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NeuroQuantology | September 2018 | Volume 16 | Issue 9 | Page 1-4 | doi: 10.14704/nq.2018.16.9.1368
Caponigro M., Ontology of Quantum Entangled States No Needs Philosophy
eISSN 1303-5150 www.neuroquantology.com
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Ontology of Quantum Entangled States
No Needs Philosophy
Michele Caponigro
ABSTRACT
When we understand, usually philosophical debates disappear. In this brief paper, starting from some works, we
argue that quantum entangled states could be considered as primitive. We analyze from a conceptual point of view
this basic question: can the nature of quantum entangled states be interpreted ontologically or epistemologically?
According to some works, the degrees of freedom (and the tool of quantum partitions) of quantum systems permit
us to establish a possible classification between factorizable and entangled states. We suggest, that the "choice" of
degree of freedom (or quantum partitions), even if mathematically justified introduces an epistemic element, not
only in the systems but also in their classification. We retain, instead, that there are not two classes of quantum
states, entangled and factorizable, but only a single class of states: the entangled states. In fact, the factorizable
states become entangled for a different choice of their degrees of freedom (i.e. they are entangled with respect to
other observables). In the same way, there are no partitions of quantum systems which have an ontologically
superior status with respect to any other. For all these reasons, both mathematical tools utilize(i.e quantum
partitions or degrees of freedom) are responsible for creating an improper classification of quantum systems.
Finally, we argue that we cannot speak about a classification of quantum systems: all quantum states exhibit a
uniquely objective nature, they are all entangled states. In this framework we think that Rovelli’s interpretation of
QM based on the relational approach could be considered a good candidate to understand the nature of physical
reality.
Key Words: Ontology of Quantum Entanglement, Subsystems (Partitions and Factorizables States), Epistemic vs
Ontic Elements, Quantum Reality
DOI Number: 10.14704/nq.2018.16.9.1368 NeuroQuantology 2018; 16(9):1-4
Introduction
Partitions and Systems
In spite of continuous progress, the current state
of entanglement theory is still marked by a
number of outstanding unresolved problems.
These problems range from the complete
classification of mixed-state bipartite
entanglement to entanglement in systems with
continuous degrees of freedom, and the
classification and quantification of multipartite
entanglement for arbitrary quantum states.
In this paper, starting form two
important works,
1) Torre et al.,(2010) and 2) Zanardi
(2001), we will analyze the possible relationship
among these elements:
1. the degrees of freedom of the quantum system
2. the partitions of the quantum system
3. the epistemic elements introduced from the
procedures (1) and (2)
As we know, the relationship between
quantum systems (QS) and their possible
quantum entangled systems (QES) is not a trivial
question. There are many efforts to understand
this dynamics. Zanardi (2001) in his paper argues
that the partitions of a possible system do not
Corresponding author: Michele Caponigro
Address: ISHTAR, Bergamo University
e-mail  michele.caponigro@unibg.it
Relevant conflicts of interest/financial disclosures: The authors declare that the research was conducted in the absence of any
commercial or financial relationships that could be construed as a potential conflict of interest.
Received: 16 April 2018; Accepted: 27 July 2018

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have an ontologically superior status with respect
to any other: according Zanardi given a physical
system , the way to subdivide it in subsystems is
in general by no means unique. We will analyze
his conclusion in the following sections below.
According Zanardi the consequences of
the non uniqueness of the decomposition of a
given system into subsystems imply (at the
quantum level), a fundamental ambiguity about
the very notion of entanglement that accordingly
becomes a relative one. The concept of "relative"
for an entangled system, has been developed by
Viola and Barnun (2010). They concentrate their
efforts on this fundamental question: how can
entanglement be understood in an arbitrary
physical system, subject to arbitrary constraints
on the possible operations one may perform for
describing, manipulating, and observing its
states? In their papers, the authors proposed that
entanglement is an inherently relative concept,
whose essential features may be captured in
general in terms of the relationships between
different observers (i.e. expectations of quantum
observables in different, physically relevant sets).
They stressed how the the role of the observer
must be properly acknowledged in determining
the distinction between entangled and
unentangled states.
Quantum Entanglement: brief overview
From a phenomenological point of view, the
phenomenon of entanglement is quite simple.
When two or more physical systems form an
interaction, some correlation of a quantum nature
is generated between the two of them, which
persists even when the interaction is switched off
and the two systems are spatially separated.
Quantum entanglement describes a non-separable
state of two or more quantum objects and has
certain properties which contradict common
physical sense. While the concept of entanglement
between two quantum systems, which was
introduced by E. Schrödinger (1936) is well
understood, its generation and analysis still
represent a substantial challenge. Moreover, the
problem of quantification of entangled states, is a
long standing issue debated in quantum
information theory. Today the bipartite
entanglement (two-level systems, i.e. qubits) is
well understood and has been prepared in many
different physical systems. The mathematical
definition of entanglement varies depending on
whether we consider only pure states or a general
set of mixed states (see Giannetto 1995: where it
is discussed the reason why entanglement
generally requires a density matrix formalism). In
the case of pure states, we say that a given a state
| ⟩ of parties is entangled if it is not a tensor
product of individual states for each one of the
parties, that is,
| ⟩ ≠ | ⟩ ⊗ | ⟩ ⊗ ⋯ ⊗ | ⟩ . (1)
For instance, in the case of 2 qubits and
(sometimes called "Alice" and "Bob") the
quantum state
| ⟩ =
√
[ |0⟩ ⊗ |0⟩ + |1⟩ ⊗ |1⟩ ] (2)
is entangled since | ⟩ ≠ | ⟩ ⊗ | ⟩ .
On the contrary, the state
| ⟩ = [ |0⟩ ⊗ |0⟩ + |1⟩ ⊗ |0⟩ + |0⟩ ⊗
|1⟩ + |1⟩ ⊗ |1⟩ ] (3)
is not entangled, since
| ⟩ =
√
|0⟩ + |1⟩ ! ⊗
√
|0⟩ + |1⟩ !
(4)
A pure state like the one from Eq.2 is
called a maximally entangled state of two qubits, or
a Bell pair, whereas a pure state like the one from
Eq.4 is called separable. In the general case of
mixed states, we say that a given state " of
constituent states is entangled if it is not a
probabilistic sum of tensor products of individual
states for each one of the subconstituents, that is,
" ≠ ∑ $%
% "%
⊗ "%
⊗ ⋯ ⊗ "%
, (5)
with {$%} being some probability
distribution. Otherwise, the mixed state is called
separable. The essence of the above definition of
entanglement relies on the fact that entangled
states of constituents cannot be prepared by
acting locally on each one of them, together with
classical communication among them.
Entanglement is a genuinely quantum-mechanical
feature which does not exist in the classical world.
It carries non-local correlations between the
different systems in such a way that they cannot
be described classically.

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Are quantum states all Entangled?
As mentioned above, the recent work by Torre
(2010) is a fundamental paper which gives us the
possibility to speculate about the nature and the
classification of entangled states. The paper
demonstrates that a state is factorizable in the
Hilbert space corresponding to some choice of
degrees of freedom, and that this same state
becomes entangled for a different choice of
degrees of freedom. Therefore, entanglement is
not a special case, but is ubiquitous in quantum
systems. According to the authors, one may
erroneously think that there are two classes of
states for the QS: 1) factorizable and 2) entangled,
which correspond to qualitative difference in the
behaviour of the system, close to classical in one
case and with strong quantum correlations in the
other. They argue that this is indeed wrong
because factorizable states also exhibit
entanglement with respect to other observables.
In this sense, all states are entangled;
entanglement is not an exceptional feature of
some states but is ubiquitous in QM..
To sum up this conceptual analysis by
Torre and Zanardi, we think that there is an
unclear relationship among these elements:
1. factorizable states (Torre)
2. entangled states
3. the (choice) of partitions of quantum
system (Zanardi)
4. the role of the observer (in determining
the distinction between entangled and
unentangled states)
We think that all points (except the second
point) introduce epistemic elements in the
analysis and in the classification of the quantum
systems. We suggest that the second point is the
key to understand the nature of the underlying
physical reality. We argue in the next sections,
that the conceptual analysis of Torre and Zanardi
differs from what we suggest concerning the
epistemic elements introduced in their papers.
Factorizability of a state as Epistemic property?
An important question is related at the property
of factorizability of quantum state. Is the
factorizability tool an objective property? Briefly
stored, is factorizability an objective property of
the system or is it a feature of (our) description of
system (i.e. an epistemic property)?. With
reference to Torre’s paper (2010), the authors
show that factorizability and entanglement are
not preserved in a change of the degrees of
freedom used to describe the system, they
demonstrate in detailed case that the
factorizability of a state is a property that is not
invariant under a change of the degrees of
freedom that we use in order to describe the
system. From mathematical point of view[1], they
consider a quantum system with two subsystems
= , that may correspond to two degrees
of freedom and . The state of the system
belongs then to the Hilbert space ℋ = ℋ ⊗ ℋ
and the two degrees of freedom are represented
by operators ⊗ * and * ⊗ . Suppose it is given
that the system has a factorizable, non entangled,
state Ψ = Ψ ⊗ Ψ with Ψ and Ψ arbitrary
states (not necessarily eigenvectors of and ) in
the spaces ℋ and ℋ . Then there exists a
transformation of the degrees of freedom , =
, , and - = - , that suggests a different
factorization, ℋ = ℋ. ⊗ ℋ/, where the state is
no longer factorizable: Ψ ≠ Ψ. ⊗ Ψ/ with Ψ. ∈
ℋ. and Ψ/ ∈ ℋ/. The state becomes entangled
with respect to the new degrees of freedom; the
factorizability of states is not invariant under a
different factorization of the Hilbert space. To
conclude, they have shown that for any system in
a factorizable state, it is possible to find different
degrees of freedom that suggest a different
factorization of the Hilbert space where the same
state becomes entangled; for this reason they
argued that every state, even for those
factorizable, it is possible to find pairs of
observables that will violate Bell’s inequalities.
The authors analyze also the inverse problem: the
fact that the appearance of entanglement depends
on the choice of degrees of freedom can find an
interesting application in the "disentanglement"
of a state; one can, sometimes, transform an
entangled state into a factorizable one by a
judicious choice of the degrees of freedom.
To conclude, we think that the epistemic
element is inherent in the possibility to "choose"
the degrees of freedom of the quantum system:
this possibility affects the classification of
quantum states in entangled or factorizables. In
fact, it is simple to ask these epistemological
questions: a)what are the degrees of freedom for
a quantum system? b) Is it a complete set that
describe all quantum properties? Can be a particle
entangled in one context be factorizables in
another context?
The partitions of quantum system as Epistemic
property?
As we have seen, given a quantum system, the
way to subdivide (to partition) it in subsystems in

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not unique. We call this first phase "epistemic",
as in fact we are able to decide how to partition
the quantum system. The conclusion of this
operation is most important of its premise: in fact
if we find (in the subsystems) an entangled state,
this state has an ontological nature but only if
referred to that kind of particular partition
carried. We have, in other words, an objective
entangled state for an epistemic partitions! For
these reasons, the notions of an entangled state
becomes a relative concept and the relativity of
this concept is linked, to us, at the choice of
partitions or degrees of freedom. At the same
time, the property of the entangled state is
objective.
Conclusions
We have seen that quantum systems admit a
variety of tensor product structures depending on
the complete system of commuting observables
chosen for the analysis; as a consequence we have
different notions of entanglement associated with
these different tensor product.
We notice that, in the determination of
whether a state is factorizable or entangled, the
factorization of the Hilbert space is crucial and
this factorization depends on the choice of the
observables corresponding to the degrees of
freedom. In the same way, as Zanardi stressed,
given a quantum system, the way to subdivide it
(via partitions) in subsystems it is not unique;
the partitions of a possible system have not an
ontologically superior status with respect to any
other. Based on these points, we argue, that the
criteria of partitions and factorizability (or
partitions) contain an a-priori epistemic element.
In conclusion, we suggest that all quantum system
exhibit an objective nature that is entangled, at
basic level the underlying physical reality is
entangled. A quantum state could be non-
entangled if and only if it would be factorizable for
every possible partition or choice of degrees of
freedom, but this can never occur. The epistemic
level emerges with the öbserver" (partitions or
degree of freedom), the physicists and
philosophers should consider these arguments in
their debates. In this framework we think that
Rovelli’s interpretation of QM based on the
relational approach could be considered a good
candidate to understand the nature of physical
reality.
Acknowledgement
I would like to thank my supervisor Prof. Enrico
Giannetto (Bergamo University).
References
De la Torre AC, Goyeneche D, Leitao L. Entanglement for all
quantum states. European Journal of Physics 2010;
31(2):325-32.
Zanardi P. Virtual quantum subsystems. Physical Review
Letters 2001; 87(7): 077901.
Viola L, Barnum H. Entanglement and subsystems,
entanglement beyond subsystems, and all that.
Philosophy of Quantum Information and Entanglement;
Bokulich, A., Jaeger, G., Eds. 2010: 16-43.
Schrödinger E. Discussion of Probability Relations Between
Separated Systems. Proceedings of the Cambridge
Philosophical Society 1936; 31(4): 555-63.
Giannetto E. Some remarks on non-separability, in the
foundations of quantum mechanics. C. Garola, A. Rossi
(eds.), Dordrecht: Kluwer, 1995: 315-24.

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