Do spins have directions?
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DOI: 10.1007/s0050001519130
 Cite this article as:
 Aerts, D. & Sassoli de Bianchi, M. Soft Comput (2017) 21: 1483. doi:10.1007/s0050001519130
Abstract
The standard Bloch sphere representation has been recently generalized to describe not only systems of arbitrary dimension, but also their measurements, in what has been called the extended Bloch representation of quantum mechanics. This model, which offers a solution to the longstanding measurement problem, is based on the hiddenmeasurement interpretation of quantum mechanics, according to which the Born rule results from our lack of knowledge of the measurement interaction that each time is actualized between the measuring apparatus and the measured entity. In this article, we present the extended Bloch model and use it to investigate, more specifically, the nature of the quantum spin entities and of their relation to our threedimensional Euclidean theater. Our analysis shows that spin eigenstates cannot generally be associated with directions in the Euclidean space, but only with generalized directions in the Blochean space, which apart from the special case of spin onehalf entities, is a space of higher dimensionality. Accordingly, spin entities have to be considered as genuine nonspatial entities. We also show, however, that specific vectors can be identified in the Blochean theater that are isomorphic to the Euclidean space directions, and therefore representative of them, and that spin eigenstates always have a predetermined orientation with respect to them. We use the details of our results to put forward a new view of realism, that we call multiplex realism, providing a specific framework with which to interpret the human observations and understanding of the component parts of the world. Elements of reality can be represented in different theaters, one being our customary Euclidean space, and another one the quantum realm, revealed to us through our sophisticated experiments, whose elements of reality, in the quantum jargon, are the eigenvalues and eigenstates. Our understanding of the component parts of the world can then be guided by looking for the possible connections, in the form of partial morphisms, between the different representations, which is precisely what we do in this article with regard to spin entities.
Keywords
Spin eigenstate Spin measurement Extended Bloch sphere Hiddenmeasurement interpretation Nonspatiality Multiplex realism1 Introduction
According to the hypothesis of realism, reality is out there, available to us to be experienced. Therefore, what we know about reality comes from our experiences of it, and from the worldviews that we are able to construct when we order these experiences into a possibly consistent map of relations.
No doubts, the human process of creation of a worldview started a long time ago, in precultural and prescientific times, when our experiences of reality were only of the “ordinary kind,” i.e., obtained using exclusively our human macroscopic body as a measuring apparatus, plus a few very basic tools. From the ordering of these experiences, a first “clothing and decoration” of reality resulted, consisting in the identification of those portions of it that were characterizable by aggregates of sufficiently stable properties, such as having a certain amount of weight, a given size, shape, state of movement and orientation. These “clusters of properties” are what we today call classical entities: the macroscopic objects of our everyday life, and the astronomical objects that we can observe moving in the sky, such as the Moon and the Sun.
We can say that this “clothing and decoration” of reality happened (and happens) typically in two different “directions of penetration.” The first one is a penetration in depth, which precisely corresponds to the process we have just mentioned of identifying the entities forming “stable aggregates of elements of reality.” The second one is a penetration in width, corresponding to a process of organization of the relations existing between the different entities, i.e., between those aggregates of properties that appear to be separate (Aerts 2002) (not influencing each other in a significant way). This penetration in width of our reality can be considered as an ordering process giving rise to a space, and more specifically to our Euclidean threedimensional space, which therefore should be considered, first of all, a space of relations.
According to the above, it is clear that when a given ensemble of experiences is properly ordered and organized, a specific theater of reality will emerge, inside of which a given typology of actors can perform and relate, in predetermined ways. Our Euclidean physical space, in that sense, should not be understood as an external “container,” accommodating the play of these classical actors, but really as the manifestation of the structure of their possible relations. It is then also clear that when an ensemble of new experiences becomes all of a sudden available, our first attempt will be that of trying to find a place for them in the theater of reality that we have already constructed. This also because, as time passes by, it is easy to forget about the process of its construction and start believing that all of reality should necessarily fit into it, as the theater and its content, and reality, would just be one and the same thing.
As is known, an amazing ensemble of new experiences that came shaking the structure of our classical theater are those that were obtained in the investigation of the microscopic layer of our reality, in confirmation of that corpus of knowledge known as quantum mechanics. Examples are the famous experiments of Gerlach and Stern (1922) with silver atoms, of Rauch et al. (1974), (Rauch 1975, 1988) with neutrons, and of Aspect (1982, 1999) with entangled photons. (Another important ensemble of experiments that have undermined the structure of our classical theater are those that have led to the discovery of relativity theory, but we shall not be concerned by them in this article). Thanks to these very sophisticated experiments, we could access elements of reality that in the past were totally beyond our reach, because of our too coarse physical senses, but also because our natural environment did not provide the controlled conditions of a modern laboratory. These experiments revealed to us that although these new elements of reality were in part similar to those we had so far experienced, at the same time they were sufficiently different to spoil our efforts to incorporate them in our classical Euclidean theater.
A typical example is that of spins, which we will discuss more particularly in this article. A spin is usually described as an intrinsic angular momentum carried by a microscopic entity, such as an electron, a proton, a neutron, or by a combination of elementary entities (composite entity), such as an atomic nuclei. However, although it possesses the same physical dimension of an angular momentum carried by a macroscopic body (the dimension of the Planck constant \(\hbar \)), it cannot be associated with any specific rotation in space. For instance, if the spin is calculated on a classical basis, such a rotational movement would yield a superluminal velocity along the microscopic particles’ periphery (Hentschel 2009), in violation of the relativistic limit. Also, in the case of fractionary spins, we know that a \(360^{\circ }\) rotation will not bring the spin entity back in the same state (Rauch 1975), as is instead the case for the actors of the classical theater.
This impossibility of understanding a spin as a state of rotation in the threedimensional space can be further evidenced by observing that it cannot generally and consistently be represented as a threedimensional vector, as is the case for the angular momentum of a macroscopic object. This is so because in quantum mechanics a spin (and more generally an angular momentum) is described by an operator, and therefore cannot be simply drawn as a threedimensional vector quantity whose components would be real numbers.
When a spins entity is, say, in the eigenstate \(\mu ,\hat{\mathbf{x}}_3\rangle \) of \(S_3\equiv \mathbf{S}\cdot \hat{\mathbf{x}}_3\), we can try to classically describe the situation by drawing a vector of length \(\hbar \sqrt{s(s+1)}\) (according to \(S^2\mu ,\hat{\mathbf{x}}_3\rangle = \hbar ^2s(s+1)\mu ,\hat{\mathbf{x}}_3\rangle \)), whose projection onto \(\hat{\mathbf{x}}_3\) is precisely \({\hbar \mu }\) (according to \(S_3\mu ,\hat{\mathbf{x}}_3\rangle =\hbar \mu \mu ,\hat{\mathbf{x}}_3\rangle \)). In fact, there is an entire collection of possible vectors of length \(\hbar \sqrt{s(s+1)}\), whose projections onto \(\hat{\mathbf{x}}_3\) give the value \({\hbar \mu }\): they form a cone of height \(\hbar \mu \), lateral height \(\hbar \sqrt{s(s+1)}\), and radius \(\sqrt{\hbar ^2 s(s+1)  \hbar ^2 \mu ^2}\) (see Fig. 1, for the \(s={1\over 2}\) case).
The vectors belonging to the cone, however, cannot consistently represent the spin of a quantum entity in the state \(\mu ,\hat{\mathbf{x}}_3\rangle \). Indeed, we know that if we measure the spin along any other direction, all the values \(\hbar s,\dots ,\hbar s\) are in principle obtainable. This, however, is incompatible with the geometry of the cone. To see this, let us just consider for simplicity the spin\({1\over 2}\) case. If we take one of the cone’s vectors as the direction of a spin measurement (here understood in the classical sense), then by orthogonally projecting all the cone’s vectors along that specific direction, we obtain the interval of possible values \([{\hbar \over 2\sqrt{3}}, {\hbar \sqrt{3}\over 2}]\) (see Fig. 1), which is manifestly incompatible with the experimental data, as it does not contain the eigenvalue \({\hbar \over 2}\).
The above constitutes a very simple nogo argument showing that it is impossible to directly associate a quantum spin eigenstate (and a fortiori, of course, a superposition of spin eigenstates) with an angular momentum vector having a specific direction in space, not even if instead of a single direction we accept to associate to it an entire collection of indeterminate vectors lying on a cone. This means that not only a spin eigenstate cannot be understood as an entity belonging to our threedimensional spatial theater, it neither can be understood as a higher dimensional classicallike entity characterized by multiple directions, that would simply “cast a threedimensional shadow” onto it (in a sort of view à la Edwin A. Abbott) (Fig. 2).
So, coming back to our initial discussion, we see that through our quantum experiments we have recently (in the time scale of human evolution) opened a window to an entire new sector of our reality, which requires us to construct a brand new theater, to find a suitable (nonspatial) “place” for the new quantum actors, in which their show can be represented. This of course will be quite a nonordinary theater according to our classical criteria, as these quantum actors can behave in a way that is technically impossible to the classical ones. But considering the existence of these two theaters, the quantum and the classical, two questions arise.
The first question is about the nature of their interactions. Indeed, quantum entities, although not in space (and possibly also not in time), are nevertheless available to leave traces into it. If this would not be the case, we would never have learned about their existence, as our measuring apparatuses are macroscopic objects, extensions of our human bodies, which means that we can only explore the content of the quantum theater as from the Euclidean one. In quantum mechanics, the effects of these interactions are described by the Born rule and the associated projection postulate, so the question is: How does a quantum entity interact with a classical one (a measuring apparatus) during a measurement process, so as to produce the different possible outcomes, in accordance with the predictions of the Born rule? Considering that a measurement process cannot be understood as a mere process of discovery of a shadow that a quantum entity would cast onto the Euclidean stage, how can we describe the process through which it leaves a detectable trace onto it?
The second question is about finding what are the possible relations between the elements of reality characterizing the entities living in the quantum (nonspatial) theater, and those characterizing the classical entities (spatial objects) located in our human classical theater. In other terms, it is about finding the possible connections between what our sophisticated experimental apparatuses indicate to us to be real, and what we know (or think) to be real from our direct prescientific human experience. A more specific formulation of this question, in relation to spin entities, would be the following: What is the general relation between the nonspatial spin eigenstates and the space directions in the Euclidean space, and is it possible to have a precise geometrical description of such relation?
The purpose of this article is to offer an answer to the above two questions. For this, the article is organized as follows. In Sect. 2, we introduce the standard Bloch sphere representation, recalling that for spin\({1\over 2}\) entities there is a direct (onetoone) correspondence between spin states and space directions (each state being oriented along a different space direction). In Sect. 3, we show how to extend the Bloch result to also include the measurements, and in Sect. 4 we explain how the model can be naturally generalized to describe entities of arbitrary dimension N, answering in this way the first question. Then, in Sect. 5, we use the extended Bloch representation to show that the spin eigenstates, beyond the spin\({1\over 2}\) case, are not anymore oriented along space directions, and therefore should be considered to be genuine nonspatial elements of our reality (not belonging to our classical theater). However, we also show that the different eigenstates are oriented in a very specific way with respect to the space directions describing the measurements for which they are eigenstates, within the extended Bloch sphere (the quantum theater), answering in this way the second question. In Sect. 6, we show that this relation between spin eigenstates and space directions remains valid also for composite spin entities, and in Sect. 7 we explain how the correspondence between the elements of reality in our classical theater, and those in the quantum one, should be understood, and what are the limitations with regard to spins. Finally, in Sect. 8, we introduce the notion of multiplex realism and offer some concluding remarks.
2 The Bloch sphere
In the previous section, we have shown that the representation of quantum spin eigenstates of different magnitudes by means of cones of vectors in a threedimensional space is inconsistent, and therefore misleading, as it conveys the wrong idea that a spin eigenstate could be put into full correspondence with an intrinsic classical angular momentum, although of an indeterminate nature. On the other hand, if we renounce associating spin eigenstates with classical angular momenta, it is certainly possible, at least in the special case of spin\({1\over 2}\) entities, to characterize each state by means of a specific direction of space. However, for spin entities of magnitude \(s>{1\over 2}\), this will no longer be possible, as to characterize the different eigenstates, a \(4s(s+1)\)dimensional quantum theater will be required.
But let us start by considering the simple spin\({1\over 2}\) situation. Let \(\pm {1\over 2},\mathbf{n}\rangle \in {\mathbb C}^2\) be the two eigenvectors of the spin operator \(S_\mathbf{n}\), oriented along an arbitrary direction \(\mathbf{n}\), for the eigenvalues \(\pm {\hbar \over 2}\). In the following, for simplicity, we will simply write: \(\pm ,\mathbf{n}\rangle \equiv \pm {1\over 2},\mathbf{n}\rangle \). Then, also the vectorstates \(e^{i\alpha }\pm ,\mathbf{n}\rangle \), with \(\alpha \in {\mathbb R}\), are eigenvectors of \(S_\mathbf{n}\), for the same eigenvalues. In other terms, the vectors \(\pm ,\mathbf{n}\rangle \) and \(e^{i\alpha }\pm ,\mathbf{n}\rangle \) represent the same physical states, in accordance with the known fact that no measurement can determine a global phase factor.
The reason for this is that a measurement is a process that is precisely designed to single out a given entity, to specifically measure some of its properties. This means that a measurement is a context conceived to only act on the entity which is meant to be measured, and not, say, on combinations of that entity with other entities. This isolation of the measured entity in the measurement context thus excludes the possibility that a global phase factor would be able to produce interference effects, and therefore become observable.
A way to make fully explicit this isolation of an entity in a measurement context, causing the global phase of its vectorstate \(+,\mathbf{n}\rangle \) to become irrelevant, is to describe its state by means of a onedimensional projection operator \(P(\mathbf{n})=+,\mathbf{n}\rangle \langle +,\mathbf{n}\). Indeed, knowing \(P(\mathbf{n})\), one can always reconstruct \(+,\mathbf{n}\rangle \), but only up to a global phase factor. To show this, we write \(+,\mathbf{n}\rangle = a_++,\hat{\mathbf{x}}_3\rangle + a_,\hat{\mathbf{x}}_3\rangle \), so that (to simplify the notation, we set: \(\pm \rangle \equiv \pm ,\hat{\mathbf{x}}_3\rangle \)): \(P(\mathbf{n})= a_+^2 +\rangle \langle + + a_^2 \rangle \langle + (a_+a_^*+\rangle \langle  + \mathrm{c.c.})\). We assume that \(a_+\ne 0\), and we fix the global phase of \(+,\mathbf{n}\rangle \) in a way that \(a_+\) is a real strictly positive number. Then, since \(\langle +P(\mathbf{n})+\rangle = a_+^2 = a_+^2\), we have \(a_+=\langle +P(\mathbf{n})+\rangle ^{1\over 2}\). Also, \(\langle +P(\mathbf{n})\rangle =a_+a_^*\), so that \(a_=\langle P(\mathbf{n})+\rangle \langle +P(\mathbf{n})+\rangle ^{{1\over 2}}\). And of course, if \(a_+=0\), then \(a_\) is simply a phase factor.
So, when we go from the vectorstate representation \(+,\mathbf{n}\rangle \), to the associated (rank1 projection) operatorstate representation \(P(\mathbf{n})\), the only information we loose is that relative to the vector’s global phase, which plays no role in a measurement. Therefore, the operatorstate \(P(\mathbf{n})\) can be considered to be the proper mathematical object representing the state in which the spin entity (and more generally any quantum entity) is prepared and isolated in view of being subjected to a measurement process.
Our next step is to show how to represent the spin states \(P(\mathbf{n})\) as space directions. This is a wellknown representation that dates back to 1892, when the French physicist and mathematician Henri Poincaré discovered that a surprisingly simple representation of the polarization states of electromagnetic radiation could be obtained by representing the polarization ellipse on a complex plane, and then further projecting such plane onto a sphere (Poincaré 1892). In 1946, this representation was adapted by the Swiss physicist Bloch (1946) to represent the states of twolevel quantum systems, such as spin\({1\over 2}\) entities, in what is today known as the Poincaré sphere or Bloch sphere.
The spin eigenstates \(P(\mathbf{n})\) of a spin\({1\over 2}\) entity can thus be fully determined by specifying the unit vectors \(\mathbf{n}\) defining the space direction for which the associated spin observable \(S_\mathbf{n}\), if measured, would produce with certainty the outcome \({\hbar \over 2}\). This representation (see Fig. 3) is however very different from the (incomplete) cone representation described in the previous section, as is clear that none of the classical spin vectors lying on the cone point to the same direction as the unit vector representative of the state. Apart from this difference, the Bloch sphere allows us to maintain a onetoone correspondence between spin states and space directions, and is valid for arbitrary measurements. However, such direct correspondence will be lost when considering spin entities of magnitude greater than \({1\over 2}\), in the sense that the directions of space will not anymore be sufficient to characterize all the possible relations between states and measurements, which can only be represented in an extended space of higher dimensionality. However, each spin eigenstate will maintain a very specific relation (orientation) with respect to the space direction characterizing its measurement. But before showing this, we need to extend the Bloch representation.
3 The extended Bloch sphere
In the previous section, we have described the wellknown Bloch sphere representation for spin\({1\over 2}\) entities (also valid for general qubit systems). This is a representation that works for the states, and not for the measurements. However, it is possible to extend it to also include a modelization of the different possible measurements, as was shown by one of us forty years after the work of Bloch, providing at the same time a plausible explanation of the origin of the Born rule (Aerts 1986, 1987).
Before showing in this section how this can be done, it is worth mentioning that the perspective we are taking here is that which consists in taking the quantum formalism very seriously. By this, we mean that we consider that a Hilbertian vectorstate, or (when its global phase becomes irrelevant) its associated operatorstate, does actually describe the real state of the physical entity under investigation, and not just our knowledge or our beliefs about its condition. Accordingly, we also consider that a measurement process is an objective physical process, bringing the entity from an initial premeasurement state to a final postmeasurement state, in a way that cannot be determined in advance. However, what can be determined in advance are the probabilities of the different transitions, producing the different possible outcomes of the measurement.
Once the quantum formalism is taken seriously in the above sense, we have of course to face the challenge of understanding what happens during the quantum measurement process. Indeed, as is well known, measurements are not explained in the standard quantum formalism, as the Born rule of correspondence, with which the different probabilities are calculated, is just postulated.
What we will now show is how the measurement of an observable, for example, the spin observable \(S_3\), producing the probabilities (6), can be modeled within the Bloch sphere, in what is called the extended Bloch representation (Aerts and Sassoli de Bianchi 2014). To do so, we will need more (pure) states than those that are usually considered in the standard formalism. Indeed, to describe the indeterministic dynamics characterizing a quantum measurement, and derive from it the Born rule, we need to go inside the Bloch sphere, i.e., to also consider nonunit representative vectors.
We start by specifying to what kind of operators the nonunit vectors \(\mathbf{r}\), living inside the sphere, are associated with. Let \(D(\mathbf{r})={1\over 2}(\mathbb {I} + \mathbf{r}\cdot \varvec{\sigma })\). Clearly, even if \(\Vert \mathbf{r}\Vert < 1\), we still have \(\mathrm{Tr}\, D(\mathbf{r})=1\). Also, considering that the Pauli matrices have eigenvalues \(\pm 1\), it follows that \(D(\mathbf{r})\) has eigenvalues \({1\over 2}(1\pm \Vert \mathbf{r}\Vert )\), and therefore is a positive semidefinite operator. Being also a manifestly selfadjoint operator (\(\mathbf{r}\) is a real vector), it corresponds to what is usually called a density matrix and is generally interpreted as a classical statistical mixture of pure states. However, such interpretation is not without difficulties, as a same density matrix can have arbitrarily many different representations as a mixture of onedimensional projection operators (Hughston et al. 1993; Aerts and Sassoli de Bianchi 2014). Our assumption here is that density matrices are able to describe not only classical mixtures of states, but also (extended) pure states, and precisely those pure states that describe how an entity, during a measurement process, approaches the measurement’s “region of potentiality,” triggering in this way the actualization of a specific interaction.
This line segment, or diameter, associated with \(S_3\), precisely corresponds to that potentiality region characterizing the measurement context, which is responsible for the indeterministic “collapse” (as we shall see in a moment). But in order for this to occur, the entity has first to enter into contact with such region. This means that the measurement process has to involve a preparatory deterministic phase, through which the premeasurement state \(P(\mathbf{n})\) is brought into contact with the latter. This process corresponds to the immersion of a point particle associated with the vector \(\mathbf{n}\), representative of the entity’s premeasurement state, from the surface of the sphere to its interior, to reach the line segment representing the potentiality region, along an orthogonal path (see Fig. 4). If we denote \(\mathbf{n}^\parallel \) the point on the line segment obtained in this way, that is, \(\mathbf{n}^\parallel =(\mathbf{n}\cdot \hat{\mathbf{x}}_3)\,\hat{\mathbf{x}}_3 = \cos \theta \, \hat{\mathbf{x}}_3\), we can describe this deterministic movement of approach of the potentiality region by means of a parameter \(\tau \), which is varied from 0 to 1: \(\mathbf{r}(\tau ) = (1\tau )\,\mathbf{n} + \tau \, \cos \theta \, \hat{\mathbf{x}}_3\). Clearly, \(\mathbf{r}(0) = \mathbf{n}\) is the initial condition, and \(\mathbf{r}(1)=\mathbf{n}^\parallel =\cos \theta \, \hat{\mathbf{x}}_3\) is the final condition on the potentiality region.
Before explaining in the next section how this representation can be further generalized to an arbitrary number of dimensions, a few remarks are in order. What we have described is clearly a measurement of the first kind. Indeed, once the point particle has reached one of the two outcome positions \(\pm \hat{\mathbf{x}}_3\), if subjected again to the same measurement, being already located in one of the two anchor points of the elastic, we have \(\mathbf{n} = \mathbf{n}^\parallel = \pm \hat{\mathbf{x}}_3\), so that its position cannot be further changed by its collapse.
The disintegration points \(\varvec{\lambda }\) can be interpreted as variables specifying the measurement interactions. Thus, the model provides a consistent hiddenmeasurement interpretation of the quantum probabilities, as epistemic quantities characterizing our lack of knowledge regarding the measurement interaction that is actualized between the measured entity and the measuring apparatus, at each run of the experiment.
It is worth observing that almost each measurement interaction \(\varvec{\lambda }\) gives rise to a purely deterministic process, changing the state of the entity from \(\mathbf{n}^\parallel \) to either \(\hat{\mathbf{x}}_3\) or \(\hat{\mathbf{x}}_3\), depending whether \(\varvec{\lambda }\in A_+\), or \(\varvec{\lambda }\in A_\). We say ‘almost’ because when \(\varvec{\lambda }=\mathbf{n}^\parallel \), that is, when \(\varvec{\lambda }\) coincides with the point separating region \(A_\) from region \(A_+\), we have a situation of classical unstable equilibrium. This point of classical instability is at the origin of the distinction between the two outcomes, but being of measure zero, it does not contribute to the values of the probabilities associated with them. In other terms, although the border points \(\varvec{\lambda }=\mathbf{n}^\parallel \) are the “source of the possibilities,” they do not contribute to the values of the probabilities that are associated with them.
As a last remark, we observe that according to the extended Bloch model, since quantum probabilities would be related to a situation of lack of knowledge (not about the state of the entity, but about the deterministic interaction which is each time selected during a measurement), similarly to classical probabilities their nature would be epistemic, and not ontic, thus compatible with the view of a completely deterministic world as a whole. However, this is not in conflict with the idea of an ontological indeterminism, as the lack of knowledge subtended by quantum probabilities would in fact not be eliminable. This because if we try to remove the randomness incorporated in the quantum (and more generally, quantumlike) observational processes, acquiring over them a better control, necessarily we will have to alter them in such a radical way that in the end they will not correspond anymore to the test of the same property, or to the measurement of the same physical quantity (for a more specific discussion of this point, see for instance Sassoli de Bianchi 2015).
4 Modeling a general measurement
The preliminary 1986 study that allowed to extend the threedimensional Bloch sphere representation to also include measurements (as shown in the previous section) generated over the years a number of works, further exploring the explicative power contained in this modelization, also called the hiddenmeasurement interpretation of quantum mechanics (see Aerts and Sassoli de Bianchi 2014 and the references cited therein). According to it, a quantum measurement is an experimental context characterized by a lack of knowledge not about the state of the measured entity, but about the interaction between the measuring apparatus and the measured entity. And it is precisely through a mechanism of actualization of potential interactions (or of spatialization of nonspatial interactions) that the nonordinary entities belonging to the quantum theater would leave their ephemeral traces into our classical spatial theater (the first question we addressed in the Introduction).
For some time this hiddenmeasurement extension of the Bloch sphere, called the \(\epsilon \)model (Aerts 1998, 1999; Aerts and Sassoli de Bianchi 2014) (which also allowed an exploration of the intermediary region between classical and quantum) was taken in serious consideration only by a small number of physicists working in the foundations of physical theories. This is probably because of the existence of the socalled nogo theorems, such as those of Gleason (1957) and Kochen and Specker (1967), which were known to be valid only for Ndimensional Hilbert spaces with \(N>2\), hence not for the special case of twolevel systems (qubit), which therefore were considered to be pathological. Accordingly, the extended Bloch representation was mostly considered as a mathematical curiosity, but this was a misconception, as the hiddenmeasurement interpretation has little to do with a classical hiddenvariable theory, as is clear that the lack of knowledge is not associated with the state of the measured entity, but with its measurement interactions (so that the nogo theorems do not apply).
Despite this prejudice that no hiddenmeasurement modelizations could be given for dimensions \(N>2\), new results became available over the years, showing that the hiddenmeasurement mechanism was by no means restricted to twodimensional situations (Aerts et al. 1997; Coecke 1995a, b). These promising results were however not totally convincing, as what was still lacking was a natural generalization of the Bloch sphere representation beyond the twodimensional situation. Things changed in more recent times, when some important mathematical results became available, precisely providing this muchsought generalized Bloch sphere representation, which exploits the properties of the socalled generators of SU(N), the special unitary group of degree N (Arvind et al. 1997; Kimura 2003; Byrd and Khaneja 2003; Kimura and Kossakowski 2005; Bengtsson and Życzkowski 2006, 2013).
Thanks to these results, very recently we could extend the generalized Bloch construction to also include a full description of the measurements (including the degenerate ones), offering in this way what we think is a general and convincing solution to the measurement problem (Aerts and Sassoli de Bianchi 2014). In this section, we will explain how a Noutcome measurement can precisely be described, and therefore explained, in this extended Bloch modelization. In other terms, we will now consider an entity whose Hilbert state is \(\mathcal{H}_N={\mathbb C}^N\) (if it is a spins entity, then \(s={N1\over 2}\)).
What we need to observe is that, as already mentioned, the threedimensional Blochsphere representation is based on the property of the Pauli matrices and the identity operator of generating a basis for all linear operators on \({\mathbb C}^2\), so allowing to expand any operatorstate on them, in the form: \(D(\mathbf{r}) = {1\over 2}\left( \mathbb {I} + \mathbf{r}\cdot \varvec{\sigma }\right) \). This correspondence between \(2\times 2\) operatorstates and real vectors in the threedimensional unit ball is the expression of the wellknown homomorphism between SU(2) and SO(3). Such homomorphism cannot be extended beyond the \(N=2\) situation, but one can still represent operatorstates as real vectors in a unit sphere, which however will not be anymore threedimensional, nor completely filled with states.
Similar to the \(N=2\) situation, states can therefore be represented as vectors living in a unit ball. On its surface, one finds the rankone projection operators, and in its interior the density matrices, i.e., the operatorstates. The important difference with the standard Bloch representation is that the ball is now \((N^21)\)dimensional, i.e., it is not anymore representable in \({\mathbb R}^{3}\), and, more importantly, only a small portion of it contains states. Indeed, if it is true that for every operatorstate D one can always find a vector \(\mathbf{r}\in B_1({\mathbb R}^{N^21})\) such that D can be written in the form (15), the converse will not be generally true: given a vector \(\mathbf{r}\in B_1({\mathbb R}^{N^21})\), the operator (15) does not necessarily describe a state. The reason for this is that \(D(\mathbf{r})\), to be a state, needs not only to be of unit trace and selfadjoint, but also positive semidefinite, which is not automatically guaranteed for an operator written as the real linear combination (15). Just to give an example, consider the unit vector \(\mathbf{r}=(0,\dots ,0,1)^T\). Then, \(D(\mathbf{r})= {1\over N}\left( \mathbb {I} + c_N\Lambda _{N^21}\right) \), and according to (13) \(\Lambda _{N^21}=W_{N1}\), so that \(D(\mathbf{r}) = {N2\over N} b_N\rangle \langle b_N+ {2\over N} \sum _{j=1}^{N1} b_j\rangle \langle b_j\), which for \(N\ge 3\) is clearly a matrix with a strictly negative eigenvalue \({N2\over N}\), and therefore cannot be positive semidefinite.
What is important to observe is that although \(B_1({\mathbb R}^{N^21})\) is only partially filled with states, and that the shape of the region containing the states is rather complex, it is however a closed convex region. This follows from the wellknown fact that a convex linear combination of operatorstates is again an operatorstate. Thus, if \(\mathbf{r} = a_1\, \mathbf{r}_1 + a_2\, \mathbf{r}_2\), with \(a_1+a_2=1\), \(a_1,a_2\ge 0\), and \(\mathbf{r}_1\) and \(\mathbf{r}_2\) are representative of two operatorstates, from (15) we immediately obtain that \(D(\mathbf{r})=a_1D(\mathbf{r}_1)+a_2D(\mathbf{r}_2)\) is again an operatorstate, being a convex linear combination of operatorstates . We thus conclude that a vector \(\mathbf{r}\), which is a convex linear combination of two vectors \(\mathbf{r}_1\) and \(\mathbf{r}_2\), representative of states, is also representative of a bona fide state \(D(\mathbf{r})\).
We now explain how measurements can be represented within \(B_1({\mathbb R}^{N^21})\), so generalizing the previous twodimensional measurement model. We start by observing that also in the general Ndimensional situation a vectorstate \(\psi \rangle \) can always be represented, in a measurement context, by a onedimensional projection operator \(P_\psi = \psi \rangle \langle \psi \), as the only information one looses in the passage from \(\psi \rangle \) to \(P_\psi \) is the global phase of the former. To show this, we write: \(\psi \rangle = \sum _{i=1}^N b_i b_i\rangle \) and \(P_\psi =\sum _{i,j=1}^N b_i b_j^* b_i\rangle \langle b_j\), so that \(\langle b_kP_\psi b_\ell \rangle =b_kb_\ell ^*\). For \(k=\ell =1\), we have \(\langle b_1P_\psi b_1\rangle =b_1^2\). Assuming that \(b_1\ne 0\) (if this is not the case we can reason with \(b_2\), and so on), we can always fix the global phase of \(\psi \rangle \) so as to have \(b_1>0\). Then, \(b_1=\langle b_1P_\psi b_1\rangle ^{1\over 2}\), and we obtain: \(b_\ell =\langle b_\ell P_\psi b_1\rangle \langle b_1P_\psi b_1\rangle ^{{1\over 2}}\), \(\ell =1,\dots ,N\). Thus, except for a global phase factor, we can fully reconstruct \(\psi \rangle \) from \(P_\psi \).
Our goal is to explain how the values of the transition probabilities \(\mathcal{P}(P_\psi \rightarrow P_{a_i}) = \mathrm{Tr}\, P_\psi P_{a_i} =\langle a_i\psi \rangle ^2\) can be derived by means of a hiddenmeasurement mechanism, where \(P_{a_i}= a_i\rangle \langle a_i\), and the orthonormal vectors \(a_1\rangle ,\dots ,a_N\rangle \) are the eigenvectors of the spectral decomposition of an arbitrary observable \(A=\sum _{i=1}^N a_i P_{a_i}\), which for the moment we shall assume to be nondegenerate (the eigenvalues \(a_i\) are all distinct).
If the initial state is an eigenstate \(P_{a_j}\), then we know that \(\mathcal{P}(P_{a_j} \rightarrow P_{a_i}) = \delta _{ji}\). It follows from (16) that \(\cos \theta (\mathbf{n}_j,\mathbf{n}_i)={1\over N1}\), that is: \(\theta (\mathbf{n}_j,\mathbf{n}_i) = \theta _N\equiv \cos ^{1} ({1\over N1})\), for all \(i\ne j\). This means that the N unit vectors \(\mathbf{n}_i\), representative of the eigenstates \(P_{a_i}\), \(i=1\dots ,N\), are the vertices of a \((N1)\)dimensional simplex\(\triangle _{N1}\), inscribed in the unit ball, with edges of length \(\Vert \mathbf{n}_i \mathbf{n}_j\Vert = \sqrt{2(1\cos \theta _N)} = \sqrt{2N\over N1}\). Also, considering that \(\triangle _{N1}\) is a convex set of vectors, it immediately follows that all points contained in it are representative of operatorstates, in accordance with the fact that the states in \(B_1(\mathbb {R}^{N^21})\) form a closed convex subset.
For \(N=2\), \(\theta _1 = \pi \), and \(\triangle _{1}\) is a line segment of length 2 inscribed in a threedimensional ball, as we have seen in the previous section. For \(N=3\), \(\theta _2 = {\pi \over 3}\), and \(\triangle _{2}\) is an equilateral triangle of area \({3\sqrt{3}\over 4}\). For \(N=4\), \(\theta _3 \approx 0.6\,\pi \), and \(\triangle _{3}\) is a tetrahedron of volume \({1\over 3}({4\over 3})^{3\over 2}\). For \(N=5\), \(\theta _4 \approx 0.58\,\pi \), and \(\triangle _{4}\) is a pentachoron; and so on.
To see how all this works, we will only describe here, for simplicity, the \(N=3\) situation, as the general situation proceeds according to the same logic and is therefore a straightforward generalization (Aerts and Sassoli de Bianchi 2014). So, the measurement context is in this case represented by a twodimensional triangular elastic membrane inscribed in a eightdimensional ball, and the three possible outcomes are the eigenstates associated with its three vertex vectors \(\mathbf{n}_1\), \(\mathbf{n}_2\) and \(\mathbf{n}_3\). If the initial, premeasurement state \(P(\mathbf{n})\) is associated with a unit vector \(\mathbf{n}\in B_1({\mathbb R}^{8})\), the entity proceeds first with a deterministic movement \(\mathbf{r}(\tau ) = (1\tau )\,\mathbf{n} + \tau \, \mathbf{n}^\parallel \), \(\tau \in [0,1]\), which brings the state of the entity from the point at the surface \(\mathbf{r}(0) = \mathbf{n}\), to the onmembrane point \(\mathbf{r}(1)=\mathbf{n}^\parallel \), along a path that is orthogonal to \(\triangle _{2}\) (see Fig. 6).
Once the particle has reached its onmembrane position \(\mathbf{n}^\parallel \), it defines three different triangular subregions \(A_1\), \(A_2\) and \(A_3\), delineated by the line segments connecting the particle’s position with the three vertex points (see Fig. 6). One should think of these line segments as “tension lines” altering the functioning of the membrane, in the sense of making it less easy to disintegrate along them. Then, at some moment, the membrane disintegrates, at some unpredictable point \(\varvec{\lambda }\), belonging to one of these three subregions. The disintegration then propagates inside that specific subregion, but not into the other two subregions, because of the presence of the tension lines. This causes the two anchor points of the disintegrating subregion to tear away, producing the detachment of the membrane, which being elastic contracts towards the only remaining anchor point, drawing to that position also the particle that is attached to it, which in this way reaches its final destination (state), corresponding to the outcome of the measurement (see Fig. 6).
Reasoning in the same way as we did in the \(N=2\) case, it is clear that the transition probability \(\mathcal{P}(P_\psi \rightarrow P_{a_i})\) is given by the probability \(\mathcal{P}(\varvec{\lambda }\in A_i)\) that the disintegration point \(\varvec{\lambda }\) belongs to region \(A_i\). Considering that \(\triangle _{2}\) is an equilateral triangle of area \({3\sqrt{3}\over 4}\), we thus have \(\mathcal{P}(\varvec{\lambda }\in A_i)= {4\over 3\sqrt{3}}\mu (A_i)\). Let us consider for instance \(A_1\). It is a triangle with vertices \(\mathbf{n}_2\), \(\mathbf{n}_3\) and \(\mathbf{n}^\parallel = \sum _{i=1}^3 n^\parallel _i \mathbf{n}_i\). Using the explicit coordinates of the three vertices of \(A_1\), we can easily calculate its area. For this, we can use a system of coordinate directly in the plane of the triangle, such that \(\mathbf{n}_2=(0,1)\equiv (x_1,x_2)\), \(\mathbf{n}_3=({\sqrt{3}\over 2},{1\over 2})\equiv (y_1,y_2)\), and \(\mathbf{n}_1=({\sqrt{3}\over 2},{1\over 2})\equiv (z_1,z_2)\). To calculate the area, we can then use the general formula: \(\mu (A_1)={1\over 2}\left y_1x_2 +z_1x_2+x_1y_2z_1y_2x_1z_2+y_1z_2 \right \). After a calculation without difficulties one finds, using (17): \(\mu (A_1)={3\sqrt{3}\over 4}n^\parallel _1\). Doing similar calculation for \(A_2\) and \(A_3\), one obtains that \(\mu (A_i)={3\sqrt{3}\over 4}n^\parallel _i\), \(i=1,2,3\), so that \(\mathcal{P}(\varvec{\lambda }\in A_i)= {4\over 3\sqrt{3}}\mu (A_i)=n^\parallel _i\), \(i=1,2,3\), in accordance with the predictions of the quantum mechanical Born rule (18).
The hiddenmeasurement membrane mechanism that we have described for the \(N=2\) case in the previous section, and for the \(N=3\) case in the present section, generalizes in a natural way to an arbitrary number of dimensions N, and we refer the reader to Aerts and Sassoli de Bianchi (2014) for all the mathematical details. The membrane mechanism can also be used to describe measurements of degenerate observables. Then, the subregions associated with the degenerate eigenvalues are fused together and form bigger composite subregions, so that when the initial disintegration point \(\varvec{\lambda }\) takes place inside one of them, the process draws the particle not to a vertex point of \(\triangle _{N1}\), but to one of its subsimplexes. The collapse of the elastic substance remains, however, compatible with the predictions of the Lüdersvon Neumann projection formula, but to complete the process the particle also has to reemerge from the subsimplex region, to deterministically reach its final position, at the surface of the unit ball.
In other words, in the general situation, a measurement is to be understood as a tripartite process formed by (1) an initial deterministic decoherencelike process, corresponding to the particle reaching the “onmembrane region of potentiality;” (2) a subsequent indeterministic collapselike process, corresponding to the disintegration of the elastic substance filling the simplex, with the particle being drawn to some of its peripheral points; and (3) a possible final deterministic purificationlike process, bringing again the particle to a unit distance from the center of the sphere (Aerts and Sassoli de Bianchi 2014).
5 Spin eigenstates and space directions
In the previous two sections, we have shown that not only the standard Bloch sphere representation of twostate systems, like spin\({1\over 2}\) entities, can be generalized to describe general Nstate systems, like spins entities with \(s={N1\over 2}\), but also that it can be extended—by means of a hiddenmeasurement mechanism—to include a full representation of all possible measurements. We have therefore answered the question regarding the nature of the interaction between the quantum and classical theaters (an answer which constitutes a solution to the measurement problem). In this section, we want to answer the second question we have addressed in the Introduction, regarding the general relation between spin eigenstates and space directions, by exploiting the descriptive power of the extended Bloch model.
We have already observed that the classicallike image of a quantum spin, as a set of undetermined classical angular momentum vectors lying on a cone, is inconsistent, but we have also seen that for a spin\({1\over 2}\) entity there is a direct onetoone correspondence between spin eigenstates and directions of space. These directions, however, are not anymore those characterizing classical angular momenta, but those for which the states produce a predetermined outcome, in spin measurements along those same directions.
Now, even beyond the \(s={1\over 2}\) case, eigenstates will continue to be associated with space directions, as is clear that they are the eigenstates of spin observables that can only be defined by specifying a space direction (the orientation of the Stern–Gerlach apparatus). However, we cannot expect anymore each eigenstate to be entirely characterized by a single space direction, as is clear that more than a single eigenvalue can be observed for each direction. In addition to that, we know that in the extended Bloch model, states are not anymore represented by threedimensional vectors, beyond the special spin\({1\over 2}\) situation. So, how do spin eigenstates relate, in general, to the directions of our threedimensional space, that is, to the directions defining the spin observables for which they are the eigenstates? The answer to this question is contained in the following:
Proposition 1
This does not mean, however, that when \(\mathbf{v}\) spans a threedimensional subball of \(B_1({\mathbb R}^{N^21})\), correspondingly to as \(\hat{\mathbf{n}}\) spans \(B_1({\mathbb R}^3)\), the vertex vectors \(\mathbf{n}_\mu \) will also span the surfaces of threedimensional subballs. Indeed, there are no unit subballs filled with states within \(B_1({\mathbb R}^{N^21})\) (see below).
The “space” vectors \(\mathbf{v}\) have a simple expression in the \(N=2\) and \(N=3\) cases. Indeed, for \(N=2\), we have: \(\mathbf{v} = {1\over 2}(\mathbf{n}_{1\over 2}\mathbf{n}_{{1\over 2}})\), and for \(N=3\), we have: \(\mathbf{v} = {1\over \sqrt{3}}(\mathbf{n}_{1}\mathbf{n}_{1})\) (see Fig. 7). In other terms, for elementary spin\({1\over 2}\) fermions and spin1 bosons, the space directions \(\mathbf{v}\) are simply given by the difference of the two eigenvectors of extremal eigenvalues. For spin\({1\over 2}\) entities, this is so because there are only two eigenvectors available, whereas for spin1 entities it is because one of the eigenvectors is associated with a zero eigenvalue, and therefore cannot contribute to the sum (19). Note that the fact that \(\mathbf{v}\) is given by the normalized difference of two eigenvectors means that the associated edge of the measurement simplex is oriented like \(\mathbf{v}\).
6 Composite spin entities
In the present section, we explore the relation between spin eigenstates and space directions in the case of a spin entity formed by two subentities. We have the following:
Proposition 2
The following remarks are in order. In the previous section, we have observed that, for noncomposite spin entities, the \(s={1\over 2}\) and \(s=1\) cases were special, in the sense that the difference of the two extremal eigenstates (corresponding to one of the edges of the measurement simplex) is oriented like \(\mathbf{v}\). When we combine two spin\({1\over 2}\) entities (\(s_1=s_2={1\over 2}\)), we have four eigenstates, but two of them have zero eigenvalue: the singlet state \(\mathbf{n}_{0,0}\) and the triplet state \(\mathbf{n}_{1,0}\), which therefore cannot contribute to the sum (29). This means that also in this case \(\mathbf{v}\) is simply given by the difference of the two extremal spin states: \(\mathbf{v}= \sqrt{3\over 8}(\mathbf{n}_{1,1}\mathbf{n}_{1,1})\).
We also observe that: \(\mathbf{v}\cdot \mathbf{n}_{s,\mu _s} = {Nd_{N_1N_2}\over N1}\,\mu _s\), which means that the eigenvectorstates \(\mathbf{n}_{s,\mu _s}\), when orthogonally projected onto \(\mathbf{v}\), give rise also in this case to equally spaced points. However, since there are now degenerate eigenvalues, although all the vectors \(\mathbf{n}_{s,\mu _s}\) are oriented differently within \(B_1({\mathbb R}^{N^21})\), as they are the vertex vectors of a \((N1)\)simplex, those associated with the same eigenvalue will produce the same spot, when projected onto the \(\mathbf{v}\)axis (see Fig. 9).
7 Connecting classical and quantum elements of reality
In the introductory section, we have explained that because of our human condition within reality, we can only explore its content from the perspective of our Euclidean theater. Therefore, our situation is in a sense similar to that described by the philosopher Plato in his famous allegory of the cave. However, Plato opposed the material world known to us through our senses (described as the shadows on the cave’s wall) to a world of immutable “ideas,” possessing an ultimate reality. Different from Plato, the view we bring forward in this article is that although our Euclidean theater is like a Plato’s cave, being the expression of a limited perspective, different caves possibly exist, corresponding to the different vantage points on reality that in principle can be adopted.
What we have called the “quantum theater” is in fact nothing but another cave, one not directly inhabited by us humans, and we could generally say that the whole of reality is a construction about the different possible caves (or theaters, we use these two terms here as synonyms). Our work, as investigators and participators of reality, is then not only that of identifying the content of the cave we inhabit, and of the other possible caves, but also the relations (the partial morphisms) that exist between their different elements of reality. This means that we do not necessarily ask reality to be fully contained in a single fundamental theater, although of course we cannot exclude that such an ultimate stage could not one day be identified.
We neither cannot exclude that there may be elements of reality belonging to one cave that cannot find a direct correspondence in another cave. We already know that certain caves are, in a sense, bigger than others. For instance, most physicists believe that the quantum cave should contain the classical one, but this cannot be taken for granted, as we know that there are severe structural shortcomings in the quantum representation. For example, in the standard quantum formalism, entities that are fully separated in experimental terms cannot be described, whereas there are plenty of separated entities in the classical theater (Aerts 1982, 1984, 1999, 2000). Also, there are elements of reality which appear to be of a genuine intermediate nature, which cannot be represented neither in the classical nor in the quantum caves, as they belong to a truly intermediate representation, which so far has not received much attention (Aerts 1998, 1999; Aerts and Sassoli de Bianchi 2015a, b). And it may very well be that the many failed tentative to unify gravitational and quantum elements of reality, within a same consistent big theater, could be due to the fact that, for structural reasons, a single “quantum & gravitational” theater cannot be constructed.
This last remark is crucial to understand how the correspondences between elements of reality belonging to different caves should be understood. In that respect, let us make precise that what we mean here by the notion of “element of reality” is exactly what was meant by Einstein, Podolsky and Rosen, in their famous 1935 article (Einstein et al. 1935). An element of reality is a state of prediction: a property of an entity that we know is actual, in the sense that, should we decide to observe it (i.e., to test its actuality), the outcome of the observation would be certainly successful. So, what we have to consider is that the way we observe a classical entity is different form the way we observe a quantum one. More specifically, and coming back now to the example of spins, which more particularly concerns us in this article, when we compare a quantum spin with a classical spin, i.e., with an angular momentum understood as a “state of rotation in the threedimensional Euclidean space,” we need not to forget that their elements of reality are operationally defined in a different way. Indeed, for a quantum spin, an observation can be invasive (as it can produce a change of state), unless the entity is precisely in an eigenstate of the observation in question, whereas for a classical spin the observation is always noninvasive (as it never produce a change of state), independently of the state of the entity.
In other terms, when we look for those elements of reality belonging to the quantum cave that can be put in a correspondence with the elements of reality belonging to the classical one, only eigenstates can be considered, and more precisely eigenstates in association with their specific measurements. This because only for eigenstates the observational process is of the pure discovery kind, as is the case for all classical elements of reality, and of course it would make no sense to try to compare quantum potential properties, described by superpositions of eigenstates, with classical actual properties, as superpositions of eigenstates truly describe elements of the quantum cave that have no direct counterpart with what is contained in the classical one.
On the other hand, the correspondence is not anymore complete for the Bloch vectors \(\mathbf{n}_{1,0}\) and \(\mathbf{n}_{0,0}\), which are representative of the entangled (nonproduct) states \(\hat{\mathbf{n}},1,0\rangle = {1\over \sqrt{2}}(\hat{\mathbf{n}},+{1\over 2}\rangle \otimes \hat{\mathbf{n}},{1\over 2}\rangle +\hat{\mathbf{n}},{1\over 2}\rangle \otimes \hat{\mathbf{n}},+{1\over 2}\rangle )\) and \(\hat{\mathbf{n}},0,0\rangle = {1\over \sqrt{2}}(\hat{\mathbf{n}},+{1\over 2}\rangle \otimes \hat{\mathbf{n}},{1\over 2}\rangle \hat{\mathbf{n}},{1\over 2}\rangle \otimes \hat{\mathbf{n}},+{1\over 2}\rangle )\), as is clear that they are not anymore eigenvectors of the oneentity spin observables \(S^{(1)}_{\hat{\mathbf{n}}}\otimes \mathbb {I}^{(2)}\) and \(\mathbb {I}^{(1)}\otimes S^{(2)}_{\hat{\mathbf{n}}}\). It is worth emphasizing, however, that the lack of a full correspondence between the quantum and classical theaters can also manifest when the composite system is in a state which is a product of eigenstates. For example, the two states \(\hat{\mathbf{n}},\pm {1\over 2}\rangle \otimes \hat{\mathbf{n}},\mp {1\over 2}\rangle \), although they are eigenstates of the oneentity operators \(S_{\hat{\mathbf{n}}}^{(1)}\otimes \mathbb {I}^{(2)}\), \((S^{(1)})^2\otimes \mathbb {I}^{(2)}\), \(\mathbb {I}^{(1)}\otimes S_{\hat{\mathbf{n}}}^{(2)}\), \(\mathbb {I}^{(1)}\otimes (S^{(2)})^2\), and of the total spin operator \(S_{\hat{\mathbf{n}}}\), they are not however eigenstates of \(S^2\), and therefore cannot be represented as classical threedimensional vectors, as this would imply a well defined value for \(S^2\).
This difficulty, of obtaining a fully consistent correspondence for all spin eigenstates, is related to the wellknown fact that we can either choose \(\{(S^{(1)})^{ 2}\otimes \mathbb {I}^{(2)}, \mathbb {I}^{(1)}\otimes (S^{(2)})^{ 2}, S^2, S_{\hat{\mathbf{n}}}\}\), as a complete set of commuting observables, or \(\{S_{\hat{\mathbf{n}}}^{(1)}\otimes \mathbb {I}^{(2)}, (S^{(1)})^2\otimes \mathbb {I}^{(2)}, \mathbb {I}^{(1)}\otimes S_{\hat{\mathbf{n}}}^{(2)}, \mathbb {I}^{(1)}\otimes (S^{(2)})^2, S_{\hat{\mathbf{n}}}\}\). In the first case, we have the basis of spin eigenvectors \(\{\hat{\mathbf{n}},1,1\rangle , \hat{\mathbf{n}},1,1\rangle , \hat{\mathbf{n}},1,0\rangle , \hat{\mathbf{n}},0,0\rangle \}\), and we can always exhibit a correspondence for the total spin, but not always a correspondence for its components. In the second case, we have the basis of spin eigenvectors \(\{ \hat{\mathbf{n}},+{1\over 2}\rangle \otimes \hat{\mathbf{n}},+{1\over 2}\rangle , \hat{\mathbf{n}},{1\over 2}\rangle \otimes \hat{\mathbf{n}},{1\over 2}\rangle , \hat{\mathbf{n}},+{1\over 2}\rangle \otimes \hat{\mathbf{n}},{1\over 2}\rangle , \hat{\mathbf{n}},{1\over 2}\rangle \otimes \hat{\mathbf{n}},+{1\over 2}\rangle \}\), and we can always exhibit a correspondence for the composing spins, but not always a correspondence for the total spin. Note that the above two eigenbases share the first two of their vectors, which are precisely those eigenstates that admit a full representation, within the spatial theater, for both the total spin and for its components.
According to the above, it is clear that an alternative, equivalent version of Proposition 2 can be written, using a different set of spin eigenvectors to defne the space vectors \(\mathbf{v}\). More precisely, we have:
8 Concluding remarks
In this article, we have presented the extended Bloch representation of quantum mechanics and the associated hiddenmeasurement interpretation, and used it to explore spin states and spin measurements. The extended Bloch representation is a candidate for a refined version of quantum theory, in which operatorstates (density matrices) can also play a role as pure states in the description of what goes on during a measurement process. We refer the reader to (Aerts and Sassoli de Bianchi 2014) for a more detailed and complete description of the model, and for a further generalization of it (see also Aerts and Sassoli de Bianchi 2015 for a general application of the model to multipartite systems).
Let us mention that it is also possible to relax the hypothesis that the substance filling the measurement simplexes is uniform (in the sense of having a uniform probability density of disintegrating in a point), and still be able to derive the Born rule, if a (universal) average over all possible nonuniform substances is considered. Also, one can show that nonuniform substances, when not averaged out, describe physical entities of a genuine intermediate nature, which cannot be described by the classical or quantum formalisms (their elements of reality being neither contained in the classical nor in the quantum theaters) (Aerts and Sassoli de Bianchi 2014, 2015a, b).
The extended Bloch representation elucidates the mechanism through which quantum entities interact with classical measurement apparatus. The interaction is invasive and governed by (nonspatial) fluctuations, which by definition cannot be controlled by the experimenter, and therefore produce a genuine indeterministic change of the state of the measured entity. According to this interpretation, a quantum measurement is not only a process of discovery, but also, and above all, a process of creation.
The extended Bloch representation also allows us to gain some insight into how the vaster quantum world relates to our ordinary “niche” reality, made of macroscopic objects which are always present in the threedimensional Euclidean theater. In that respect, we have shown that quantum spins cannot be considered as intrinsic angular momenta, as no space directions can be directly associated to them, beyond the \(s={1\over 2}\) situation. On the other hand, they always remain in a specific relation to the space directions, as expressed by the fact that they possess a predetermined orientation with respect to the vectors (19) and (29)–(34), which are the representative of the space directions within the Blochean theater.
It is worth observing that even though spin\({1\over 2}\) entities can be characterized, when isolated, by single space directions, not for this they should be considered to be generally in space, as is clear that SU(2) is a double cover of SO(3), which means that \(2\pi \) and \(4\pi \) rotations, although they correspond to two distinct elements of SU(2), are mapped onto the same element of SO(3). In the Bloch sphere, this twotoone correspondence cannot be seen, as global phase factors are not represented into it. However, when a spin is coupled with the translational degrees of freedom of a microscopic entity, the distinction between \(2\pi \) and \(4\pi \) rotations can become observable, in the suitable experimental context.
Rauch’s experiment has shown us two things: the first one is that, as we said, even spin\({1\over 2}\) entities are not in space, considering that a \(2\pi \) rotation is not sufficient to bring them back exactly in the same state, when not in an isolated condition. In fact, this was already clear from our discussion in the Introduction, considering that we could not find classical elements of reality able to account for spin observations along all possible directions. And this also emerged in our analysis of the combination of two spin–\({1\over 2}\) entities, which although they can be in a welldefined eigenstate within the composite entity (when in a product state), not for this they can always be associated with a welldefined threedimensional classical angular momentum vectors.
Of course, if, strictly speaking, spins are not in space, it means that it is not totally correct to interpret the action of the magnetic field on the spinor component of the wave function in Rauch’s experiments as a true rotation in space. And in fact, it has been proposed (also by one of us) that a minimal interpretation of the experiment is as a longitudinal Stern–Gerlach effect, i.e., that the (apparent) Larmor precession would only result from the interference between the two Stern–Gerlach states in the weak field limit (Mezei 1988; Martin and Sassoli de Bianchi 1994).
Rauch’s experiment has also shown us that the notion of nonspatiality equally applies to the configuration space part of the wave function. Indeed, it shows that although a neutron can only be detected in one place, it nevertheless can be acted upon, simultaneously, from different separated places, in a way that can produce a measurable effect. And if we reflect attentively, such a possibility can only be understood if we admit that a neutron is an entity which is not permanently present in space, but only possibly enters it when it interacts with a measuring apparatus, which in a sense is able to “drag” it into a spatial state (Aerts 1999).
From a conceptual point of view, we should certainly distinguish the notion of space direction from the notion of space location. Elementary entities are generally to be considered not in space in both senses: because they do not generally have an actual position in space, but only a potential one, and because their spins (if they are not zero) are not oriented toward a space direction, apart from the special case of isolated spin\({1\over 2}\) entities (where the term “isolated” means here not coupled with other systems, including the translational degrees of freedom of the very entity carrying the spin). Now, as we said, microscopic entities are able to acquire a location in space, in specific contexts, as is clear that we can localize them in space with arbitrary precision. On the other hand, their spins, with the exception of the \(s={1\over 2}\) case, remain always out of alignment with respect to space, as is clear that all spin eigenvectors within the Bloch sphere are oriented in directions which are different from those of the vectors \(\mathbf{v}\), defined in Proposition 1, 2 and 2bis, which are the referents of the space directions within the quantum theater.
In the special \(s={1\over 2}\) situation, the space directions and the directions of the spin eigenstates coincide, but as we have seen the correspondence between spin eigenvectors and angular momentum vectors is incomplete: because it can only be established for a single spin measurement at a time, and because it does not fully work when composite entities are considered. Therefore, even spin\({1\over 2}\) should not be considered to be in space (or fully in space), i.e., to be representable by one (or many) threedimensional vectors of suitable length and orientation.
But although generally not in space, quantum spin entities, when in an eigenstate, are nevertheless always in a specific relation to space, as evidenced by the fact that the eigenvectors in the Bloch sphere always maintain fixed and predetermined orientations with respect to the “spatial” vectors \(\mathbf{v}\). These orientations are precisely those that allow the eigenvectors to always be positioned at the vertices of the measurement simplexes, so that they cannot be affected by the collapse of the corresponding membranes. This is why we can predict in advance the outcome of the measurements, and bring them into correspondence with the classical angular momentum elements of reality of the classical theater. It is in that sense, and only in that sense, that we can justify the assertion that, say, an electron has a spin of a given magnitude along a given direction.
However, as emphasized many times, when we combine spins, the classical picture breaks down, as is clear that there is no simple relation between the elements of reality described by two threedimensional Bloch spheres, associated with two separated spin\({1\over 2}\) entities, and the 15dimensional Bloch sphere describing the states emerging from their quantum combinations. This because no sum of threedimensional vectors will ever be able to account for the emergence of a 15dimensional Bloch sphere from two threedimensional ones, and of a tetrahedron (3simplex) from two line segments (1simplexes).
A final comment is in order. As we explained in Sect. 7, not all the elements of reality that are present in the quantum theater do necessarily have a correspondence, however partial, with some of the elements of reality present in the classical theater. Quantum spins can individually be observed in the classical theater also because they are all in a specific relation to space directions, i.e., to classical elements of reality. But this need not be the case for other quantum properties, such as, for instance, the color charge properties of individual quarks, which, as far as we know, and contrary to spins, have no relations to space directions, which may be one of the reasons of their confinement within the quantum cave.
 (1)
Due to our presence, for hundreds of thousands of years, in that specific niche of reality which is the surface of our planet Earth, surrounded by material objects that with good approximation obey the Newton’s laws, we have constructed an Euclidean theater to stage our relations and interactions with these classical entities.
 (2)
In more recent times, we became aware of the existence of quantum and relativistic entities, whose reality could only be put in a partial correspondence with the properties and behaviors of the previously known Euclidean entities, that is, we could only find partial morphisms, and not perfect isomorphisms. For instance, quantum eigenstates could be linked, to some extent, to classical elements of reality, but not their superpositions.
 (3)
Although we have good reasons to believe that the quantum and relativistic theaters are the expression of more advanced theories of reality, we nonetheless continue to be strongly influenced by our Euclidean theater, also because we are ordinarily not aware that it has been constructed by our ancestors, in particular as regards the primitive notions of space and time. Therefore, we have interest in becoming fully aware of this ‘construction aspect,’ taking into consideration its effect in the way we conceive, and try to understand, the more recently discovered nonEuclidian quantum and relativistic elements of reality.
Our emphasis on this constructive aspect, on the importance of the EPR reality criterion, on the distinction between processes of creation and discovery, and on the specific role played by our Euclidean theater (for historical reasons related to our evolution on this planet), is what typically distinguish multiplex realism from other approaches where realism has been understood in a pluralistic sense, as for instance in Heisenberg’s notion of closed theories, or in Cartwright’s metaphysical nomological realism (see for instance (Bokulich 2008) and the references cited therein). However, a comprehensive analysis of the differences and similarities of multiplex realism with respect to the pluralistic views that have been proposed to date would go beyond the scope of this article, but we plan to come back to this philosophical discussion in future works.
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Vrije Universiteit Brussel 

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