Hard Problem and Free Will: an information-theoretical approach
HHard Problem and Free Will:an information-theoretical approach
Giacomo Mauro D’Ariano and Federico Faggin
We are such stuff as dreams are made on, and our little life is rounded with a sleep.
William Shakespeare
Abstract
We explore definite theoretical assertions about consciousness, startingfrom a non-reductive psycho-informational solution of David Chalmers’s hard prob-lem , based on the hypothesis that a fundamental property of "information" is itsexperience by the supporting "system". The kind of information involved in con-sciousness needs to be quantum for multiple reasons, including its intrinsic privacyand its power of building up thoughts by entangling qualia states. As a result wereach a quantum-information-based panpsychism, with classical physics superven-ing on quantum physics, quantum physics supervening on quantum information, andquantum information supervening on consciousness.We then argue that the internally experienced quantum state, since it correspondsto a definite experience–not to a random choice–must be pure, and we call it on-tic . 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 anexternal observer) which we call epistemic and is generally mixed. Purity of theontic state requires an evolution that is purity preserving, namely a so-called atomic quantum operation. The latter is generally probabilistic, and its particular outcome isinterpreted as the free will, which is unpredictable even in principle since quantumprobability cannot be interpreted as lack of knowledge. We also see how the samepurity of state and evolution allow solving the well-known combination problem ofpanpsychism.Quantum state evolution accounts for a short-term buffer of experience and containsitself quantum-to-classical and classical-to-quantum information transfers. Long
Giacomo Mauro D’ArianoDipartimento di Fisica, University of Pavia, via Bassi 6, 27100 Pavia, e-mail: [email protected]
Federico FagginFederico and Elvia Faggin Foundation 1 a r X i v : . [ qu a n t - ph ] J a n Giacomo Mauro D’Ariano and Federico Faggin term memory, on the other hand, is classical, and needs memorization and recallprocesses that are quantum-to-classical and classical-to-quantum, respectively. Suchprocesses can take advantage of multiple copies of the experienced state re-preparedwith "attention", and therefore allowing for a better quality of classical storing.Finally, we explore the possibility of experimental tests of our theory in cognitivesciences, including the evaluation of the number of qubits involved, the existence ofcomplementary 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 con-venient black-box approach of the OPT, along with its methodological robustness inseparating objective from theoretical elements, guaranteeing experimental controland falsifiability. We finally synthetically compare the mathematical postulates andtheorems 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.
In his book
The Character of Consciousness [1] David Chalmers states what he callsthe hard problem of consciousness , namely the issue of explaining our experience –sensorial, bodily, mental, and emotional, including any stream of thoughts. Chalmerscontrasts 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 problemof 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 aconscious 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 somebiological property of life [2]; 2) the
Panpsychist view –with consciousness as afundamental feature of the world that all entities have. What is proposed here is:Panpsychism with consciousness as a fundamental feature of “information”, andphysics supervening on information.The idea that physics is a manifestation of pure information processing has beenstrongly advocated by John Wheeler [3] and Richard Feynman [4, 5], along with ard Problem and Free Will 3 several other authors, among which David Finkelstein [6], who was particularly fondof this idea [7]. Only quite recently, however, the new informational paradigm forphysics has been concretely established. This program achieved: 1) the derivationof quantum theory as an information theory [8, 9, 10], and 2) the derivation offree quantum field theory as emergent from the nontrivial quantum algorithm withdenumerable systems with minimal algorithmic complexity [11, 12]. In additionto such methodological value, the new information-theoretic derivation of quantumfield theory is particularly promising for establishing a theoretical framework forquantum gravity as emergent from the quantum information processing, as alsosuggested by the role played by information in the holographic principle [14, 15]. Insynthesis: the physical world emerges from an underlying algorithm, and the kind ofinformation that is processed beneath is quantum.The idea that quantum theory (QT) could be regarded as an information theory isa relatively recent one [16], and originated within the field of quantum information [17]. Meanwhile what we name “information theory” has largely evolved, from itsorigins as a communication theory [18], toward a general theory of “processing” ofinformation, 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 con-nected with quantum information), the theoretical framework for all informationtheories emerged in the physics literature in terms of the notion of
OperationalProbabilistic Theory (OPT) [9, 10, 19]. an isomorphic framework that emergedwithin computer science in terms of
Category Theory [20, 21, 22, 23, 24, 25].Indeed, the mathematical framework of an information theory is precisely that ofthe OPT, whichever information theory we consider–either classical, quantum, or"post-quantum". The main structure of OPT is reviewed in the Appendix. Among the information theories, classical theory (CT) plays a special role. Infact, besides being itself an OPT, CT enters the operational framework in termsof objective outcomes of the theory, which for causal OPTs (as QT and CT) canbe 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 givenoutcome can be regarded as a quantum-to-classical information exchange, whereasconditioning constitutes a classical-to-quantum information exchange. We concludethat we should regard the physical world faithfully ruled by both quantum andclassical theories together, with information transforming between the two types. The literature on the informational derivation of free quantum field theory is extensive, and,although not up to date, we suggest the review [7] written by one of the authors in memoriamof David Finkelstein. The algorithmic paradigm has opened for the first time the possibility ofavoiding physical primitives in the axioms of the physical theory, allowing a re-foundation of thewhole physics over logically solid grounds [13]. The reader who is not familiar with the notion of OPT can regard the OPT as the mathematicalformalization of the rules for building quantum information circuits. For a general idea about OPTit is recommended to read the appendix. The reader is supposed to be familiar with elementarynotions, such as state and transformation with finite dimensions. Giacomo Mauro D’Ariano and Federico Faggin
This theoretical description of reality should be contrasted with the usual viewof reality as being quantum, creating a fallacy of misplaced concreteness. The mostpragmatic point of view is to regard QT and CT together as the correct informationtheory to describe reality, without incurring any logical paradox. We will use thisidea 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 informationand physics supervening on quantum information:
Quantum-Information Panpsy-chism .In place of QT, one may consider a post-quantum
OPT–e.g. RQT (QT on realHilbert spaces), FQT (Fermionic QT), PRB (an OPT build on Popescu-Rohlich boxes[26]), etc. [10], yet some of the features of the present consciousness theory can betranslated into the other OPT, as long as the notion of “entanglement” survives inthe considered OPT.
The fundamental nature of the solution to the hard problem proposed here has beensuggested by David Chalmers as satisfying the following requirements [1]Chalm : Consciousness as fundamental entity, not explained in terms of anythingsimpler...
Chalm : ... a non reductive theory of experience will specify basic principles thattell us how basic experience depends on physical features of the world. Chalm : 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 infor-mation our principle will be psychoinformational (see Chalm ). The non-reductive(Chalm ) psycho-informational principle that is proposed here is the following:P1: psychoinformational principle: Consciousness is the information-system’s expe-rience of its own information state and processing.
As we will see soon, it is crucial that the kind of information that is directlyexperienced be quantum.Principle P1 may seem ad hoc , but the same happens with the introduction ofany fundamental quantity (Chalm , ) in physics, e.g. the notions of inertial mass , electric charge , etc. Principle P1 asserts that experience is a fundamental featureof information , hence also of physics, which supervenes on it. P1 is not reductive(Chalm ), and it does not affect physics (Chalm ), since the kind of informationinvolved in physics is quantum+classical. On the other hand, P1 supplements physics(Chalm ), since the latter supervenes on information theory. ard Problem and Free Will 5 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 informationon those objects: colour, position, shape,... Information is supported by a successionof systems: the light modes, followed by the retina, then the optical nerves, andfinally 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 inspirecrucial experiments. OPT has the power of being a black-box method that does notneed the detailed "physical" specification of the systems, and this is a great advan-tage! And, indeed, our method tackles the problem in terms of a pure in-principlereasonings, independent of the "hardware" supporting information, exactly as we doin information theory. Such hardware independence makes the approach particularlysuited 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 com-patible with a theory of consciousness. Our experience is indeed not sharable: thisis a fact. We hypothesize that that non shareability of experience holds in principle ,not just as a technological limitation. A crucial fact is that information shareability is equivalent to information readability with no disturbance . Recently it has beenproved that the only theory where any information can be extracted without dis-turbance is classical information theory [28]. We conclude that P2 implies that atheory of consciousness needs nonclassical information theory, namely QT, or elsea “post-quantum” OPT.Here we will consider the best known instance of OPT, namely QT. As we willsee, such a choice of theory turns out to be very powerful in accounting for all themain features of consciousness. We state this choice of theory as a principle:P2’: Quantumness of experience:
The information theory of consciousness is quan-tum theory.
We remind that classical theory is always accompanying any OPT, thus there will beexchanges and conversions of classical and quantum information.We now introduce a third principle: It may be possible to know which systems are involved in a particular experience, as considered inRef. [27]. However, we could never know the experience itself, since it corresponds to non sharablequantum information. In 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 readinginformation without disturbing it . Indeed, if one could read information without disturbing it, hecould read the information as many times as needed in order to acquire all its different comple-mentary sides, technically performing a tomography of the information. And once he knows all theinformation he can prepare copies of it at will. Viceversa, if one could clone quantum information,he could read clones keeping the original untouched. Giacomo Mauro D’Ariano and Federico Faggin
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 agreeon, something we can be sure of. It is definite , in the sense that one has only a singleexperience at a time–not a probability distribution of experiences. The latter woulddescribe the knowledge of somebody else’s experience . Consider that the experienceof a blurred coin is just as definite as the experience of a sharp coin since they aretwo 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 berepresented in probabilistic terms, e.g. by the mixture-state TAIL+ HEAD, basedon a fair-coin hypothesis (see Fig. 1). We thus can understand that by the samedefinition the notion of mixture is an epistemic one, based on our prior knowledgeabout the coin having two possible states. In synthesis:1.
Experience is described by an ontic quantum state , which is pure .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 . Ontic State
Experienced by the system
Epistemic State I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) p ( | "i ⌦ |•i + | ) = p ( |"i + | ) = + I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) p ( | "i ⌦ |•i + | ) = p ( |"i + | ) = + + Predicted by an outside observer
Fig. 1
Illustration of the notions of ontic and epistemic states for a system given by a classicalbit, here represented by a coin, with states 0=HEAD or 1=TAIL. We call ontic the “actual” stateof the system, which is pure and generally unknown, except as an unsharable experience (in thefigure it is the coin state HEAD covered by the hand). We call epistemic the state that represents theknowledge about the system of an outside observer, e.g. the state of the unbiased coin correspondsto HEAD+ 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 consciousexperience. Their existence has so far been inexplicable in the traditional scientificframework. Their privacy and ineffability are notorious. Qualia are structured : some ard Problem and Free Will 7 of them are more fundamental than others, e.g. spectrum colours, sound pitch, the fivedifferent kinds of taste buds (sweet, salty, sour, bitter, and umami), somatic sensations(pain, pleasure, ...), basic emotions (sadness, happiness, ...). Qualia are compositionalwith 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 fruitionof a very structured kind of quantum information (we are the system supporting suchfinal information), then, according to principle P1, qualia correspond to informationstates “felt” by the aware system, and according to principles P2 and P2’ such statesare 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 organizingnontrivial qualia. The qualia space thus would be the Hilbert space of the systemsinvolved in the quale.Classical information coming from the environment through the senses is ul-timately converted into quantum information which is experienced as qualia. Theinverse quantum-to-classical transformation is also crucial in converting structureswithin qualia into logical and geometric representations expressible with classicalinformation to communicate free-will choices to classical structures. For such pur-poses, the interaction with the memory of past experiences may be essential, for, aswe 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 manifestationsof quantum information. Since the manifestation is classical, the possible choices ofthe conscious agent are in principle jointly perfectly discriminable . We then state what we mean by free .S2: Will is free if its unpredictability by an external observer cannot be interpretedin terms of lack of knowledge. In particular, statement S2 implies that the actual choice of the entity exercising itsfree 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 cannotbe interpreted as such. In the usual classical information theory the convex set of states is a simplex, and its extremalstates are jointly discriminable. Here there is a clear distinction between the "willing agent" and the observers of it. The word unpredictability applies only to the observers. The impossibility of interpretation of quantum randomness as lack on knowledge fits quantumcomplementarity. Indeed we cannot by any means know both values of two complementary vari-ables. One may argue that both values may exist anyway, even if we cannot know them, but suchargument 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 Giacomo Mauro D’Ariano and Federico Faggin
According to Statement S1 qualia are described by ontic quantum states and, beingsuch states pure , we can represent them by normalised vectors | 𝜓 (cid:105) ∈ H A in theHilbert space H A of system A. For the sake of illustration we consider two simplecases of qualia: direction and colour . These examples should not be taken literally, but only for the sake of illustration of the concept that the linear combinations ofqualia can give rise to new complex qualia. Fig. 2
Illustration of how qualia superpose to make new qualia.
Left figure: the Bloch sphereillustrates 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 samecombinations 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 | + | 𝑏 | =
1) we get the qualecorresponding to a generic direction, corresponding to a generic pure state | 𝜓 (cid:105) . Right figure: summations of two of the three colour basis red, green , and blue (RGB) and subtraction of two ofthe three colours cyan, magenta and yellow (CMY). As explained in the main text, these examplesare only for the sake of illustration of the notion of linear combination of quantum ontic states tomake new ontic states corresponding to new qualia. A third case of superposition is that of soundswith precise frequency that combine with addition and subtraction into timbers and chords. Noticethat 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 anentangled way produces new kinds of qualia. Indeed, consider the superposition of a distribution for the values of complementary observables, assumption that obviously is violated byquantum complementarity. Others can argue (and this is the most popular argument) that one mustuse different local random variables depending on remote settings, which leads to the interpretationof the CHSH correlations in terms of nonlocality : such interpretation, however, is artificial, whereasthe most natural one iw that the measurement outcome is created by the measurement . We emphasize that the above examples are only for the sake of illustration of concepts. Theexample about color qualia in Fig. 2 would be a faithful one if the colours were monochromatic andthe summation or subtraction were made with wave amplitude, not intensity, which is the actualcase. The case of direction qualia based on the Bloch sphere is literally correct. The directions arethose of the state representations on the Block sphere.ard Problem and Free Will 9 red up-arrow with a green down-arrow. This is not a yellow right-arrow as one mayexpect, since the latter corresponds to the independent superposition of directionand colour, as in the following equation: I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) . I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) (1)We would have instead I ( X : Y ) : = H ( p XY || p X p Y ) = Â i X , j Y p ij l og p ij p i p j p X : = { p i } i X p ij : = r i A a j r X = : Â i X r i , p i : = T r r i c ( { r i } i X ) : = S ( r X ) Â i X ( T r r i ) S ⇣ r i T r r i ⌘ S ( r ) : = T r ( r l og r ) S ( r k s ) : = T r [ r ( l og r l og s ) ] p ( | "i ⌦ |•i + | ) = | !i ⌦ |•i = ( | "i + | ) ⌦ ( |•i + |•i ) p ( | "i ⌦ |•i + | ) = p ( |"i + | ) = ↵⇣ p ⌘ N [ . . . ⌦ ( | a i ⌦ | b i ± | a i ⌦ | b i ) ⌦ ( | "i ⌦ |•i ± | ) ⌦ ( | r i ⌦ | s i ± i | r i ⌦ | s i ) ⌦ . . . ] µ [ . . . ⌦ ( | a i ⌦ | b i ⌦ | c i ± | a i ⌦ | b i ⌦ | c i ) ⌦ ( | "i ⌦ |•i ± | ) ⌦ ( | r i ⌦ | s i ± i | r i ⌦ | s i ) ⌦ . . . ] , (2)where the ket with the blue star represents a completely new qualia. In a more generalcase, we have a state vector with triple or quadruple or more entanglement in a factorof a tensor product, and more generally, every system is entangled, e.g. . . . ⊗ (| 𝑎 (cid:105) ⊗ | 𝑏 (cid:105) ⊗ | 𝑐 (cid:105) ⊗ | 𝑑 (cid:105) ± | 𝑎 (cid:105) ⊗ | 𝑏 (cid:105) ⊗ | 𝑐 (cid:105) ⊗ | 𝑑 (cid:105)) ⊗ (| ↑(cid:105) ⊗ |•(cid:105) ± | ↓(cid:105) ⊗ |•(cid:105)) ⊗ . . . (3)We can realise how in this fashion one can achieve new kinds of qualia whose numbergrows exponentially with the number of systems. In fact, the number of differentways of entangling 𝑁 systems corresponds to the number of partitions of 𝑁 intointegers, multiplied by the number of permutations of the systems, and therefore itgrows as 𝑁 ! 𝑒 √ 𝑁 / , and this without considering the variable vectors that can beentangled!Since qualia correspond to pure states of the conscious systems, their Hilbert spacecoincides with the multidimensional Hilbert space of the system. Experimentally onemay be able to locate the systems in terms of neural patterns, but will never be ableto read the encoded information without destroying the person’s experience, whileat best gaining only a single complementary side of the qualia, out of exponentiallymany. In synthesis: • System identification is possible, but not the "experience" within.The fact that it is possible to identify the information system proves that theidentity of the observer/agent is public and is thus correlated with its "sense ofself," which is instead private. This is a crucial requirement for a unified theoryof consciousness and free will, namely that the observer/agent be identifiable bothprivately–from within and through qualia–and publicly–from without and throughinformation. Within QT (or post-quantum OPTs) this is possible.
Since the quantum state of a conscious system must be pure at all times, the onlyway to guarantee that the evolving system state remains pure is that the evolutionitself is pure (technically it is atomic , namely its CP-map T = (cid:205) 𝑖 𝑇 𝑖 · 𝑇 † 𝑖 has a singleKrauss operator 𝑇 𝑖 ). We remind that both states and effects are also transformations from trivial to non trivial systems and viceversa, respectively. In Table 1 we reportthe theoretical representations of the three kinds of ontic transformations, includingthe special cases of state and effect. Circuit symbol Symbol Map Operator domain-codomaintransformation A T B T T = 𝑇 · 𝑇 † 𝑇 Bnd (H A → H B ) Effect A 𝛼 ( 𝛼 | Tr [·| 𝛼 (cid:105) (cid:104) 𝛼 |] (cid:104) 𝛼 | Bnd (H A → C ) State 𝜓 A | 𝜓 ) | 𝜓 (cid:105) (cid:104) 𝜓 | | 𝜓 (cid:105) Bnd ( C → H A ) Table 1
Notations for the three kinds of ontic transformations (for the meaning of symbols seeTable 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 anontic state 𝜔 𝑡 at time 𝑡 evolved by the one-step ontic transformation O ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 withoutcome 𝐹 𝑡 , depending on classical input 𝑥 𝑡 from senses and memory 𝜔 𝑡 A 𝑡 O ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 A 𝑡 + = : 𝜔 𝑡 + 𝑡 + . (4)The epistemic transformation would be the sum of all ontic ones corresponding toall possible outcomes: E ( 𝑡,𝑥 𝑡 ) : = ∑︁ 𝐹 𝑡 O ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 . (5)The outcome 𝐹 𝑡 is a classical output, and we identify it with the free will of theexperiencing system.It is a probabilistic outcome that depends on the previous history of qualia of thesystem. Its kind of randomness is quantum, which means that it cannot be interpretedas lack of knowledge , and, as such, it is free . Notice that both mathematically andliterally the free will is the outcome of a transformation that corresponds to a changeof experience of the observer/agent. The information conversion from quantum toclassical can also take into account a stage of "knowledge of the will" correspondingto "intention/purpose", namely "understanding" of which action is taken.We may need to provide a more refined representation of the one-step ontictransformation of the evolution in terms of a quantum circuit, for example: ABC O ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 M = A 𝛼 𝑖 𝜓 ( 𝑖 ) D R ( 𝑘 ) G A ( 𝑥 𝑡 ) MB E 𝑘 E H B N Λ ( 𝑙 ) 𝑗 C F V 𝑙 L C O (6) As we will see, memory is classical.ard Problem and Free Will 11
Generally for each time 𝑡 we have a different circuit. In the example in circuit (6) wesee that a single step can contain also states and effects, and the output system ofthe whole circuit (M in the present case) is generally different (not even isomorphic)to the input ones ABC. Following the convention used for the ontic transformation O ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 the lower index is a random outcome, and the upper index ( 𝑡, 𝑥 𝑡 ) is aparameter from which the transformation generally depends. Overall in circuit (6)we have free will 𝐹 𝑡 = { 𝑖, 𝑘, 𝑙, 𝑗 } , whereas the transformations 𝜓 ( 𝑖 ) , R ( 𝑘 ) , B , and C are deterministic (they have no lower index), and do not contribute to the free will,and the same for the transformation A ( 𝑥 𝑡 ) , which depends on the sensorial input 𝑥 𝑡 .In circuit (6) we can see that we have also classical information at work , since e.g.the transformation R ( 𝑘 ) depends on the outcome 𝑘 of the transformation E 𝑘 , andsimilarly the choice of test { Λ ( 𝑙 ) 𝑗 } depends on the outcome 𝑙 of V 𝑙 , and similarlythe state 𝜓 ( 𝑖 ) depends on the effect outcome 𝛼 𝑖 . Each element of the circuit is ontic,i.e. atomic, and atomicity of sequential and parallel composition guarantee that thewhole transformation is itself atomic. This means that its Krauss operator 𝑂 𝑡 can bewritten as the product of the Krauss operators of the component transformations asfollows 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 = ( 𝐼 𝑀 ⊗ (cid:104) Λ ( 𝑙 ) 𝑗 |) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) MNO → M ( 𝐴 ( 𝑥 𝑡 ) ⊗ 𝐵 ⊗ 𝐶 ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) GHL → MNO ( 𝑅 ( 𝑘 ) ⊗ 𝑉 𝑙 ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) DEF → GHL (| 𝜓 𝑖 (cid:105)(cid:104) 𝑎 𝑖 | ⊗ 𝐸 𝑘 ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ABC → DEF , (7)where O ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 = 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 · 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 † . Notice that in the expression for the operator 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 in Eq. (7) the operators from the input to the output are written from theright to the left–the way we compose operators on Hilbert spaces. Moreover, theexpression (7) of 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 is not unique, since it depends on the choice of foliation ofthe circuit, namely the way you cover all wires with leaves to divide the circuit intoinput-output sections. For example, Eq. (7) would correspond to the foliation in Fig.3 r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. One can realise how in this fashion one can achieve a number of new kinds of qualia that isexponential with the number of systems. In fact, the number of different ways of entangling N systems corresponds to the number of partitions of the N into integers times the permutations ofthe systems, and this will grows as N ! e p N/ , and this without considering the different vectorsthat are entangled. In this ways thoughts correspond to pure states in the tensor product of thetensor-product Hilbert space of the involved systems. In short, such Hilbert space describes themultidimensional qualia space. Experimentally one may be able to locate the systems in terms ofa neural patterns, but not to read the information encoded on them, if not destroying the person’sthought for just gaining only one of the exponentially many complementary sides of the thought.
4. Consciousness Evolution and Free Will
Since the quantum state of consciousness must be pure at all times, the only way to guaranteethat the evolved consciousness state remain pure is that the evolution is atomic , namely thecorresponding CP-map has a single Krauss operator. We remind that both states and effects arealso transformations, from trivial to non trivial system and viceversa, respectively. In Table 1 wereport the theoretical representations of the three kinds of ontic transformations: transformation,effect, and state. Circuit symbol Symbol Operator map Operator domain-codomaintransformation A T B T T = T · T † T Bnd( H A ! H B ) Effect A ↵ ( ↵ | Tr[ ·| ↵ ih ↵ | ] h ↵ | Bnd( H A ! C ) State A | ) | ih | | i Bnd( C ! H A ) Table 1.
Notations for the three kinds of ontic transformations.
Let’s consider now the general scenario of the conscious composite system in an ontic state ! t at time t evolved by the one-step ontic transformation O ( t,x t ) F t with outcome F t , depending onclassical input from senses x t ! t A t O ( t,x t ) F t A t +1 =: ! t +1 A t +1 (4.1)The epistemic transformation would be the sum over all possible outcomes of the ontic one: E ( t,x t ) := X F t O ( t,x t ) F t . (4.2)We may need to provide a more refined representation of the one-step ontic transformation of theevolution in terms of a quantum circuit, for example: ABC O ( t,x t ) F t M = A ↵ i ( i ) D R ( k ) G A ( x t ) MB E k E H B N ⇤ ( l ) j C F V l L C O . (4.3)Generally for each time t we have a different circuit. We see that in the example in Eq. 4.3the circuit describing a single step can contain also states and effects, and the output systemsof the whole circuit are generally different (not even isomorphic) to the input ones. Followingthe convention used for the ontic transformation O ( t,x t ) F t the lower index is a random outcomeand the upper index is a parameter from which the transformation depends. Overall in Eq. 4.3 Fig. 3
Quantum circuit foliation corresponding to Eq. (7)
A different foliation, for example, is the one reported in Fig. 4, corresponding to theexpression for 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 The outcome is random for an observer other than the conscious system, for which, instead, it isprecisely known.2 Giacomo Mauro D’Ariano and Federico Faggin 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 = ( 𝐼 𝐶 ⊗ (cid:104) Λ ( 𝑙 ) 𝑗 |) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) MNO → M ( 𝐴 ( 𝑥 𝑡 ) ⊗ 𝐵 ⊗ 𝐼 O ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) GHO → MNO ( 𝑅 ( 𝑘 ) ⊗ 𝐶𝑉 𝑙 ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) DEF → GHO (| 𝜓 𝑖 (cid:105) ⊗ 𝐸 𝑘 ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) BC → DEF ((cid:104) 𝑎 𝑖 | ⊗ 𝐼 BC ) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) ABC → BC . (8) r s t a . r o y a l s o c i e t y pub li s h i ng . o r g P h il . T r an s . R . S o c . A .................................................................. One can realise how in this fashion one can achieve a number of new kinds of qualia that isexponential with the number of systems. In fact, the number of different ways of entangling N systems corresponds to the number of partitions of the N into integers times the permutations ofthe systems, and this will grows as N ! e p N/ , and this without considering the different vectorsthat are entangled. In this ways thoughts correspond to pure states in the tensor product of thetensor-product Hilbert space of the involved systems. In short, such Hilbert space describes themultidimensional qualia space. Experimentally one may be able to locate the systems in terms ofa neural patterns, but not to read the information encoded on them, if not destroying the person’sthought for just gaining only one of the exponentially many complementary sides of the thought.
4. Consciousness Evolution and Free Will
Since the quantum state of consciousness must be pure at all times, the only way to guaranteethat the evolved consciousness state remain pure is that the evolution is atomic , namely thecorresponding CP-map has a single Krauss operator. We remind that both states and effects arealso transformations, from trivial to non trivial system and viceversa, respectively. In Table 1 wereport the theoretical representations of the three kinds of ontic transformations: transformation,effect, and state. Circuit symbol Symbol Operator map Operator domain-codomaintransformation A T B T T = T · T † T Bnd( H A ! H B ) Effect A ↵ ( ↵ | Tr[ ·| ↵ ih ↵ | ] h ↵ | Bnd( H A ! C ) State A | ) | ih | | i Bnd( C ! H A ) Table 1.
Notations for the three kinds of ontic transformations.
Let’s consider now the general scenario of the conscious composite system in an ontic state ! t at time t evolved by the one-step ontic transformation O ( t,x t ) F t with outcome F t , depending onclassical input from senses x t ! t A t O ( t,x t ) F t A t +1 =: ! t +1 A t +1 (4.1)The epistemic transformation would be the sum over all possible outcomes of the ontic one: E ( t,x t ) := X F t O ( t,x t ) F t . (4.2)We may need to provide a more refined representation of the one-step ontic transformation of theevolution in terms of a quantum circuit, for example: ABC O ( t,x t ) F t M = A ↵ i ( i ) D R ( k ) G A ( x t ) MB E k E H B N ⇤ ( l ) j C F V l L C O . (4.3)Generally for each time t we have a different circuit. We see that in the example in Eq. 4.3the circuit describing a single step can contain also states and effects, and the output systemsof the whole circuit are generally different (not even isomorphic) to the input ones. Followingthe convention used for the ontic transformation O ( t,x t ) F t the lower index is a random outcomeand the upper index is a parameter from which the transformation depends. Overall in Eq. 4.3 Fig. 4
Quantum circuit foliation corresponding to the expression for 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 Eq. (8)
We remind that all operators (including kets and bras as operators from and tothe trivial Hilbert space 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 itselfa contraction. Contractivity for operator 𝑋 can be conveniently expressed as 𝑋 ∈ Bnd (H 𝑖𝑛 → H 𝑜𝑢𝑡 ) , 𝑋 † 𝑋 (cid:54) 𝑃 𝑆 𝑋 (cid:54) 𝐼 𝑖𝑛 , 𝑃 𝑆 𝑋 : = Proj Supp 𝑋. (9)Now, since the evolution of consciousness must be atomic at all times, we can writea whole consciousness history as the product of the Krauss operators 𝑂 𝑡 ≡ 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 at all previous times 𝑡 Ω 𝑡 : = 𝑂 𝑡 𝑂 𝑡 − . . . 𝑂 , 𝑂 𝑡 : = 𝑂 ( 𝑡,𝑥 𝑡 ) 𝐹 𝑡 , (10)and apply the history operator to the wavevector of the initial ontic state-vector | 𝜔 (cid:105)| 𝜔 𝑡 (cid:105) = Ω 𝑡 | 𝜔 (cid:105) . (11)The squared norm || 𝜔 𝑡 || of vector 𝜔 𝑡 is the probability of the full history of consciousstates { 𝜔 , 𝜔 , 𝜔 , . . . , 𝜔 𝑡 } and equally of the free-will history { 𝐹 , 𝐹 , . . . , 𝐹 𝑡 } 𝑝 ( 𝜔 , 𝜔 , 𝜔 , . . . , 𝜔 𝑡 ) = 𝑝 (∅ , 𝐹 , . . . , 𝐹 𝑡 ) = || 𝜔 𝑡 || = || Ω 𝑡 𝜔 || . (12) The ontic evolution of the consciousness state, though it maintains coherence, cankeep very limited quantum memory of experience. The latter, due to contractivityof 𝑂 , will go down very fast as || Ω 𝑡 || (cid:39) || 𝑂 || − 𝑡 , i.e. it will decrease exponentiallywith the number of time-steps. The quantum memory intrinsic in the ontic evolutioninstead works well as a short-term buffer to build up a fuller experience–e.g. of a ard Problem and Free Will 13 landscape, or of a detailed object, or even to detect motion. How many qubits willmake such a single-step buffer? The answer is: not so many. Indeed, if we openour eyes for just a second to look at an unknown scene, and thereafter we are askedto answer binary questions, we would get only a dozen of right answers better thanchance. Tor Nørretranders writes [31]:
The bandwidth of consciousness is far lowerthan the bandwidth of our sensory perceptors. ...
Consciousness consists of discardedinformation far more than information present. There is hardly information left inour consciousness. We reasonably deduce that there is actually no room for long-term memory inconsciousness, and we conclude that:S3: Memory is classical. Only the short-term buffer to collect each experience isquantum.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 informa-tion from quantum to classical in the memorization process, and from classical toquantum in the recollection process. The first process, namely transferring quantumexperience to classical memory, must be necessarily incomplete, otherwise it wouldviolate the quantum no-cloning theorem. The Holevo theorem [33] establishes thatthe maximal amount of classical information that can be extracted from a quantumsystem is a number of bits equal to the number of qubits that constitute the quantumsystem. Obviously, such maximal classical information is infinitesimal compared to For example, the amount of visual information is significantly degraded as it passes from the eye tothe visual cortex. Marcus E. Raichle says [29]:
Of the virtually unlimited information available fromthe environment only about bits/sec are deposited in the retina. Because of a limited numberof axons in the optic nerves (approximately 1 million axons in each) only ∗ bits/sec leave theretina and only make it to layer IV of V1 [30, 31]. These data clearly leave the impression thatvisual cortex receives an impoverished representation of the world, a subject of more than passinginterest to those interested in the processing of visual information [32]. Parenthetically, it shouldbe noted that estimates of the bandwidth of conscious awareness itself (i.e. what we "see") are inthe range of 100 bits/sec or less [30, 31]. We believe that the inability to recall much information contained in one second of visualexperience, when the actual experience is felt to be quite rich, should not be construed to diminishthe importance of consciousness. In fact, experience is quantum while memory is classical, andalthough not much classical information appears to have been memorized, the actual experience hasthe cardinality of the continuum in Hilbert space. Consciousness is about living the experience in itsunfolding and understanding what is happening, so to make the appropriate free-will decisions whennecessary. Recalling specific information in full detail is unnecessary. Consciousness is focusedon the crucial task of getting the relevant meaning contained in the flow of experience. The scarceconscious memory of specific objects and relationships between objects should not be an indicationthat consciousness is a "low bandwidth" phenomenon, but that what is relevant to consciousnessmay not be what the interrogator believes should be relevant.4 Giacomo Mauro D’Ariano and Federico Faggin the continuum of classical information needed to communicate classically a quantumstate!From a single measurement one can extract classical information about just oneof the continuum of complementary aspects of the quantum state, e.g. along a givendirection for a spin. Moreover, the state after the measurement would be necessarilydisturbed , due to the information-disturbance tradeoff . So, which measurementshould one perform for collecting the best classical information from a quantumsystem?The answer is easy for a single qubit. Just perform the usual von Neumann mea-surement along a random direction! This has been proved in Ref. [35]. For largerdimensions it is in principle possible to generalise this result, however, the probabil-ity space turns out to be geometrically much more complicated than a sphere, anda scheme for random choice is unknown. For this reason we can consider the largerclass of observation tests called informationally complete ("infocomplete"), which,unlike the case of the optimal measurement of spin direction, can be taken as discreteand with a finite number of outcomes. Such a measurement would correspond to thefollowing epistemic transformation For example, for the spin originally oriented horizontally and measured vertically a la vonNeumann, the final state would be vertical up or down, depending on the measurement outcome. There are many ways of regarding the information-disturbance tradeoff, depending on the specificcontext and the resulting definitions of information and disturbance . The present case of atomicmeasurements with the "disturbance" defined in terms of the probability of reversing the measure-ment transformation has been analysed in the first part of Ref. [34]. In the same reference it isalso shown that a reversal of the measurement would provide a contradicting information whichnumerically cancels the information achieved from the original measurement, thus respecting thequantum principle of no information without disturbance. Let’s consider the case of a single qubit realised with a particle spin. The usual observable ofvon 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 weprepare 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 measurethe spin along a parallel direction we find the spin always pointing north-east! So the spin isindeed (ontically!) pointing north-east along the same diagonal! Now it would be a legitimatequestion 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 isusually called POVM, the acronym of Positive Operator Valued Measure). However, it turns outthat the measured direction is a fake, since such a quantum measurement with a continuous setof outcomes is realised as a continuous random choice of a von Neumann measurement [35]. Inconclusion the optimal measurement of the spin direction is realised by a customary Stern-Gerlachexperiment in which the magnetic field is randomly oriented! The same method can be used toachieve an informationally complete measurement to perform a quantum tomography [36] of thestate, by suitably processing the outcome depending on the orientation of the spin measurement. The convex set of deterministic states for a qutrit (i.e. 𝑑 = What in OPT language is called observation test is the same of what in the quantum informationliterature is called discrete POVM. An observation test is infocomplete for system A when it spansthe linear space of effects Eff R ( A ) .ard Problem and Free Will 15 M ∈
Trn ( A → B ) , M = ∑︁ 𝑗 ∈ 𝐽 𝑀 𝑗 𝜌𝑀 † 𝑗 , | 𝐽 | ≥ dim (H A ) , (13)which is the coarse graining of the infocomplete quantum test {M 𝑗 } 𝑗 ∈ 𝐽 , with M 𝑗 = 𝑀 𝑗 · 𝑀 † 𝑗 where {| 𝑀 | 𝑗 } 𝑗 ∈ 𝐽 is an infocomplete observation test . "Infocomplete" meansthat, in the limit of infinitely many usages over the same reprepared state, themeasurement allows to perform a full tomography [36] of the state.The infocomplete measurement (or the random observables seen before) is bet-ter suited to extract classical information from the quantum buffer for long-termmemory, since is does not privilege a particular observable. It is also likely that theconscious act of memorising an experience may be achieved by actually repreparingthe ontic state multiple times in the quantum buffer , and performing the infocom-plete test multiple times, thus with the possibility of memorising a (possibly partial)tomography of the state. Notice that, generally, the ontic state 𝑀 𝑗 𝜌𝑀 † 𝑗 for a given 𝑗 still depends on 𝜌 . However, the test will disturb it, and the less disturbance, thesmaller the amount of classical information that can be extracted [34]. Transferring classical memory to quantum experience
We have seen that an infocomplete measurement is needed to (approximately) storean experience in the classical long-term memory. By definition, the recovery ofan experience requires a reproduction of it, meaning that the corresponding onticstate is (approximately) reprepared from the classical stored data. The memory,being classical, will be read without disturbance, thus left available to followingrecollections. In order to transfer classical to quantum information, methods of state-preparation have been proposed in terms of quantum circuit schemes [38].A possible benchmark for the memory store-and-recall process is the maximalfidelity achievable in principle with a measure-and-reprepare scheme that optimisesthe fidelity between the experienced state and the recalled one. We will consider 𝑀 ≥ attention to the experience. Theoptimal fidelity between the experienced state and the recalled one averaged overall possible experiences (i.e. input states) is given by [39] 𝐹 ( 𝐴, 𝑀, 𝑑 ) = 𝑀 + 𝑀 + 𝑑 , (15) Particularly symmetric types of measurements are those made with a SIC POVM (Symmetricinformationally complete POVM) [37], where 𝑀 𝑗 = | 𝜓 𝑗 (cid:105) (cid:104) 𝜓 𝑗 | are 𝑑 projectors on pure stateswith equal pairwise fidelity | (cid:104) 𝜓 𝑗 | 𝜓 𝑘 (cid:105) | = 𝑑 𝛿 𝑗𝑘 + 𝑑 + . (14)The projectors | 𝜓 𝑗 (cid:105) (cid:104) 𝜓 𝑗 | defining the SIC POVM in dimension 𝑑 form a ( 𝑑 − ) -dimensionalregular simplex in the space of Hermitian operators. The fidelity 𝐹 between two pure states corresponding to state vectors | 𝜓 (cid:105) and | 𝜑 (cid:105) , respectively,is defined as | (cid:104) 𝜑 | 𝜓 (cid:105) | .6 Giacomo Mauro D’Ariano and Federico Faggin where 𝑑 is the dimension of the experiencing system. Such upper bound for fidelity may be used to infer an effective dimension 𝑑 of thesystem involved in consciousness, e.g. in very focused restricted experiences as thoseinvolved in masking conscious perception [40]. Information transfer from body to consciousness and vice versa
We would expect that most of the operation of the human body is automatic anduses classical information that is never translated into quantum information to beexperienced by consciousness. The portion of the classical information producedby the body that needs to be converted into quantum should only be the salientinformation that supports the qualia perception and comprehension necessary to "livelife" and make appropriate free-will choices. The amount of quantum information tobe translated into classical for the purpose of free-will control of the body top-downshould be relatively small.
The combination problem concerns the issue about if and how the fundamentalconscious minds come to compose, constitute, or give rise to some other, additionalconscious mind [41]. By definition, the problem becomes crucial for panpsychism:if cosciousness is everywhere, what is the criterion to select novel conscious in-dividuals? 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 presenttheoretical approach provides a precise individuation criterion. The criterion derivesfrom 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 experi-ence . Such a statement may be immediately obvious to some readers. However, forthose who may not agree, we propose a thought experiment. The optimal fidelity in Eq. (15) is achieved by an observation test with atomic effects | Φ 𝑖 (cid:105) (cid:104) Φ 𝑖 | with | Φ 𝑖 (cid:105) = √︁ 𝑤 𝑖 𝑑 𝑀 | 𝜓 𝑖 (cid:105) ⊗ 𝑀 , 𝑑 𝑀 = (cid:18) 𝑀 + 𝑑 − 𝑑 − (cid:19) , (16)where { | 𝜓 𝑖 (cid:105) } 𝐴𝑖 = are pre-specified pure state-vectors, and { 𝑤 𝑖 } 𝐴𝑖 = is a probability vector satisfyingthe identity 𝐴 ∑︁ 𝑖 = 𝑤 𝑖 | 𝜓 𝑖 (cid:105) (cid:104) 𝜓 𝑖 | ⊗ 𝑀 = (cid:68) | 𝜓 (cid:105) (cid:104) 𝜓 | ⊗ 𝑀 (cid:69) . (17)where the notation (cid:104) . . . (cid:105) denotes averaging over the prior of pure-state vectors, taking as the priorthe Haar measure on the unit sphere in C 𝑑 .ard Problem and Free Will 17 Consider a futuristic “quantum teleportation experiment”, meaning that the quan-tum state of a system is substituted to that of a remote isomorphic system. The matter(electrons, protons, ...) of which a teleported person is composed is available at thereceiver location and is in-principle indistinguishable from the same matter at thetransmission location: indeed, it is only its quantum state that is teleported. Theresulting individual would be the same as the original one, including his thoughtsand memory. 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 ontictransformation, with no subsystem as such.In Figs. 5 and 6 we illustrate the use of the criterion in two paradigmatic cases. Herewe emphasise that the role of interactions is crucial for the criterion. entangled marginal states are epistemic factorised ontic states
Fig. 5 The combination problem.
Stated generally, the problem is about how the fundamentalconscious minds come to compose, constitute, or give rise to some further conscious mind. Theontic-state principle P1 provides a partial criterion to exclude some situations, e.g. (figure on theleft) two entangled systems cannot separately be conscious entities, since each one is in a marginalstate of an entangled one, hence it is necessarily mixed. On the other hand, (figure on the right) iftwo systems are in a factorized pure state, each system is in an ontic state, and they are two singleindividuals, and remain so depending on their following interactions (see Fig. 6).
This section has described what is necessary to form a combination of consciousentities, thus removing the difficulties encountered with panpsychist models basedon classical physics. The key idea is that the states of the combining systems andtheir transformations be ontic and that such systems interact quantumly. These re-quirements assure that the combined entity is also in a pure state and that none of itssubsets are conscious . Clearly, the interactions between the conscious entity and theenvironment (including other entities) can only be classical, otherwise a larger entity Teleportation will need the availability of shared entanglement and classical communication, andtechnically would use a Bell measurement at the sender and a conditioned unitary transformationat the receiver. [42] Of course, this would violate the no cloning theorem, if at the transmission point the originalindividual would not be destroyed. Indeed, according to the quantum information-disturbancetradeoff [34], teleportation cannot even make a bad copy leaving the original untouched.8 Giacomo Mauro D’Ariano and Federico Faggin would be created and would persist for as long as a disentangling transformationdoes not occur, as in the case illustrated in Fig. 6.
Fig. 6 The combination problem.
The general individuation criterion in statement S4 requiresfull quantum coherence, namely the ontic nature of both states and transformations. In the casedepicted 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 onthe left represent two separate individuals. The box in the middle merges the two individual intoa 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 oftwo individuals necessarily needs a quantum interaction.
Proposing feasible experiments about the quantum nature of consciousness is avery exciting challenge. In principle quantumness could be established throughexperimental demonstration of one of the two main nonclassical features of quantumtheory: nonlocality and complementarity . The two notions are not independent, sincein order to prove nonlocality we need complementary observations, in addition toshared entanglement.In order to prove nonlocality of consciousness we need measurements at two sepa-rate points sufficiently far apart to exclude causal connection, and such a requirementis very challenging, since it demands very fast measurements, considering that fora distance of 3 cm between the measurements points a time-difference of a tenth ofnanosecond is sufficient to have signalling. ard Problem and Free Will 19
About nonlocality together with complementarity, we speculate that they togethermay be involved in 3D vision, and take the opportunity to suggest that such line ofresearch may anyway be a fruitful field of experimental research on consciousness. For example, a genuine case of complementarity in 3D experience is occurring asa result of looking at
Magic Eye images published in a series of books [44]. Theseimages feature autostereograms , which allow most people to see 3D images byfocusing on 2D patterns that seem to have nothing in common with the 3D imagethat emerges from them. The viewer must diverge or converge his eyes in orderto see a hidden three-dimensional image within the patterns. The 3D image thatshows up in the experience is like a glassy object that, depending on convergenceor divergence of the eyes, shows up as either concave or convex. Clearly the twoalternative 3D views–convex and concave–are truly complementary experiences,and each experience has an intrinsic wholeness.Regarding complementarity alone, speculative connections with contrasting oropposite dimensions of human experience have been considered in the literature,e.g. "analysis" vs "synthesis", and "logic" vs "intuition" [45]. Here, however, weare interested in complementariy in experiences possibly reproducible in differentindividuals , of the kind of gedanken-experiments a la Heisenberg , e.g. one experienceincompatible with another and/or disturbing each other. Having something morethan two complementary observables, namely an informationally complete set ofmeasurements such as all three Pauli matrices for a single consciousness qubit,would allow us to make a quantum tomography [36] of the qubit state, assumingthis is reprepared many times, e.g. through an intense prolonged attention. To ourknowledge, the feasibility assessment of such a kind of experiment is of a difficultycomparable to that of a nonlocality experiment.Finally, apart from proving the quantumness of experience, we can at least ex-perimentally infer some theoretical parameters in the quantum theoretical approachconsistent with observations. This is the case e.g. of the experiment already men-tioned in discussing the upper bound (15) in memory-recall fidelity, which can beused in inferring an effective dimension 𝑑 of the system involved in consciousness invery focused restricted experiences, as for masking conscious perception [40]. Webelieve that memory-recall experiments versus variables such as attention, memory-recovery delay, and variable types of qualia, may be helpful to make a preliminarymapping of the dimensionality of the spaces of conscious systems involved.We conclude with a few considerations about the feasibility of a simulation ofa conscious process like the ones proposed here. As already mentioned, due tothe purity of the ontic process, a simulation just needs multiplications of generallyrectangular matrices, which for matrices with the same dimension 𝑑 is essentially a Θ ( 𝑑 ) 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 probabilitydistribution of each ontic step, one needs to evaluate the conditional probability A 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 reversalcontrolled by the principles of quantum mechanics [43]. However, the 3D experience of the Neckercube does not require binocular disparity since monocular vision also produces the same effect.0 Giacomo Mauro D’Ariano and Federico Faggin for the output state of the previous step, and this needs the multiplication for allpossible outcomes (the free will) of the corresponding Kraus operator of the lastontic transformation. With a RAM of the order of Gbytes one could definitelyoperate with a dozen qubits at a time. This should be compared with 53 qubits ofthe Google or IBM quantum computer, the largest currently available, likely to shareclassical information in tandem with a large classical computer. However, it is notexcluded that some special phenomena could be already discovered/analyzed with alaptop.
We have presented a theory of consciousness, based on principles, assumptions, andkey concepts that we consider crucial for the robustness of the theory and the removalof the limitations of most panpsychist theories [41]. We believe that conferring innerreality and agency to quantum systems in pure quantum state, with conversionof information from classical to quantum and vice versa, is unprecedented, withmajor philosophical and scientific consequences. In the present approach free willand consciousness go hand in hand, allowing a system to act based on its qualiaexperience by converting quantum to classical information, and thus giving causalpower to subjectivity–something that until now has been highly controversial, if notconsidered impossible.The theory provides that a conscious agent may intentionally convert quantuminformation into a specific classical information to express its free will, a classicaloutput that is in principle unpredictable due to its quantum origin. The theory wouldbe incoherent without the identification of the conscious system in terms of purity andinseparability of the quantum state, which is identified with the systems experience.The purity of non-deterministic quantum evolution identifies consciousness withagency through its outcome.Metaphysically the proposed interpretation that a pure, non-separable quantumstate is a state of consciousness, could be turned on its head by assuming the on-tology of consciousness and agency as primary, whereas physics is emergent fromconsciousness and agency. This same interpretation would then consider classi-cal physics the full reification (objectification) of quantum reality as quantum-to-classical agency corresponding to the free will of conscious entities existing entirelyin the quantum realm. The ontology deriving by the acceptance of consciousness asfundamental, would be that objectivity and classical physics supervene on quantumphysics, quantum physics supervenes on quantum information, and quantum infor-mation supervenes on consciousness. If we were to accept this speculative view,physics could then be understood as describing an open-ended future not yet existingbecause the free will choices of the conscious agents have yet to be made. In thisperspective, we, as conscious beings, are the co-creators of our physical world. Wedo so individually and collectively, instant after instant and without realizing it, byour free-will choices. ard Problem and Free Will 21
Appendices about general OPTs
In these appendices we provide the general operational probabilistic framework forfuture post-quantum explorations for a theory of consciousness. We report just ashort illustration of what is an operational probabilistic theory (OPT), of whichQuantum Theory and Classical Theory are the most relevant istances. As the readermay appreciate, the OPT provides a framework much reacher, general, flexible, andmathematically rigorous as compared to other theoretical frameworks, such as thecausal approach of Tononi [46].
The operational probabilistic theory (OPT) framework
It is not an overstatement to say that the OPT framework represents a new Galileanrevolution 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 physicsand possibly in other sciences. Such rules constitute what is called the operationalframework . Its rigour is established by the simple fact that the OPT is just "meta-mathematics", since it is a chapter of category theory [47, 48]. To be precise thelargest class of OPTs corresponds to a monoidal braided category . The fact that thesame categoric framework is used in computer science [49, 50, 51, 52, 53] gives anidea of the thoroughness and range of applicability of the rules of the OPT.Lucien Hardy in several seminal papers [54, 55, 56] introduced a heuristic frame-work that can be regarded as a forerunner of the OPT, which made its first appearancein Refs. [9, 8], and soon was connected to the categorical approach in computer sci-ence [20, 22, 23, 24, 25, 21]. As already mentioned, both QT and CT are OPTs[10], but one can build up other OPTs, such as variations of QT e.g. fermionic QT[57], or QT on real Hilbert space [10], or QT with only qubits, but also CT withentanglement [58] or without local discriminability [59]. Also other toy-theories,such as the PR-Boxes, are believed to be completable to OPTs (see e.g. [60]).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 circuitsand for attaching to them a joint probability: in such a way the OPT literally becomesan extension of probability theory.
The general idea
How does an OPT works? It associates to each joint probability of multiple eventsa closed directed acyclic graph (CDAG) of input-output relations as in Fig. 7. Each event (e.g. E 𝑚 in figure) is an element of a complete test ( {E 𝑚 } 𝑚 ∈ M in figure)with normalized marginal probability (cid:205) 𝑚 ∈ M 𝑝 (E 𝑚 ) =
1. The graph tells us that themarginal probability distribution of any set of tests still depends on the marginal-ized set, e.g. the marginal probability distribution of test {E 𝑚 } 𝑚 ∈ M depends on the full graph of tests to which it is connected, although it has been partially or fullymarginalised. As a rule, disconnected graphs (as 𝛾 and 𝛾 in Fig. 7) are statisticallyindependent, namely their probability distributions factorize. The oriented wires de-
An OPT associates to each joint probability distribution of multiple tests/events aclosed directed acyclic graph (CDAG) of output-input relations (see text). As a rule, uncon-nected graph components are statistically independent, e.g. in the case in the figure, one has 𝑝 ( A 𝑖 , E 𝑚 , . . . , X 𝑟 , . . . | 𝛾 ∪ 𝛾 ) = 𝑝 ( A 𝑖 , E 𝑚 , . . . | 𝛾 , ) 𝑝 (X 𝑟 , . . . | 𝛾 ) . noting output-input connections between the tests (labeled by Roman letters in figure7), 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 ofparticles all with spin up (namely in the state 𝜌 = | ↑(cid:105)(cid:104)↑ | ). At the output we havetwo von Neumann measurements in cascade, the first one Σ 𝛼 measuring 𝜎 𝛼 , and A von Neumann measurement of e.g. 𝜎 𝑧 has two outcomes "up" and "down", and the outputparticle will be in the corresponding eigenstate of 𝜎 𝑧 .ard Problem and Free Will 23 the second one Θ 𝛽 measuring 𝜎 𝛽 , where 𝛼 and 𝛽 can assume either of the two values 𝑥, 𝑧 . The setup is represented by the graph shown in the figure, where A representsthe system corresponding to the particle spin, 𝑒 the deterministic test that simplydiscards the particle, and the two tests Σ 𝛼 , Θ 𝛽 ( 𝛼, 𝛽 = 𝑥, 𝑧 ) the two von Neumanmeasurements. Now, clearly, for 𝛼 = 𝑧 one has the marginal probability distribution 𝑝 ( Σ 𝑧 ) = ( , ) and 𝑝 ( Σ 𝑥 )( , ) independently of the choice of the test Θ 𝛽 . On theother hand, for the second test one has marginal probability 𝑝 ( Θ 𝑥 ) = ( , ) for Σ 𝑧 and 𝑝 ( Θ 𝑥 ) = ( , ) for Σ 𝑥 . We conclude that the marginal probability of Σ 𝛼 isindependent of the choice of the test Θ 𝛽 , whereas the marginal probability of Θ 𝛽 depends of the choice of Σ 𝛼 . Thus, the marginal probability of Θ 𝛽 generally dependson the choice of Σ 𝛼 , and this concept goes beyond the content of joint probability, andneeds the OPT graph. Theoretically, we conclude that there is "something flying"from test Σ 𝛼 to test Θ 𝛽 (although we cannot see it!): this is what we theoreticaldescribe as a spinning particle! This well illustrates the notion of system : a theoreticalconnection between tested events. A A A ⌃ ↵
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 mathe-matical 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. C i A ! B M j C ! D = C U A C i B V DE E P j Y AB = Y AB , Y A a B b = Y A a B b A C i B = Y C U C c i B A e A B (4) A T i B = A U B s F E Z i Y A a i AB b k AA { A a } a A A U A { C c } c C AB { B b } b B B B { D d } d D BA a i A U A a j AB b k B B b l B Y A a i AB b k A Y i A A j B C l C E n D G q E F D m GH B k L M F p NO P Y i , A j , B k , D m BGMP C l , E n , F p , G q Fig. 9 OPT finer and coarser descriptions.
The OPT is a black-box approach . It can be mademore or less detailed, as in in the box on the left, and even at so fine a level that it is equivalent to afield-theoretical description, as in the right figure.4 Giacomo Mauro D’Ariano and Federico Faggin
Notice that although any graph can be represented in 2D (e.g. using crossing ofwires), one can more suitably design it in 3D (2D + in-out), as in Fig. 10.
Fig. 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 mostgeneral goal of science, namely to connect objective facts happening (the events),devising a theory of such "connections" (the systems), thus allowing making predic-tions for future occurrences in terms of joint probabilities of events depending ontheir connections.One of the main methodologically relevant features of the OPT is that it makesperfectly 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 eachtest. What is theoretical is the graph of connections between the tests, along withthe mathematical representations of both systems and tests. The OPT frameworkdictates the rules that the mathematical description should satisfy, and the specificOPT gives the particular mathematical representation of systems and tests/eventsand 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 distinctionbetween what is experimental and what is theoretical. Though this would seem atrivial statement, such a confusion happens to be often the source of disagreementbetween scientists. Though the description of the apparatus is generally intermingledwith theoretical notions, the pure experimental datum must have a conventionally ard Problem and Free Will 25 defined "objectivity status", corresponding to "openly known" information, namelyshareable by any number of different observers. Then both the theoretical languageand the framework must reflect the theory-experiment distinction, by indicatingexplicitly which notions are assigned the objectivity status. Logic, with the Booleancalculus of events, is an essential part of the language, and Probability Theory canbe regarded as an extension of logic, assigning probabilities to events. The notionthat is promoted to the objectivity status is that of "outcome of a test", announcingwhich event of a given test has occurred. The OPT framework thus represents anextension of probability theory, providing a theoretical connectivity between events,the other theoretical ingredients being the mathematical descriptions of systems andtests.
The OPT as a general information theory ```
ABC D E FGHL M NOP ABCDE F PGH OL NM {A i } i I
Equivalence between the CDAG and the quantum information circuit or, equivalently, anyrun-diagram of a program.
We can immediately realise that a CDAG is exactly the same graph of a quantumcircuit 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 representsa subroutine, and the wires represent the registers through which the subroutinescommunicate data. Indeed, the OPT can be regarded as the proper framework forinformation science in general.For a recent complete presentation of the OPT framework, the reader is addressedto the the work [9, 10] or the more recent thorough presentation [19].
Notation and abbreviations
Bnd + (H) bounded positive operators over H CP (cid:54) trace-non increasing completely positive mapCP = trace-preserving completely positive map H Hilbert space over C Cone ( S ) conic hull of SCone (cid:54) ( S ) convex hull of { S ∪ } Conv ( S ) convex hull of S Eff ( A ) set of effects of system AEff ( A ) set of deterministic effects of system AMrkv (cid:54) normalization-non-incressing right-stochastic Markov matricesMrkv normalization-preserving right-stochastic Markov matricesPrm ( 𝑛 ) 𝑛 × 𝑛 permutation matrices ( R 𝑛 ) + (cid:54) { x ∈ R 𝑛 A | x ≥ , x (cid:54) } : simplex S 𝑛 A + ( R 𝑛 ) + { x ∈ R 𝑛 A | x ≥ , || x || = } : simplex S 𝑛 A St ( A ) set of states of system ASt ( A ) set of deterministic states of system AT (H) trace-class operators over H T + (H) trace-class positive operators over H T + (cid:54) (H) positive sub-unit-trace operators over H T + = (H) positive unit-trace operators over H Trn ( A → B ) set of transformations from system A to system BTrn ( A → B ) set of deterministic transformations from system A to system B U (H) unitary group over H Special cases corollaries T ( C ) = C , T + ( C ) = R + , T + (cid:54) ( C ) = [ , ] , T + = ( C ) = { } CP ( T (H) → T ( C )) = P ( T (H) → T ( C )) = { Tr [· 𝐸 ] , 𝐸 ∈ Bnd + (H) } CP ( T ( C ) → T (H)) = P ( T ( C ) → T (H)) = T + (H) CP (cid:54) ( T ( C ) → T (H)) ≡ T + (cid:54) (H) CP (cid:54) ( T (H) → T ( C )) ≡ { 𝜖 ( ·) = Tr [· 𝐸 ] , (cid:54) 𝐸 (cid:54) 𝐼 } Mrkv (cid:54) ( 𝑛, ) = ( R 𝑛 ) + (cid:54) Mrkv ( 𝑛, ) = ( R 𝑛 ) + = Abbreviations
CT Classical TheoryOPT Operational Probabilistic TheoryPR boxes Popescu-Rohrlich boxesQT Quantum Theory
Table 2
Notation, special-cases corollaries, and common abbreviations.ard Problem and Free Will 27
Quantum theory
A minimal mathematical axiomatisation of Quantum Theory as an OPT is providedin 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 rulesof compositions of transformations and their respective systems are provided by theOPT framework. The reader who is not familiar with such framework can simplyuse the intuitive construction of quantum circuits. In Table 4 we report the maintheorems following from the axioms. The reader interested in the motivations for thepresent axiomatization is addressed to Ref. [61].
Quantum Theory system A H A system composition AB H AB = H A ⊗ H B transformation T ∈
Trn ( A → B ) T ∈ CP (cid:54) ( T (H A ) → T (H B )) Born rule 𝑝 ( T) = Tr T T ∈
Trn ( I → A ) Table 3 Mathematical axiomatisation of Quantum Theory.
As given in the table, in QuantumTheory to each system A we associate a Hilbert space over the complex field H A . To the compo-sition of systems A and B we associate the tensor product of Hilbert spaces H AB = H A ⊗ H B .Transformations from system A to B are described by trace-nonincreasing completely positive (CP)maps from traceclass operators on H A to traceclass operators on H B . Special cases of transforma-tions are those with input trivial system I corresponding to states, whose trace is the preparationprobability, the latter providing an efficient Born rule from which one can derive all joint probabil-ities of any combination of transformations. Everything else is simply special-case corollaries andone realisation theorem: these are reported in Table 4.8 Giacomo Mauro D’Ariano and Federico Faggin Quantum theorems trivial system I H I = C reversible transf. U = 𝑈 · 𝑈 † 𝑈 ∈ U (H A ) determ. transformation T ∈
Trn ( A → B ) T ∈ CP (cid:54) ( T (H A ) → T (H B )) parallel composition T ∈ Trn ( A → B ) , T ∈ Trn ( C → D ) T ⊗ T sequential composition T ∈ Trn ( A → B ) , T ∈ Trn ( B → C ) T T states 𝜌 ∈ St ( A ) ≡ Trn ( I → A ) 𝜌 ∈ T + (cid:54) (H A ) 𝜌 ∈ St ( A ) ≡ Trn ( I → A ) 𝜌 ∈ T + = (H A ) 𝜌 ∈ St ( I ) ≡ Trn ( I → I ) 𝜌 ∈ [ , ] 𝜌 ∈ St ( I ) ≡ Trn ( I → I ) 𝜌 = 𝜖 ∈ Eff ( A ) ≡ Trn ( A → I ) 𝜖 ( ·) = Tr A [· 𝐸 ] , (cid:54) 𝐸 (cid:54) 𝐼 𝐴 𝜖 ∈ Eff ( A ) ≡ Trn ( A → I ) 𝜖 = Tr A Transformations asunitary interaction+von Neumann-Luders A T 𝑖 B = A U B 𝜎 F E P 𝑖 T 𝑖 𝜌 = Tr E [ 𝑈 ( 𝜌 ⊗ 𝜎 ) 𝑈 † ( 𝐼 B ⊗ 𝑃 𝑖 ) ] Table 4 Corollaries and a theorem of Quantum Theory, starting from Table 3 axiomatization.
The first corollary states that the trivial system I in order to satisfy the composition rule IA = AI = Amust be associated to the one-dimensional Hilbert space H I = C , since it is the only Hilbert spacewhich trivializes the Hilbert space tensor product. The second corollary states that the reversibletransformations are the unitary ones. The third corollary states that the deterministic transforma-tions are the trace-preserving ones. Then the fourth and fifth corollaries give the composition oftransformations 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 onthe system Hilbert space, having trace bounded by one; 2) the deterministic states correspond tounit-trace positive operator; 3) the states of trivial system are just probabilities; 4) The only trivialsystem deterministic state is the number 1. We then have two corollaries for effects, as special casesof transformation toward the trivial system: 1) the effect is represented by the partial trace over thesystem Hilbert space of the multiplication with a positive operator bounded by the identity over thesystem Hilbert space; 2) the only deterministic effect is the partial trace over the system Hilbertspace. Finally, we have the realization theorem for transformations in terms of unitary interaction U = 𝑈 · 𝑈 † with an environment F and a projective effect-test {P 𝑖 } over environment E, with P 𝑖 = 𝑃 𝑖 · 𝑃 𝑖 , { 𝑃 𝑖 } being a complete set of orthogonal projectors.ard Problem and Free Will 29 Classical theory
Classical Theory system A R 𝑛 A system composition AB R 𝑛 AB = R 𝑛 A ⊗ R 𝑛 B transformation T ∈
Trn ( A → B ) T ∈ Mrkv (cid:54) ( R 𝑛 A , R 𝑛 B ) Table 5 Mathematical axiomatisation of Classical Theory.
To each system A we associate areal Euclidean space R 𝑛 A . To composition of systems A and B we associate the tensor productspaces R 𝑛 A ⊗ R 𝑛 B . Transformations from system A to system B are described by substochasticMarkov matrices from the input space to the output space. Everything else are simple special-casecorollaries: these are reported in Table 6. Classical theorems trivial system I H I = R reversible transformations P P ∈
Prm ( 𝑛 A ) transformation T ∈
Trn (cid:54) ( A → B ) T ∈ Mrkv (cid:54) ( R 𝑛 A , R 𝑛 B ) determ. transformation T ∈
Trn ( A → B ) T ∈ Mrkv ( R 𝑛 A , R 𝑛 B ) parallel composition T ∈ Trn ( A → B ) , T ∈ Trn ( C → D ) T ⊗ T sequential composition T ∈ Trn ( A → B ) , T ∈ Trn ( B → C ) T T states x ∈ St ( A ) ≡ Trn ( I → A ) x ∈ ( R 𝑛 A ) + (cid:54) x ∈ St ( A ) ≡ Trn ( I → A ) x ∈ ( R 𝑛 A ) + = 𝑝 ∈ St ( I ) ≡ Trn ( I → I ) 𝑝 ∈ [ , ] 𝑝 ∈ St ( I ) ≡ Trn ( I → I ) 𝑝 = 𝜖 ∈ Eff ( A ) ≡ Trn ( A → I ) 𝜖 ( ·) = · x , (cid:54) x (cid:54) 𝜖 ∈ Eff ( A ) ≡ Trn ( A → I ) 𝜖 = · . The first corollarystates that the trivial system I in order to satisfy the composition rule IA = AI = A must beassociated to the one-dimensional space R , since it is the only real linear space that trivialises thetensor product. The second corollary states that the reversible transformations are the permutationmatrices. The third states that transformations are substochastic Markov matrices. The fourth statesthat the deterministic transformations are stochastic Markov matrices. Then the fifth and sixthcorollaries give the composition of transformations in terms of composition of matrices. We thenhave 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 elementsbounded by one; 2) the deterministic states correspond to normalised probability vectors; 3) thecase of trivial output-system correspond to just probabilities; 4) The only trivial output-systemdeterministic state is the number 1. We then have two corollaries for effects, as special cases oftransformation toward the trivial system: 1) the effect is represented by scalar product with a vectorwith components in the unit interval; 2) the only deterministic effect is the scalar product with thevector with all unit components.0 Giacomo Mauro D’Ariano and Federico Faggin Acknowledgements
The authors acknowledge helpful conversations and encouragement by DonHoffman, and interesting discussions with Chris Fields. Giacomo Mauro D’Ariano acknowledgesinteresting discussions with Ramon Guevarra Erra.This work has been sponsored by Elvia and Federico Faggin foundation through Silicon ValleyCommunity 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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