# Relational Quantum Entanglement Beyond
Non-Separable and Contextual Relativism

Christian de Ronde1,2,3,4 and César Massri5,6

###### Abstract

In this paper we address the relativist-perspectival nature of the orthodox
definition of quantum entanglement in terms of preferred factorizations. We
also consider this aspect aspect within the generalized definition of
entanglement proposed by Barnum et al. [6, 7] in terms of preferred
observables. More specifically, we will discuss the non-separable relativism
implied by the orthodox definition of entanglement, the contextual relativism
implied by its generalization as well as some other serious problems presently
discussed within the specialized literature. In the second part of this work,
we address a recently proposed objective-invariant definition of entanglement
understood as the actual and potential coding of effective and intensive
relations [32]. Through the derivation of two theorems we will show explicitly
how this new objective definition of entanglement is able to escape both non-
separable relativism and contextual relativism. According to these theorems,
within this proposed relational definition, all possible subsets of
observables as well as all possible factorizations can be globally considered
as making reference to the same (potential) state of affairs. The conclusion
is that, unlike with the orthodox definitions, this new objective-relational
notion of entanglement is able to bypass relativism right from the start
opening the door to a realist understanding of quantum correlations.
Key-words: Entanglement, separability, contextuality, relativism, objectivity,
relationalism.

1\. Philosophy Institute Dr. A. Korn, University of Buenos Aires - CONICET
2\. Institute of Engineering - National University Arturo Jauretche, ,
Argentina.
3\. Center Leo Apostel for Interdisciplinary Studies - Vrije Universiteit
Brussel, Belgium.
4\. Federal University of Santa Catarina - Brazil.
5\. Institute of Mathematical Investigations Luis A. Santaló, UBA - CONICET
6\. University CAECE, Argentina.

##  1 The Strange (Hi)Story of Quantum Entanglement

The notion of entanglement (Verschränkung) was introduced in Quantum Mechanics
(QM) by Albert Einstein, Boris Podolsky and Nathan Rosen in their famous 1935
“EPR paper” [45]. That same year, Erwin Schödinger explicitly defined and
named the new born concept [67]. The main purpose of the introduction of
entanglement by Einstein and Schrödinger was to expose the inconsistencies
present in what had already become the orthodox “collapse” interpretation of
QM.111The paradox enlightened by EPR had been conceived by Einstein some years
before in discussion with Schrödinger. This earlier Gedankenexperiment, which
Einstein regarded as more clear, consisted in entangled particles encapsulated
in two different boxes and was finally published in 1948 [44]. Both Einstein
and Schrödinger were trying to expose why “quantum jumps” precluded the
possibility of the theory of quanta to refer to an objectively represented
state of affairs. Something, they feared, had become regarded by the community
of physicists as a sort of chimera. As remarked by Alisa Bokulich and Gregg
Jaeger, “[in the EPR paper] the possibility of such a phenomenon [of
entanglement] in quantum mechanics was taken to be a reductio ad absurdum
showing that there is a fundamental flaw with the theory.” In one of his
papers from 1935 Schödinger also remarked an astonishing consequence of the
orthodox collapse interpretation of QM, namely, that when two systems get
entangled through a known interaction, the knowledge we have of the parts
might anyhow decrease.

> “If two separated bodies, each by itself known maximally, enter a situation
> in which they influence each other, and separate again, then there occurs
> regularly that which I have just called entanglement of our knowledge of the
> two bodies. The combined expectation-catalog consists initially of a logical
> sum of the individual catalogs; during the process it develops causally in
> accord with known law (there is no question of measurement here). The
> knowledge remains maximal, but at the end, if the two bodies have again
> separated, it is not again split into a logical sum of knowledges about the
> individual bodies. What still remains of that may have become less than
> maximal, even very strongly so.—One notes the great difference over against
> the classical model theory, where of course from known initial states and
> with known interaction the individual states would be exactly known.” [67,
> p. 161]

Einstein’s and Schrödinger’s criticisms to QM through the notion of
entanglement were in fact targeting directly the artificial “collapse” of the
quantum wave function. A new “quantum jump” that had been introduced by Dirac
just a few years before in order to bridge the gap between quantum
superpositions and single measurement outcomes [42]. Similar in nature to the
original “quantum jumps” that Bohr had introduced in his famous 1913 model of
the atom, this new ad hoc process was also completely irrepresentable within
the theory. Einstein, was clearly uncomfortable with this unjustified
addition. Targeting the collapse, he is quoted by Hugh Everett [47, p. 88] to
have said that he “could not believe that a mouse could bring about drastic
changes in the universe simply by looking at it”. He also shared his fears
with Wolfgang Pauli who would later recall:

> “Einstein’s opposition to [QM] is again reflected in his papers which he
> published, at first in collaboration with _Rosen_ and _Podolsky_ , and later
> alone, as a critique of the concept of reality in quantum mechanics. We
> often discussed these questions together, and I invariably profited very
> greatly even when I could not agree with _Einstein_ ’s view. ‘Physics is
> after all the description of reality’ he said to me, continuing, with a
> sarcastic glance in my direction ‘or should I perhaps say physics is the
> description of what one merely imagines?’ This question clearly shows
> _Einstein_ ’s concern that the objective character of physics might be lost
> through a theory of the type of quantum mechanics, in that as a consequence
> of a wider conception of the objectivity of an explanation of nature the
> difference between physical reality and dream or hallucination might become
> blurred.” [60, p. 122]

Schrödinger was also disgusted by the introduction of this “jump” to which he
would refer with great sarcasm:

> “But jokes apart, I shall not waste the time by tritely ridiculing the
> attitude that the state-vector (or wave function) undergoes an abrupt
> change, when ‘I’ choose to inspect a registering tape. (Another person does
> not inspect it, hence for him no change occurs.) The orthodox school wards
> off such insulting smiles by calling us to order: would we at last take
> notice of the fact that according to them the wave function does not
> indicate the state of the physical object but its relation to the subject;
> this relation depends on the knowledge the subject has acquired, which may
> differ for different subjects, and so must the wave function.” [47, p. 95]

Regardless of the strong and clear arguments presented by the two rebels,
mainly due to the triumph of Bohr’s anti-realist interpretation of QM sealed
later by the post-war coming into power of U.S. instrumentalism, the notion of
quantum entanglement remained buried for almost half a century. As Jeffrey Bub
[16] explains: “Most physicists attributed the puzzling features of entangled
quantum states to Einstein’s inappropriate ‘detached observer’ view of
physical theory, and regarded Bohr’s reply to the EPR argument (Bohr, 1935) as
vindicating the Copenhagen interpretation.” Physicists were not taught about
entanglement in Universities and the notion became considered as an anomaly
arising from old metaphysical discussions, a ghost which had been silently
forgotten by more down-to-earth and useful physical applications.222As
recalled by Lee Smolin [65, p. 312]: “When I learned physics in the 1970s, it
was almost as if we were being taught to look down on people who thought about
foundational problems. When we asked about the foundational issues in quantum
theory, we were told that no one fully understood them but that concern with
them was no longer part of science. The job was to take quantum mechanics as
given and apply it to new problems. The spirit was pragmatic; ‘Shut up and
calculate’ was the mantra. People who couldn’t let go of their misgivings over
the meaning of quantum theory were regarded as losers who couldn’t do the
work.” This was at least until during the first years of the 1980s Alain
Aspect and his group in Orsay were able to experimentally test in the lab the
strange correlations exposed in the old EPR thought-experiment [3]. Since it
was not possible to measure the correlation between single outcomes —as
originally presupposed in the EPR paper—, Aspect used a reconfigured
experiment proposed by John Bell who a few decades before, during his spare
time from his real work at C.E.R.N. in particle physics, derived an inequality
for the correlations imposed by classical probability —something that Boole
had already done one century before.333The relation between the famous Bell’s
inequalities from 1964 and Boole’s less known inequalities from 1862, derived
almost exactly one century before, has been explicitly considered by Itamar
Pitowsky in [62]. The existence of a violation in the Boole-Bell inequality
within an EPR type-experiment implied a violation of classical correlations.
This only meant that classical correlations could not represent consistently
the data found within the experiment. Period. It did not logically imply —as
it is still presupposed in great part of the physical and philosophical
literature—, the existence of entanglement or quantum correlations. Something
that should have been obvious to everyone from the fact that Boole-Bell
inequalities are a statistical statement about classical probability, not
about QM.

Aspect’s experiment became a huge boost for the young discipline called
“philosophy of QM”. But its influence was not restricted to philosophy. The
post-war generation of physicists which had been trained with instrumentalist
values and a purely pragmatic understanding of physics suddenly became aware
of the fact that quantum correlations were much stronger than classical ones.
It became evident —to some of them— that the existence of entanglement in
quantum correlations could be used as a resource of information transfer. As
described by Bub [16], “[…] it was not until the 1980s that physicists,
computer scientists, and cryptographers began to regard the non-local
correlations of entangled quantum states as a new kind of non-classical
resource that could be exploited, rather than an embarrassment to be explained
away.” In this new context, full of new technical possibilities, the “shut up
and calculate!” widespread instrumentalist attitude of physicists had to
unwillingly allow the reopening of foundational and philosophical debates
about QM. Very soon, even physicists would come to regard entanglement —like
Schrödinger had done many decades before— as the most essential feature of QM
itself. During the 1990s, even a new sub-discipline was created in order to
discuss the phenomena of entanglement, namely, Foundations of Quantum
Mechanics. As part of it, Quantum Information would also became one of the
most important lines of research in physics worldwide. After half a century of
hibernation, the problems and notions addressed by Einstein and Schrödinger
were beginning to be openly discussed not only by philosophers but also by
physicists. However, as remarked by Bokulich and Jaeger [9, p. xi] in the
introduction to their book, Philosophy of Quantum Information and
Entanglement, “[During the last decades] a large body of literature has
emerged in physics, revealing many new dimensions to our concepts of
entanglement and non-locality, particularly in relation to information.
Regrettably, however, only a few philosophers have followed these more recent
developments, and many philosophical discussions still end with Bell’s work.”
One decade after the publication of this book, the situation does not seem to
have changed and —apart from few exceptions— philosophers of physics have not
engaged —as they certainly could— in this ongoing technological revolution
that is taking place today.

In this work we attempt to critically address the orthodox definition of
quantum entanglement in terms of non-factorizability, as well as its recent
generalization in terms of observables [6, 7]. While the first definition is
relative to a preferred factorization of the state of the quantum system, the
latter definition of entanglement is relative to the choice of a subset of
preferred observables. Regardless of their differences, both of these notions
are explicitly non-objective, in the sense that they are unable to provide a
global, consistent and invariant representation of a state of affairs. The
subject must be necessarily introduced in order to choose between different
non-consistent factorizations and observables. After critically discussing
these two definitions as well as some of their problems already addressed in
the specialized literature we continue to discuss a newly proposed objective
definition of quantum entanglement grounded on the actual and potential coding
of effective and intensive relations [32]. We attempt to show, through the
explicitly derivation of two theorems, that within this new relational
definition of entanglement, all possible factorizations as well as all
possible subsets of observables can be conceived as making reference to the
same (potential) state of affairs in a global, consistent and operationally
invariant manner.

##  2 Orthodox Entanglement and Non-Separable Relativism

The contemporary definition and interpretation of quantum entanglement is
intrinsically linked to the triumph, that took place during the 20th Century,
of a purely pragmatic understanding of physics. This victory was made
possible, firstly, by the unspoken alliance between Niels Bohr —maybe the most
influential physicist of the century— and positivists, and secondly, by the
post-war convergence of these anti-realist forces into an even more radical
anti-realist scheme called “instrumentalism”. At the beginning of the 1960s
Karl Popper, one of the major figures within the field of epistemology and
philosophy of science, wrote in Conjectures and Refutations [63] that the
realist account of physics —the idea that physical theories make reference to
physis (or reality)— had been finally defeated: “Today the view of physical
science founded by Osiander, Cardinal Bellarmino, and Bishop Berkeley, has won
the battle without another shot being fired. Without any further debate over
the philosophical issue, without producing any new argument, the
instrumentalist view (as I shall call it) has become an accepted dogma. It may
well now be called the ‘official view’ of physical theory since it is accepted
by most of our leading theorists of physics (although neither by Einstein nor
by Schrödinger). And it has become part of the current teaching of physics.”
Popper argued there were two main reasons for having reached this result. The
first reason was the impact of Bohr’s interpretation of QM in the physics
community: “In 1927 Niels Bohr, one of the greatest thinkers in the field of
atomic physics, introduced the so-called principle of complementarity into
atomic physics, which amounted to a ‘renunciation’ of the attempt to interpret
atomic theory as a description of anything.”444It is interesting to notice
that pointing his finger to Bohr, the neo-Kantian, Popper seems to have
overlooked the deep influence of his own tradition within the instrumentalist
path. In fact, logical and empirical positivism —just like Bohr— had also
shifted the center of analysis from the theory to the subject, from
representation to observation and from explanation to prediction. The second
reason mentioned by Popper was related to “the spectacular practical success
of [quantum mechanical] applications.” According to Popper: “Instead of
results due to the principle of complementarity other and more practical
results of atomic theory were obtained, some of them with a big bang. No doubt
physicists were perfectly right in interpreting these successful applications
as corroborating their theories. But strangely enough they took them as
confirming the instrumentalist creed.” Of course, one could also argue in
favor of instrumentalism that what physicists had seemingly proved —specially
in the influential Manhattan project— was that spectacular instrumental
applications —such as the atomic bomb— could be actually developed without
entering philosophical debates about the meaning and the reference of the
theory. However, this —as entanglement would show— was a very limited
conclusion to be drawn.

As it has been argued in detail in [25, 26], the anti-realist approach to
physics produced an essentially inconsistent discourse, still today firmly in
place, grounded in two contradictory claims about the theory of quanta. On the
one hand, QM was regarded as a “tool” that should be used by agents in order
to predict measurement outcomes. As famously argued by Chris Fuchs and Asher
Peres: [48, p. 70]: “[…] quantum theory does not describe physical reality.
What it does is provide an algorithm for computing probabilities for the
macroscopic events (‘detector clicks’) that are the consequences of
experimental interventions. This strict definition of the scope of quantum
theory is the only interpretation ever needed, whether by experimenters or
theorists.” But on the other hand, it was also sustained —even by extreme
anti-realists— that QM made reference to a microscopic realm of unseen and
irrepresentable “elementary particles”. This inconsistent discourse mixing an
extreme form of pragmatism together with a fictional microscopic realm was
constructed by no one else than Niels Bohr himself. In many different
occasions Bohr systematically introduced pictures, principles and concepts
which had no link whatsoever to the mathematical formalism of QM nor the
experience observed in the lab. A very good example of this procedure is the
introduction of “quantum jumps” in his own model of the Hydrogen atom. An idea
he also applied in his interpretation of Schrödinger’s wave mechanics. Bohr’s
line of reasoning was exposed during his meeting in 1926 with Erwin
Schrödinger in which the existence of “quantum jumps” within the atom was a
kernel point of disagreement. Under the attentive gaze of Werner Heisenberg,
Schrödinger presented many arguments exposing not only the lack of any
explanatory power of this “magical process” but also the contradictions
reached when introducing such a-causal “jumps”. The complete lack of
conceptual and theoretical support allowed Schrödinger to conclude that “the
whole idea of quantum jumps is sheer fantasy.” [50, p. 74] But while the
Austrian physicist was unwilling to accept the “jumps” —a critical position
that would be shared not only by Einstein but also, as the years passed by, by
an elder Heisenberg—, the Danish physicist was ready to blame the lack of
representation to the theory of quanta itself. Instead of providing a reply to
the many criticism presented by Schrödinger, Bohr simply embraced them and
turned them completely upside-down as showing the limits of the theory of
quanta itself. Like a great Judo Master Bohr didn’t try to fight against
Schrödinger’s strong arguments; instead, by inverting the direction of their
force, he used them as weapons against his enemy:

> “What you say is absolutely correct. But it does not prove that there are no
> quantum jumps. It only proves that we cannot imagine them, that the
> representational concepts with which we describe events in daily life and
> experiments in classical physics are inadequate when it comes to describing
> quantum jumps. Nor should we be surprised to find it so, seeing that the
> processes involved are not the objects of direct experience.” [50, p. 74]

According to Bohr, QM went beyond our classical “manifest image” of the world,
and thus it came with no surprise that our (classical) concepts were incapable
of explaining what was really going on in the quantum world. We can only
explain what we see in terms of classical physics because experience is, and
will always be, classical. That is the way things are, and there is nothing to
be done but accept the limitations we have reached within science, limitations
which —Bohr declared— are not only technological but also ontological. QM had
shown us the limitations in the possibilities of representation of Nature
itself. It is in this way that Bohr was able to turn his own incapacity to
develop a consistent representation of QM into a proof of the theory’s own
difficulties and limits. In the second half of the 20th Century this
inconsistent discourse was sedimented and widespread within both physics and
philosophy communities investigating QM. Ever since, the idea that QM makes
reference to “elementary particles” has stood as a dogma which, quite
regardless of the lack of theoretical and experimental support, cannot be
questioned. Bohr also devised an escape route, to be used when pushed to
address the theoretical representation of this microscopic realm, which has
been learned by both physicists and philosophers. When confronted to such type
of questioning about quantum particles you can always declare that —in fact—
this was “just a way of talking”. This inconsistent discourse has had also
deep consequences in the constitution of the debates about quantum
entanglement.

During the mid-1990s, the physicist and philosopher Abner Shimony [68] gave
one of the first contemporary definitions of entanglement: “A quantum state of
a many-particle system may be ‘entangled’ in the sense of not being a product
of single-particle states.” Of course, this uncritical application of the
notion of “particle” is not something that can be exclusively blamed to
Shimony. During the 1990s the discussions about entanglement were taking place
in an instrumentalist environment with physicists accustomed to talk about
“particles” in a meaningless fashion, just as “a way of talking” —or, as
philosophers preferred to term it, as a “useful fiction”. It is true that both
Einstein and Schrödinger used the notion of ‘particle’ in order to discuss
about the phenomena of entanglement, but —let us not forget— they did so as a
reduction ad absurdum, to show a problem, not a solution.555In fact, neither
Einstein nor Schrödinger believed in the need to make reference to
‘particles’. For example, in a letter to Max Born, Einstein wrote the
following: “We are, to be sure, all of us aware of the situation regarding
what will turn out to be the basic foundational concepts in physics: the
point-mass or the particle is surely not among them.” It is also true that
Bell talked about “particles” in the 1960s, but in this case he was arguing in
favor of finding a classical representation to describe quantum phenomena —an
idea that Einstein found “to cheap”.666In a letter to Max Born [47, p. 44] he
commented: “Have you noticed that Bohm believes (as de Broglie did, by the
way, 25 years ago) that he is able to interpret the quantum theory in
deterministic terms? That way seems too cheap to me.” And this possibility
—contrary to Bell’s intuition— was ruled out by Aspect’s experiments. Today,
the unjustified application of particle metaphysics has reached an unspoken
consensus within the specialized literature about entanglement which can be
witnessed in the introduction of most papers on the subject, all of which make
explicit reference not only to “particles” but also to “separability”,
“locality” and —even— “purity”. It is then commonly argued by specialists in
the field that entanglement is an “holistic property of compound quantum
systems, which involves nonclassical correlations between subsystems” [53, p.
865]. For example, in the introduction to the just mentioned book edited by
Bokulich and Jaeger, Philosophy of Quantum Information and Entanglement, the
concept is presented in the following manner:

> “Consider two particles, $A$ and $B$, whose (pure) states can be represented
> by the state vectors $\psi_{A}$ and $\psi_{B}$. Instead of representing the
> state of each particle individually, one can represent the composite two-
> particle system by another wavefunction, $\Psi_{AB}$. If the two particles
> are unentangled, then the composite state is just the tensor product of the
> states of the components: $\Psi_{AB}=\psi_{A}\otimes\psi_{B}$; the state is
> then said to be factorable (or separable). If the particles are entangled,
> however, then the state of the composite system cannot be written as such a
> product of a definite state for $A$ and a definite state for $B$. This is
> how an entangled state is defined for pure states: a state is entangled if
> and only if it cannot be factored: $\Psi_{AB}\neq\psi_{A}\otimes\psi_{B}$.”
> [9, p. xiii]

The fact that none of these notions —i.e., particle, separability, locality
and purity— have been adequately defined —neither formally nor conceptually—
in the context of quantum theory has remained almost unnoticed to the general
physicist community which tends to regard, at least when necessary (i.e., when
understanding is required), this same discourse as devoid of reference, “just
as a way of talking” (see for a detailed discussion [35]). This controversial
definition and picture of entanglement has not stopped here. It has been also
extended to the case of mixed states, another very controversial notion
introduced within orthodox textbook QM during the 1940s and 1950s. According
to orthodoxy, mixed states are convex sums of pure states. Extending the non-
separability definition of pure states, entangled mixed state are those which
cannot be written as a convex combination of products.

$\rho_{mix}=\sum_{i}p_{i}\ \rho_{i}^{pure}=\sum_{i}p_{i}\
|\Psi_{i}\rangle\langle\Psi_{i}|$

Mixtures imply an ignorance interpretation about the purity of states which,
in turn, introduces a distinction between states which are “purely quantum”
and those which are not, namely, mixtures (of pure states). While the convex
elements representing pure states are the “quantum” ones, mixtures are
understood as being derived from them. In fact, the orthodox understanding of
mixtures applies an ignorance interpretation which is forbidden in the general
case of quantum probability.

The introduction of convex sets in order to describe pure states and mixtures
has very serious drawbacks and problems which have been addressed in detail in
[33, 34]. But even leaving behind these deeper criticisms against the
tenability of the whole pure-mixture interpretation of QM, there are serious
technical problems coming even from within the orthodox literature about
entanglement. One of the main recognized difficulties is the fact that the
reference to entanglement is dependent on the chosen factorization of the
quantum system. As Alejandro de la Torre et al. have shown explicitly in [22]:
“for any system in a factorizable state, we can find different degrees of
freedom that suggest a different factorization of the Hilbert space where the
same state becomes entangled.” Or put more dramatically by Thirring et al.
[70]: “for a given quantum state, it is our freedom of how to factorize the
algebra to which a density matrix refers. Thus we may choose! Via global
unitary transformations we can switch from one factorization to the other,
where in one factorization the quantum state appears entangled, however, in
the other not. Consequently, entanglement or separability of a quantum state
depends on our choice of factorizing the algebra of the corresponding density
matrix.” Let us see this in some detail. In their article [22], the authors
prove that given a decomposition of the Hilbert space
$\mathcal{H}=\mathcal{H}_{A}\otimes\mathcal{H}_{B}$ and a factorizable state
$\Psi=\Psi_{1}\otimes\Psi_{2}$, there exist transformations $F=A\otimes
I+I\otimes B$ and $G=A\otimes I-I\otimes B$ such that
$\mathcal{H}=\mathcal{H}_{F}\otimes\mathcal{H}_{G}$ and $\Psi$ is not
factorizable. In order to prove this result, the authors use the Quantum
Covariant Function _QCF_ ,

$QCF(X,Y,\Psi):=\langle\Psi,XY\Psi\rangle-\langle\Psi,X\Psi\rangle\langle\Psi,Y\Psi\rangle.$

Notice that a factorizable state $\Psi$ with respect to $A\otimes I$ and
$I\otimes B$ satisfies $QCF=0$,

$QCF(A\otimes I,I\otimes
B,\Psi_{1}\otimes\Psi_{2})=\langle\Psi_{1}\otimes\Psi_{2},A\otimes
B\Psi_{1}\otimes\Psi_{2}\rangle-\langle\Psi_{1},A\Psi_{1}\rangle\|\Psi_{1}\|^{2}\|\Psi_{2}\|^{2}\langle\Psi_{2},B\Psi_{2}\rangle=0.$

But the same state $\Psi$ with respect to $F$ and $G$ satisfies
$QCF(F,G,\Psi)\neq 0$. In particular, $\Psi$ is entangled in
$\mathcal{H}=\mathcal{H}_{F}\otimes\mathcal{H}_{G}$ and this shows
mathematically that the notion of entanglement depends on the factorization of
the Hilbert space, in other words, it depends on the chosen basis for
$\mathcal{H}$. This result has introduced some unease in the foundational
literature. According to John Earman, there are two broad reactions to this
“threatened rampant ambiguity” of quantum entanglement:

> “Realism. The Realist claims that the kind of entanglement we should be
> concerned with is entanglement over subalgebras that correspond to real as
> opposed to virtual or fictitious subsystems. Obviously, this distinction
> cannot be drawn in terms of purely formal conditions on the algebras but
> must come from background physics. […] But where the hope is utterly dashed
> and there is no principled way to draw the distinction, the Realist holds
> that entanglement ought to be regarded as a moot topic.
>
> Pragmatism. The Pragmatist asserts that there is no metaphysically valid way
> to draw a line between subsystems that are ‘real’ and those that are
> ‘fictitious’ or ‘virtual.’ Any decomposition of the system algebra into
> subalgebras meeting appropriate formal criteria is as good a way as any
> other for defining subsystems. Thus, contrary to Realism, there is no
> metaphysically valid distinction to be drawn between ‘real’ entanglement
> (i.e. entanglement over ‘real’ subsystems) and ‘faux’ entanglement (i.e.
> entanglement over ‘fictitious’ or ‘virtual’ subsystems).” [43]

This relativist dependence of the notion of entanglement with respect to
different factorizations can be resumed in the following manner.

Non-Separable Relativism: Given a pure state, the existence of entanglement is
relative to the chosen factorization. Consequently, the same state will be
entangled or not depending on the factorization considered.

Due to its relative nature, the orthodox definition of quantum entanglement
cannot be interpreted as a property of quantum systems. Instead, it must be
regarded as intrinsically relative to the factorization of systems into
subsystems. In this context, entanglement becomes more a way in which the
observer chooses to describe systems, than a property of the systems
themselves. This idea seems to flagrantly contradict the original
understanding of entanglement as a (real) property characterizing the relation
between quantum particles, presenting instead a rather controversial reference
to the way in which subjects or agents choose to describe the fictitious (or
virtual) interaction between quantum systems. Embracing this anti-realist
account even more explicitly, a generalized notion of entanglement has been
put forward by Barnum, Knill, Ortiz, Somma and Viola in a series of papers to
which we now turn our attention.

##  3 Generalized Entanglement and Contextual Relativism

The maturity of perspectival-relativism in QM and its generalized acceptance
by the physics community seems to have taken place when Niels Bohr was finally
declared the indisputable champion of the EPR battle in 1935. Forced by
Einstein into an uncomfortable debate about the definition of the physical
reality of entangled particles, Bohr had to state more emphatically than ever
his perspectival-relativism with respect to measurement situations and
outcomes. Bohr’s principle of complementarity, originally created in order to
bring together the inconsistent representation of quantum objects as ‘waves’
and ‘particles’, was now applied to the mathematical formalism of the quantum
theory itself in order to ascribe the value of properties as relative to the
choice of the experimental set up. As Bohr [11, p. 700] famously argued, the
value of complementary properties could not be considered as prior to the
determination of the actual measurement set up since, he argued, “there is
essentially the question of an influence on the very conditions which define
the possible types of predictions regarding the future behavior of the
system.” The Danish physicist was applauded by a physics community already in
tune with the anti-foundational postmodern Zeitgeist of the 20th Century in
which reality and truth were regarded as metaphysical hallucinations.
Portrayed as a mystic figure of wisdom and knowledge Bohr was contra posed to
Einstein, an old and lonely man who seemed incapable of understanding —what
Bohr claimed to be— the deep revolutionary consequences of QM.

Bohr’s solution to the EPR paradox was not an objective one, it was an
intersubjective solution which had the purpose of erasing any reference of the
theory of quanta beyond the choice of particular measurement situations and
outcomes. Let us be very clear about this point. While ‘objectivity’ meant —in
the terms inherited from Kantian philosophy— the possibility to represent
consistently a moment of unity in categorical conceptual terms, in the neo-
Kantian terms of Bohrians and positivists, ‘objectivity’ was translated either
as ‘faithful observation’ or simply as ‘intersubjectivity’.777This also
becomes explicit in Bohr’s understanding of the word phenomena. While in the
Kantian scheme the term phenomena implied the necessary reference to a
conceptual moment of unity, for Bohr ‘clicks’ in detectors and ‘spots’ in
photographic plates could be also labeled as phenomena. As explained by
Hilgevoord and Uffink in their excellent entry of the Stanford Encyclopedia on
Heisenberg’s relations [52]: “Central in Bohr’s considerations is the language
we use in physics. No matter how abstract and subtle the concepts of modern
physics may be, they are essentially an extension of our ordinary language and
a means to communicate the results of our experiments. These results, obtained
under well-defined experimental circumstances, are what Bohr calls the
‘phenomena’. A phenomenon is ‘the comprehension of the effects observed under
given experimental conditions’ (Bohr 1939: 24).” The essential point is that
intersubjectivity did not required the conceptual unity presupposed within
objective representations. Bohr had shifted from objectivity to
intersubjectivity, but he was not willing to abandon the term ‘objective’.
Thus, he simply renamed ‘intersubjective statements’ as ‘objective
statements’. Stressing the claim that his account of QM was as “objective” as
classical physics he [36, p. 98] argued that: “The description of atomic
phenomena has […] a perfectly objective character, in the sense that no
explicit reference is made to any individual observer and that therefore… no
ambiguity is involved in the communication of observation.” Bernard D’Espagnat
explains this quotation in the following manner: “That Bohr identified
objectivity with intersubjectivity is a fact that the quotation above makes
crystal clear. In view of this, one cannot fail to be surprised by the large
number of his commentators, including competent ones, who merely half-agree on
this, and only with ambiguous words. It seem they could not resign themselves
to the ominous fact that Bohr was not a realist.” In Bohr’s scheme the
experiences acquired by different agents were simply detached from any common
objective reference and representation.888In his book, [36], D’Espagnat
clearly distinguishes between objective statements and Bohr’s intersubjective
statements, which he calls: weakly objective statements. This also implied a
silent replacement of the notion of object —a conceptual moment of unity
categorically constituted through the general principles of existence, non-
contradiction and identity—; by that of event, actual observation, or even,
measurement outcome. These steps were perfectly aligned with Bohr’s
instrumentalist understanding of the mathematical formalism of QM as making
exclusive reference to observable predictions.

> “The entire formalism is to be considered as a tool for deriving
> predictions, of definite or statistical character, as regards information
> obtainable under experimental conditions described in classical terms and
> specified by means of parameters entering into the algebraic or differential
> equations of which the matrices or the wave-functions, respectively, are
> solutions. These symbols themselves, as is indicated already by the use of
> imaginary numbers, are not susceptible to pictorial interpretation; and even
> derived real functions like densities and currents are only to be regarded
> as expressing the probabilities for the occurrence of individual events
> observable under well-defined experimental conditions.” [12, p. 314]

Bohr stressed repeatedly that the most important lesson we should learn from
QM was an epistemological one; namely, that we are not only spectators, but
also actors in the great drama of (quantum) existence. This idea went in line
with his pragmatic understanding of physics according to which [13]: “Physics
is to be regarded not so much as the study of something a priori given, but
rather as the development of methods of ordering and surveying human
experience. In this respect our task must be to account for such experience in
a manner independent of individual subjective judgement and therefore
objective in the sense that it can be unambiguously communicated in ordinary
human language.” All these ideas can be condensed in the orthodox widespread
claim that in QM “the properties of a system are different whether you look at
them or not” (e.g., [17]).

When taking for granted —as Bohr did— that experience must be necessarily
described in terms of classical concepts there seems to exist only two
possible ways out, either we must change completely the mathematical formalism
(as in Bohm’s theory or GRWs proposal), or we must avoid to talk about
physical reality beyond measurement outcomes (as QBism has explicitly
proposed). This latter anti-realist position to which Bohr contributed so much
can be also re-farmed in co-relational terms, something that has been
extensively proposed (e.g., [46, 64]) and has led many [72] to believe that “a
quantum state is relative in nature. That is, an observer independent quantum
state is not necessarily the basic description of a quantum system.” It is
this possibility which has been investigated by Howard Barnum et al. in a
series of papers [6, 7, 8] which attempt to generalize the notion of quantum
entanglement beyond any objective reference. As recalled by them [8]: “The
standard definition of quantum entanglement requires a preferred partition of
the overall system into subsystems —that is, mathematically, a factorization
of the Hilbert space as a tensor product. Even within quantum mechanics, there
are motivations for going beyond such subsystem-based notions of
entanglement.” In order to address quantum entanglement from a more general
perspective they propose the following [8]: “the key idea behind GE is that
entanglement is an observer-dependent concept, whose properties are determined
by the expectations of a distinguished subspace of observables of the system
of interest, without reference to a preferred subsystem decomposition.” Thus,
avoiding the reference to different factorizations [71]: “Generalized
Entanglement (GE) of a quantum state relative to the distinguished set may
then be defined without reference to a decomposition of the overall system
into subsystems (15; 16).” However, GE reintroduces relativism even more
explicitly by making reference to a preferred set of observables (or context)
[71]: “entanglement is an inherently relative concept, whose essential
features may be captured in general in terms of the relationships between
different observers”. Let us see how this works in some detail.

In [8], the authors give a definition of entangled state using the notion of
cones. Let us start by recalling some of the definitions. A _convex set_ $C$
is a set in a real vector space $V$ closed under convex combinations, that is,
if $x,y\in C$ then $tx+(1-t)y\in C$ for all $0\leq t\leq 1$. An element $x\in
C$ is called _extremal_ if it cannot be written as a nontrivial convex
combination. In other words, if $x=tx_{1}+(1-t)x_{2}$ with $x_{1},x_{2}\in C$
and $0<t<1$, then $x$ is not extremal. A _positive cone_ is a proper subset
$K$ of a real vector space $V$ closed under multiplication by nonnegative
scalars. It is called _regular cone_ if it is:

  * •

convex (equivalently, closed under addition: $K+K=K$),

  * •

generating ($K-K=V$, equivalently $K$ linearly generates $V$)

  * •

pointed ($K\cap-K=\\{0\\}$, so that it contains no non-null subspace of $V$),
and

  * •

topologically closed (in the norm topology).

A _ray_ in a cone $K$ is a set given by the intersection of a one-dimensional
subspace with $K$. An _extreme ray_ $R$ of a cone $K$ is a ray such that no
$y\in R$ can be written as a convex combination of elements of $K$ that are
not in $R$. The authors show that a cone is convexly generated by its extreme
rays.

######  Definition 3.1 (Def. 3.3 in [8])

Let $K_{1}\subseteq V_{1}$ and $K_{2}\subseteq V_{2}$ be two cones. Let
$\pi:V_{1}\to V_{2}$ be a linear map such that $\pi(K_{1})=K_{2}$. An element
$x\in K_{1}$ is called _generalized unentangled_ if either (a) $x$ and
$\pi(x)$ belong to extreme rays, or (b) $x$ is a positive linear combination
(or a limit of such combinations) of elements satisfying (a). Then, an element
$x$ is called _generalized entangled_ if it is not _generalized unentangled_.
Both notions are relative to $K_{2}$ and $\pi$.

Let us show that the standard definition of entangled state is a particular
case of a generalized entangled state. Let $\mathcal{H}$ be a Hilbert space
and consider the real vector space of Hermitian operators $\mathcal{A}$.
Inside $\mathcal{A}$, we have the cone $Pos$ of positive operators. The
extreme rays of $Pos$ are in bijection with pure states (rank one density
matrices). Assume now that we have a decomposition
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. Then, the partial traces
define a map $\pi:\mathcal{A}\to\mathcal{A}_{1}\oplus\mathcal{A}_{2}$. The
evaluation of $\pi$ at a factorizable state $|v\rangle\langle
v|\otimes|w\rangle\langle w|$ gives a pair of pure states

$\pi(|v\rangle\langle v|\otimes|w\rangle\langle w|)=(|v\rangle\langle
v|,|w\rangle\langle w|)\in\mathcal{A}_{1}\oplus\mathcal{A}_{2}.$

Then, according to the previous definition, the factorizable states (and
separable states) are generalized unentangled. In particular, the previous
definition generalizes to a more wider situation. It is true that, as remarked
by Viola and Barum [71, p. 18], “In spite of suggestive points of contact, our
approach differs from the [orthodox one] in (at least) two important ways:
physically, the need for a decomposition into distinguishable subsystems is
bypassed altogether; mathematically, the GE notion rests directly (and solely)
on extremality properties of quantum states in convex sets which are
associated to different observers.” However, as we mentioned already, both
definitions (i.e., the orthodox and its generalization) are relative to a
chosen decomposition $\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$. As
proved in [22], if we change the decomposition, we arrive at a situation where
a (generalized) separable state becomes (generalized) entangled. Finally, let
us also remark that it is possible to give equivalent definitions of
generalized entanglement with convex sets instead of cones (see [8]).

Perspectival-relativism has been always —since the Ancient Greek sophists—
related to anti-realist trends of thought (see [27]). However, Barnum [5, p.
348] seems to propose what we consider to be a very strange idea according to
which relativism in QM should be understood as supporting a realist
interpretation of the theory: “I view the relative state interpretation less
as a way of getting the classical world to emerge from Hilbert space, and more
as a way of giving a realistic interpretation to Hilbert space structure in
the presence of additional structures such as preferred bases or subsystem
decompositions that represent other aspects of physics.” In this case,
relativism seems to be considered as part of a general realist project which
attempts to justify the existence of preferred factorizations and bases.
According to this viewpoint, the realist should content herself with finding a
way to justify the existence of these preferred perspectives. But while in the
case of Earman it is argued that these conditions should come from “background
physics”, Barnum [4], advances an even stranger step further —at lest, for a
supposedly realist position— and argues in favor of considering the role of
consciousness: “As I see it, the correct way, in this view, to account for the
appearance that there is a single measurement result is the idea that the
experience of a conscious history is associated with definite measurement
results, so that consciousness forks when a quantum measurement is made.” We
will come back to these essential points in the following sections. For now we
content ourselves to provide an explicit definition of this relativist
contextual understanding of entanglement.

Contextual Relativism: Given a pure state, the existence of entanglement is
relative to a preferred subset of observables. Thus, the same state will be
entangled or not depending on the choice of the specific subset.

To sum up, while the orthodox notion of entanglement presents a definition
relative to the factorization of systems, the notion of generalized
entanglement replaces this type of relativism by a contextual one. This
contextual type of relativism in which observers must choose between the
actual existence of different and incompatible states of affairs, all of them
contained within the same mathematical representation, is of course highly
problematic from a representational realist understanding of physics which
understand theories as providing an objective-invariant formal-conceptual
representation of a state of affairs —let it be classical or quantum. But
let’s keep this on hold. Regardless of these criticisms —which might be
considered by many physicists as merely “philosophical”— there are also
serious technical problems recognized in the orthodox foundational and
informational literature which we would like to address in some detail.

##  4 The Many Problems of Orthodox Entanglement

The exponential growth in the research being done in quantum entanglement due
to explosion during the last decades of quantum information processing as one
of the most important lines of research worldwide has also lead to the
findings of several essential technical difficulties —and even
inconsistencies— which expose in a different context the serious difficulties
which hunt the definitions discussed in the previous sections. Let us mention
at least some of them.

To start with, the presupposed relation between entanglement and non-locality
has been shown to be unsatisfactory —to say the least. As resumed by Bokulich
and Jaeger:

> “There are […] limitations to using a violation of Bell’s inequality as a
> general measure of entanglement. First, there are Bell-type inequalities
> whose largest violation is given by a non-maximally entangled state (Acín et
> al. 2002), so entanglement and non-locality do not always vary
> monotonically. More troublingly, however, Reinhard Werner (1989) showed that
> there are some mixed states (now referred to as Werner states) that, though
> entangled, do not violate Bell’s inequality at all; that is, there can be
> entanglement without non-locality. In an interesting twist, Sandu Popescu
> (1995) has shown that even with these local Werner states one can perform a
> non-ideal measurement (or series of ideal measurements) that ‘distills’ a
> non-local entanglement from the initially local state. In yet a further
> twist, the Horodecki family (1998) subsequently showed that not all
> entanglement can be distilled in this way —there are some entangled states
> that are ‘bound’. These bound entangled states are ones that satisfy the
> Bell inequalities (i.e., they are local) and cannot have maximally entangled
> states violating Bell’s inequalities extracted from them by means of local
> operations. Not only can one have entanglement without non-locality, but
> also, as Bennett et al. (1999) have shown, one can have a kind of ‘non-
> locality without entanglement’. There are systems that exhibit a type of
> non-local behavior even though entanglement is used neither in the
> preparation of the states nor in the joint measurement that discriminates
> the states (see also Niset and Cerf (2006))” [9, pp. xvii-xviii]

As we pointed out earlier, these results are not strange once we recognize
that the addressed relation between entanglement and non-locality comes from a
widespread misunderstanding regarding the meaning of the violation of Boole-
Bell type inequalities in EPR type experiments. Even today, while many
physicists still tend to believe that such a violation implies the existence
of “something truly quantum and non-local”, the correct conclusion —known by
the philosophy of QM community— to be drawn by the violation of Boole-Bell
inequalities is that such correlations cannot be represented by a classical
theory. Period. The uncritical application of locality in the context of QM is
also linked to the notion of separability and its kernel relation to the
orthodox definition of entanglement in terms of non-separable states. Of
course, Einstein himself understood very clearly the difficulties of
introducing the principle of separability in QM. As he [14, p. 172] himself
remarked in a letter directed to Max Born: “quantum theory nowhere makes
explicit use of this [the separability] requirement.” Regardless of Einstein’s
warning, in the contemporary literature separability has been uncritically
translated as (mathematical) factorizability. This idea is not only
ungrounded, it is simply absurd. The factorizability of Hilbert spaces cannot
be consistently interpreted as the separability (or dissection) of systems
into subsystems. The product of subspaces cannot be understood as their
addition. The product of two rays (${\cal{H}}_{1}\otimes{\cal{H}}_{2}$)
generates a whole plane (${\cal{H}}_{12}={\cal{H}}_{1}\otimes{\cal{H}}_{2}$)
which obviously contains more points than the ones present when considering
the addition of the original rays. While Rob Clifton has discussed in detail
the failure of this interpretation in the context of the Schimdt (bi-
orthogonal) decomposition [20, 21], Dennis Dieks and Andrea Lubberdink have
shown that the labeling in the factorization is unrelated to the existence of
particles [41, 40]: “These physical labels do not coincide with the factor
indices occurring in the total quantum state —the latter remain associated in
the same way with all pure one-particle states even in the classical limit,
and therefore cannot refer to individual particles.” Furthermore, as it is
well known in quantum logic due to a famous theorem by Diederik Aerts [1], the
conjunction in QM is not the same as the union of sets and quantum systems
must be regarded essentially as non-separable. It is far from clear what is
the physical meaning of factorizability in QM and what exactly is its
operational content. In fact, as Aerts has argued, one is forced to adapt the
mathematical formalism of QM if one is still willing to address separable
systems [2].

The situation is even worse when considering density operators or matrices. It
is well known fact that mixed states are a necessary extension beyond pure
states for measuring matrices (of rank $\neq$ 1) in the lab. But as explained
by Earman [43]: “Since different convex linear combinations can produce the
same mixed state, it is not easy to determine whether or not a given mixed
state is indecomposable.” As also remarked by Thirring et al. [70]: “For mixed
states, however, the situation is much more complex [than for pure states]
(see, e.g., Ref. [17]). The reason is that the maximal mixed state, the
tracial state $\frac{1}{D}\mathds{1}_{D}$, is separable for any factorization
and therefore a sufficiently small neighborhood D of it is separable too.” It
has becomes a difficult technical problem in the specialized literature to
determine weather or not a mixed state is entangled or not [57]. At a more
profound level, one will also need to deal with the serious inconsistencies,
analyzed in [33], present within the definition of pure state itself —also
essential in the orthodox definition of entanglement. These inconsistencies
also threaten the extension of entanglement to the case of mixtures [34].

Also, the orthodox definition of quantum entanglement remains linked to the
existence of “collapses” which still, almost one century after its ad hoc
introduction, remains with no formal nor experimental support [49]. As
remarked by Dennis Dieks [39, p. 120]: “Collapses constitute a process of
evolution that conflicts with the evolution governed by the Schrödinger
equation. And this raises the question of exactly when during the measurement
process such a collapse could take place or, in other words, of when the
Schrödinger equation is suspended. This question has become very urgent in the
last couple of decades, during which sophisticated experiments have clearly
demonstrated that in interaction processes on the sub-microscopic, microscopic
and mesoscopic scales collapses are never encountered.” The bottom line of
this is an essential one. If collapses do not really exist, then the notion of
entanglement, which was created by Einstein and Schrödinger in order to target
the subjective appearance of collapses in measurement, —obviously— needs to be
seriously reconsidered.

Not only the definition, but also the measure of the degree of entanglement in
the lab faces serious technical problems when attempting to follow its
orthodox understanding in terms of non-separable pure states. It has been even
argued that the degree of entanglement is a physically ill-posed problem [61].
Mercin Pawowski and Marek Czachor have analyzed “an example of a photon in
superposition of different modes, and ask what is the degree of their
entanglement with vacuum. The problem turns out to be ill-posed since we do
not know which representation of the algebra of canonical commutation
relations (CCR) to choose for field quantization. Once we make a choice, we
can solve the question of entanglement unambiguously.” The way to characterize
and measure the degree of entanglement remains today one of the most serious
open problems within the foundational literature. Finally, we might remark
that the survey of various operational and non-operational criteria of
entanglement by Dagmar Bruß [15] points to the serious difficulty that the
growing number of competing definitions —all of them grounded on particle
metaphysics and collapses— threaten to fragment the literature about
entanglement in smaller and smaller, compartmented debates with no contact
between each other. In fact, for the specific case of indistinguishable
particles there is even no consensus regarding the way to define entanglement.
As remarked by Rosario Lo Franco and Giuseppe Compagno [58]: “Quantum
entanglement of identical particles is essential in quantum information
theory. Yet, its correct determination remains an open issue hindering the
general understanding and exploitation of many-particle systems” (see also
[40, 43]).

We might conclude that if no consensus is reached in the field regarding the
very basic definition and meaning of quantum entanglement, the literature
might end up constructing a tower of Babel in which researchers working in the
same field will become unable to understanding each other. Unfortunately, we
have already witnessed this fragmentation in understanding within the
interpretational debate about quantum theory which has been recently
characterized by Ádan Cabello as “a map of madness” [18].

##  5 An Objective Relational Definition of Quantum Entanglement

Perspectival-relativism is a special type of co-relationalism which stresses
the specific relation between a perceiving subject and its surrounding.
Obviously, this particular type of relation goes hand in hand with those
positions which assume as a standpoint an anti-realist subject-perspective of
analysis —i..e, subject-object or subject-observation relations. Sophistry,
empiricism, positivism, Bohrian philosophy and instrumentalism are all co-
relational schemes which take as a standpoint the subjective act of perception
given within the process of measurement. As positivists from the Vienna circle
justified in their influential manifesto [19]: “In science there are no
‘depths’; there is surface everywhere: all experience forms a complex network,
which cannot always be surveyed and, can often be grasped only in parts.
Everything is accessible to man; and man is the measure of all things. Here is
an affinity with the Sophists, not with the Platonists; with the Epicureans,
not with the Pythagoreans; with all those who stand for earthly being and the
here and now.” Starting from a perceptive perspective, observation and
measurement become then the necessary foundation of what is considered to be
“scientific”. Observation, considered as an unproblematic given of “common
sense” experience, comes always first and physical theories are just
mathematical models (with some additional rules) designed to predict future
experience. As explained by David Deutsch:

> “during the twentieth century, most philosophers, and many scientists, took
> the view that science is incapable of discovering anything about reality.
> Starting from empiricism, they drew the inevitable conclusion (which would
> nevertheless have horrified the early empiricists) that science cannot
> validly do more than predict the outcomes of observations, and that it
> should never purport to describe the reality that brings those outcomes
> about. This is known as instrumentalism. It denies that what I have been
> calling ‘explanation’ can exist at all. It is still very influential. In
> some fields (such as statistical analysis) the very word ‘explanation’ has
> come to mean prediction, so that a mathematical formula is said to ‘explain’
> a set of experimental data. By ‘reality’ is meant merely the observed data
> that the formula is supposed to approximate. That leaves no term for
> assertions about reality itself, except perhaps ‘useful fiction’.” [37, p.
> 15]

Realism, on the other hand, is intrinsically related to the origin of both
physics and philosophy, both of which begun with the ancient Greek idea
according to which there is something, a fundament, we call physis (or
reality) which comprises the whole of existence. Everything there is,
absolutely everything —including ourselves—, is part of physis. The second
essential presupposition is that physis is not chahotic, it has a certain
order or what the Greeks called a logos. Such a logos is expressible through
the creation of theories. It is a difficult task to expose the true logos
since, as remarked by Heraclitus, “physis loves to hide.” [f. 123 DK]. Doing
so requires hard work and sensibility, but —following Heraclitus— the latter
can be revealed in the former. In a particular logos one can “listen”
something that exceeds it, that is not only that personal discourse but the
logos of physis: “Listening not to me but to the logos it is wise to agree
that all things are one” [f. 50 DK]. We are thus able to represent physis, to
exhibit its logos.999As shown by Heraclitus, the logos of theories was never
supposed to mirror the logos of physis. One allows to reach the other.
However, the idea that realism implies a a one-to-one correspondence relation
between theory and reality has been systematically applied by anti-relaists in
order to create a straw-realist they can fight and easily destroy. By being
able to subsume the many different phenomena within a theoretical unity, the
claim made by the first realists was that theories were able to capture or
express, in some way, aspects of reality. As we all know, this idea was
strongly confronted by sophists, giving rise to an endless war between
realists and anti-relaists.

Thus, we might regard the main point of disagreement between realists and
anti-realists as a mater of perspective. As Thomas Nagel described, the anti-
realist is a perspective from now here in contraposition to the realist which
he ironically referred to as a perspective from no where (see also [5, 69]).
While the first group assumed the down to earth perspective of a perceiving
subject, the latter assumed the theoretical perspective of physis. This makes
all the difference. For realists, theoretical representation comes always
before observation. The conditions of observation are derived conceptually and
formally from a theory, not the other way around. As Einstein famously said to
Heisenberg: “it is only the theory which can tell you what can be observed.”
Only adequate concepts are capable to explain what has been observed. As
Heisenberg [51, p. 264] himself would make the point: “For an understanding of
the phenomena the first condition is the introduction of adequate concepts.
Only with the help of correct concepts can we really know what has been
observed.” From a realist perspective, observation can never affect the
representation of a state of affairs. As remarked by Einstein [38, p. 175]:
“[…] it is the purpose of theoretical physics to achieve understanding of
physical reality which exists independently of the observer, and for which the
distinction between ‘direct observable’ and ‘not directly observable’ has no
ontological significance.” Observation and measurement are nonstarters for any
true realist. Realism is not a subjective belief in an interpretation, it is a
praxis, a specific procedure in which physicists seek to create invariant-
objective (subject-independent) theoretical (formal-conceptual)
representations of states of affairs (see for a more detailed discussion [25,
26]). Examples of such theoretical productions are Newtonian classical
mechanics, Maxwell’s theory of electromagnetism or Einstein’s Relativity. The
program implies the need to produce global formal-conceptual representations
in which all reference frames and observational perspectives are consistently
considered allowing in this way to detach the subject from the represented
course of events.

The specific manner in which theories are able to create (subject-independent)
representations of states of affairs is intrinsically related to two kernel
notions of modern physics, namely, objectivity and operational-invariance.
While the first is one of a conceptual nature, the latter is one of a
mathematical or formal character. One is the counterpart of the other. Both
act as preconditions of consistency regarding the global representations
constructed by the realist. Of course, within such representations there is no
room for preferred reference frames (bases) nor preferred observational or
experimental perspectives. Let us discuss this in some more detail.
Objectivity means the provision of a categorical scheme which is able to
account for a multiplicity of phenomena in terms of a conceptual moment of
unity. Following Kant, the notion of ‘object’ must be regarded as a conceptual
machinery capable not only of creating a representational realm in which
thinking of experience becomes possible but also qualified to unify in a
consistent manner the different phenomena observed by empirical subjects. In
physics, all perspectives must be considered as equally acceptable. This is in
fact the democratic pre-condition of scientific objectivity. Realism imposes
the need that all representations from different viewpoints must be subsumed
within the consistent account of the same state of affairs. Even though there
might exist a multiplicity of different perspectives of analysis, reality must
be considered as one and the same —independent of any particular perspective.
Operational-invariance in physical theories capture exactly this requirement
in mathematical terms. In this case, the mathematical representation of the
state of a system (i.e., the object in a specific situation) is given from a
specific reference frame (or basis) which can be consistently translated into
any other reference frame (or basis) (see for a detailed analysis [29, 30,
66]). While in the case of classical mechanics this translation is provided by
the Galilean transformations, in the case of relativity theory this is given
through the Lorentz transformations. This is what defines the realist program:
the attempt to produce a consistent, coherent and unified theory, which is
also the attempt to produce an invariant-objective (formal-conceptual)
representation of both reality and experience.

In the particular case of QM, the possibility of such a realist development
has been investigated by the logos categorical approach. An approach which,
avoiding metaphysical prejudices, attempts to follow the realist procedure of
theory construction. According to this procedure, since in QM we already have
an operational mathematical formalism capable to operationally account for
experience, the focus must be centered in the development of a new conceptual
scheme which is able to provide a consistent account of the mathematical
formalism. We already know that the classical atomist representation does not
work for this mathematical scheme; thus, we need to produce a set of
completely new non-classical concepts which match the formalism in an adequate
manner —and not the other way around. The thread of Ariadna which has allowed
us to escape the Bohrian-positivist labyrinth has been the well known
operational-invariance of the theory of quanta itself, namely, the Born rule.
It is this mathematical expression of the theory which captures the essential
connection between the theory, particular reference frames (or bases) and
experience. The Born rule provides the operational invariant content of QM
which, leaving behind dogmatic (atomist) metaphysics and certain (binary)
observations, can be related to a new understanding and generalization of
EPR’s famous element of physical reality. According to it, the probabilistic
information supplied by the Born rule should be considered as providing
objective knowledge of a state of affairs [23] —instead of the probabilistic
prediction of a single measurement outcome. This can be done given we abandon
the binary reference to actual properties and certain observations and advance
into an intensive representation of a state of affairs. Escaping relativism,
we have proven in [30] that our scheme is capable of bypassing the Kochen-
Specker theorem through the introduction of a global intensive valuation
which, in turn, allows us also to produce an objective (subject-independent)
invariant (basis-independent) account of the orthodox mathematical formalism.
Let us see this in some more detail.

Let us begin by recalling some results from [30] which allow us to introduce
some essential concepts. A _Global Intensive Valuation_ (GIV) is a function
from a graph to the closed interval $[0,1]$. A _Global Binary Valuation_ (GBV)
is a function from a graph to the set $\\{0,1\\}$. The graph that we are
interested in, is the graph of projection operators which we term intensive
powers. Let $\mathcal{H}$ be a Hilbert space and let
$\mathcal{G}=\mathcal{G}(\mathcal{H})$ be the set of observables. We give to
$\mathcal{G}$ a graph structure by assigning an edge between observables $P$
and $Q$ if and only if $[P,Q]=0$. We call this graph, _the graph of powers_.
Among all global intensive valuations we are interested in the particular
class of PSA.

###### Definition 5.1

Let $\mathcal{H}$ be a Hilbert space. A _Potential State of Affairs_ is a
global intensive valuation $\Psi:\mathcal{G}(\mathcal{H})\to[0,1]$ from the
graph of powers $\mathcal{G}(\mathcal{H})$ such that $\Psi(I)=1$ and

$\Psi(\sum_{i=1}^{\infty}P_{i})=\sum_{i=1}^{\infty}\Psi(P_{i})$

for any piecewise orthogonal projections $\\{P_{i}\\}_{i=1}^{\infty}$. The
numbers $\Psi(P)\in[0,1]$, are called intensities or potentia and the nodes
$P$ are called _powers_. Hence, a PSA assigns a potentia to each power.

Intuitively, we can picture a PSA as a table,

$\Psi:\mathcal{G}(\mathcal{H})\rightarrow[0,1],\quad\Psi:\left\\{\begin{array}[]{rcl}P_{1}&\rightarrow&p_{1}\\\
P_{2}&\rightarrow&p_{2}\\\ P_{3}&\rightarrow&p_{3}\\\
&\vdots&\end{array}\right.$

###### Theorem 5.1

Let $\mathcal{H}$ be a separable Hilbert space, $\dim(\mathcal{H})>2$ and let
$\mathcal{G}$ be the graph of immanent powers with the commuting relation
given by QM.

  * •

Any positive semi-definite self-adjoint operator of the trace class $\rho$
determines in a bijective way a PSA $\Psi:\mathcal{G}\to[0,1]$.

  * •

Any GIV determines univocally a GBV such that the set of powers are considered
as potentially existent.

Proof: See [30].

###### Definition 5.2

Let $\mathcal{G}$ be a graph. We define a _context_ as a complete subgraph (or
aggregate) inside $\mathcal{G}$. For example, let $P_{1},P_{2}$ be two
elements of $\mathcal{G}$. Then, $\\{P_{1},P_{2}\\}$ is a contexts if $P_{1}$
is related to $P_{2}$, $P_{1}\sim P_{2}$. Saying it differently, if there
exists an edge between $P_{1}$ and $P_{2}$. In general, a collection of
elements $\\{P_{i}\\}_{i\in I}\subseteq\mathcal{G}$ determine a context if
$P_{i}\sim P_{j}$ for all $i,j\in I$. Equivalently, if the subgraph with nodes
$\\{P_{i}\\}_{i\in I}$ is complete. A _maximal_ context is a context not
contained properly in another context. If we do not indicate the opposite,
when we refer to contexts we will be implying maximal contexts.

For the graph of powers, the notion of context coincides with the usual one; a
complete set of commuting operators. However, all projection operators can be
assigned a consistent value bypassing in this way the famous Kochen-Specker
theorem.

###### Theorem 5.2

(Intensive Non-Contextuality Theorem) If $\mathcal{H}$ is a Hilbert space,
then a PSA is possible.

Proof: See [30].

This theorem restores the possibility of an objective physical representation
of any quantum wave function $\Psi$. Contrary to the orthodox interpretation
of QM in terms of systems with properties which imply a binary valuation, our
conceptual representation of quantum physical reality is not relative to any
particular context, it is global and essentially intensive.

In turn, our logos approach has allowed us to reconsider the notion of
entanglement from a purely mathematical standpoint in terms of the actual and
potential coding of effective and intensive relations between different
screens [32]. Let us see this in some detail.

Effective Relations: The relations determined by a difference of possible
actual effectuations. Effective relations discuss the possibility of an
actualist definite coding. It involves the path from intensive relations to
definite correlated (or anti-correlated) outcomes. They are determined by a
binary valuation of the factorized context in which only one correlated node
is considered as true in each subgraph, while the rest are considered as
false.

Intensive Relations: The relations determined by the intensity of different
powers. Intensive relations imply the possibility of a potential intensive
coding. They are determined by the correlation of intensive valuations.

These relations provide an intuitive grasp of what can be done in a lab and
what type of relations are at play. The following definitions provide a new
account of entanglement which rests on the analysis of relational intensive
and effective correlations.

######  Definition 5.1 (Quantum Entanglement)

Given $\Psi_{1}$ and $\Psi_{2}$ two PSAs, if $\Psi_{1}$ and $\Psi_{2}$ are
related intensively and effectively we say there exists _quantum entanglement_
between $\Psi_{1}$ and $\Psi_{2}$.

According to this definition entanglement relates to the potential coding of
intensive and effective relations between two distant measuring set-ups. We
also have the possibility to provide an intuitive non-spatial definition of
separability which relates to the lack of correlations between two distant
screens.

######  Definition 5.2 (Relational Separability)

Given $\Psi_{1}$ and $\Psi_{2}$ two PSAs, if $\Psi_{1}$ and $\Psi_{2}$ are not
related intensively nor effectively we say there is _relational separability_
between $\Psi_{1}$ and $\Psi_{2}$.

It is interesting to notice that our definitions of potential coding in terms
of intensive and effective relations allows us to address a third possibility
which considers the cases in which there are only intensive relations involved
but not effective ones.

######  Definition 5.3 (Intensive Correlation)

Given $\Psi_{1}$ and $\Psi_{2}$ two PSAs, if $\Psi_{1}$ and $\Psi_{2}$ are
related intensively but not effectively we say there exists an _intensive
relation_ between $\Psi_{1}$ and $\Psi_{2}$.

This new approach shows not only that it is possible to derive an objective
representation for the theory of quanta but also that metaphysical
considerations are essential for the analysis of operational data. As we have
discussed in detail in [28], the reference to intensive data provides a
completely new perspective of analysis for quantum information processing. The
essential relevance of intensive relations in QM has been completely bypassed
within the orthodox definition of entanglement which has been exclusively
focused on the dogmatic belief in the existence of invisible and
irrepresentable particles which collapse when observed [35]. This dogmatic
beliefs have acted in the philosophical and foundational communities as
epistemological obstructions which have boycotted the possibility of thinking
about entanglement in new original and consistent ways. It must be clear that
since the orthodox notion of entanglement is essentially grounded on both
‘particle metaphysics’ and the existence of ‘collapses’, the rejection of
these concepts also implies the rejection of the present definition of quantum
entanglement. Our redefinition beyond particles and ad hoc rules hopes to open
a necessary debate about the notion of quantum entanglement.

In the following section we attempt to show that the logos definition of
entanglement is capable to bypass both the non-separable relativism (section
2) and the contextual relativism (section 3) which hunt the orthodox
definitions.

##  6 Relational Quantum Entanglement Beyond Relativism

During the 1980s the Finish physicist Kalervo Laurikainen focused his research
in philosophical issues about QM and, more specifically, the thought of
Wolfgang Pauli, a key figure standing just in between Niels Bohr and Albert
Einstein. According to Laurikainen:

> “It is not generally known that there was a profound difference in the
> philosophical attitudes of Niels Bohr and Wolfgang Pauli (Laurikainen 1985b,
> section 3). In his address at the Second Centenary of Columbia University in
> 1954, ‘The Unity of Knowledge’, Bohr claimed that the observer even in
> quantum mechanics can be considered ‘detached’ provided we understand the
> observation in the right way (Bohr 1955, p. 83). An observation includes a
> detailed description of all the experimental arrangements which can have an
> influence upon the phenomenon under investigation, and it is finished only
> when a registered result is obtained which everybody can verify afterwards.
> In this sense, Bohr said, an observation is quite objective (which for Bohr
> means ‘intersubjective’), and the observer does not have any influence on
> the result in any other way than by choosing the method of observation. The
> result is explicitly associated with a given method of observation. If
> physics is understood as a system which makes it possible to govern such
> objective observational results —which, however, is only possible in
> probabilistic sense— then physics, according to Bohr, can even in atomic
> physics be considered quite objective and the observer is ‘detached’ in
> exactly the same way as in classical physics.” [56, p. 42]

Bohr had sent the manuscript of the paper to Pauli in order to receive his
critics and comments. In his reply, dated February 15, 1955, Pauli tried to
explain him —in line with Einstein— that the role of the observer in classical
mechanics was essentially different from the one presented in the orthodox
collapse interpretation of QM.

> “[…] it seems to me quite appropriate to call the conceptual description of
> nature in classical physics, which Einstein so emphatically wishes to
> retain, ‘the ideal of the detached observer’. To put it drastically the
> observer has according to this ideal to disappear entirely in a discrete
> manner as hidden spectator, never as actor, nature being left alone in a
> predetermined course of events, independent of the way in which phenomena
> are observed. ‘Like the moon has a definite position’ Einstein said to me
> last winter, ‘whether or not we look at the moon, the same must also hold
> for the atomic objects, as there is no sharp distinction possible between
> these and macroscopic objects. Observation cannot create an element of
> reality like position, there must be something contained in the complete
> description of physical reality which corresponds to the possibility of
> observing a position, already before the observation has been actually
> made.’ I hope, that I quoted Einstein correctly; it is always difficult to
> quote somebody out of memory with whom one does not agree. It is precisely
> this kind of postulate which I call the ideal of the detached observer.
>
> In quantum mechanics, on the contrary, an observation hic et nunc changes in
> general the ‘state’ of the observed system, in a way not contained in the
> mathematical formulated laws, which only apply to the automatical time
> dependence of the state of a closed system. I think here of the passage to a
> new phenomenon of observation which is taken into account by the so-called
> ‘reduction of the wave packets’. As it is allowed to consider the
> instruments of observation as a kind of prolongation of the sense organs of
> the observer, I consider the impredictable change of the state by a single
> observation —in spite of the objective character of the results of every
> observation and notwithstanding the statistical laws of frequencies of
> repeated observation under equal conditions— to be _an abandonment of the
> idea of the isolation (detachment) of the observer from the course of
> physical events outside himself._
>
> To put it in nontechnical common language one can compare the role of the
> observer in quantum theory with that of a person, who by his freely chosen
> experimental arrangements and recordings brings forth a considerable
> ‘trouble’ in nature, without being able to influence its unpredictable
> outcome and results which afterwards can be objectively checked by
> everyone.” [55, p. 60]

Bohr’s misunderstanding with respect to the meaning of objectivity went hand
in hand with his misunderstanding of the fact that Einstein’s relativity
theory did not imply any sort of perspectival-relativism. Just like Bohr had
confused the notion of intersubjectivity with that of objectivity, Bohr also
confused in several occasions his relativist-perspectival interpretation of QM
with Einstein’s relativity theory. In his Commo paper from 1929 he wrote the
following:

> “While the theory of relativity reminds us of the subjective character of
> all physical phenomena, a character which depends essentially upon the state
> of motion of the observer, so does the linkage of the atomic phenomena and
> their observation, elucidated by the quantum theory, compel us to exercise a
> caution in the use of our means of expression similar to that necessity in
> psychological problems where we continually come upon the difficulty of
> demarcating the objective content.” [10, p. 116]

This fragment shows the deep failure of Bohr not only to understand the
physics behind Einstein’s theory of relativity, but more importantly, the role
of objectivity and invariance in physics as the main conditions for a
“detached subject” representation —as contra-posed to his own perspectival-
relativist approach.101010This essential mistake is also present in Bohr’s
famous reply to EPR [11, p. 702]: “The dependence on the reference system, in
relativity theory, of all readings of scales and clocks may even be compared
with the essentially uncontrollable exchange of momentum or energy between the
objects of measurements and all instruments defining the space-time system of
reference, which in quantum theory confronts us with the situation
characterized by the notion of complementarity. In fact this new feature of
natural philosophy means a radical revision of our attitude as regards
physical reality, which may be paralleled with the fundamental modification of
all ideas regarding the absolute character of physical phenomena brought about
by the general theory of relativity.”

Unfortunately, this confusion has become completely widespread in many
relativist-perspectival interpretations of QM which regardless of their
impossibility to provide a global representation of the theory still claim
—following Bohr’s rhetorics— to be both realist and objective [27]. Let us try
to be very clear about this essential point. It is obviously true that
physical theories require in order to account for any given situation the
specification of a reference frame. This is maybe the very first lesson that
one must learn as a physicist, namely, that the representation of a state of
affairs always requires the explicit specification of the formal perspective
from which the situation is being described. Even in classical mechanics, the
position or velocity of a body have meaning only with respect (i.e., relative)
to a frame of reference. But of course, this relativism with respect to
reference frames has no ontological content. This relativism evaporates
immediately when considering the group of transformations which allows us to
shift the representation of the state of affairs from one reference frame to
another in a consistent (invariant) manner.111111As shown in [29], this
consistency in the translation of values of properties for different reference
frames is what perspectival interpretations of QM lack completely.
Perspectival or relativist interpretations of QM have no global
representational unity, and that is the reason measurement set-ups and
preferred bases become essential. And it is exactly this same operational-
invariance —of the values of the properties relative to different reference
frames— which allows us to detach all reference frames (or empirical subjects)
from the particular representation of a (real) state of affairs. In fact, the
whole point of talking about “something real”, is to talk about something
independent of any preferred (formal or empirical) perspective. This is the
essentially democratic nature of science which goes against any one claiming
to have a preferred access to reality. The realist cannot accept the existence
of a preferred basis or a preferred context in order to consider the
representation of a state of affairs.

Going back to our quotation, the analogy that Bohr attempted to make between
relativity and QM is obviously wrong: Relativity theory is in no way different
from classical mechanics with respect to invariance and objectivity. None of
them imply anything like the existence of the “subjective character of all
physical phenomena”. On the contrary, both theories are objective and
operationally invariant allowing a detached subject representation of any
state of affairs. The only difference between them is that while in classical
mechanics it is the Galilean transformations which allows us to consider all
reference frames as consistently referring to the same real state of affairs,
in the case of relativity theory this is done through the Lorentz
transformations. As Max Jammer [54, p. 132] remarked about our just quoted
passage: “Bohr overlooked that the theory of relativity is also a theory of
invariants and that, above all, its notion of ‘events,’ such as the collision
of two particles, denotes something absolute, entirely independent of the
reference frame of the observer and hence logically prior to the assignment of
metrical attributes.” The only thing that is “subjective” here is Bohr’s own
interpretation of QM, where there is no global consistency of the (binary)
values of projection operators considered from different contexts.121212The
impossibility of a global binary valuation was later on explicitly proved by
the now famous Kochen-Specker theorem. See for a detailed analysis [24]. As
Bohr famously argued in his reply to the EPR paper, the condition to consider
a subset of observables as definite valued was the specification of the
reference frame (or basis). Thus, in Bohr’s approach —unlike in Einstein’s
relativity theory— the description of a state of affairs would become
intrinsically relative to the agent’s choice of a single (preferred) context
(or basis). Of course, this goes in line with Bohr’s claim that in the great
drama of quantum existence the subject should be regarded not only a spectator
but also as actor —something untenable in the theory of relativity.

As we have shown explicitly in [29, 30] the inconsistency regarding the value
invariance of properties is not a consequence of the mathematical formalism of
QM, it is only specific to the consideration of binary valuation imposed by a
dogmatically presupposed actualist (metaphysical) understanding of physics. As
we mentioned above, the failure to understand the consequences of relativism
seem also present in Barnum’s considerations [5, p. 348] who restricts the
role of realism —like Earman also does— to the mere justification of the
existence of preferred bases and factorizations: “An important point brought
out by the attempt at a relative state interpretation of quantum mechanics is
the need to bring in, in addition to Hilbert space, notions of preferred
subsystems (‘experimenter’ and ‘system’ perhaps also the ‘rest of the world’)
or preferred orthogonal subspace decompositions (choice of ‘pointer basis’
(Zurek, 1981)).” As we have attempted to make clear, one of the essential
points of being a realist is to reject any preferential perspective within the
representation of a state of affairs. It is exactly this independence from
preferred viewpoints, reference frames or basis, which has been accomplished
in the logos formulation of QM through the consideration of a Global Intensive
Valuation [30]. If the notion of entanglement is to be understood as a feature
of the real word, and not merely a way in which subjects make choices about
fictitious entities, then we must be able to provide an objective account of
its relations independently of the choice of reference frames (bases) or
mathematical decompositions (factorizations). In the following, we derive two
theorems which show explicitly how the logos definition of entanglement is
capable to escape the relativism that hunts the orthodox definitions.

By reinterpreting a result from [30] and using our previous considerations we
can derive the following important theorem.

###### Theorem 6.1

(Basis Invariance Theorem) Let $\mathcal{H}$ be a Hilbert space, $\mathcal{G}$
its graph of powers and let $\Psi:\mathcal{G}\to[0,1]$ be a PSA. Let
$\mathcal{C}_{1},\mathcal{C}_{2}\subseteq\mathcal{G}$ be two contexts. The
intensities of $\Psi$ over $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ may be
different, but both determine in a unique way the whole PSA. In other words,
the GIV defined by $\Psi$ does not depend on the basis for $\mathcal{H}$. In
particular, since effective and intensive relations are defined globally, the
definitions of quantum entanglement, relational separability and intensive
correlation are non-contextual.

As a consequence of the Basis Invariance Theorem just stated, the relational
definition of entanglement is shown to be independent of the choice of any
particular basis or context. The choice of the basis or context becomes in
this scheme just the choice of a viewpoint of analysis which is compatible
with the choice of any other viewpoint.

Let us now analyze the factorization problem posed in [22]. Let us denote by
${\cal PSA}(H)$ the set of all possible PSAs on the Hilbert space $H$. We can
identify this set (after fixing a basis) with the space of density matrices
over $H$. In fact, given any completely positive trace-preserving map
$T:B(H_{1})\to B(H_{2})$, we obtain a map $T_{*}:{\cal PSA}(H_{1})\to{\cal
PSA}(H_{2})$. An example of a completely positive trace-preserving map is the
partial trace. Notice that for each factorization $H=H_{1}\otimes H_{2}$, we
have a different partial trace, $T:B(H)\to B(H_{1})$.

###### Definition 6.1

Let $T:B(H)\to B(H^{\prime})$ be a completely positive trace-preserving map
between (trace-class) operators on Hilbert spaces $H,H^{\prime}$ and let
$\Psi:\mathcal{G}(H)\to[0,1]$ be a PSA over $H$. We say that
$\Psi^{\prime}:\mathcal{G}(H^{\prime})\to[0,1]$ is a _shadow_ of $\Psi$ if
$\Psi^{\prime}=T_{*}(\Psi)$. Specifically, if $\Psi$ is given by a density
matrix $\rho$. Then, $\Psi^{\prime}$ is given by $\rho^{\prime}=T(\rho)$.

The relevance of this definition is that it allows us to avoid the
factorization problem posed by De la Torre et al.

###### Theorem 6.2

(Factorization Invariance Theorem) Let $\mathcal{H}$ be a Hilbert space with
factorizations
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}=\mathcal{H}_{1}^{\prime}\otimes\mathcal{H}_{2}^{\prime}$.
Let $T:B(\mathcal{H})\to B(\mathcal{H}_{1})$ and $T^{\prime}:B(\mathcal{H})\to
B(\mathcal{H}_{1}^{\prime})$ be the partial traces of these factorizations.
Let $\Psi$ be a PSA over $\mathcal{H}$ and let $\Psi_{1}=T_{*}(\Psi)$ and
$\Psi_{1}^{\prime}=T_{*}^{\prime}(\Psi)$ the corresponding PSAs over
$\mathcal{H}_{1}$ and $\mathcal{H}_{1}^{\prime}$. Assume there exists a
completely positive trace-preserving map $U:B(\mathcal{H}_{1})\to
B(\mathcal{H}_{1}^{\prime})$ such that $UT=T^{\prime}$, or diagrammatically,

$\textstyle{B(\mathcal{H})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{T}$$\scriptstyle{T^{\prime}}$$\textstyle{B(\mathcal{H}_{1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{U}$$\textstyle{B(\mathcal{H}_{1}^{\prime})}$

Then, $\Psi_{1}^{\prime}=U_{*}(\Psi_{1})$.

Proof: Straightforward,
$\Psi_{1}^{\prime}=T^{\prime}_{*}(\Psi)=(UT)_{*}(\Psi)=U_{*}T_{*}(\Psi)=U_{*}(\Psi_{1})$.

The previous theorem implies that all factorizations are consistent and that
all corresponding shadows of the PSA are all compatible between each other. In
fact, if we have a compatibility $U$ between the factors $\mathcal{H}_{1}$ and
$\mathcal{H}_{1}^{\prime}$, then this compatibility translates itself in a
compatibility between $\Psi_{1}$ and $\Psi_{1}^{\prime}$.

###### Corollary 6.1

All factorizations are compatible with respect to the same PSA.

Proof: Follows from the previous Theorem.

###### Remark 6.1

Notice that there is a deep relation between factorizing a Hilbert space as
$\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ and choosing a basis for
$\mathcal{H}$. In fact, given a basis $\\{v_{i}\\}$ for $\mathcal{H}_{1}$ and
$\\{w_{j}\\}$ for $\mathcal{H}_{2}$, we can construct a basis for
$\mathcal{H}$ as $\\{v_{i}\otimes w_{j}\\}$. Then, we can say that a
factorization determines a basis for $\mathcal{H}$ from the basis of its
factors.

Thus, the relational definition of entanglement is immune to the contextual
relativism discussed in [22]. The choice of the factorization has no incidence
in the intensive and effective relations already contained in the considered
PSA. In the logos approach the choice of the factorization makes reference to
the number of intensive powers considered within a PSA. All choices of
different factorizations remain consistent with the same global account of the
PSA.

To sum up, we might conclude that the non-separable and contextual relativisms
found in the orthodox interpretations is bypassed in the new objective
definition of entanglement. The choice of bases and factorizations does not
change the intensive and effective relations already contained within the
given (potential) state of affairs described by QM in terms of powers with
definite potentia. Thus, all powers and potentia in a given (potential) state
of affairs can be considered as objectively existent, invariant and
independent of any particular measurement.

##  7 Final Remarks

Erwin Schrödinger wrote in 1935 that quantum entanglement should be regarded
as “the characteristic trait of quantum mechanics, the one that enforces its
entire departure from classical lines of thought.” It took more than fifty
years for the community of physics to acknowledge this fact. As John Earman
[43] makes the point: “With the rise of quantum computing and quantum
information theory there has been a sea change in attitude towards
entanglement: it is not something to be feared but rather is a resource to be
exploited.” However, as pointed out by Alisa Bokulich and Greeg Jaeger [9, p.
xiv]: “Despite the fact that the phenomenon of entanglement was recognized
very early on in the development of quantum mechanics, it remains one of the
least understood aspects of quantum theory.” We believe that the deep
misunderstandings surrounding this kernel notion in the orthodox literature
are intrinsically related, in part to the widespread instrumentalist approach
of physics and in part to the dogmatic reference to atomist metaphysics. Both
instrumentalism and metaphysical atomism act today as a epistemological
obstacles which preclude the possibility of a truly (non-classical) conceptual
approach to the theory of quanta. In this same respect, the rise of
entanglement in the new millennia might be taken as a lesson of the essential
role of philosophical conceptual analysis within physics. It is our conviction
that it will not be possible to continue this revolution without truly
comprehending the meaning of entanglement. Such a comprehension implies the
need to go not only beyond the instrumental operational prediction of ‘clicks’
in detectors but also beyond the misleading discourse about “small particles”
living in an unobservable microscopic world.

## Acknowledgements

Christian de Ronde wants to thank Diederik Aerts and Décio Krause for
discussions on related subjects. This work was partially supported by the
following grants: FWO project G.0405.08 and FWO-research community W0.030.06.
CONICET RES. 4541-12 and the Project PIO-CONICET-UNAJ (15520150100008CO)
“Quantum Superpositions in Quantum Information Processing”.

## References

  * [1] Aerts, D., 1981, The one and the many: towards a unification of the quantum a classical description of one and many physical entities, Doctoral dissertation, Brussels Free University, Brussels.
  * [2] Aerts, D., 1984, “How do we have to change quantum mechanics in order to describe separated systems”, in The Wave-Particle Dualism pp. 419-431, S. Diner, D. Fargue, G. Lochak and F. Selerri (Eds.), Springer, Dordrecht.
  * [3] Aspect A., Grangier, P. $\&$ Roger, G., 1981, “Experimental Tests on Realistic Local Theories via Bell’s Theorem”, Physical Review Letters, 47, 725-729.
  * [4] Barnum, H., 1990, “The Many-Worlds Interpretation of Quantum Mechanics: Psychological versus Physical Bases for the Multiplicity of ‘Worlds’. ”, preprint. (PittPhilsciArchive:2647)
  * [5] Barnum, H., 2003, “Quantum information processing, operational quantum logic, convexity, and the foundations of physics”, Studies in History and Philosophy of Modern Physics, 34, 343-379.
  * [6] Barnum, H., Knill, E., Ortiz, G., Somma, R. $\&$ L. Viola, L., 2003, “Generalizations of entanglement based on coherent states and convex sets”, Physical Review A, 68, 032308.
  * [7] Barnum, H., Knill, E., Ortiz, G., Somma, R. $\&$ L. Viola, L., 2004, “A subsystem independent generalization of entanglement”, Physical Review Letters, 92, 107902.
  * [8] Barnum, H. $\&$ L. Viola, L., 2005, “Generalization of entanglement to convex operational theories: Entanglement relative to a subspace of observables”, International Journal of Theoretical Physics, 44, 2127-2145.
  * [9] Bokulich, A. $\&$ Jaeger, G., 2010, Philosophy of Quantum Information and Entanglement, Cambridge University Press, Cambridge.
  * [10] Bohr, N., 1934, Atomic theory and the description of nature, Cambridge University Press, Cambridge.
  * [11] Bohr, N., 1935, “Can Quantum Mechanical Description of Physical Reality be Considered Complete?”, Physical Review, 48, 696-702.
  * [12] Bohr, N., 1948, “On the notions of causality and complementarity”, Dialectica, 2, 312-319.
  * [13] Bohr, N., 1960, The Unity of Human Knowledge, In Philosophical writings of Neils Bohr, vol. 3., Ox Bow Press, Woodbridge.
  * [14] Born, M., 1971, The Born-Einstein Letters, Walker and Company, New York.
  * [15] Bruß, D., 2002, “Characterizing entanglement”, Journal of Mathematical Physics, 43, 4237-4251.
  * [16] Bub, J., 2017, “Quantum Entanglement and Information”, The Stanford Encyclopedia of Philosophy (Spring 2017 Edition), E.N. Zalta (ed.), URL = https://plato.stanford.edu/archives/spr2017/entries/qt-entangle/.
  * [17] Butterfild, J., 2017, interview: Jeremy Butterfield: What is contextuality?, (published 3rd of March, 2017) https://www.youtube.com/watch?v=ZJAnixX8T4U
  * [18] Cabello, A., 2017, “Interpretations of quantum theory: A map of madness”, in What is Quantum Information?, pp. 138-143, O. Lombardi, S. Fortin, F. Holik and C. López (eds.), Cambridge University Press, Cambridge.
  * [19] Carnap, H., Hahn, H. $\&$ Neurath, O., 1929, “The Scientific Conception of the World: The Vienna Circle”, Wissendchaftliche Weltausffassung.
  * [20] Clifton, R.K., 1995, “Why Modal Interpretations of Quantum Mechanics must Abandon Classical Reasoning About Physical Properties”, International Journal of Theoretical Physics, 34, 1302-1312.
  * [21] Clifton, R.K., 1996, “The Properties of Modal Interpretations of Quantum Mechanics”, British Journal for the Philosophy of Science, 47, 371-398.
  * [22] de la Torre, A.C., Goyeneche, D. $\&$ Leitao, L., 2010, “Entanglement for all quantum states”, European Journal of Physics, 31, 325-332.
  * [23] de Ronde, C., 2016, “Probabilistic Knowledge as Objective Knowledge in Quantum Mechanics: Potential Immanent Powers instead of Actual Properties”, in Probing the Meaning of Quantum Mechanics: Superpositions, Semantics, Dynamics and Identity, pp. 141-178, D. Aerts, C. de Ronde, H. Freytes and R. Giuntini (Eds.), World Scientific, Singapore.
  * [24] de Ronde, C., 2020, “Unscrambling the Omelette of Quantum Contextuality (Part I): Preexistent Properties or Measurement Outcomes?”, Foundations of Science, 25, 55-76.
  * [25] de Ronde, C., 2020, “The (Quantum) Measurement Problem in Classical Mechanics”, in Probing the Meaning of Quantum Mechanics, D. Aerts, J. Arenhart, C. de Ronde and G. Sergioli (Eds.), World Scientific, Singapore, in press.
  * [26] de Ronde, C., 2020, “Understanding Quantum Mechanics (Beyond Metaphysical Dogmatism and Naive Empiricism)”, preprint.
  * [27] de Ronde, C. $\&$ Fernández Mouján, R., 2018, “Epistemological vs. Ontological Relationalism in Quantum Mechanics: Relativism or Realism?”, in Probing the Meaning of Quantum Mechanics, pp. 277-318, D. Aerts, M.L. Dalla Chiara, C. de Ronde and D. Krause (Eds.), World Scientific, Singapore.
  * [28] de Ronde, C., Freytes, H. $\&$ Sergioli, G., 2019, “Quantum probability: A reliable tool for an agent or a source of reality”, Synthese, DOI:10.1007/s11229-019-02177-x.
  * [29] de Ronde, C. $\&$ Massri, C., 2017, “Kochen-Specker Theorem, Physical Invariance and Quantum Individuality”, Cadernos da Filosofia da Ciencia, 2, 107-130.
  * [30] de Ronde, C. $\&$ Massri, C., 2018, “The Logos Categorical Approach to Quantum Mechanics: I. Kochen-Specker Contextuality and Global Intensive Valuations.”, International Journal of Theoretical Physics, DOI: 10.1007/s10773-018-3914-0.
  * [31] de Ronde, C. $\&$ Massri, C., 2019, “The Logos Categorical Approach to Quantum Mechanics: II. Quantum Superpositions and Measurement Outcomes.”, International Journal of Theoretical Physics, 58, 1968-1988.
  * [32] de Ronde, C. $\&$ Massri, C., 2019, “A New Objective Definition of Quantum Entanglement as Potential Coding of Intensive and Effective Relations”, Synthese, DOI: 10.1007/s11229-019-02482-5.
  * [33] de Ronde, C. $\&$ Massri, C., 2019, “Against the Tyranny of Pure States in Quantum Theory”, sent. (quant-ph/arXive:1902.01667).
  * [34] de Ronde, C. $\&$ Massri, C., 2020, “Beyond Purity and Mixtures in Categorical Quantum Mechanics”, sent. (quant-ph/arXive:2002.04423).
  * [35] de Ronde, C. $\&$ Massri, C., 2020, “Against ‘Particles’ and ’Collapses’ in Quantum Entanglement”, forthcoming. (quant-ph/arXive:1911.10990).
  * [36] D’Espagnat, B., 2006, On Physics and Philosophy, Princeton University Press, Princeton.
  * [37] Deutsch, D., 2004, The Beginning of Infinity. Explanations that Transform the World, Viking, Ontario.
  * [38] Dieks, D., 1988, “The Formalism of Quantum Theory: An Objective description of reality”, Annalen der Physik, 7, 174-190.
  * [39] Dieks, D., 2010, “Quantum Mechanics, Chance and Modality”, Philosophica, 83, 117-137.
  * [40] Dieks, D., 2020, “Identical Quantum Particles, Entanglement, and Individuality”, Entropy, 22, 134-140.
  * [41] Dieks, D. $\&$ Lubberdink, A., 2011, “How Classical Particles Emerge From the Quantum World”, Foundations of Physics, 41, 1051-1064.
  * [42] Dirac, P.A.M., 1974, The Principles of Quantum Mechanics, 4th Edition, Oxford University Press, London.
  * [43] Earman, J., 2015, “Some Puzzles and Unresolved Issues About Quantum Entanglement”, Erkenntnis, 80, 303-337.
  * [44] Einstein, A., 1948, “Quanten-Mechanik und Wirklichkeit”, Dialectica, 2, 320-324.
  * [45] Einstein, A., Podolsky, B. $\&$ Rosen, N., 1935, “Can Quantum-Mechanical Description be Considered Complete?”, Physical Review, 47, 777-780.
  * [46] Everett, H., 1957, “Relative State Formulation of Quantum Mechanics”, Review of Modern Physics, 29, 454.
  * [47] Freire Jr., O. 2015, The Quantum Dissidents. Rebuilding the Foundations of Quantum Mechanics (1950-1990), Springer, Berlin.
  * [48] Fuchs, C.A. $\&$ Peres A., 2000, “Quantum theory needs no ‘interpretation”’, Physics Today 53, 70-71.
  * [49] Gurtovoi, V.L., Il’in A.I. $\&$ Nikulov, A.V., 2020, “Experimental investigations of the problem of the quantum jump with the help of superconductor nanostructures”, Physics Letters A, 26.
  * [50] Heisenberg, W., 1971, Physics and Beyond, Harper & Row, New York.
  * [51] Heisenberg, W., 1973, “Development of Concepts in the History of Quantum Theory”, in The Physicist’s Conception of Nature, pp. 264-275, J. Mehra (Ed.), Reidel, Dordrecht.
  * [52] Hilgevoord, J. $\&$ Uffink, J., 2001, “The Uncertainty Principle”, The Stanford Encyclopedia of Philosophy (Winter 2001 Edition), E. N. Zalta (Ed.), URL: http://plato.stanford.edu/archives/win2001/entries/qt-uncertainty/.
  * [53] Horodecki, R., Horodecki, P., Horodecki, M. $\&$ Horodecki, K. “Quantum entanglement”, Reviews of Modern Physics, 81, 865-942.
  * [54] Jammer, M., 1974, The Philosophy of Quantum Mechanics, John Wiley and sons, New York.
  * [55] Laurikainen, K.V., 1988, Beyond the Atom, The Philosophical Thought of Wolfgang Pauli, Springer-Verlag, Berlin.
  * [56] Laurikainen, K.V., 1998, The Message of the Atoms, Essays on Wolfgang Pauli and the Unspeakable, Springer-Verlag, Berlin.
  * [57] Li, J.-L. $\&$ Qiao, C.-F., 2018, “A Necessary and Sufficient Criterion for the Separability of Quantum State”, Scientific Reports, 8, 1442.
  * [58] Lo Franco, R. $\&$ Compagno, G., 2016, “Quantum entanglement of identical particles by standard information-theoretic notions.”, Scientific Repoprts, 6, 20603.
  * [59] Maudlin, T., 2019, Philosophy of Physics. Quantum Theory, Princeton University Press, Princeton.
  * [60] Pauli, W., 1994, Writings on Physics and Philosophy, Enz, C. and von Meyenn, K. (Eds.), Springer-Verlag, Berlin.
  * [61] Pawowski M. $\&$ Czachor, M., 2006, “Degree of entanglement as a physically ill-posed problem: The case of entanglement with the vacuum”, Physical Review A, 73 042111\.
  * [62] Pitowsky, I., 1994, “George Boole’s ‘Conditions of Possible Experience’ and the Quantum Puzzle", The British Journal for the Philosophy of Science, 45, 95-125.
  * [63] Popper, K.R., 1963, Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge Classics, London.
  * [64] Rovelli, C., 1996, “Relational Quantum Mechanics”, International Journal of Theoretical Physics, 35, 1637-1678.
  * [65] Smolin, L., 2007, The trouble with physics. The rise of string theory, the fall of a science, and what comes next, Mariner books, New York.
  * [66] Scholz, E., 2018, “Weyl’s search for a difference between ‘physical’ and ‘mathematical’ automorphisms’.” Studies in History and Philosophy of Modern Physics, 61, 57-67.
  * [67] Schrödinger, E., 1935, “The Present Situation in Quantum Mechanics”, Naturwiss, 23, 807-812. Translated to english in Quantum Theory and Measurement, J.A. Wheeler and W.H. Zurek (Eds.), 1983, Princeton University Press, Princeton.
  * [68] Shimony, A., 1995, “Degree of entanglement”, Annals New York Academy of Science, 755, 675-679.
  * [69] Sudbery, A., 2016, “Time, Chance and Quantum Theory”, in Probing the Meaning and Structure uantum Mechanics: Superpositions, Semantics, Dynamics and Identity, pp. 324-339, D. Aerts, C. de Ronde, H. Freytes and R. Giuntini (Eds.), World Scientific, Singapore.
  * [70] Thirring, W., Bertlmann R. A., Köhler, P. $\&$ Narnhofer, H., 2011, “Entanglement or separability: the choice of how to factorize the algebra of a density matrix”, The European Physical Journal D, 64, 181-196.
  * [71] Viola, L. $\&$ Barnum, H., 2010, “Entanglement and subsystems, Entanglement beyond subsystems, and All That”, in Philosophy of Quantum Information and Entanglement, pp. 16-43, A. Bokulich and G. Jaeger (Eds.), Cambridge University Press, Cambridge.
  * [72] Yang, J.M., 2018, “A Relational Formulation of Quantum Mechanics”, Scientific Reports, 8, 13305.