the OPT as the mathematical formalization of the
rules for building quantum information circuits. For a general idea about OPT
it is recommended to read the appendix. The reader is supposed to be familiar
with elementary notions, such as state and transformation with finite
dimensions.

Among the information theories, classical theory (CT) plays a special role. In
fact, besides being itself an OPT, CT enters the operational framework in
terms of objective outcomes of the theory, which for causal OPTs (as QT and
CT) can be used for conditioning the choice of a following transformation
within a set. Clearly this also happens in the special case of QT. Thus, the
occurrence of a given outcome can be regarded as a quantum-to-classical
information exchange, whereas conditioning constitutes a classical-to-quantum
information exchange. We conclude that we should regard the physical world
faithfully ruled by both quantum and classical theories together, with
information transforming between the two types.

This theoretical description of reality should be contrasted with the usual
view of reality as being quantum, creating a fallacy of misplaced
concreteness. The most pragmatic point of view is to regard QT and CT together
as the correct information theory to describe reality, without incurring any
logical paradox. We will use this idea in the rest of the chapter. Due to the
implicit role played by CT in any OPT, when we mention CT we intend to
designate the special corresponding OPT.

We will call the present view, with consciousness as fundamental for
information and physics supervening on quantum information: Quantum-
Information Panpsychism.

In place of QT, one may consider a post-quantum OPT–e.g. RQT (QT on real
Hilbert spaces), FQT (Fermionic QT), PRB (an OPT build on Popescu-Rohlich
boxes Popescu1992 ), etc. CUPDCP , yet some of the features of the present
consciousness theory can be translated into the other OPT, as long as the
notion of “entanglement” survives in the considered OPT.

##  2 A non-reductive psycho-informational solution: general principles

The fundamental nature of the solution to the hard problem proposed here has
been suggested by David Chalmers as satisfying the following requirements
Chalmers2010

  * Chalm1: Consciousness as fundamental entity, not explained in terms of anything simpler…

  * Chalm2: … a non reductive theory of experience will specify basic principles that tell us how basic experience depends on physical features of the world.

  * Chalm3: These psychophysical principles will not interfere with physical laws (closure of physics). Rather they will be a supplement to physical theory.

In an information-theoretic framework, in which physics supervenes on
information our principle will be psychoinformational (see Chalm3). The non-
reductive (Chalm2) psycho-informational principle that is proposed here is the
following:

  * P1:

psychoinformational principle: Consciousness is the information-system’s
experience of its own information state and processing.

As we will see soon, it is crucial that the kind of information that is
directly experienced be quantum.

Principle P1 may seem ad hoc, but the same happens with the introduction of
any fundamental quantity (Chalm1,2) in physics, e.g. the notions of inertial
mass, electric charge, etc. Principle P1 asserts that experience is a
fundamental feature of information, hence also of physics, which supervenes on
it. P1 is not reductive (Chalm2), and it does not affect physics (Chalm3),
since the kind of information involved in physics is quantum+classical. On the
other hand, P1 supplements physics (Chalm3), since the latter supervenes on
information theory.

It is now natural to ask: which are the systems? Information, indeed, is
everywhere: light strikes objects and thereafter reaches our eyes, providing
us with information on those objects: colour, position, shape,… Information is
supported by a succession of systems: the light modes, followed by the retina,
then the optical nerves, and finally the several bottom-up and top-down visual
processes occurring in the brain. Though the final answer may have to come
from neuroscience, molecular biology, and cognitive science experiments, we
can use the present OPT approach to inspire crucial experiments. OPT has the
power of being a black-box method that does not need the detailed ”physical”
specification of the systems, and this is a great advantage! And, indeed, our
method tackles the problem in terms of a pure in-principle reasonings,
independent of the ”hardware” supporting information, exactly as we do in
information theory. Such hardware independence makes the approach particularly
suited to address a problem so fundamental as the problem of consciousness.

We now proceed with the second principle:

  * P2:

Privacy principle: Experience is not sharable, even in principle.

Principle P2 plays a special role in selecting which information theories are
compatible with a theory of consciousness. Our experience is indeed not
sharable: this is a fact. We hypothesize that that non shareability of
experience holds in principle, not just as a technological limitation.333It
may be possible to know which systems are involved in a particular experience,
as considered in Ref. Loorits2014 . However, we could never know the
experience itself, since it corresponds to non sharable quantum information.A
crucial fact is that information shareability is equivalent to information
readability with no disturbance.444In fact, information shareability is
equivalent to the possibility of making copies of it–technically cloning
information. The possibility of cloning information, in turn, is equivalent to
that of reading information without disturbing it. Indeed, if one could read
information without disturbing it, he could read the information as many times
as needed in order to acquire all its different complementary sides,
technically performing a tomography of the information. And once he knows all
the information he can prepare copies of it at will. Viceversa, if one could
clone quantum information, he could read clones keeping the original
untouched. Recently it has been proved that the only theory where any
information can be extracted without disturbance is classical information
theory tosini2020quantum . We conclude that P2 implies that a theory of
consciousness needs nonclassical information theory, namely QT, or else a
“post-quantum” OPT.

Here we will consider the best known instance of OPT, namely QT. As we will
see, such a choice of theory turns out to be very powerful in accounting for
all the main features of consciousness. We state this choice of theory as a
principle:

  * P2’:

Quantumness of experience: The information theory of consciousness is quantum
theory.

We remind that classical theory is always accompanying any OPT, thus there
will be exchanges and conversions of classical and quantum information.

We now introduce a third principle:

  * P3:

Psycho-purity principle: The state of the conscious system is pure.

P3 may seem arbitrary if one misidentifies the experience with the knowledge
of it. The actual experience is ontic and definite. It is ontic, à la
Descartes (Cogito, ergo sum …). The existence of our experience is surely
something that everybody would agree on, something we can be sure of. It is
definite, in the sense that one has only a single experience at a time–not a
probability distribution of experiences. The latter would describe the
knowledge of somebody else’s experience. Consider that the experience of a
blurred coin is just as definite as the experience of a sharp coin since they
are two different definite experiences. It is the knowledge of the coin side
that is definite, which only occurs if the image of the coin is sufficiently
sharp, otherwise it would be represented in probabilistic terms, e.g. by the
mixture-state $\tfrac{1}{2}$TAIL+$\tfrac{1}{2}$HEAD, based on a fair-coin
hypothesis (see Fig. 1). We thus can understand that by the same definition
the notion of mixture is an epistemic one, based on our prior knowledge about
the coin having two possible states. In synthesis:

  1. 1.

Experience is described by an ontic quantum state, which is pure.

  2. 2.

Knowledge is described by an epistemic quantum state, which is generally
mixed.

We will see later how principle P3 is crucial for solving the combination
problem.

Figure 1:  Illustration of the notions of ontic and epistemic states for a
system given by a classical bit, here represented by a coin, with states
0=HEAD or 1=TAIL. We call ontic the “actual” state of the system, which is
pure and generally unknown, except as an unsharable experience (in the figure
it is the coin state HEAD covered by the hand). We call epistemic the state
that represents the knowledge about the system of an outside observer, e.g.
the state of the unbiased coin corresponds to
$\tfrac{1}{2}$HEAD+$\tfrac{1}{2}$TAIL.

Let’s now state our fourth principle, which is theory independent:

  * P4:

Qualia principle: Experience is made of structured qualia.

Qualia (phenomenal qualitative properties) are the building blocks of
conscious experience. Their existence has so far been inexplicable in the
traditional scientific framework. Their privacy and ineffability are
notorious. Qualia are structured: some of them are more fundamental than
others, e.g. spectrum colours, sound pitch, the five different kinds of taste
buds (sweet, salty, sour, bitter, and umami), somatic sensations (pain,
pleasure, …), basic emotions (sadness, happiness, …). Qualia are compositional
with internal structures that generally partially determine their qualitative
nature. They are connected by different relations, forming numerous complex
structures, such as thoughts and emotions. Since we regard consciousness as
the direct fruition of a very structured kind of quantum information (we are
the system supporting such final information), then, according to principle
P1, qualia correspond to information states “felt” by the aware system, and
according to principles P2 and P2’ such states are quantum states of such
systems.

  * S1:

Qualia are described by ontic pure quantum states.

We will see in Subsect. 3 how quantum entanglement can account for organizing
nontrivial qualia. The qualia space thus would be the Hilbert space of the
systems involved in the quale.

Classical information coming from the environment through the senses is
ultimately converted into quantum information which is experienced as qualia.
The inverse quantum-to-classical transformation is also crucial in converting
structures within qualia into logical and geometric representations
expressible with classical information to communicate free-will choices to
classical structures. For such purposes, the interaction with the memory of
past experiences may be essential, for, as we shall see, memory is classical.

Finally we come to the problem of free will. First we must specify that free
will, contrary to consciousness, produces public effects which are classical
manifestations of quantum information. Since the manifestation is classical,
the possible choices of the conscious agent are in principle jointly perfectly
discriminable.555In the usual classical information theory the convex set of
states is a simplex, and its extremal states are jointly discriminable.

We then state what we mean by free.

  * S2:

Will is free if its unpredictability by an external observer cannot be
interpreted in terms of lack of knowledge.666Here there is a clear distinction
between the ”willing agent” and the observers of it. The word unpredictability
applies only to the observers.

In particular, statement S2 implies that the actual choice of the entity
exercising its free will cannot be in principle predicted with certainty by
any external observer. One can immediately recognise that if one regards the
choice as a random variable, it cannot be a classical one, which can always be
interpreted as lack of knowledge. On the other hand, it fits perfectly the
case of quantum randomness, which cannot be interpreted as such.777The
impossibility of interpretation of quantum randomness as lack on knowledge
fits quantum complementarity. Indeed we cannot by any means know both values
of two complementary variables. One may argue that both values may exist
anyway, even if we cannot know them, but such argument disagrees with the
violation of the CHSH bound (the most popular Bell-like bound), which is
purely probabilistic, and is based on the assumption of the existence of a
joint probability distribution for the values of complementary observables,
assumption that obviously is violated by quantum complementarity. Others can
argue (and this is the most popular argument) that one must use different
local random variables depending on remote settings, which leads to the
interpretation of the CHSH correlations in terms of nonlocality: such
interpretation, however, is artificial, whereas the most natural one iw that
the measurement outcome is created by the measurement.

##  3 Qualia: the role of entanglement

According to Statement S1 qualia are described by ontic quantum states and,
being such states pure, we can represent them by normalised vectors
$|\psi\rangle\in{\mathcal{H}}_{\rm A}$ in the Hilbert space
${\mathcal{H}}_{\rm A}$ of system ${\rm A}$. For the sake of illustration we
consider two simple cases of qualia: direction and colour. These examples
should not be taken literally,888We emphasize that the above examples are only
for the sake of illustration of concepts. The example about color qualia in
Fig. 2 would be a faithful one if the colours were monochromatic and the
summation or subtraction were made with wave amplitude, not intensity, which
is the actual case. The case of direction qualia based on the Bloch sphere is
literally correct. The directions are those of the state representations on
the Block sphere. but only for the sake of illustration of the concept that
the linear combinations of qualia can give rise to new complex qualia.

|
---
|
---
Figure 2:  Illustration of how qualia superpose to make new qualia. Left
figure: the Bloch sphere illustrates the case of direction qualia. Summing and
subtracting the quale “up” with the quale “down” we obtain the qualia “right”
and “left”, respectively. Similarly, if we make the same combinations with the
imaginary unit $i$ in front of “down” we obtain the qualia “front” and “back”.
Summing with generic complex amplitudes $a$ and $b$ (with $|a|^{2}+|b|^{2}=1$)
we get the quale corresponding to a generic direction, corresponding to a
generic pure state $|\psi\rangle$. Right figure: summations of two of the
three colour basis red, green, and blue (RGB) and subtraction of two of the
three colours cyan, magenta and yellow (CMY). As explained in the main text,
these examples are only for the sake of illustration of the notion of linear
combination of quantum ontic states to make new ontic states corresponding to
new qualia. A third case of superposition is that of sounds with precise
frequency that combine with addition and subtraction into timbers and chords.
Notice that all three cases fit the wave aspect of reality, not the particle
one.

We have said that qualia combine to make new qualia, and thoughts and emotions
themselves are structured qualia. Superimposing two different kinds of qualia
in an entangled way produces new kinds of qualia. Indeed, consider the
superposition of a red up-arrow with a green down-arrow. This is not a yellow
right-arrow as one may expect, since the latter corresponds to the independent
superposition of direction and colour, as in the following equation:

(1)

We would have instead

(2)

where the ket with the blue star represents a completely new qualia. In a more
general case, we have a state vector with triple or quadruple or more
entanglement in a factor of a tensor product, and more generally, every system
is entangled, e.g.

$\ldots\otimes(|a_{1}\rangle\otimes|b_{1}\rangle\otimes|c_{1}\rangle\otimes|d_{1}\rangle\pm|a_{2}\rangle\otimes|b_{2}\rangle\otimes|c_{2}\rangle\otimes|d_{2}\rangle)\otimes(|\uparrow\rangle\otimes|{\color[rgb]{1,0,0}\bullet}\rangle\pm|\downarrow\rangle\otimes|{\color[rgb]{0,1,0}\bullet}\rangle)\otimes\ldots$
(3)

We can realise how in this fashion one can achieve new kinds of qualia whose
number grows exponentially with the number of systems. In fact, the number of
different ways of entangling $N$ systems corresponds to the number of
partitions of $N$ into integers, multiplied by the number of permutations of
the systems, and therefore it grows as $N!e^{\sqrt{2N/3}}$, and this without
considering the variable vectors that can be entangled!

Since qualia correspond to pure states of the conscious systems, their Hilbert
space coincides with the multidimensional Hilbert space of the system.
Experimentally one may be able to locate the systems in terms of neural
patterns, but will never be able to read the encoded information without
destroying the person’s experience, while at best gaining only a single
complementary side of the qualia, out of exponentially many. In synthesis:

  * •

System identification is possible, but not the ”experience” within.

The fact that it is possible to identify the information system proves that
the identity of the observer/agent is public and is thus correlated with its
”sense of self,” which is instead private. This is a crucial requirement for a
unified theory of consciousness and free will, namely that the observer/agent
be identifiable both privately–from within and through qualia–and
publicly–from without and through information. Within QT (or post-quantum
OPTs) this is possible.

##  4 Evolution of Consciousness and Free Will

Since the quantum state of a conscious system must be pure at all times, the
only way to guarantee that the evolving system state remains pure is that the
evolution itself is pure (technically it is atomic, namely its CP-map
$\mathcal{T}=\sum_{i}T_{i}\cdot T_{i}^{\dagger}$ has a single Krauss operator
$T_{i_{0}}$). We remind that both states and effects are also transformations
from trivial to non trivial systems and viceversa, respectively. In Table 1 we
report the theoretical representations of the three kinds of ontic
transformations, including the special cases of state and effect.

| Circuit symbol | Symbol | Map | Operator | domain-codomain
---|---|---|---|---|---
transformation | $\textstyle{{{}^{{\rm A}}_{\phantom{{\rm A}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{T}}$$\textstyle{{{}^{{\rm B}}_{\phantom{{\rm B}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$ | $\mathcal{T}$ | $\mathcal{T}=T\cdot T^{\dagger}$ | $T$ | $\mathrm{Bnd}({\mathcal{H}}_{\rm A}\to{\mathcal{H}}_{\rm B})$
Effect | $\textstyle{{{}^{{\rm A}}_{\phantom{{\rm A}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\alpha}$ | $(\alpha|$ | $\operatorname{Tr}[\cdot|\alpha\rangle\langle\alpha|]$ | $\langle\alpha|$ | $\mathrm{Bnd}({\mathcal{H}}_{\rm A}\to\mathbb{C})$
State | $\textstyle{\psi}$ $\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{}^{{\rm A}}_{\phantom{{\rm A}}}}}$ | $|\psi)$ | $|\psi\rangle\langle\psi|$ | $|\psi\rangle$ | $\mathrm{Bnd}(\mathbb{C}\to{\mathcal{H}}_{\rm A})$
Table 1: Notations for the three kinds of ontic transformations (for the
meaning of symbols see Table 2). For the list of Quantum Theory axioms and its
main theorems see Tables 3 and 4.

Let’s consider now the general scenario of a conscious composite system in an
ontic state $\omega_{t}$ at time $t$ evolved by the one-step ontic
transformation $\mathcal{O}^{(t,x_{t})}_{F_{t}}$ with outcome $F_{t}$,
depending on classical input $x_{t}$ from senses and memory999As we will see,
memory is classical.

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The epistemic transformation would be the sum of all ontic ones corresponding
to all possible outcomes:

$\mathcal{E}^{(t,x_{t})}:=\sum_{F_{t}}\mathcal{O}^{(t,x_{t})}_{F_{t}}.$ (9)

  * The outcome $F_{t}$ is a classical output, and we identify it with the free will of the experiencing system.

It is a probabilistic outcome that depends on the previous history of qualia
of the system. Its kind of randomness is quantum, which means that it cannot
be interpreted as lack of knowledge, and, as such, it is free. Notice that
both mathematically and literally the free will is the outcome of a
transformation that corresponds to a change of experience of the
observer/agent. The information conversion from quantum to classical can also
take into account a stage of ”knowledge of the will” corresponding to
”intention/purpose”, namely ”understanding” of which action is taken.

We may need to provide a more refined representation of the one-step ontic
transformation of the evolution in terms of a quantum circuit, for example:

$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{}^{{\rm
A}{\rm B}{\rm C}}_{\phantom{{\rm A}{\rm B}{\rm
C}}}}}$$\textstyle{\mathcal{O}^{(t,x_{t})}_{F_{t}}}$$\textstyle{\ignorespaces\ignorespaces\ignorespaces\ignorespaces{{}^{\;{\rm
M}}_{\phantom{\;{\rm M}}}}}$$\textstyle{\;\;=\;}$ $\textstyle{{{}^{{\rm
A}}_{\phantom{{\rm
A}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\alpha_{i}}$
$\textstyle{\psi^{(i)}}$ $\textstyle{{{}^{{\rm D}}_{\phantom{{\rm
D}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathcal{R}^{(k)}}$$\textstyle{{{}^{{\rm G}}_{\phantom{{\rm
G}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{A}^{(x_{t})}}$$\textstyle{{{}^{{\rm
M}}_{\phantom{{\rm
M}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{{{}^{{\rm
B}}_{\phantom{{\rm B}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\mathcal{E}_{k}}$$\textstyle{{{}^{{\rm E}}_{\phantom{{\rm
E}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{{{}^{{\rm H}}_{\phantom{{\rm
H}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{B}}$$\textstyle{{{}^{{\rm
N}}_{\phantom{{\rm N}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{\Lambda_{j}^{(l)}}$$\textstyle{{{}^{{\rm C}}_{\phantom{{\rm
C}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\textstyle{{{}^{{\rm F}}_{\phantom{{\rm
F}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{V}_{l}}$$\textstyle{{{}^{{\rm
L}}_{\phantom{{\rm
L}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{C}}$$\textstyle{{{}^{{\rm
O}}_{\phantom{{\rm O}}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
(10)

Generally for each time $t$ we have a different circuit. In the example in
circuit (10) we see that a single step can contain also states and effects,
and the output system of the whole circuit (${\rm M}$ in the present case) is
generally different (not even isomorphic) to the input ones ${\rm A}{\rm
B}{\rm C}$. Following the convention used for the ontic transformation
$\mathcal{O}^{(t,x_{t})}_{F_{t}}$ the lower index is a random outcome,101010
The outcome is random for an observer other than the conscious system, for
which, instead, it is precisely known. and the upper index $(t,x_{t})$ is a
parameter from which the transformation generally depends. Overall in circuit
(10) we have free will $F_{t}=\\{i,k,l,j\\}$, whereas the transformations
$\psi^{(i)},\mathcal{R}^{(k)},\mathcal{B}$, and $\mathcal{C}$ are
deterministic (they have no lower index), and do not contribute to the free
will, and the same for the transformation $\mathcal{A}^{(x_{t})}$, which
depends on the sensorial input $x_{t}$. In circuit (10) we can see that we
have also classical information at work, since e.g. the transformation
$\mathcal{R}^{(k)}$ depends on the outcome $k$ of the transformation
$\mathcal{E}_{k}$, and similarly the choice of test $\\{\Lambda_{j}^{(l)}\\}$
depends on the outcome $l$ of $\mathcal{V}_{l}$, and similarly the state
$\psi^{(i)}$ depends on the effect outcome $\alpha_{i}$. Each element of the
circuit is ontic, i.e. atomic, and atomicity of sequential and parallel
composition guarantee that the whole transformation is itself atomic. This
means that its Krauss operator $O_{t}$ can be written as the product of the
Krauss operators of the component transformations as follows

$O^{(t,x_{t})}_{F_{t}}=\underbrace{(I_{M}\otimes\langle\Lambda^{(l)}_{j}|)}_{{\rm
M}{\rm N}{\rm O}\to{\rm M}}\underbrace{(A^{(x_{t})}\otimes B\otimes C)}_{{\rm
G}{\rm H}{\rm L}\to{\rm M}{\rm N}{\rm O}}\underbrace{(R^{(k)}\otimes
V_{l})}_{{\rm D}{\rm E}{\rm F}\to{\rm G}{\rm H}{\rm
L}}\underbrace{(|\psi_{i}\rangle\langle a_{i}|\otimes E_{k})}_{{\rm A}{\rm
B}{\rm C}\to{\rm D}{\rm E}{\rm F}},$ (11)

where $\mathcal{O}^{(t,x_{t})}_{F_{t}}=O^{(t,x_{t})}_{F_{t}}\cdot
O^{(t,x_{t})}_{F_{t}}{}^{\dagger}$. Notice that in the expression for the
operator $O^{(t,x_{t})}_{F_{t}}$ in Eq. (11) the operators from the input to
the output are written from the right to the left–the way we compose operators
on Hilbert spaces. Moreover, the expression (11) of $O^{(t,x_{t})}_{F_{t}}$ is
not unique, since it depends on the choice of foliation of the circuit, namely
the way you cover all wires with leaves to divide the circuit into input-
output sections. For example, Eq. (11) would correspond to the foliation in
Fig. 3

Figure 3: Quantum circuit foliation corresponding to Eq. (11)

A different foliation, for example, is the one reported in Fig. 4,
corresponding to the expression for $O^{(t,x_{t})}_{F_{t}}$

$O^{(t,x_{t})}_{F_{t}}=\underbrace{(I_{C}\otimes\langle\Lambda^{(l)}_{j}|)}_{{\rm
M}{\rm N}{\rm O}\to{\rm M}}\underbrace{(A^{(x_{t})}\otimes B\otimes I_{\rm
O})}_{{\rm G}{\rm H}{\rm O}\to{\rm M}{\rm N}{\rm
O}}\underbrace{(R^{(k)}\otimes CV_{l})}_{{\rm D}{\rm E}{\rm F}\to{\rm G}{\rm
H}{\rm O}}\underbrace{(|\psi_{i}\rangle\otimes E_{k})}_{{\rm B}{\rm C}\to{\rm
D}{\rm E}{\rm F}}\underbrace{(\langle a_{i}|\otimes I_{{\rm B}{\rm C}})}_{{\rm
A}{\rm B}{\rm C}\to{\rm B}{\rm C}}.$ (12)

Figure 4: Quantum circuit foliation corresponding to the expression for
$O^{(t,x_{t})}_{F_{t}}$ Eq. (12)

We remind that all operators (including kets and bras as operators from and to
the trivial Hilbert space $\mathbb{C}$) are contractions, namely have norm
bounded by 1, corresponding to marginal probabilities not greater than 1.
Thus, also $O^{(t,x_{t})}_{F_{t}}$ is itself a contraction. Contractivity for
operator $X$ can be conveniently expressed as

$X\in\mathrm{Bnd}({\mathcal{H}}_{in}\to{\mathcal{H}}_{out}),\quad
X^{\dagger}X\leqslant P_{S_{X}}\leqslant I_{in},\quad
P_{S_{X}}:=\mathsf{Proj}\,\mathsf{Supp}\,X.$ (13)

Now, since the evolution of consciousness must be atomic at all times, we can
write a whole consciousness history as the product of the Krauss operators
$O_{t}\equiv O^{(t,x_{t})}_{F_{t}}$ at all previous times $t$

$\Omega_{t}:=O_{t}O_{t-1}\ldots O_{1},\quad O_{t}:=O_{F_{t}}^{(t,x_{t})},$
(14)

and apply the history operator to the wavevector of the initial ontic state-
vector $|\omega_{0}\rangle$

$|\omega_{t}\rangle=\Omega_{t}|\omega_{0}\rangle.$ (15)

The squared norm $|\\!|\omega_{t}|\\!|^{2}$ of vector $\omega_{t}$ is the
probability of the full history of conscious states
$\\{\omega_{0},\omega_{1},\omega_{2},\ldots,\omega_{t}\\}$ and equally of the
free-will history $\\{F_{1},F_{2},\ldots,F_{t}\\}$

$p(\omega_{0},\omega_{1},\omega_{2},\ldots,\omega_{t})=p(\emptyset,F_{1},\ldots,F_{t})=|\\!|\omega_{t}|\\!|^{2}=|\\!|\Omega_{t}\omega_{0}|\\!|^{2}.$
(16)

##  5 Memory

The ontic evolution of the consciousness state, though it maintains coherence,
can keep very limited quantum memory of experience. The latter, due to
contractivity of $O$, will go down very fast as
$|\\!|\Omega_{t}|\\!|^{2}\simeq|\\!|O|\\!|^{-2t}$, i.e. it will decrease
exponentially with the number of time-steps. The quantum memory intrinsic in
the ontic evolution instead works well as a short-term buffer to build up a
fuller experience–e.g. of a landscape, or of a detailed object, or even to
detect motion. How many qubits will make such a single-step buffer? The answer
is: not so many.111111For example, the amount of visual information is
significantly degraded as it passes from the eye to the visual cortex. Marcus
E. Raichle says Raichle2010 : Of the virtually unlimited information available
from the environment only about $10^{10}$ bits/sec are deposited in the
retina. Because of a limited number of axons in the optic nerves
(approximately 1 million axons in each) only $6*10^{6}$ bits/sec leave the
retina and only $10^{4}$ make it to layer IV of V1 Anderson2005 ;
norretranders1998user . These data clearly leave the impression that visual
cortex receives an impoverished representation of the world, a subject of more
than passing interest to those interested in the processing of visual
information Olshausen2005 . Parenthetically, it should be noted that estimates
of the bandwidth of conscious awareness itself (i.e. what we ”see”) are in the
range of 100 bits/sec or less Anderson2005 ; norretranders1998user . Indeed,
if we open our eyes for just a second to look at an unknown scene, and
thereafter we are asked to answer binary questions, we would get only a dozen
of right answers better than chance. Tor Nørretranders writes
norretranders1998user : The bandwidth of consciousness is far lower than the
bandwidth of our sensory perceptors. … Consciousness consists of discarded
information far more than information present. There is hardly information
left in our consciousness.121212We believe that the inability to recall much
information contained in one second of visual experience, when the actual
experience is felt to be quite rich, should not be construed to diminish the
importance of consciousness. In fact, experience is quantum while memory is
classical, and although not much classical information appears to have been
memorized, the actual experience has the cardinality of the continuum in
Hilbert space. Consciousness is about living the experience in its unfolding
and understanding what is happening, so to make the appropriate free-will
decisions when necessary. Recalling specific information in full detail is
unnecessary. Consciousness is focused on the crucial task of getting the
relevant meaning contained in the flow of experience. The scarce conscious
memory of specific objects and relationships between objects should not be an
indication that consciousness is a ”low bandwidth” phenomenon, but that what
is relevant to consciousness may not be what the interrogator believes should
be relevant.

We reasonably deduce that there is actually no room for long-term memory in
consciousness, and we conclude that:

  * S3:

Memory is classical. Only the short-term buffer to collect each experience is
quantum.

By “short” we mean comparable to the time in which we collect the full
experience, namely of the order of a second.

#### Transferring quantum experience to classical memory

If we have quantum experiences and classical memory, we need to convert
information from quantum to classical in the memorization process, and from
classical to quantum in the recollection process. The first process, namely
transferring quantum experience to classical memory, must be necessarily
incomplete, otherwise it would violate the quantum no-cloning theorem. The
Holevo theorem Holevo1973BoundsFT  establishes that the maximal amount of
classical information that can be extracted from a quantum system is a number
of bits equal to the number of qubits that constitute the quantum system.
Obviously, such maximal classical information is infinitesimal compared to the
continuum of classical information needed to communicate classically a quantum
state!

From a single measurement one can extract classical information about just one
of the continuum of complementary aspects of the quantum state, e.g. along a
given direction for a spin. Moreover, the state after the measurement would be
necessarily disturbed131313For example, for the spin originally oriented
horizontally and measured vertically a la von Neumann, the final state would
be vertical up or down, depending on the measurement outcome., due to the
information-disturbance tradeoff .141414There are many ways of regarding the
information-disturbance tradeoff, depending on the specific context and the
resulting definitions of information and disturbance. The present case of
atomic measurements with the ”disturbance” defined in terms of the probability
of reversing the measurement transformation has been analysed in the first
part of Ref. d2003heisenberg . In the same reference it is also shown that a
reversal of the measurement would provide a contradicting information which
numerically cancels the information achieved from the original measurement,
thus respecting the quantum principle of no information without disturbance.
So, which measurement should one perform for collecting the best classical
information from a quantum system?

The answer is easy for a single qubit. Just perform the usual von Neumann
measurement along a random direction! This has been proved in Ref.
chiribella2007continuous .151515Let’s consider the case of a single qubit
realised with a particle spin. The usual observable of von Neumann corresponds
to a measurement of the orientation of the spin along a given direction, e.g.
up or down along the vertical, or left or right along the horizontal
direction. But when we prepare a spin state (e.g. by Rabi techniques), we can
put the spin in a very precise direction, e.g. pointing north-east along the
diagonal from south-west to north-east, and, indeed if we measure the spin
along a parallel direction we find the spin always pointing north-east! So the
spin is indeed (ontically!) pointing north-east along the same diagonal! Now
it would be a legitimate question to ask: how about measuring the direction
itself of the spin? This can be done–not exactly, but optimally–using a
continuous observation test constituted of a continuous of effects (what is
usually called POVM, the acronym of Positive Operator Valued Measure).
However, it turns out that the measured direction is a fake, since such a
quantum measurement with a continuous set of outcomes is realised as a
continuous random choice of a von Neumann measurement chiribella2007continuous
. In conclusion the optimal measurement of the spin direction is realised by a
customary Stern-Gerlach experiment in which the magnetic field is randomly
oriented! The same method can be used to achieve an informationally complete
measurement to perform a quantum tomography DAriano2000  of the state, by
suitably processing the outcome depending on the orientation of the spin
measurement. For larger dimensions it is in principle possible to generalise
this result, however, the probability space turns out to be geometrically much
more complicated than a sphere,161616The convex set of deterministic states
for a qutrit (i.e. $d=3$), defined by algebraic inequalities, has eight
dimensions, and has a boundary made of a continuous of balls. and a scheme for
random choice is unknown. For this reason we can consider the larger class of
observation tests called informationally complete (”infocomplete”),171717What
in OPT language is called observation test is the same of what in the quantum
information literature is called discrete POVM. An observation test is
infocomplete for system ${\rm A}$ when it spans the linear space of effects
$\mathrm{Eff}_{\mathbb{R}}({\rm A})$. which, unlike the case of the optimal
measurement of spin direction, can be taken as discrete and with a finite
number of outcomes. Such a measurement would correspond to the following
epistemic transformation

$\mathcal{M}\in\mathrm{Trn}_{1}({\rm A}\to{\rm
B}),\qquad\mathcal{M}=\sum_{j\in J}M_{j}\rho
M_{j}^{\dagger},\quad|J|\geq\dim({\mathcal{H}}_{\rm A})^{2},$ (17)

which is the coarse graining of the infocomplete quantum test
$\\{\mathcal{M}_{j}\\}_{j\in J}$, with $\mathcal{M}_{j}=M_{j}\cdot
M_{j}^{\dagger}$ where $\\{|M|^{2}_{j}\\}_{j\in J}$ is an infocomplete
observation test. ”Infocomplete” means that, in the limit of infinitely many
usages over the same reprepared state, the measurement allows to perform a
full tomography DAriano2000  of the state.

The infocomplete measurement (or the random observables seen before) is better
suited to extract classical information from the quantum buffer for long-term
memory, since is does not privilege a particular observable. It is also likely
that the conscious act of memorising an experience may be achieved by actually
repreparing the ontic state multiple times in the quantum buffer, and
performing the infocomplete test multiple times, thus with the possibility of
memorising a (possibly partial) tomography of the state. Notice that,
generally, the ontic state $M_{j}\rho M_{j}^{\dagger}$ for a given $j$ still
depends on $\rho$. However, the test will disturb it, and the less
disturbance, the smaller the amount of classical information that can be
extracted d2003heisenberg .181818Particularly symmetric types of measurements
are those made with a SIC POVM (Symmetric informationally complete POVM)
Fuchs2017 , where $M_{j}=|\psi_{j}\rangle\langle\psi_{j}|$ are $d^{2}$
projectors on pure states with equal pairwise fidelity
$|\langle\psi_{j}|\psi_{k}\rangle|^{2}=\frac{d\delta_{jk}+1}{d+1}.$ (18) The
projectors $|\psi_{j}\rangle\langle\psi_{j}|$ defining the SIC POVM in
dimension $d$ form a $(d^{2}-1)$-dimensional regular simplex in the space of
Hermitian operators.

#### Transferring classical memory to quantum experience

We have seen that an infocomplete measurement is needed to (approximately)
store an experience in the classical long-term memory. By definition, the
recovery of an experience requires a reproduction of it, meaning that the
corresponding ontic state is (approximately) reprepared from the classical
stored data. The memory, being classical, will be read without disturbance,
thus left available to following recollections. In order to transfer classical
to quantum information, methods of state-preparation have been proposed in
terms of quantum circuit schemes PhysRevA.83.032302 .

A possible benchmark for the memory store-and-recall process is the maximal
fidelity achievable in principle with a measure-and-reprepare scheme that
optimises the fidelity between the experienced state and the recalled one. We
will consider $M\geq 1$ input copies of the same state to quantify attention
to the experience. The optimal fidelity191919The fidelity $F$ between two pure
states corresponding to state vectors $|\psi\rangle$ and $|\varphi\rangle$,
respectively, is defined as $|\langle\varphi|\psi\rangle|^{2}$. between the
experienced state and the recalled one averaged over all possible experiences
(i.e. input states) is given by PhysRevA.72.032325

$F(A,M,d)=\frac{M+1}{M+d},$ (19)

where $d$ is the dimension of the experiencing system.202020The optimal
fidelity in Eq. (19) is achieved by an observation test with atomic effects
$|\Phi_{i}\rangle\langle\Phi_{i}|$ with
$|\Phi_{i}\rangle=\sqrt{w_{i}d_{M}}{|\psi_{i}\rangle^{\otimes M}},\qquad
d_{M}={{M+d-1}\choose{d-1}},$ (20) where $\\{|\psi_{i}\rangle\\}_{i=1}^{A}$
are pre-specified pure state-vectors, and $\\{w_{i}\\}_{i=1}^{A}$ is a
probability vector satisfying the identity
$\sum_{i=1}^{A}w_{i}|\psi_{i}\rangle\langle\psi_{i}|^{\otimes
M}=\Big{\langle}|\psi\rangle\langle\psi|^{\otimes M}\Big{\rangle}.$ (21) where
the notation $\langle\ldots\rangle$ denotes averaging over the prior of pure-
state vectors, taking as the prior the Haar measure on the unit sphere in
$\mathbb{C}^{d}$.

Such upper bound for fidelity may be used to infer an effective dimension $d$
of the system involved in consciousness, e.g. in very focused restricted
experiences as those involved in masking conscious perception
dehaene2014consciousness .

### Information transfer from body to consciousness and vice versa

We would expect that most of the operation of the human body is automatic and
uses classical information that is never translated into quantum information
to be experienced by consciousness. The portion of the classical information
produced by the body that needs to be converted into quantum should only be
the salient information that supports the qualia perception and comprehension
necessary to ”live life” and make appropriate free-will choices. The amount of
quantum information to be translated into classical for the purpose of free-
will control of the body top-down should be relatively small.

##  6 On the combination problem

The combination problem concerns the issue about if and how the fundamental
conscious minds come to compose, constitute, or give rise to some other,
additional conscious mind Chalmers2016-CHATCP-6 . By definition, the problem
becomes crucial for panpsychism: if cosciousness is everywhere, what is the
criterion to select novel conscious individuals? Is the union of two conscious
beings a conscious being? If this is true, then any subset of a conscious
being can also be a conscious being. The present theoretical approach provides
a precise individuation criterion. The criterion derives from principle P3
about the purity of quantum conscious states and, consequently, the need for
ontic transformations. Let’s see how it works.

It is reasonable to say that an individual is defined by the continuity of its
experience. Such a statement may be immediately obvious to some readers.
However, for those who may not agree, we propose a thought experiment.

Consider a futuristic “quantum teleportation experiment”, meaning that the
quantum state of a system is substituted to that of a remote isomorphic
system.212121Teleportation will need the availability of shared entanglement
and classical communication, and technically would use a Bell measurement at
the sender and a conditioned unitary transformation at the receiver. tele  The
matter (electrons, protons, …) of which a teleported person is composed is
available at the receiver location and is in-principle indistinguishable from
the same matter at the transmission location: indeed, it is only its quantum
state that is teleported. The resulting individual would be the same as the
original one, including his thoughts and memory.222222Of course, this would
violate the no cloning theorem, if at the transmission point the original
individual would not be destroyed. Indeed, according to the quantum
information-disturbance tradeoff d2003heisenberg , teleportation cannot even
make a bad copy leaving the original untouched.

The above thought experiment suggests the following individuation criterion
within our theoretical approach:

  * S4:

A conscious mind is a composite system in an ontic state undergoing an ontic
transformation, with no subsystem as such.

In Figs. 5 and 6 we illustrate the use of the criterion in two paradigmatic
cases. Here we emphasise that the role of interactions is crucial for the
criterion.

Figure 5: The combination problem. Stated generally, the problem is about how
the fundamental conscious minds come to compose, constitute, or give rise to
some further conscious mind. The ontic-state principle P1 provides a partial
criterion to exclude some situations, e.g. (figure on the left) two entangled
systems cannot separately be conscious entities, since each one is in a
marginal state of an entangled one, hence it is necessarily mixed. On the
other hand, (figure on the right) if two systems are in a factorized pure
state, each system is in an ontic state, and they are two single individuals,
and remain so depending on their following interactions (see Fig. 6).

This section has described what is necessary to form a combination of
conscious entities, thus removing the difficulties encountered with
panpsychist models based on classical physics. The key idea is that the states
of the combining systems and their transformations be ontic and that such
systems interact quantumly. These requirements assure that the combined entity
is also in a pure state and that none of its subsets are conscious. Clearly,
the interactions between the conscious entity and the environment (including
other entities) can only be classical, otherwise a larger entity would be
created and would persist for as long as a disentangling transformation does
not occur, as in the case illustrated in Fig. 6.

Figure 6: The combination problem. The general individuation criterion in
statement S4 requires full quantum coherence, namely the ontic nature of both
states and transformations. In the case depicted in this figure, every
transformation (including effects and states as special cases) is pure, and we
suppose each multipartite transformation is not factorizable. Then the first
two boxes on the left represent two separate individuals. The box in the
middle merges the two individual into a single one, whereas the immediately
following effects convert quantum to classical information, and separate again
the single individual into the original separate ones. Notice that a merging
of two individuals necessarily needs a quantum interaction.

##  7 Theory experimentability and simulability

Proposing feasible experiments about the quantum nature of consciousness is a
very exciting challenge. In principle quantumness could be established through
experimental demonstration of one of the two main nonclassical features of
quantum theory: nonlocality and complementarity. The two notions are not
independent, since in order to prove nonlocality we need complementary
observations, in addition to shared entanglement.

In order to prove nonlocality of consciousness we need measurements at two
separate points sufficiently far apart to exclude causal connection, and such
a requirement is very challenging, since it demands very fast measurements,
considering that for a distance of 3 cm between the measurements points a
time-difference of a tenth of nanosecond is sufficient to have signalling.

About nonlocality together with complementarity, we speculate that they
together may be involved in 3D vision, and take the opportunity to suggest
that such line of research may anyway be a fruitful field of experimental
research on consciousness.232323A paradigmatic case of superposition between
incompatible 3D views is that of the Necker cube, which some authors regard as
neuro-physiological transformation leading to perceptual reversal controlled
by the principles of quantum mechanics Caglioti-Benedek-Neckercube . However,
the 3D experience of the Necker cube does not require binocular disparity
since monocular vision also produces the same effect.  For example, a genuine
case of complementarity in 3D experience is occurring as a result of looking
at Magic Eye images published in a series of books Magiceye . These images
feature autostereograms, which allow most people to see 3D images by focusing
on 2D patterns that seem to have nothing in common with the 3D image that
emerges from them. The viewer must diverge or converge his eyes in order to
see a hidden three-dimensional image within the patterns. The 3D image that
shows up in the experience is like a glassy object that, depending on
convergence or divergence of the eyes, shows up as either concave or convex.
Clearly the two alternative 3D views–convex and concave–are truly
complementary experiences, and each experience has an intrinsic wholeness.

Regarding complementarity alone, speculative connections with contrasting or
opposite dimensions of human experience have been considered in the
literature, e.g. ”analysis” vs ”synthesis”, and ”logic” vs ”intuition”
Jahn2007 . Here, however, we are interested in complementariy in experiences
possibly reproducible in different individuals, of the kind of gedanken-
experiments a la Heisenberg, e.g. one experience incompatible with another
and/or disturbing each other. Having something more than two complementary
observables, namely an informationally complete set of measurements such as
all three Pauli matrices for a single consciousness qubit, would allow us to
make a quantum tomography DAriano2000  of the qubit state, assuming this is
reprepared many times, e.g. through an intense prolonged attention. To our
knowledge, the feasibility assessment of such a kind of experiment is of a
difficulty comparable to that of a nonlocality experiment.

Finally, apart from proving the quantumness of experience, we can at least
experimentally infer some theoretical parameters in the quantum theoretical
approach consistent with observations. This is the case e.g. of the experiment
already mentioned in discussing the upper bound (19) in memory-recall
fidelity, which can be used in inferring an effective dimension $d$ of the
system involved in consciousness in very focused restricted experiences, as
for masking conscious perception dehaene2014consciousness . We believe that
memory-recall experiments versus variables such as attention, memory-recovery
delay, and variable types of qualia, may be helpful to make a preliminary
mapping of the dimensionality of the spaces of conscious systems involved.

We conclude with a few considerations about the feasibility of a simulation of
a conscious process like the ones proposed here. As already mentioned, due to
the purity of the ontic process, a simulation just needs multiplications of
generally rectangular matrices, which for matrices with the same dimension $d$
is essentially a $\Theta(d^{3})$ process. For sparse matrices (as it is often
the case in a quantum simulation) the process can be speeded up considerably.
However, to determine the probability distribution of each ontic step, one
needs to evaluate the conditional probability for the output state of the
previous step, and this needs the multiplication for all possible outcomes
(the free will) of the corresponding Kraus operator of the last ontic
transformation. With a RAM of the order of Gbytes one could definitely operate
with a dozen qubits at a time. This should be compared with 53 qubits of the
Google or IBM quantum computer, the largest currently available, likely to
share classical information in tandem with a large classical computer.
However, it is not excluded that some special phenomena could be already
discovered/analyzed with a laptop.

##  8 Philosophic implications

We have presented a theory of consciousness, based on principles, assumptions,
and key concepts that we consider crucial for the robustness of the theory and
the removal of the limitations of most panpsychist theories
Chalmers2016-CHATCP-6 . We believe that conferring inner reality and agency to
quantum systems in pure quantum state, with conversion of information from
classical to quantum and vice versa, is unprecedented, with major
philosophical and scientific consequences. In the present approach free will
and consciousness go hand in hand, allowing a system to act based on its
qualia experience by converting quantum to classical information, and thus
giving causal power to subjectivity–something that until now has been highly
controversial, if not considered impossible.

The theory provides that a conscious agent may intentionally convert quantum
information into a specific classical information to express its free will, a
classical output that is in principle unpredictable due to its quantum origin.
The theory would be incoherent without the identification of the conscious
system in terms of purity and inseparability of the quantum state, which is
identified with the systems experience. The purity of non-deterministic
quantum evolution identifies consciousness with agency through its outcome.

Metaphysically the proposed interpretation that a pure, non-separable quantum
state is a state of consciousness, could be turned on its head by assuming the
ontology of consciousness and agency as primary, whereas physics is emergent
from consciousness and agency. This same interpretation would then consider
classical physics the full reification (objectification) of quantum reality as
quantum-to-classical agency corresponding to the free will of conscious
entities existing entirely in the quantum realm. The ontology deriving by the
acceptance of consciousness as fundamental, would be that objectivity and
classical physics supervene on quantum physics, quantum physics supervenes on
quantum information, and quantum information supervenes on consciousness. If
we were to accept this speculative view, physics could then be understood as
describing an open-ended future not yet existing because the free will choices
of the conscious agents have yet to be made. In this perspective, we, as
conscious beings, are the co-creators of our physical world. We do so
individually and collectively, instant after instant and without realizing it,
by our free-will choices.

## Appendices about general OPTs

In these appendices we provide the general operational probabilistic framework
for future post-quantum explorations for a theory of consciousness. We report
just a short illustration of what is an operational probabilistic theory
(OPT), of which Quantum Theory and Classical Theory are the most relevant
istances. As the reader may appreciate, the OPT provides a framework much
reacher, general, flexible, and mathematically rigorous as compared to other
theoretical frameworks, such as the causal approach of Tononi Tononi2008 .

#### The operational probabilistic theory (OPT) framework

It is not an overstatement to say that the OPT framework represents a new
Galilean revolution for the scientific method. In fact, it is the first time
that a theory-independent set of rules is established on how to build up a
theory in physics and possibly in other sciences. Such rules constitute what
is called the operational framework. Its rigour is established by the simple
fact that the OPT is just ”metamathematics”, since it is a chapter of category
theory maclane ; selinger2011survey . To be precise the largest class of OPTs
corresponds to a monoidal braided category. The fact that the same categoric
framework is used in computer science coecke ; practicingcoecke ; coeckerev ;
kinder ; picturialism  gives an idea of the thoroughness and range of
applicability of the rules of the OPT.

Lucien Hardy in several seminal papers hardy2001 ; causaloid ; duotenz
introduced a heuristic framework that can be regarded as a forerunner of the
OPT, which made its first appearance in Refs. purification ; QUIT-Arxiv , and
soon was connected to the categorical approach in computer science CoeckePV ;
Coecke2012 ; Coecke2016 ; abramsky_heunen_2016 ; Coecke2018 ; TullPV . As
already mentioned, both QT and CT are OPTs CUPDCP , but one can build up other
OPTs, such as variations of QT e.g. fermionic QT supersel , or QT on real
Hilbert space CUPDCP , or QT with only qubits, but also CT with entanglement
PhysRevA.101.042118  or without local discriminability CTnolocdiscr . Also
other toy-theories, such as the PR-Boxes, are believed to be completable to
OPTs (see e.g. barrett ).

The connection of OPT with computer science reflects the spirit of the OPT,
which essentially was born on top of the new field of quantum information.
Indeed, the OPT framework is the formalization of the rules for building up
quantum circuits and for attaching to them a joint probability: in such a way
the OPT literally becomes an extension of probability theory.

#### The general idea

How does an OPT works? It associates to each joint probability of multiple
events a closed directed acyclic graph (CDAG) of input-output relations as in
Fig. 7. Each event (e.g. $\mathcal{E}_{m}$ in figure) is an element of a
complete test ($\\{\mathcal{E}_{m}\\}_{m\in{\rm M}}$ in figure) with
normalized marginal probability $\sum_{m\in{\rm M}}p(\mathcal{E}_{m})=1$. The
graph tells us that the marginal probability distribution of any set of tests
still depends on the marginalized set, e.g. the marginal probability
distribution of test $\\{\mathcal{E}_{m}\\}_{m\in{\rm M}}$ depends on the full
graph of tests to which it is connected, although it has been partially or
fully marginalised. As a rule, disconnected graphs (as $\gamma_{1}$ and
$\gamma_{2}$ in Fig. 7) are statistically independent, namely their
probability distributions factorize.

---
Figure 7:  An OPT associates to each joint probability distribution of
multiple tests/events a closed directed acyclic graph (CDAG) of output-input
relations (see text). As a rule, unconnected graph components are
statistically independent, e.g. in the case in the figure, one has
$p(\mathcal{A}_{i},\mathcal{E}_{m},\ldots,\mathcal{X}_{r},\ldots|\gamma_{1}\cup\gamma_{2})=p(\mathcal{A}_{i},\mathcal{E}_{m},\ldots|\gamma_{1},)p(\mathcal{X}_{r},\ldots|\gamma_{2})$.

The oriented wires denoting output-input connections between the tests
(labeled by Roman letters in figure 7), are the so called systems of the
theory.

#### A paradigmatic quantum example

A paradigmatic example of OPT graph is given in Fig.8. There, we have a source
of particles all with spin up (namely in the state
$\rho=|\uparrow\rangle\langle\uparrow|$). At the output we have two von
Neumann measurements242424A von Neumann measurement of e.g. $\sigma_{z}$ has
two outcomes ”up” and ”down”, and the output particle will be in the
corresponding eigenstate of $\sigma_{z}$. in cascade, the first one
$\Sigma_{\alpha}$ measuring $\sigma_{\alpha}$, and the second one
$\Theta_{\beta}$ measuring $\sigma_{\beta}$, where $\alpha$ and $\beta$ can
assume either of the two values $x,z$. The setup is represented by the graph
shown in the figure, where ${\rm A}$ represents the system corresponding to
the particle spin, $e$ the deterministic test that simply discards the
particle, and the two tests $\Sigma_{\alpha},\Theta_{\beta}$
($\alpha,\beta=x,z$) the two von Neuman measurements. Now, clearly, for
$\alpha=z$ one has the marginal probability distribution $p(\Sigma_{z})=(1,0)$
and $p(\Sigma_{x})(\tfrac{1}{2},\tfrac{1}{2})$ independently of the choice of
the test $\Theta_{\beta}$. On the other hand, for the second test one has
marginal probability $p(\Theta_{x})=(1,0)$ for $\Sigma_{z}$ and
$p(\Theta_{x})=(\tfrac{1}{2},\tfrac{1}{2})$ for $\Sigma_{x}$. We conclude that
the marginal probability of $\Sigma_{\alpha}$ is independent of the choice of
the test $\Theta_{\beta}$, whereas the marginal probability of
$\Theta_{\beta}$ depends of the choice of $\Sigma_{\alpha}$. Thus, the
marginal probability of $\Theta_{\beta}$ generally depends on the choice of
$\Sigma_{\alpha}$, and this concept goes beyond the content of joint
probability, and needs the OPT graph. Theoretically, we conclude that there is
”something flying” from test $\Sigma_{\alpha}$ to test $\Theta_{\beta}$
(although we cannot see it!): this is what we theoretical describe as a
spinning particle! This well illustrates the notion of system: a theoretical
connection between tested events.

Figure 8: A simple OPT graph. A paradigmatic example for the sake of
illustration and for motivation (see text).

#### A black-box approach

Finally, the OPT is a black-box approach, where each test is described by a
mathematical object which can be ”actually achieved” by a very specific
physical device. However, nobody forbids to provide a more detailed OPT
realisation of the test, e.g. as in Fig. 9.

Figure 9:  OPT finer and coarser descriptions. The OPT is a black-box
approach. It can be made more or less detailed, as in in the box on the left,
and even at so fine a level that it is equivalent to a field-theoretical
description, as in the right figure.

Notice that although any graph can be represented in 2D (e.g. using crossing
of wires), one can more suitably design it in 3D (2D + in-out), as in Fig. 10.

|
---|---
Figure 10: An (open) DAG whose topology is suitably representable in 3D.

#### The OPT and the goal of science

One can soon realise that the OPT framework precisely allows to express the
most general goal of science, namely to connect objective facts happening (the
events), devising a theory of such ”connections” (the systems), thus allowing
making predictions for future occurrences in terms of joint probabilities of
events depending on their connections.

One of the main methodologically relevant features of the OPT is that it makes
perfectly distinct what is the ”objective datum” from what is ”a theoretical
element”. What is objective is which tests are performed, and what is the
outcome of each test. What is theoretical is the graph of connections between
the tests, along with the mathematical representations of both systems and
tests. The OPT framework dictates the rules that the mathematical description
should satisfy, and the specific OPT gives the particular mathematical
representation of systems and tests/events and of their compositions (in
sequence and in parallel) to build up the CDAG.

#### The OPT and the scientific method

One of the main rules of the scientific method is to have a clearcut
distinction between what is experimental and what is theoretical. Though this
would seem a trivial statement, such a confusion happens to be often the
source of disagreement between scientists. Though the description of the
apparatus is generally intermingled with theoretical notions, the pure
experimental datum must have a conventionally defined ”objectivity status”,
corresponding to ”openly known” information, namely shareable by any number of
different observers. Then both the theoretical language and the framework must
reflect the theory-experiment distinction, by indicating explicitly which
notions are assigned the objectivity status. Logic, with the Boolean calculus
of events, is an essential part of the language, and Probability Theory can be
regarded as an extension of logic, assigning probabilities to events. The
notion that is promoted to the objectivity status is that of ”outcome of a
test”, announcing which event of a given test has occurred. The OPT framework
thus represents an extension of probability theory, providing a theoretical
connectivity between events, the other theoretical ingredients being the
mathematical descriptions of systems and tests.

#### The OPT as a general information theory

---
Figure 11:  Equivalence between the CDAG and the quantum information circuit
or, equivalently, any run-diagram of a program.

We can immediately realise that a CDAG is exactly the same graph of a quantum
circuit as it is drawn in quantum information science. The quantum circuit, in
turn, can be interpreted as the run-diagram of a program, where each test
represents a subroutine, and the wires represent the registers through which
the subroutines communicate data. Indeed, the OPT can be regarded as the
proper framework for information science in general.

For a recent complete presentation of the OPT framework, the reader is
addressed to the the work purification ; CUPDCP  or the more recent thorough
presentation Chiribella2016 .

## Notation and abbreviations

$\mathrm{Bnd}^{+}({\mathcal{H}})$ | bounded positive operators over ${\mathcal{H}}$
---|---
$\mathrm{CP}_{\leqslant}$ | trace-non increasing completely positive map
$\mathrm{CP}_{=}$ | trace-preserving completely positive map
${\mathcal{H}}$ | Hilbert space over $\mathbb{C}$
$\mathsf{Cone}({\mathsf{S}})$ | conic hull of ${\mathsf{S}}$
$\mathsf{Cone}_{\leqslant 1}({\mathsf{S}})$ | convex hull of $\\{{\mathsf{S}}\cup{0}\\}$
$\mathsf{Conv}({\mathsf{S}})$ | convex hull of ${\mathsf{S}}$
$\mathrm{Eff}({\rm A})$ | set of effects of system ${\rm A}$
$\mathrm{Eff}_{1}({\rm A})$ | set of deterministic effects of system ${\rm A}$
$\mathrm{Mrkv}_{\leqslant}$ | normalization-non-incressing right-stochastic Markov matrices
$\mathrm{Mrkv}_{1}$ | normalization-preserving right-stochastic Markov matrices
$\mathrm{Prm}(n)$ |  $n\times n$ permutation matrices
$(\mathbb{R}^{n})^{+}_{\leqslant 1}$ |  $\\{{\bf x}\in\mathbb{R}^{n_{\rm A}}|{\bf x}\geq 0,{\bf x}\leqslant\bf 1\\}$: simplex ${\bf S}^{n_{\rm A}+1}$
$(\mathbb{R}^{n})^{+}_{1}$ |  $\\{{\bf x}\in\mathbb{R}^{n_{\rm A}}|{\bf x}\geq 0,|\\!|{\bf x}|\\!|_{1}=1\\}$: simplex ${\bf S}^{n_{\rm A}}$
$\rm{St}({\rm A})$ | set of states of system ${\rm A}$
$\rm{St}_{1}({\rm A})$ | set of deterministic states of system ${\rm A}$
$\mathrm{T}({\mathcal{H}})$ | trace-class operators over ${\mathcal{H}}$
$\mathrm{T}^{+}({\mathcal{H}})$ | trace-class positive operators over ${\mathcal{H}}$
$\mathrm{T}_{\leqslant 1}^{+}({\mathcal{H}})$ | positive sub-unit-trace operators over ${\mathcal{H}}$
$\mathrm{T}_{=1}^{+}({\mathcal{H}})$ | positive unit-trace operators over ${\mathcal{H}}$
$\mathrm{Trn}({\rm A}\to{\rm B})$ | set of transformations from system ${\rm A}$ to system ${\rm B}$
$\mathrm{Trn}_{1}({\rm A}\to{\rm B})$ | set of deterministic transformations from system ${\rm A}$ to system ${\rm B}$
${\mathbb{U}}({\mathcal{H}})$ | unitary group over ${\mathcal{H}}$
| Special cases corollaries
|  $\mathrm{T}(\mathbb{C})=\mathbb{C}$,
$\mathrm{T}^{+}(\mathbb{C})=\mathbb{R}^{+}$, $\mathrm{T}_{\leqslant
1}^{+}(\mathbb{C})=[0,1]$, $\mathrm{T}_{=1}^{+}(\mathbb{C})=\\{1\\}$
|
$\mathrm{CP}(\mathrm{T}({\mathcal{H}})\to\mathrm{T}(\mathbb{C}))=\mathrm{P}(\mathrm{T}({\mathcal{H}})\to\mathrm{T}(\mathbb{C}))=\\{\operatorname{Tr}[\cdot
E],\,E\in\mathrm{Bnd}^{+}({\mathcal{H}})\\}$
|
$\mathrm{CP}(\mathrm{T}(\mathbb{C})\to\mathrm{T}({\mathcal{H}}))=\mathrm{P}(\mathrm{T}(\mathbb{C})\to\mathrm{T}({\mathcal{H}}))=\mathrm{T}^{+}({\mathcal{H}})$
|
$\mathrm{CP}_{\leqslant}(\mathrm{T}(\mathbb{C})\to\mathrm{T}({\mathcal{H}}))\equiv\mathrm{T}^{+}_{\leqslant
1}({\mathcal{H}})$
|
$\mathrm{CP}_{\leqslant}(\mathrm{T}({\mathcal{H}})\to\mathrm{T}(\mathbb{C}))\equiv\\{\epsilon(\cdot)=\operatorname{Tr}[\cdot
E],\,0\leqslant E\leqslant I\\}$
| $\mathrm{Mrkv}_{\leqslant}(n,1)=(\mathbb{R}^{n})^{+}_{\leqslant 1}$
| $\mathrm{Mrkv}_{1}(n,1)=(\mathbb{R}^{n})^{+}_{=1}$
| Abbreviations
CT | Classical Theory
OPT | Operational Probabilistic Theory
PR boxes | Popescu-Rohrlich boxes
QT | Quantum Theory
Table 2: Notation, special-cases corollaries, and common abbreviations.

## Quantum theory

A minimal mathematical axiomatisation of Quantum Theory as an OPT is provided
in Table 3. For an OPT we need to provide the mathematical description of
systems, their composition, and transformations from one system to another.
Then all rules of compositions of transformations and their respective systems
are provided by the OPT framework. The reader who is not familiar with such
framework can simply use the intuitive construction of quantum circuits. In
Table 4 we report the main theorems following from the axioms. The reader
interested in the motivations for the present axiomatization is addressed to
Ref. DAriano2020 .

Quantum Theory
---
system | ${\rm A}$ | ${\mathcal{H}}_{\rm A}$
system composition | ${\rm A}{\rm B}$ | ${\mathcal{H}}_{{\rm A}{\rm B}}={\mathcal{H}}_{\rm A}\otimes{\mathcal{H}}_{\rm B}$
transformation | $\mathcal{T}\in\mathrm{Trn}({\rm A}\to{\rm B})$ | $\mathcal{T}\in\mathrm{CP}_{\leqslant}(\mathrm{T}({\mathcal{H}}_{\rm A})\to\mathrm{T}({\mathcal{H}}_{\rm B}))$
Born rule | $p(\mathcal{T})=\operatorname{Tr}\mathcal{T}$ | $\mathcal{T}\in\mathrm{Trn}({\rm I}\to{\rm A})$
Table 3: Mathematical axiomatisation of Quantum Theory. As given in the table,
in Quantum Theory to each system ${\rm A}$ we associate a Hilbert space over
the complex field ${\mathcal{H}}_{\rm A}$. To the composition of systems ${\rm
A}$ and ${\rm B}$ we associate the tensor product of Hilbert spaces
${\mathcal{H}}_{{\rm A}{\rm B}}={\mathcal{H}}_{{\rm
A}}\otimes{\mathcal{H}}_{{\rm B}}$. Transformations from system ${\rm A}$ to
${\rm B}$ are described by trace-nonincreasing completely positive (CP) maps
from traceclass operators on ${\mathcal{H}}_{\rm A}$ to traceclass operators
on ${\mathcal{H}}_{\rm B}$. Special cases of transformations are those with
input trivial system ${\rm I}$ corresponding to states, whose trace is the
preparation probability, the latter providing an efficient Born rule from
which one can derive all joint probabilities of any combination of
transformations. Everything else is simply special-case corollaries and one
realisation theorem: these are reported in Table 4.  Quantum theorems
---
trivial system | ${\rm I}$ | ${\mathcal{H}}_{\rm I}=\mathbb{C}$
reversible transf. | $\mathcal{U}=U\cdot U^{\dagger}$ | $U\in{\mathbb{U}}({\mathcal{H}}_{\rm A})$
determ. transformation | $\mathcal{T}\in\mathrm{Trn}_{1}({\rm A}\to{\rm B})$ | $\mathcal{T}\in\mathrm{CP}_{\leqslant 1}(\mathrm{T}({\mathcal{H}}_{\rm A})\to\mathrm{T}({\mathcal{H}}_{\rm B}))$
parallel composition |  $\mathcal{T}_{1}\in\mathrm{Trn}({\rm A}\to{\rm B})$, $\mathcal{T}_{2}\in\mathrm{Trn}({\rm C}\to{\rm D})$ | $\mathcal{T}_{1}\otimes\mathcal{T}_{2}$
sequential composition |  $\mathcal{T}_{1}\in\mathrm{Trn}({\rm A}\to{\rm B})$, $\mathcal{T}_{2}\in\mathrm{Trn}({\rm B}\to{\rm C})$ | $\mathcal{T}_{2}\mathcal{T}_{1}$
states | $\rho\in\rm{St}({\rm A})\equiv\mathrm{Trn}({\rm I}\to{\rm A})$ | $\rho\in\mathrm{T}^{+}_{\leqslant 1}({\mathcal{H}}_{\rm A})$
$\rho\in\rm{St}_{1}({\rm A})\equiv\mathrm{Trn}_{1}({\rm I}\to{\rm A})$ | $\rho\in\mathrm{T}^{+}_{=1}({\mathcal{H}}_{\rm A})$
$\rho\in\rm{St}({\rm I})\equiv\mathrm{Trn}({\rm I}\to{\rm I})$ | $\rho\in[0,1]$
$\rho\in\rm{St}_{1}({\rm I})\equiv\mathrm{Trn}({\rm I}\to{\rm I})$ | $\rho=1$
effects | $\epsilon\in\mathrm{Eff}({\rm A})\equiv\mathrm{Trn}({\rm A}\to{\rm I})$ | $\epsilon(\cdot)=\operatorname{Tr}_{\rm A}[\cdot E],\;0\leqslant E\leqslant I_{A}$
$\epsilon\in\mathrm{Eff}_{1}({\rm A})\equiv\mathrm{Trn}_{1}({\rm A}\to{\rm I})$ | $\epsilon=\operatorname{Tr}_{\rm A}$
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Table 4: Corollaries and a theorem of Quantum Theory, starting from Table 3
axiomatization. The first corollary states that the trivial system ${\rm I}$
in order to satisfy the composition rule ${\rm I}{\rm A}={\rm A}{\rm I}={\rm
A}$ must be associated to the one-dimensional Hilbert space
${\mathcal{H}}_{\rm I}=\mathbb{C}$, since it is the only Hilbert space which
trivializes the Hilbert space tensor product. The second corollary states that
the reversible transformations are the unitary ones. The third corollary
states that the deterministic transformations are the trace-preserving ones.
Then the fourth and fifth corollaries give the composition of transformations
in terms of compositions of maps. We then have four corollaries about states:
1) states are transformations starting from the trivial system and, as such,
are positive operators on the system Hilbert space, having trace bounded by
one; 2) the deterministic states correspond to unit-trace positive operator;
3) the states of trivial system are just probabilities; 4) The only trivial
system deterministic state is the number 1. We then have two corollaries for
effects, as special cases of transformation toward the trivial system: 1) the
effect is represented by the partial trace over the system Hilbert space of
the multiplication with a positive operator bounded by the identity over the
system Hilbert space; 2) the only deterministic effect is the partial trace
over the system Hilbert space. Finally, we have the realization theorem for
transformations in terms of unitary interaction $\mathcal{U}=U\cdot
U^{\dagger}$ with an environment ${\rm F}$ and a projective effect-test
$\\{\mathcal{P}_{i}\\}$ over environment ${\rm E}$, with
$\mathcal{P}_{i}=P_{i}\cdot P_{i}$, $\\{P_{i}\\}$ being a complete set of
orthogonal projectors.

## Classical theory

Classical Theory
---
system | ${\rm A}$ | $\mathbb{R}^{n_{\rm A}}$
system composition | ${\rm A}{\rm B}$ | $\mathbb{R}^{n_{{\rm A}{\rm B}}}=\mathbb{R}^{n_{\rm A}}\otimes\mathbb{R}^{n_{\rm B}}$
transformation | $\mathcal{T}\in\mathrm{Trn}({\rm A}\to{\rm B})$ | $\mathcal{T}\in\mathrm{Mrkv}_{\leqslant}(\mathbb{R}^{n_{\rm A}},\mathbb{R}^{n_{\rm B}})$
Table 5: Mathematical axiomatisation of Classical Theory. To each system ${\rm
A}$ we associate a real Euclidean space $\mathbb{R}^{n_{\rm A}}$. To
composition of systems ${\rm A}$ and ${\rm B}$ we associate the tensor product
spaces $\mathbb{R}^{n_{\rm A}}\otimes\mathbb{R}^{n_{\rm B}}$. Transformations
from system ${\rm A}$ to system ${\rm B}$ are described by substochastic
Markov matrices from the input space to the output space. Everything else are
simple special-case corollaries: these are reported in Table 6.  Classical
theorems
---
trivial system | ${\rm I}$ | ${\mathcal{H}}_{\rm I}=\mathbb{R}$
reversible transformations | $\mathcal{P}$ | $\mathcal{P}\in\mathrm{Prm}(n_{\rm A})$
transformation | $\mathcal{T}\in\mathrm{Trn}_{\leqslant}({\rm A}\to{\rm B})$ | $\mathcal{T}\in\mathrm{Mrkv}_{\leqslant}(\mathbb{R}^{n_{\rm A}},\mathbb{R}^{n_{\rm B}})$
determ. transformation | $\mathcal{T}\in\mathrm{Trn}_{1}({\rm A}\to{\rm B})$ | $\mathcal{T}\in\mathrm{Mrkv}_{1}(\mathbb{R}^{n_{\rm A}},\mathbb{R}^{n_{\rm B}})$
parallel composition |  $\mathcal{T}_{1}\in\mathrm{Trn}({\rm A}\to{\rm B})$, $\mathcal{T}_{2}\in\mathrm{Trn}({\rm C}\to{\rm D})$ | $\mathcal{T}_{1}\otimes\mathcal{T}_{2}$
sequential composition |  $\mathcal{T}_{1}\in\mathrm{Trn}({\rm A}\to{\rm B})$, $\mathcal{T}_{2}\in\mathrm{Trn}({\rm B}\to{\rm C})$ | $\mathcal{T}_{2}\mathcal{T}_{1}$
states | ${\bf x}\in\rm{St}({\rm A})\equiv\mathrm{Trn}({\rm I}\to{\rm A})$ | ${\bf x}\in(\mathbb{R}^{n_{\rm A}})^{+}_{\leqslant 1}$
${\bf x}\in\rm{St}_{1}({\rm A})\equiv\mathrm{Trn}_{1}({\rm I}\to{\rm A})$ | ${\bf x}\in(\mathbb{R}^{n_{\rm A}})^{+}_{=1}$
$p\in\rm{St}({\rm I})\equiv\mathrm{Trn}({\rm I}\to{\rm I})$ | $p\in[0,1]$
$p\in\rm{St}_{1}({\rm I})\equiv\mathrm{Trn}({\rm I}\to{\rm I})$ | $p=1$
effects | $\epsilon\in\mathrm{Eff}({\rm A})\equiv\mathrm{Trn}({\rm A}\to{\rm I})$ | $\epsilon(\cdot)=\cdot{\bf x},\;{\bf 0}\leqslant{\bf x}\leqslant{\bf 1}$
$\epsilon\in\mathrm{Eff}_{1}({\rm A})\equiv\mathrm{Trn}_{1}({\rm A}\to{\rm I})$ | $\epsilon=\cdot{\bf 1}$
Table 6: Main theorems of Classical Theory, starting from axioms in Table 5.
The first corollary states that the trivial system ${\rm I}$ in order to
satisfy the composition rule ${\rm I}{\rm A}={\rm A}{\rm I}={\rm A}$ must be
associated to the one-dimensional space $\mathbb{R}$, since it is the only
real linear space that trivialises the tensor product. The second corollary
states that the reversible transformations are the permutation matrices. The
third states that transformations are substochastic Markov matrices. The
fourth states that the deterministic transformations are stochastic Markov
matrices. Then the fifth and sixth corollaries give the composition of
transformations in terms of composition of matrices. We then have four
corollaries about states: 1) states are transformations starting from the
trivial system and, as such, are sub-normalized probability vectors (vectors
in the positive octant with sum of elements bounded by one; 2) the
deterministic states correspond to normalised probability vectors; 3) the case
of trivial output-system correspond to just probabilities; 4) The only trivial
output-system deterministic state is the number 1. We then have two
corollaries for effects, as special cases of transformation toward the trivial
system: 1) the effect is represented by scalar product with a vector with
components in the unit interval; 2) the only deterministic effect is the
scalar product with the vector with all unit components.

###### Acknowledgements.

The authors acknowledge helpful conversations and encouragement by Don
Hoffman, and interesting discussions with Chris Fields. Giacomo Mauro D’Ariano
acknowledges interesting discussions with Ramon Guevarra Erra. This work has
been sponsored by Elvia and Federico Faggin foundation through Silicon Valley
Community Foundation, Grant 2020-214365 The observer: an operational
theoretical approach. For oral sources see also the Oxford podcast
http://podcasts.ox.ac.uk/mauro-dariano-awareness-operational-theoretical-
approach.

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