J. Mod. Phys., 2010, 1, 295-299
doi:10.4236/jmp.2010.15041 Published Online November 2010 (http://www.SciRP.org/journal/jmp)
Entangled States and Observables in Open Quantum
Relativity
Salvatore Capozziello1,2*, Gi usepp e Basini3,4
1Dipartimento di Scienze Fisiche, Università di Napoli “Federico II”
2INFN, Sez. di Napoli, Compl. Univ. di Monte S. Angelo, Edificio G, via Cinthia, I-80126-Napoli, Italy
3CERN, CH-1211 Geneva 23, Switzerland
4Laboratori Nazionali di Frascati, INFN via E. Fermi, Frascati I-00100, Roma, Italy
Received August 17, 2010; revised October 17, 2010; accepted October 25, 2010
Abstract
In the framework of the so called Open Quantum Relativity, we investigate a quantum universe, starting
from a minimal set of variables defining the given quantum state. Entanglement between quantum states is
the way to link different regions of the universe, even if (apparently) causally disconnected. As a conse-
quence, the concept of causality results recovered and enlarged. Besides, the observed CDM model
emerges from this picture, giving the possibility to realize a statistical and quantum interpretation of the
cosmological constant. In particular, the novelty consists in the fact that the presently observed universe
could be the result of several entanglement phenomena giving rise to a certain amount of entropy directly
related to the value of cosmological constant.
Keywords: Quantum Mechanics, Quantum Cosmology, Entanglement
1. Introduction
Up to now, several difficulties occurred in the attempts
of quantization of gravity [1-3]. Many authors have
discussed the idea that gravity cannot be quantized, due
to these problems; other authors, that present models are
effective picture and need to be generalized. All the
approaches pass through the problems of quantizing
General Relativity (GR). In other words, the common
approach is to start from a classical field theory to a
quantum one. The question is, if it is possible not to pass
through a field theory, at the beginning, and to consider,
so, only a quantum description of universe, i.e. without a
second quantization, where the chosen quantum state has
to be found by the use of observable quantities, with
minimal choice of the relevant elements of the set too.
Then, at the end of a quantum picture, considering also
the possibility to have a second quantization, from the
quantum picture. This is a peculiar characteristic of the
so-called Open Quantum Relativity [4-6]. Such a theory
[4] is based on a dynamical unification scheme of
fundamental interactions achieved by assuming a 5D
space which allows that the conservation laws are always
and absolutely valid as a natural necessity. What we
usually perceive as violations of conservation laws can
be described by a process of embedding and dimensional
reduction, which gives rise to an induced-matter theory
in the 4D space-time by which the usual masses, spins
and charges of particles, naturally spring out. At the
same time, it is possible to build up a covariant symplec-
tic structure directly related to general conservation laws
[7,8]. Finally, the theory leads to a dynamical explana-
tion of several paradoxes of modern physics (e.g.
entanglement of quantum states, quantum teleportation,
gamma ray bursts origin, black ho le singularities, cosmic
primary antimatter absence and a self-con sistent fit of all
the recently observed cosmological parameters [4,9-12]).
A fundamental role in this approach is the link between
the geodesic structure and the field equations of the
theory before and after the dimensional reduction
process. The emergence of an Extra Force term in the
reduction process and the possibility to recover the
masses of particles, allow to reinterpret the Equivalence
Principle as a dynamical consequence which naturally
“selects” geodesics from metric structure and vice-versa
the metric structure from the geodesics [6,13].
To start with this different point of view, we need a set
of minimal number of quantities, corresponding to a set

S. CAPOZZIELLO ET AL. 296
of classical objects, observable and measurable. From a
theoretical point of view, we have to stress that this is
also a new definition of observer. Then the set has to be
minimal, in terms of its elements and above all, it has to
represent a basis for the cosmological quantum state.
If the set of observable is represented by a certain
number of quantities, i.e. ,
(),, (),
mk
at qt, by the
hypothesis of homogeneity and isotropy, i.e. using
Friedman-Robertson-Walker (FRW) model, it would be
very easy to show that the set could be reduced and a
minimal choice would be obtained by considering only
two observable among all [14]. The relevant choice is to
consider a set based on m  and k , which appears as
a minimal one.
The ansatz is that the minimal set is sufficient to
implement a quantum state for universe. The minimal
choice may allow us to have a state but needs to be
rewritten in form of a basis; thanks to Gram-Schmidt
procedure we infer this basis; after this, it would be
possible to build a general state for universe, whose
picture is that the universe would be divided in two
space-time regions of interest, characterized by different
behaviors of scale factor )(za . GR will match the
quantum framework easily after, by considering the
definitions of important quantities [15,16], such as
),(,, za
m
 which, previously, have been consi-
dered independent from the model, in the sense that they
only are measurable astrophysical quantities [17,18].
In order to build up a reasonable cosmological
entangled state, B
n
A
n
n
 ||| , we follow the
standard construction of the minimal states by Gram-
Schmidt procedure, as suggested. We find that a
similar state is proportional to .| 










ki
mi
i
 We labeled
the states with 1,2=i for defining two regions and we
refer to entangled states ansatz hereafter of this work, for
stressing the entang led nature of the states. The first step
is to have a basis; by Gram-Schmidt is possible using a
simplest set of positions for second state

22112
21 2
12
*2=mkkmmmmm N and


22112
21 2
12
*2=kmkmkkkk N , which
are hereafter our minimal positions.
In this way we have two cosmological states of the
form of 










1,2
1,2
|
k
m
i
 referred to two different eras
1,2, with the second era having km,
 given by the
above cited minimal positions.
In this paper we describe some relevant properties of
this picture. In particular, the novelty consists in the fact
that the presently observed universe could be the result
of several entanglement phenomena giving rise to a
certain amount of entropy directly related to the value of
cosmological constant. In the next section, we will firstly
deal with mixed states considered as quantum cosmolo-
gical states. After, in the third section, we will discuss
about entangled states and about the cosmological
interpretation of the mutual information, which would be
considered as a way to analyze the connection between
regions, described by entangled states. The last section in
devoted to the conclusions and perspectives.
2. Properties of Mixed State
As we have seen, entanglement has a physical role for
the construction of a quantum state of the universe
(which is impossible considering only the standard
superimposable states); we imagine that all the universe
is correlated by this process, so recovering the causality
principle in a generalized way [5].
Furthermore, it is possible, also, to infer a cosmological
state, only by starting from mixed states so then we will
be able also to describe a correlation between states
which allows to have an expression for  . We refer to
mixed states, dealing with the possibility to have states
which behaves as 



21 |||


.
Hence, analyzing the mixed state ansatz, we notice
that, in order to describe a functional dependence of 

,
from ,mk

, for a given era, entanglement becomes not
necessary, and it will be sufficient mixed states only.
Therefore

ˆ, the density matrix for the mixed states,
is written as


2
222 2
2212 22212
12 12
2222
222122 2
12 2
1
2
ˆ
mkm
mm mm
km mm k
p
NNN N
p
p
NNN


 



  

(1)
For the sake of simplicity, we assume 0
1 k
. Moreover the invariant


1=

Tr reads
 






622 4222224222 2
12212122212121
22
mkm mmmmmk mmmm m
ppp p
 
  
and the request for 
 gives us




72 2652142 2122
12 211212221
242 222
112 221
22 21
=12
mm kmmmmm kmm
mmm kmm
pp p

 

 

(2)

S. CAPOZZIELLO ET AL. 297
written with the physical, matter dominance hypothesis,
i.e. mi kj
. Note that the trace and   are written
with simplest positions and the rule nm
nmk j
kj

holds.
The eigenvalues of density matrix are(3) and their
expressions allows to write a form of entropy for uni-
verse as follows.
=lnln,S

 

 (4)
which derives from the definition of the so-called Von
Neumann entropy [19] )
ˆ
ln
ˆ
(=


TrS . This point will
be explicitly discussed in the conclusions.
Mixed states deal, so, with the possibility to have a
correlation between 
 and the observable km,

, but
it appears, in standard approach, not to be physical,
because we imagine that such a state in order to have
physical meaning, must have properties of entanglement,
necessity which is induced by the singularity. In other
words, at the Big Bang, in fact, the possible existence of
a wave function of the universe, suggests that the wave
function of universe now is derived from it but the
derivation is allowed if, in the conditions of Big Bang,
the wave function is entangled.
Then it is the singularity which implies entangled
properties of th e state of universe and, as quoted, we can
imagine that from the original state comes out from a
family of sub-states, each of them derived from the first
one and each of them able to explain features of the
modern universe, such as structure formations, inflation
and so on. This means that entanglement is the key to
infer how the correlations of the universe are
well-described by the model, while a mixed state
describes the reasons for what it is possible to write down
expressions of measurable quantities in terms of others.
This means that states apparently causally disconnected
(or considered in this way up to now) are always quan-
tistically connected.
3. Properties of Entangled States
Mixed states are physical observables by which it is
possible to build up the functional dependence


km,
=



 , but if we need an universe in which all
the regions of it are, among them, connected by a
causality principle, we must have a phenomenon of
entanglement as the genera. In fact, thel structure of the
universe. definition of entanglement, which derives from
the superposition principle, suggests us the non-separa-
bility of the state and, in particular, the non-factorizability
of the state into a singular product.
To have a suitable theory of the universe state |

,
we need a basis, from the minimal choice of the number
of elements of the state, as explained in the previous
section.
Then, we are able to infer the density matrix

ˆ, from
its definition, concerning the given state, see (5).
Together with this, we used the conditions 1=


pp
of normalized probability and the trace 1=
ˆ

Tr , together
with the two space-time eras 11 1
=1
mk Xi
i
 

and 22 2
=1.
mk Xj
j
 

In other words, the
system also if evolving, is conserving the energy.
Entropy then becomes
,lnln= 4433






S (6)


222 22
1112 222
244 222242224
22422211 222222422211 211 4
2212 221 1
11
=22
222 ,
2
mkm
mkmmkkmm
pN pN
pNN ppN pNpNpN pNpN

 
   

(3)
433 2243322
11111 112222222
322223 322223
11 11111122222222 22 22
1 2
3222233 222
11 111111222222
22 334
1111 111
ˆ|| ||
mmkmkmkm mkmkmk
mk mkmkmkmkmkmkmk
mk mkmkmkmkmkmk
kmkm kmk
NN  



  



  







23
22
22 334
2222 222
22 2222
12122 122 12
2112211222
11 2112 ** 12 21122 11222
11 2112
22 2222
12122122 12
mk
kmkm kmk
mmmmk mmk mk
mkm kmkm k
mkm mkk
mkm kmkm k
mkm mkk
kmkmkkmkkk
ppNN  










 

  



  




.








(5)

S. CAPOZZIELLO ET AL. 298
because two of the four eigenvalues are zero, i.e.
0=
1,2

and 00ln0 . The expression, given by mixed
states is quite similar for construction, in sense that here
we have two eigenvalues different from zero. Then we
can imagine a mixed state as a subcase of an entangled
state, and this is true, simply by the definition of the
mixed entangled density matrix.
3.1. Mutual Information between Eras
The basic use of the entropy S, inherent to an entangled
state, is the role of measurement of information and of
correlation between two regions of universe and, it is
surely, referred to the possible expression of entropy for
universe. The thermal entropy would really derive, so,
from the quantum expression of Von Neumann entropy,
inherent to the quantum picture proposed. In the simplest
ansatz we adopted (two regions of different space-time
epoches) we can note that, in general, the Von Neuman
entropy of Equation (6) obeys to the inequality
),()()( BA SSS   (7)
where AB  =. From that property of S, it is possible
to find the so-called mutual in fo rmation [19] as follows:
ˆˆˆˆ ˆ
()=()()()
ABA B
ISSS

 .
This relation represents the superposition of the two
regions entropies, that is to say that we have to imagine
that an universe composed by two eras, or regions, each
of them having an its own entropy value, identified by
the reduced density matrix A

ˆ, for first era and,
otherwise, by ˆ
B

, for the other one. However, this
scheme can be extended to any number or regions.
The amount of “intersection” between these eras, by
considering entropy, is quantified by the mutual infor-
mation, which appears to be regular and, so, simply by
the value of it, we infer how great is the correlation be-
tween regions.
The mutual information for our purpose is
,
1
1
ln
ln
=)
ˆ
(2
1
21
2
1
21
21
21























m
m
m
m
m
m
AB
I  (8)
in the case of spatial curvature parameter density
11 k
 and 21
k
 and 21mm

 , which has to
be considered the case of matter dominance and maxi-
mally entangled condition, i.e. 1 ==
2

. In the case
of no maximally entangled states we have
2
21
21
1
22
2
11
1
1
ln
ln
ˆ
()= (,),
1
m
m
AB m
mm
m
If













(9)
with


22 22
(,)= lnf





 which represents
the treshold of the entanglement process. Both cases are
evaluated in the simplest hypothesis of 1
=2 p. The
case of curvature dominance is specular to the case of
matter dominance, simply changing in the formula of
Equation (9) 1m

with 1k

.
The effect of curvature is that of shifting the value of
mutual information, changing its strength.
4. Conclusions
Entanglement suggests how to construct a quantum state
for the universe using the theoretical tools of Open
Quantum Relativity, i.e. considering a quantization proce-
dure at the very foundation of the theory.
In this way, GR is recovered from Entanglement and it
is described at the end of the formulation of 

, in
order to understand the role of  and of other objects
[14] in cosmology. This generality is a peculiar feature
of Open Quantum Relativity [5].
The final sense of this procedure is to find a definition
of a quantum state in terms of minimal choice of
observable quantities. The way to take into account the
state is the construction of entangled states. This allows
us to explain the reasons why   term appears to be
constant in time and avoids the coincidence problem [20].


would become, therefore, a statistical quantum
Figure 1. In this graph ic is plotted a sch eme for understanding
the exact role of mutual information ()
I
S, in the context
of two quantum regions. The degree of superposition of two
regions is directly proportional to ()IS.

S. CAPOZZIELLO ET AL. 299
result. A mixed state allows us to understand why 

is dependent from other densities (matter and curvature)
and so it is a derived emergent quantity, not a funda-
mental one.
These results can be framed in a canonical quanti-
zation scheme recovering standard results of quantum
cosmology. The procedure has been proposed in [13],
and describes a wave function, written as 



|x

,
where the state | is the entangled one, so a second
quantization would be expressed as
(,, )
=0
()= ,
Ia N
n
x
eDaDDN



 (10)
where
I
is the action. The form of the action allows
also to write down, in the semiclassical approach, the
wave function of the universe as

exp(,) .Ia

 (11)
As concluding remark, we can say that Open Quantum
Relativity is at the crossing point between GR and
Quantum Mechanics, thanks to the fact that it generalizes
the causality principle (including entanglement). The
question if a physical system can be considered existing
either with or without information is solved in the sense
that the physical system without information is not
interacting with systems outside of it. This means that
the solutions (10) and (11) with real values are what we
are ordinarily experiencing while the immaginary so lu t io n s
(the instantons) are not producing observable physical
effects unless we connect them by entanglement. The
emergence of “entropy”, in the usual sense, suggests to
take into account only one time arrow until we do not
take into account immaginary solutions and the entan gl ement
phenomenon. In such a case a backward dynamics (and
then two time arrows) have to be considered.
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