Quantum Causality

Abstract

The quantum extension of causal analysis has shown a rich picture of the subsystem causal connections, where the usual intuitive approach is hampered more commonly. The direction of causal connection is determined by the direction of irreversible information flow, and the measure of this connection, called the course of time , is determined as the velocity of such flow. The absence of causality corresponds to course of time tends to infinity , accordingly the degree of causal connection is inversely related to course of time . This formal definition of causality is valid at any time direction. The possibilities of causal analysis have been demonstrated before by series of examples of the two- and three-qubit states. In this paper we consider the new applications. The first one is the application of quantum causal analysis to the asymmetric entangled state under decoherence. Three models of decoherence: dissipation, depolarization and dephasing are studied. For the all models the strength and the direction of induced causality has been computed. It turns out that the decoherence acting along original causality destroys entanglement to a lesser degree than it acting against this causality. The second application is the interaction between a two-level atom and infinite-dimensional quantized mode of a field by Jaynes-Cummings model. An analytical solution of von Neumann equation for different initial states is examined. The filed is considered initially to be in thermal mixed state, while atom – sequentially in excited, ground or thermal states. Negativity, mutual information and causal characteristics for different temperatures are computed. It is obtained that for high temperatures distinction between behaviors of different initial states smoothes over and the state turns out to be causal, entangled and “classical” in entropic sense. And the third application is the teleportation (three-particle protocol). Contrintuitively the teleported qubit is not an effect of the original one; it proves the common effect of both two other ones. But at the same time the result of Bell measurement constitutes a cause with respect to every qubits of entangled pair just since moment of their birth. The latter is manifestation of causality in reverse time.

Full text

QUANTUM CAUSALITY
SERGEY M. KOROTAEV AND EVGENIY O. KIKTENKO
Geoelectromagnetic Research Centre of Schmidt Institute of Physics of the Earth, Russian Academy of Sciences, P.O. 30 Troitsk,
Moscow Region 142190, Russia
The quantum extension of causal analysis has shown a rich picture of the subsystem causal connections, where the usual
intuitive approach is hampered more commonly. The direction of causal connection is determined by the direction of
irreversible information flow, and the measure of this connection, called the course of time 2
c, is determined as the
velocity of such flow. The absence of causality corresponds to 2
||c, accordingly the degree of causal connection is
inversely related to 2
c. This formal definition of causality is valid at any time direction. The possibilities of causal
analysis have been demonstrated before by series of examples of the two- and three-qubit states. In this paper we consider
the new applications. The first one is the application of quantum causal analysis to the asymmetric entangled state under
decoherence. Three models of decoherence: dissipation, depolarization and dephasing are studied. For the all models the
strength and the direction of induced causality has been computed. It turns out that the decoherence acting along original
causality destroys entanglement to a lesser degree than it acting against this causality. The second application is the
interaction between a two-level atom and infinite-dimensional quantized mode of a field by Jaynes-Cummings model. An
analytical solution of von Neumann equation for different initial states is examined. The filed is considered initially to be
in thermal mixed state, while atom – sequentially in excited, ground or thermal states. Negativity, mutual information and
causal characteristics for different temperatures are computed. It is obtained that for high temperatures distinction
between behaviors of different initial states smoothes over and the state turns out to be causal, entangled and “classical”
in entropic sense. And the third application is the teleportation (three-particle protocol). Contrintuitively the teleported
qubit is not an effect of the original one; it proves the common effect of both two other ones. But at the same time the
result of Bell measurement constitutes a cause with respect to every qubits of entangled pair just since moment of their
birth. The latter is manifestation of causality in reverse time.
1. Introduction
The causality is one of the universal physical
principles. It plays the twofold role. On the one hand,
in the problems brought to the sufficient theoretical
level, this principle allows selecting of the physically
realizable solutions among a plethora of the
mathematically admissible ones. It is just the case of
relativity theory. On the other hand, the establishment
of causal-effect connections in analysis of the
complicated systems is the first step to the
construction of a phenomena model. In references to
the causality principle, usually it does not bear in mind
anything except retardation of the effect relative to the
cause. With indefinite terms the “cause” and “effect”
in the theoretical problems it may lead to the
confusions. In the complicated phenomena
investigation the rather serious mistakes are possible.
It is particularly important for the quantum entangled
states. Usually the question about possible reversal of
time ordering at quantum correlation through a
spacelike interval is avoided presupposing quantum
correlation to be causeless. But it is in conflict with
the possibility of quantum information transfer.
Although practically the conflict is damped by the fact
that for the communication purposes one should use
an ancillary classical subluminal channel, recently the
problem became relevant in connection with
macroscopic entanglement, quantum wormholes, etc.
The necessity of formal taking into account of really
existing causal connections was felt by many
researchers ([1] and references therein). In answer to
this challenge the formal method of classical causal
analysis was suggested [2]. This method had been
successfully applied before to the various
experimental problems of classical physics ([3] and
references therein). Recently it is also applied to the
experiments on macroscopic entanglement [4-8]. But
the classical approach to that quantum phenomenon is
rather limited. The quantum extension of causal
analysis has shown a richer picture of the subsystem
causal connections, where the usual intuitive approach
is hampered more commonly [9]. The direction of
causal connection is determined by the sign of
irreversible information flow, and the measure of this
connection, called the course of time, is determined as
the velocity of such flow. The absence of causality

2
2
corresponds to infinite course of time; accordingly the
degree of causal connection is inversely related to its
value. This formal definition of causality is valid at
any time direction. The independence functions used
in the causal analysis allow classification of quantum
and classical correlations of the subsystems. The
possibilities of causal analysis have been
demonstrated before by series of examples of the two-
and three-qubit states [9-13].
In this paper we consider the new applications.
Quantum mechanical development of the causality
concept turns out not only possible, but fruitful in
many respects, in particular, in solving of problem of
entanglement protection under decoherence. Next, the
quantum mechanical principle of weak causality
(suggested intuitively long ago by Cramer [14] and
formalized now in causal analysis) admits availability
of the signals in reverse time for the random
processes. It helps to understand the teleportation
process and opens way to understanding more
complicated phenomena.
In Sec. 2 the kernel of quantum causal analysis
formalism is reviewed. In Sec. 3 application of causal
analysis to the entangled states under different kinds
of decoherence is demonstrated. Sec. 4 is dedicated to
the analysis of entanglement and causality in
interaction of a two-level atom with the field. In Sec. 5
we consider three-particle teleportation protocol at
different approaches and reveal causality in reverse
time. The general results are summarized in Sec. 6.
2. Kernel of Quantum Causal Analysis
Quantum causal is an extension of classical causal
analysis [2] which operates only with classical
variables. The essence of causal analysis bases on
formalization of usual intuitive “cause” and “effect”
concepts from information-wise asymmetry of a
process without invoking time relations. The
retardation of effect relative to the cause is introduced
after their definition as an axiom.
Consider a quantum bipartite state, which
characterized by density matrix AB

, which consists
of two subsystems A and
B
with the reduced density
matrices ABAB
Tr


 and
B
AAB
Tr


 respectively.
From these matrixes we can calculate corresponding
von Neumann entropies ()SA, ()SB and ()SAB by
general formula:
2
() [ log ]
X
X
SX Tr



. (1)
Mathematical formalization of causal analysis is
founded on a pair of independence functions:
||
(|) (|)
, , [ 1,1],
() ()
BA AB
SB A SA B
iii
SB SA

(2)
where (|) ( ) ()SB A SAB S A

 and
(|) ( ) ()SAB SAB SB

 are conditional entropies.
To understand the idea of independence functions let
us consider the main demonstrative cases. |1
BA
i


(which can be realized only when |1
AB
i ) means
that we have pure entangled state: ()0SAB

,
() () 0SA SB

, that corresponds to maximal
quantum correlations between the two subsystems. If
|0
BA
i

then () ()SAB SA

and we obtain that state
B
is one-valued function of state A (notice, that
|0
BA
i

does not mean that |0
AB
i). Therefore in this
context we have maximal classical correlations. And
in case of |1
BA
i

, the
B
is independent of the A. It
is worth to mention, that generally ||
B
AAB
ii, so the
independence functions characterize one-way
correlations between two subsystems in contrast to
e.g. mutual information:
() () ( )
I
SA SB SAB

 , (3)
this characterizes total two-way correlation between
the subsystems.
It is important that for the classical variables
[0,1]i

, that is a result of the classical inequality
()max[(),()]SAB SA SB. Therefore independence
functions can indicate weather the system is
“quantum”' or “classical” in entropic sense. If at least
one |0
AB
i

or |0
BA
i

, then a system should be called
quantum. If both |0
AB
i and |0
BA
i, then a system
should be called classical. The similar definitions,
although in other terms, were proposed before in Ref.
[15] (there was considered quantum-classical' bipartite
state
A
B, where the
A
subsystem was quantum with
|0
BA
i

and the B was classical with |0
AB
i.
Causality in our consideration corresponds to
inequality ||
B
AAB
ii

. For the measure of causal
connection between subsystems A and
B
we use
2(,)cAB, called the course of time (notation follows
Kozyrev's pioneer work on causal mechanics [1]), and
derived in [9-11] as the velocity of irreversible
information flow:

3
3
||
2
||
(1 )(1 )
(,) AB BA
AB BA
ii
cAB k ii

, (4)
where /krt

 , r is an effective distance
between A and
B
, and t

is a time of
brachistochrone evolution [16]. For the orthogonal
states:
max
2( )
tE



, (5)
where max
()E is a maximal difference between
eigenvalues of the Hamiltonian.
The sign of 2(, )cAB is specified by the direction
of causal connection: 2(, ) 0cAB means that
subsystem A is a cause (information-wise source) and
B
is an effect (informational-wise sink). 2(, ) 0cAB

means that
B
is a cause and A is an effect
(22
(, ) (, )cAB cBA . The strength of the causal
connection corresponds to absolute value 2
|(,)|cAB :
the stronger is the causality, the greater is asymmetry,
the less is 2
|(,)|cAB . It is noteworthy that e.g. for all
the pure entangled states 2
|(,)|cAB that totally
conform to representation of quantum correlations as
causeless and instantaneous. But in the mixed states
the independence functions need not be equal,
therefore causality can exist.
Cramer was the first to distinguish the principles
of strong and weak causality [14]. The strong causality
corresponds to usual condition of retardation AB

 of
the effect relative to the cause:
2(,) 0 0
AB
cAB


 ,
2(,) 0 0
AB
cAB


 , (6)
2
|(,)| 0
AB
cAB


  ,
Without the axiom (6) we have weak causality, which
corresponds only to nonlocal correlations. Even as
they occur in reverse time they only relate the
unknown states (hence the “telegraph to the past”' is
impossible). Although it is not very important for the
present work scope, note that weak causality admits
the extraction of information from the future without
well known classical paradoxes. The experimental
possibility of detection of such time reversal
phenomena was theoretically predicted by Elitzur and
Dolev [17] and really proved for the intramolecular
teleportation [18] and for the macroscopic
entanglement, [4-8]. And note that we do not use the
axiom (6) anywhere in the current paper.
To keep the examples described bellow from
becoming too involved; we shall restrict ourselves by
calculations of 2
c with accuracy to 1kin Eq. (4),
since it does not qualitatively influence on the 2
c
behavior [9,10].
3. Decoherence Asymmetry and Causality
3.1. Models
We consider the models of some well known three-
qubit entangled symmetric states – GHZ and W ones
where causality originally is absent and emerges only
as a result of decoherence, and asymmetric CKW one
with finite original causality [9, 10]. The measure of
quantum causality 2
c is compared to the negativity N
as a standard measure of entanglement.
So, the model states are:
1. Greenberg- Horn-Zeilinger (GHZ) state:

1000 111 ,
2
GHZ 
(7)
2. W-state:

1001 010 100 ,
3
W
(8)
3. Coffman-Kundu-Wooters (CKW) state [19, 20]:

11
100 001 010
2
2
CKW 
. (9)
The first qubit of every state we call the
subsystem
A
, the second and third ones – the
subsystems Band C. Any two-party partitions of (7)
and (8) are equivalent. In the state (9) the party
A
sets
off from B and C, therefore only parties B, C and
A
B,
A
C are equivalent. Since all the states (7)-(9) are
pure any their two-one partitions
A
BC,
A
BC

,
etc. are causeless 2
(| | )c [9, 10].
Finite causality potentially is possible only in the
mixed subsystems
A
B

,
A
C and BC. But due
to the symmetry there is a finite causality only in the
state (9); namely the computations of Ref. [9, 10] has
yielded for the state (9): 22
( , ) ( , ) 5.30cAB cAC.
Thus
A
is the cause with respect to B and C.
The three kinds of decoherence (of 01p


degree) reduce to the following transformations [21,
22]:
Dissipation:

4
4
00 00,
11 (1 )11 0 0,
10 1 10,
01 1 01.
pp
p
p

 


(10)
Depolarization:
00 (1 )00 ,
2
11 (1 )11 ,
2
10 (1 )10,
01 (1 )01.
I
pp
I
pp
p
p
 
 


(11)
Dephasing:
10 (1 )10,
01 (1 )01.
p
p

 , (12)
We apply (10)-(12) to one of the qubits of (7)-(9).
Due to the symmetry of these states it is enough to
apply a transformation to any of qubits of (7) and (8)
and we select this qubit to be C. For the state (9) the
distinguishable results are achieved by application of a
transformation only to the qubits C and
A
(the
transformations of
B
and C are equivalent).
The resulting mixed states are the following:
Decoherence of GHZ (7):


000 000 1 111 111
1110 110 ,
2
1 000 111 111 000
dissC
GHZ
p
p
p







 


(13)

000 000 000 111
11
22
111 000 111 111
001 001 110 110 ,
2
depolC
GHZ
p
p
















(14)


000 000 111 111
1
21 000 111 111 000
dephC
GHZ p




 


(15)
Decoherence of W (8):
010 010 010 100
100 010 100 100
1(1 ) 001 001 000 000 ,
3001 010 001 100
1010 001 100 001
dissC
Wpp
p











 


















(16)

001 010 001 100
1(1 )
3010 001 100 001
1 001 001 010 010 100 100
2
000 000 011 011 011 101 ,
2101 011 101 101
depolC
Wp
p
p










  













(17)
001 001 010 010
1
3010 100 100 010 100 100
001 010 001 100
(1 ) .
010 001 100 001
dephC
W
p









 


 


(18)
Decoherence of CKW (9):

010 100
11 1
010 010
22 100 010
2
11
100 100 1 001 001 000 000
22
11
001 010 001 100
22
1,
11
010 001 100 001
22
dissC
CKW
pp
p








 






 






(19)






001 010
1
12
010 001 100 100
1
21001 100 100 001
2
1001 001 010 010
2
11
2010 100 100 010
2
1000 000 011 011
2,
1
2011 101 101 011 101 101
2
depolC
CKW p
p
p


































































(20)

5
5




1001 001 010 010
2
11
010 100 100 010
22
100 100
001 010 010 001
11;
22 001 100 100 001
dephC
CKW
p





 

















(21)

001 001 001 010
11
22 010 001 010 010
1 100 100 000 000
001 100 010 100
1,
2100 001 100 010
dissA
CKW
pp
p








 












(22)
001 100 010 100
11
2100 001 100 010
2
001 001 001 010
1
2
1 010 001 010 010
2
100 100
101 101 101 110
1
000 000 ,
22
110 101 110 110
depolA
CKW
p
p
p

















  























(23)

001 001 001 010 010 001
11
22 010 010 2 100 100
001 100 010 100
11.
100 001 100 010
2
dephA
CKW
p











 


 


(24)
From Eqs. (13)-(24) we have computed all the
marginal and conditional entropies, then – the
independence function i like (2), and at last – the
course of time 2
c like (4) for all the distinguishable
two-party partitions. For the same partitions the
negativity N, as a measure of entanglement, has been
computed too.
3.2 Causal connections at different kinds of
decoherence
Decoherence of the most symmetric GHZ state
produces the most simple causality picture shown in
Figure 1 (recall that according to our notation
2(,) 0cXY means directionality of causal
connection
X
Y, 2(,) 0cXY means YX ).
Only dissipation leads to finite causality in any
partition. If the dissipated qubit constitutes an
individual party (the partitions
A
BC and BC

)
this party always corresponds to the effect
(




22
,0, ,0cABC cAC) and with the increase
of the degree of dissipation p the causality amplifies
(20c at 1p). It is in full agreement with the
intuitive expectation – the irreversible flow of
information is directed to the dissipated particle. The
fact that




22
,,cABC cBC is explained in Ref.
[11] by stronger mixedness of the reduced state
()BC

as compared to ()
A
BC

, because mixedness
is a necessary condition of causality. In its turn
stronger mixedness of ()BC

is the consequence of
both interaction with
A
and dissipation of C i.e.
interaction with the non-controlled environment; while
mixedness of ()
A
BC

is the consequence of only the
latter. Note that in the case of dissipation of one of the
particles of two-particle counterpart of GHZ state (that
is Bell state) all the corresponding entropies and
therefore all the other parameters, including 2
c [9, 10]
exactly coincides with those of GHZ
A
BC

partition. In the partition
A
CB the behavior of
causality is nontrivial. In contrast to the above case the
couple
A
C including the dissipated particle C
constitutes the cause. The fact is dissipation of C
decreases ()SC (the states approaches to the certain
ground state according to Eq (10)). On the other hand
the dissipation of C opens the subsystem
A
C to the
environment and ()SAC increases and has the
maximum at 12p

equal to 3/2 [11], while
() 1S B const

. The particle B always corresponds
to the effect but 2
c is not monotonous: it has the
minimum at 0.594p

. To explain this fact, note that
at 0p

the state (13) is pure therefore


2,cACB; at 1p

the state (11) is maximally
mixed, but () ()SAC SB

(the fully dissipated
particle C has “disappeared”) therefore


2,cACB too. The denominator of Eq. (4) for


2,cACB: ||AC B B AC
ii

has the maximum at
0.401p

[11], while the nominator that is correlation
||
(1 )(1 )
AC B B AC
ii

 decreases as p increases,
therefore 2
min c is shifted to a higher p relative to
1/2. But by comparison with other partitions causality
in the
A
CB

one is prevailing, as it is seen from
Figure 1, at small dissipation ( 0.387p).

6
6
FIGURE 1. Causality in GHZ state with decohered qubit C.
Depolarization leads to only finite causality
A
CB (Figure 1(c)) that is with the same direction
as in the dissipation case but strength of causality
amplifies monotonously as p increases, achieving
21c at 1p.
Dephasing of GHZ state does not lead to
emergence of any causality.
W-state decoherence (Figure 2) differs from GHZ
one in that depolarization leads to finite causality in all
the three partitions, so does the dephasing in
A
CB

partition. Quantitative features of the 2
c behavior
induced by dissipation are the same as in GHZ state
and they are explained by the same reasons. The
distinction is that 2
c in
A
CB partition has the
minimum at 0.576p

and 2
c in this link is higher,
i.e. causality is weaker, than in the two other partitions
at any p.At depolarization, in contrast to dissipation,
if the depolarized qubit constitutes an individual party
(the partitions
A
BC

and BC) this party always is
the cause (


22
,0, ,0cABC cBC

) and with the
increase of the degree of dissipation p the causality
amplifies ( 20c at 1). It is also in agreement with
the intuitive expectation – the irreversible flow of
information (noise) encroaches through the
depolarized party and propagates to another one. The
fact that


22
|,||,|cABC cBC is also explained by
stronger mixedness of the reduced state ()BC

as
compared to ()
A
BC

.

7
7
FIGURE 2. Causality in W-state with decohered qubit C.
A comparison between Figures. 1(c) and 2(c)
shows that in both the cases directionality of causal
connection is
A
CB and the curves 2
c are alike. In
the W-state the dephasing also induces the causality
very similar to the depolarization case, but weaker.
The decohered CKW-state, having originally the
two causal connections
A
B and
A
C, produces
much more rich induced causality distribution. At the
beginning consider an original effect C decoherence
(Figure 3).

8
8
FIGURE 3. Causality in CKW state with decohered qubit C.

9
9
The only pair BC is originally symmetric and
therefore one should expect the same behavior of 2
c
as in the W-state. Really Figure 3(a) looks
qualitatively like Figure 2(a), but we observe stronger
causality at the depolarization. As we have seen
before, the depolarization and dissipation, acting on
one-qubit party, induce the opposite directions of
causal connection with another party. But in the pair
A
C (Figure 3(b)) all the three kinds of decoherence
amplify the original causality
A
C.The strongest
causality is observed in the intuitively expected case
of the dissipation. For the depolarization intuitively
we could expect reversal or, at least, attenuation of
original causality, but it turns out amplified, though
nonmonotonously with 2
min c at 0.427p. The
reason is that for CKWC-state () 1 maxSA and it
is impossible to reverse causal connection without
decreasing ()SA below this maximum. The
depolarization of Cat relatively small p opens more
the subsystem
A
C and amplifies the original
causality. At 1p ()SC increases up to
() 1 maxSC  and causality returns to its original
level.
In the case of partition
A
BC (Fig. 3(c)) we
have the same as for W-state (Figure 2(b)) and
intuitive expected result: the dissipated party C is the
effect with respect to
A
B, whilst the depolarized C is
the cause.
If the decohered qubit C is included in the two-
qubit party
A
C (Figure 3(d)) we observe causality
A
CBat any kind of decoherence. The variation
from W-state reduces to the stronger and
monotonously amplifying causality at the dissipation.
The case of partition
A
BC (Figure 3(e)) is close,
but at depolarization and dephasing BC A, while
at dissipation
A
BC. This peculiarity of dissipation
is clear. Indeed, at full dissipation ( 1p) the particle
C “disappears” from its two particle party and as a
result 222
(,) (,) (,)5.30cACB cABC cAB.
The original cause
A
decoherence leads to the
different causal picture (Figure 4). One may expect
that as a result of increasing dissipation of
A
, the
original causal connection
A
C will at the
beginning attenuate until disappearance at some p,
after that direction of causality will reverse with
further utmost amplification of the connection CA
as pwill tend to 1. In Figure 4(a) it is seen that indeed
2(, )cAC changes its sign at 12p. But the
variation of positive 2(, )cAC (corresponding to
directionality of the causal connection
A
C) proves
to be not monotonous; it has the intuitively
unexpected minimum equal to 5.08 at 0.103p

.
Next, in the pair
A
C (Figure 4(a)) the
depolarization leads to considerable and monotonous
amplification of causality as compared to CKWC
(Figure 3(b)). On the one hand, it is in agreement with
intuition (the depolarized
A
becomes the more
intensive information source). On the other hand, it
can easily be shown that ()SA and ()SC remain
independent of p, which demonstrates that one
should not consider the marginal entropic asymmetry
as a sufficient condition or measure of causality.
In the partition
A
BC (Figure 4(b)) the
directionality of causal connection is
A
BC at any
kind of decoherence, therewith the 2
c curves for
depolarization and dephasing are monotonous like
Figure 3(c), while for dissipation the curve has the
minimum at 0.603p

. The reason of this curve tends
to infinity at 1p is that at full dissipation the
partition
A
BC

becomes equivalent to the
symmetric one BC

. It is notable that at dissipation
22
min( ,)min(,)cABC cAC

. And there is an
interesting relation, which is valid not only in this
model [13]:






22 2
min ( , ) 1 | ( , ) | min ( , ) .
p
c AC B p c AC p c AC 
In contrast to the case when the decohered single-
party was the original effect C (Figure 3(c)), in the
case of decoherence of the original cause
A
(Figure
4(c)) only dissipation induces the causal connection,
therewith
A
becomes the effect. The monotonous
increase of negative 2(, )cABC simply reflects
amplification of causality along with increase of
dissipation of the effect
A
. At the same time
2
|( , )| 0
diss
cABC  at 1p quicker than
2
|( , )|
diss
cA BC . It reflects the influence of the original
(at 0p

) causality
A
C.

10
10
FIGURE 4. Causality in CKW state with decohered qubit
A
.
3.3 Relation between entanglement decay and
causality
All the considered states in any partition (except
pairwise one in GHZ state) are entangled. Compare
decrease of negativity N with increasing p presented
in Figures 5-8 with 2
c variation in corresponding
Figures 1-4.
If we compare N in accordance to whether the
decohered party is a cause or an effect within a given
state, we conclude that almost always (except GHZ
A
BC) N(cause) < N(effect). Further if we
compare decoherence of the causes (within a given
state) with different values of 2
c, we conclude that
decoherence in the cases of lesser 2
c (stronger
causality) leads to the stronger decrease of N. The
inverse conclusion follows from comparison of
decoherence of the effect with different values of 2
c.
Apparently we obtain a quite logical conclusion: the
cause decoherence leads to more dramatic decay of
entanglement than the effect one and the stronger
causality the stronger decay. That is causality reveals
the role of asymmetry in information propagation
(harmful for entanglement in this context). But in this
consideration we have to compare the different kinds
of decoherence. Such a consideration can not
distinguish the role of causality and decoherence
manner.

11
11
FIGURE 5. Negativity of GHZ state with decohered qubit C.
FIGURE 6. Negativity of W-state with decohered qubit C.

12
12
FIGURE 7. Negativity of CKW state with decohered qubit C.

13
13
FIGURE 8. Negativity of CKW state with decohered qubit
A
.
Another approach is the comparison of N and 2
c
at fixed both the state and the kind of decoherence.
Therefore we should consider the original cause
decoherence ()
A
and effect ()Cin CKW state.
Begin with the reduced states. Therewith the case
of dephasing is irrelevant ( dephC dephA
NN). In the
dissipated CKW state (Figures 7(b) and 8(a))
dissC dissA
NN. As we already know, dissipation of
A
leads to reversal of original causality (Figure 4(a));
dissipation of Camplifies original causality (Figure
3(b): 22
|(, )||( ,)|c A dissC c dissA C. We conclude that
dissipation, amplifying original causality, destroys
entanglement to a lower extent than dissipation, acting
against it.
In the depolarized CKW state (Figures 7(b) and
8(a)) depolC depolA
NN. And we know that
depolarization of
A
leads to the strong amplification
of the original causality (Figure 4(a); depolarization of
C only slightly varies it (Figure 3(b)):
22
|(, )||( ,)|c A depolC c depolA C. We conclude that
depolarization, amplifying original causality, destroys
entanglement to a lower extent than depolarization,
acting almost indifferently or against it.
Both the conclusions coincide. Decoherence by
the dissipation or depolarization acting along original
causality is better from viewpoint of entanglement
persistence, than acting against this causality. In other
words, for entanglement persistence one should not

14
14
“stroke the system against the grain”. As a
consequence, having compared the above inequalities
for N and 2
c, we infer that stronger entanglement
corresponds to stronger causality. Of cause, this
inference is not universal, but it shows that less
information-wise symmetric states can be more
entangled.
Now consider decoherence in the partitions where
a decohered qubit is in the party
A
C (or
A
B). That is
the party consists of both the original cause and effect.
Thus we consider influence of the “internal” causality
variation on entanglement in the partition
A
CB

in
CKW state. The corresponding curves of Figures 7(c)
and 8(b) evidence at any of three ways of decoherence
at any fixed p: decohC drcohA
NN. The inference is
nontrivial: the decohered internal effect destroys
entanglement to a lower extent than the decohered
internal cause.
4. Entanglement and Causality in Interaction of
a Two-level Atom with the Field
4.1 Interaction model
We consider a bipartite system which consists of a
two-level atom, which can be founded in the ground
state a
g
and excited state a
e, and quantized mode
of a field with possible energy states 0f, 1f,
2f,… For simplification we set detuning frequency
to zero (resonance case is considered). The interaction
is described by Jaynes-Cummings model (JCM) with
the Hamiltonian:
††
1()
2
z
ff f f
a
H
aa g e g a ga
 
   ,(25)
where

is the resonance frequency, †
f
a and f
a are
the creation and annihilation operators respectively,
g
is dipole matrix element which determines Rabi
frequency. It is helpful to write the Hamiltonian of the
full system as a sum of two commuting parts:
0
H
HV, where †
0
1
2
z
ff
H
aa
 
 is
diagonal matrix and †
()
ff
a
Vgegaga is
matrix with only off diagonal elements and
corresponds to the interaction between the subsystems.
The dynamics of the system is described by von
Neumann equation:
() ,()
af
af
t
iHt
t




, (26)
where ()
af t

is a density matrix of whole system. The
Hamiltonian (25) is time independent, the solution of
(26) is:
//
() (0)
iHt iHt
af af
te e



, (27)
If the initial state is diagonal (later we will see that
such is the case) then //
(0) (0)
iHt iHt
af af
ee



, so
the resulting solution of Eq. (26) takes the form
//
() (0)
iVt iVt
af af
te e



. (28)
In our consideration we deal only with separable
initial states:
(0) (0) (0)
af a f


, (29)
where (0)
a

and (0)
f

are the initial states of atom
and field respectively.
In the all variants we consider field initially to be
in the mixed thermal state
1
(0)
fi
f
i
Pi i



, (30)
where i
P is the probability distribution. As the field
satisfies Bose-Einstein statistics, we have
1
11
i
i
n
Pnn





, (31)
with the mean photon number
/
1
1
B
kT
ne


, (32)
where T is the temperature. As we see n
characterizes the temperature of the field.
Next one should examine a computational
problems caused by infinite dimensionality of (0)
a

.
It is evident from Eq. (31) i
P are exponentially
decaying series so that contribution of the matrix
elements if
P
ii at sufficiently high i vanishes.
Therefore we can confine series i
P at max 1iN and
estimate occurred error as

max
max 1
0
11
N
N
i
iPn n



  

. (33)
For our calculations we have set max 400N, which
gives 0.007 1%


 at the highest 80n

. At
lower n calculations are much more accurate.
For the initial states of an atom (0)
a

we
consider the pure excited and ground states:

15
15
(0) (0)
e
af f
a
ee

, (34)
(0) (0)
g
af f
a
gg


. (35)
Finally these states give two different solutions of Eq.
(28), which we will discuss further.
4.2. Computation results
With the density matrix ( )
af t

we can compute the
reduced matrices of atom and field: () ()
afaf
tTr t


and () ()
faaf
tTr t

. From these three matrices we
can get time dependent von Neumann entropies of the
whole system ()Saf and the two subsystems ()Sa
and ()Sf by Eq. (1). Then we can compute the
mutual information (3) and the independence
functions |af
i and |fa
i (2), which determine the course
of time (4). As we shall see further, it turns out
||fa af
ii, so in our consideration we use the notation
2(,)cfa to deal with the positive values. And like
before, we use the negativity N as a measure of
entanglement.
Let us start the overview of computation results
from the initial state (34), where the atom is in the
pure excited state and the field is in the thermal mixed
state. At 0n

we get the pure oscillating entangled
state vector sin( ) ,1 cos( ) , 0
e
af af af
tg i te

 |
Because of whole state purity we have ||1
af af
ii


and 2(,)cfa

.
In Figure 9(a) the dynamics of negativity for
0n

, 1n

and 10n

is presented. As we see,
the range of N variations decays and negativity
begins to fluctuate near the average value. Moreover,
the entanglement is present at 0n and 0tin
total agreement with results of Ref. [23].
FIGURE 9. Dynamics of characteristics for the initial state (34) at 1n

(thin lines) and 10n

(bold lines): (a) negativity (dashed line
corresponds to 0n; (b) information; (c) independence functions |
f
a
i (upper lines) and |af
i (lower lines); (d) causality.

16
16
The same behavior shows mutual information
I
in Figure 9(b), which corresponds to total correlations
between the subsystems. It also decays with
temperature growth and again it is positive at all times
except t = 0 (at nonzero temperature).
More detailed description of correlations
independence functions present, which are shown in
Figure 9(c). As we see ||fa af
ii, so our system is
asymmetric and the field corresponds to the cause
(information source) and the atom corresponds to the
effect (informational sink). Also it is very interesting,
that in contrast to case 1n, when our system
demonstrates quantum properties ( |af
i can be
negative), at 10n both independence functions
remain positive at t greater than about 1. It means
that system is classical in entropic sense but still is
entangled. Causality is presented in Figure 9(d). It is
particularly remarkable that that for 1n 2(,)cfa
is bounded by unit value. With temperature increase
the variation and average value of 2(,)cfa decrease,
so we see amplification of the causal connection.
It also notable that time of transfer to
quasistationary state growths with the temperature rise
(the most demonstrable is Figure 9(c). After this time
all the characteristics of the system begin to fluctuate
near some average values. The extent of such
fluctuations goes down with the temperature increase.
Next consider the case of initial state (35), where
an atom is in the ground state, while the field still is in
the thermal state (Figure 10). At 0n we have
stationary separable state vector
,0
g
af af
g
const

, which is classically
uncorrelated too: ||
1
af af
ii

.
FIGURE 10. Dynamics of characteristics for the initial state (35) at 1n

(thin lines) and 10n

(bold lines): (a) negativity; (b) information;
(c) independence functions |
f
a
i (upper lines) and |af
i (lower lines); (d) causality.

17
17
As it is seen from Figure 10(a) with a rise of n
there is an increase of the negativity: the atom in the
ground state becomes entangled with nonzero energy
states of the field ( 1,2,...
ff
). The same behavior
demonstrates information in Figure 10(b). It might be
presupposed (as the temperature is held to have a
destructive influence on correlations) that there is
some n after which an entanglement and
information would decrease. But it is not the case. The
independence functions in Figure 10(c) show that
system always is classical in entropic sense. And again
the independence functions demonstrate asymmetry
between the subsystems: ||fa af
ii (the field state still
is the cause with respect to the atom state. Causality is
presented in Figure 10 (d). As well as in previous case
it amplifies with the temperature increase.
As we have seen, all parameters of the system for
both considered initial states fluctuate near some
average values at high temperatures. It seems logically
to estimate these values as functions of n. We have
chosen time series 150 400t with time step
0.5dt

and have computed the average values
||2
,, ,,
av av f aav a f av
NIi i c for the set of mean photon
numbers 180n

. min 150t has been chosen to
avoid getting in time of transfer to quasistationary
state it is high enough for our biggest 80n. Time
step 0.5dt

has been chosen as it does not
correspond to any of system eigenfrequencies. Also
with average values we have stored minimal and
maximal values of characteristics to see variability at
our time series. The results of such time averaging are
presented in Figure 11.
FIGURE 11. Time averaged characteristics and corresponding minimal and maximal values as functions of n for different initial states: squares
and right vertical lines (34), circles and middle vertical lines (35). Averaged characteristics: (a) negativity; (b) information; (c) independence
functions |
f
a
i (empty symbols) and |af
i (filled symbols); (d) causality.

18
18
First, let us discuss the general features. As noted
above the extent of fluctuation for all the parameters
decreases with the temperature growth. Moreover the
distance between the curves for different initial states
decreases: it means that the significance of atom initial
states for the average characteristics disappears at the
high temperature. It totally corresponds to the result
that field is an information source that is a cause.
The most interesting is Figure 11 (a), which
demonstrates dependence of negativity on n. It is
expectable that for initially pure excited state of the
atom entanglement decreases with the temperature
rise, but it surprisingly does not vanish. It tends to an
asymptotic value, as well as the curve for the initially
ground state. It is intriguing that for the ground initial
atom state there is an amplification of entanglement
with growth of the temperature, so in this case the
temperature creates entanglement.
All the other characteristics also have such
asymptotic values, as it is seen in Fig. 11(b)-(d). We
can estimate that for n>>1 the averaged values are:
0.07
av
N (14% of maximal value), 0.8
av
Ibit,
|( ) 0.90
af av
i, |0.25
fa
i, 2(,) 0.25
av
cfa – the field
state is the cause with respect to the atom state.
We can summarize the time averaged results
(Figure 11) as follows. Information, reflecting total
correlations behaves similarly to the negativity. But
the independence functions are completely positive
that is classical. The atom-field state is entangled, but
correlations are apparently classical. The field state is
the cause with respect to the atom one under any
conditions. However relation between the degrees of
causality and entanglement at the low temperature
strongly depends on the initial conditions. At the high
temperature both causality and entanglement are
indifferent to them.
5. Teleportation
Teleportation is well known and amazing quantum
phenomenon. The most interesting fact is that
teleportation protocol can be considered as a process
involving hidden signal transmission in reverse time.
And it does not turn out a matter of “sophisticated”
interpretation. The experiment based on postselection
gave a direct proof of such a time reversal [18].
Another experiment demonstrated the possibility of
teleportation traveling along closed time-like curve
without the classical paradoxes [24]. At last, recently
the experiment on entanglement swapping (that is
teleportation of entanglement) has demonstrated, even
without postselection, quantum information transfer
from the future to the past; in fact it has demonstrated
a possibility of observation of the random future as the
existing reality [25]. Thus teleportation is just such a
process where determination of causality irrespective
to time direction is relevant.
We consider the standard three-particle (three-
qubit) teleportation protocol (Figure12).
FIGURE 12. Qubit
A
teleports onto qubit
B
.
The input particle, which state to be teleported by
Alice, is
A
; the EPR source produces two entangled
particles in the state

101 10
2
BC

  , (36)
one of which, B goes to Bob, another one, C goes to
Alice, who performs Bell measurement with four
possible outcomes:

100 11
2
AC

  . (40)

100 11
2
AC

 , (39)

101 10
2
AC

  , (38)

101 10
2
AC

  , (37)
If her result is (40) the output state B coincides with
input
A
, if not – Bob performs a unitary operation on
B to complete the protocol. In any case Bob needs
information about Alice result, which she sends him

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19
through an ancillary classical channel. As we are
interested in investigation of quantum information
namely, we exclude this channel. Instead Bob may
measure his particle B. Note that any measurement
(by Alice or by Bob) implies dephasing.
The state of B, accordingly commonly accepted
interpretation, changes instantaneously at the moment
of Alice joint
A
C measurement. But accordingly time
reversal formalism developed by Laforest, Baugh and
Laflamme (LBL) and the corresponding experiments
[18, 25] the B “knows” about future
A
B
measurement from very beginning. We aim to clear up
this question with causal analysis. We will do it in the
framework of usual tensor product treatment and
LBL-like time reversal treatment.
5.1. Tensor product treatment
The peculiarity of our approach is that we consider
Alice joint
A
C measurement as dephasing of degree
p accordingly to Eq. (12). One may consider it as a
soft measurement. More interesting consideration is
that dephasing is a process from 0p (measurement
without record, that is pre-measurement) to 1p

(measurement is completed). Thus p is indirect time
measure of this, certainly very fast process. The
measurement which Bob may do to get to know
something about his particle also is dephasing of a
degree 1
p; we will limit ourselves by the cases 10p

and 11p
So let the original matrix is:
0
ABC A BC A
B
C


  , (41)
Expand the matrix in Bell measurement basis:
0
,,,
ABC ijkl i j k l
B
AC
ijkl
F



, (42)
where , 1,2,3,4ij correspond to the states (37),
(38), (39) and (40) respectively; ,1,2kl correspond
to 0,1
B
B respectively; and
0
,,
ijkl i k ABC j l
F
 
  .
Alice Bell measurement of
A
C means the
replacement:
(1 )
ijkl ijkl
F
Fp at ij, (43)
while Bob measurement of B means the replacement:
1
(1 )
ijkl ijkl
F
Fp at kl. (44)
Transforming (42) according to (43) and (44), we
obtain the resulting full matrix ABC

(which explicit
expression is very longish) and can do all the
subsequent computation for causal analysis.
Consider the results for the simplest different
variants of the input states
A
. A common property of
all the variants described below turns out the fact, that
in contradiction with classical intuition, there are no
causal connections between any one-particle parties,
in particular,
A
B. It is a simple consequence of
the no-cloning theorem. Another common property is
identity of causality in the partitions
A
CB

and
A
BC

. So, below we concentrate on the partition
A
CB

.
1.
A
is in the definite state: 00
A

. Figure 13
demonstrates qualitative quite expectable result. At
0p

causality is absent ( 2(,)cACB) as the state
A
BC is pure. At finite p causality
A
CB appears
that if information goes from Alice to Bob. It is
natural that at 11p

causality is stronger since
dephased B is more definite. When Alice completes
her measurement ( 1p

), causality is most expressed:
2(,)1cACB

at any 1
p because B is already
dephased together with C.
FIGURE 13. Causality at teleportation of 00
A

 at 10p

(thin line) and 11p

(bold line).
2.
A
is in the maximally mixed state:


12 0 0 1 1
A

. Figure 14 demonstrates

20
20
stronger causality than in above case. The original
A
BC is mixed therefore the course of time is finite at
0p already: 2(,)2cACB

at 10p and
2(,)1cACB at 11p . At 1p causality
amplifies to the utmost value: 2(,)0cACB (that
means the random input completely tends to determine
a certain output, while recovery of input by output
tends to full impossibility).
FIGURE 14. Causality at teleportation of

100 11
2
A

 at
10p (thin line) and 11p (bold line).
3.
A
is in the pure equilibrium state:


12 0 0 0 1 1 0 1 1
A

  . Figure 15
demonstrates that again at 0p 2(,)cABC

 as
the state
A
BC is pure. But as 1p causality is
different at different 1
p: 2(,)1cACB at 10p

and
2(,)0cACB at 11p. The latter is a clear result of
Bob measurement of output qubit which selects a
definite state from the superposition.
FIGURE 15. Causality at teleportation of

100 01 10 11
2
A

 at 10p (thin line) and
11p

(bold line).
Although qualitatively these formal results agree
with intuition (namely Alice (
A
C) send quantum
information to Bob ( B), note that any direction of
time in the established causal link
A
CB will do.
Indeed, we nowhere specified when Bob measures
(dephases) B . Bob’s measurement may occur after
Alice’s measurement as well as before. Causality in
reverse time is allowed. But “telegraph in the past” is
impossible since a result of Alice’s measurement is
random. Instead Bob has the possibility of observation
of the random future as existing reality. Next, we saw
that Bob’s measurement can only amplify the degree
of causality
A
CB, but not generate it. That is the
result of Bell measurement constitutes a cause with
respect to every qubits of entangled pair just since
moment of their birth (like [25]). There is no a
contradiction with the above original statement: “the
EPR source produces two entangled particles in the
state (36)”. This statement in fact is conditioned on
absence of the future Bell measurement. Commonly
accepted realizing of causality as directed only from
the past to the future impeded to perceive that
conditionality before.

21
21
5.2. Time reversal treatment
We follow LBL time reversal treatment described in
detail in Ref. [18] with some simplification. The main
idea is that Bell measurement and EPR source act as
“time mirrors”. The input qubit (riding on the different
particles as the carriers) travels to Alice’s Bell
measurement device, reflects, travels backward in
time to the EPR source, reflects and goes to Bob.
Every reflection is correspondent to some
operator W: tr
i
W

, were tr is symbol of time
reversal, the components ,,|
ab
ii
Wba

 in Bell
basis are: 1
W1, 2
z
W

, 3
x
W

, 4y
Wi


.
Qubit travel and transformations are shown in Fig. 16.
The ?
W means that we do not know results of Bell
measurement; 4
W corresponds to our convention that
the source generates the state (36).
FIGURE 16. Time reversal treatment of teleportation from
A
to
B
; obs
t is time of an external observer, q
t is proper time of the
teleporting qubit.
It is not difficult to get the whole density matrix.
But we are unable to implement gradual dephasing
during Bell measurement. Instead we consider the
most important practically both the extreme cases:
measurement with ignoring of Bell measurement
record (to compare with the case 0p of tensor
product treatment) and with taking into account the
record (to compare with the case 1p of tensor
product treatment).
In the first case: (corresponding to 0p):
4
44
1
1,, ,,
4
ABC ii ii
i
WWW WWW

  

, (45)
where


is any input (
A
) state.
The second case (corresponding to 1p) is some
more complicated, so we have restrict ourselves to the
case of diagonal input states
A
that is ,jj


0,1j. We introduce to the protocol a new
object D which records the Bell measurement at
Alice site. The state of D in Bell basis is i

1, 2, 3 .4i

. As a result we have:
24
44
11
1,, , ,, ,
8
ABCD i i i i i i
ji
jWjWWj jW jWWj


 .
(46)
We have computed all the 2
c (under condition
11p

) with the following results.
In the first case: (corresponding to 0p) for
A
is in the definite state 2(,)cACB; for
A
is in the
maximally mixed state 2(,)1cACB;
A
is in the pure
equilibrium state 2(,)cACB. Thus we have
exactly the same result as in tensor product treatment.
In the second case (corresponding to 1p

) we
must take D instead of
A
B ( 2(,)cACB are the same
as in above case). For
A
is in the definite state
2(,) 1cDC

that exactly the same as in tensor product
treatment. If
A
is in the maximally mixed state
2(,)cDC is indefinite (in tensor product treatment,
having variable p we could take the limit at 1p:
2(,)0cACB).
Thus time reversal formalism gives in fact the
same mathematical results as traditional tensor
product one. But physically it proves our conclusion
about causality in reverse time in much more strait
manner. The random future D influences via
backward time traveler С on the factual result of EPR
emission.
With the recorder D we also can consider a
partition
A
DB

(which is equivalent of
A
DC

).
As a result for
A
in the maximally mixed state we
have 22
(,) (,)12cADB cADC

. This value is
minimal among all presented above finite values of
2
c, therefore causal connections
A
DB and
AD С are the strongest ones. Since Bell
measurement at Alice site occurred later (by time of
an observer obs
t) than EPR emission took place, this
result is further proof of causality in reverse time. The
EPR pair “knows” about the random future –
interaction with random
A
and random D. For
A
in
the definite state 22
(,) (,)1cADB cADC.
Therefore the greater is randomness of the future, the
stronger is time reversal causality. Obviously in the
case of deterministic future time reversal causality
must absent. It is just impossibility of “telegraph to the
past”.

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22
6. Conclusions
The quantum causal analysis is extension of the
classical one; therefore it is extension of formalized
intuitive understanding of causality. Indeed in the
simple situations our formal results are not surprising,
e.g. when dissipating particle proved to be an effect.
But even in these cases our formal approach has an
advantage over usual informal, intuitive one, because
it provides the quantitative measure of causal
connection. However in the Quantum World common
intuition often fails in rather simple systems,
consisting of a few particles. Causal analysis quite
works with any system, although its results may seem
contrintuitively. Therewith these results turns out
practically useful, e.g. in explanation of peculiarities
of entanglement decay under different kind of
decoherence or in relation between intersystem
causality and consequences of asymmetric
decoherence.
Very simple and general property of quantum
causality is that it can be finite only in the mixed
states. In our previous works [9-13] we interpreted this
fact as quantum causality can be finite only in the
open systems. But in the model of atom-field
interaction considered in this paper the system is
closed, the state mixedness, necessary for causality,
was created before, at the stage of thermal state
preparation. Therefore we have to correct
interpretation as follows: quantum causality can be
finite only in the systems, which are or were open.
The most prominent property of quantum
causality is that it can exist in direct as well as in
reverse time. Remarkably time reversal causality does
not imply the naive classical paradoxes. We have
considered such unusual causality in connection with
contemporary teleportation experiments [18, 24, 25].
But, of course its significance is much wide, e.g. for
interpretation and development of the forecasting
experiments based on macroscopic entanglement [7,
8].
Acknowledgment
This work was supported by RFBR (grant 12-05-
00001).
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