Quantum Entanglement and Entropy Page: 5
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QUANTUM ENTANGLEMENT AND ENTROPY
The critical values Q*(71,(2) and Q**(1,(2) are de-
fined as follows:Q*( 1, 2)- sup {q*(()}, (
E[ 1, 2]
Q**( 1,k2)= sup {q**(()}. (
e [ 1,2]
The auxiliary functions q*( ) and q**( ) are defined by
q*() =max{1,a*(),a (),a ()}, (:
q**() -max{ 1,ac *(), a*((), 3*(()}. (223)
P2)P(2) em)Kem + p2) e\i)(e y,
j=1(30)
24) where the set { em), ej),j= 1,2,3 } is the Bell basis set [21],
no matter what the order is. It is easy to check that these
quantum states have the following properties: (i) they belong
to the set 3, (ii) EF(P1) - EF(p()) and EF(p2) -E (2)),
and (iii) Sq(Pl) - Sq(p()) and Sq(P2) Sq(P2)).
25) Now let us introduce the transformation Ej[pB,p)],
defined by-U)
The functions ac( ), with the subscript j running from 1 to
3, are functions of the interval [ 1, 2] given by
[(Pm\] dP dP
dP()- 1 + In P In 3 d (27)
Pj d d
if the conditions Pj(e) > 0 and dPj / d> 0 apply. If these
conditions do not apply, we set aj*() -1. The functions
a *( ) are given byoP [ dP dP
1+ I In lm Inp(28)
if the conditions Pj( ) > 0 and dPj /d,< 0 apply. If these
conditions do not apply, we set aj*()= 1. The proof of this
important recipe is given in the Appendix. Note that we have
not discussed the problem of the possible divergence of
Q*(6 ,2) or Q**(, 2). We shall come back to this issue
in Sec. III E, where we shall consider, without losing any
generality, a special parametrization of the eigenvalues
within which, as we shall see, the critical index Q(pl1,p2)
will be proved to be finite.
E. Search of a critical entropy index as a function of initial
and final states
Now let us see how to use the earlier results to make
predictions in the case where the transformation and the en-
suing entanglement change are described only by the initial
and final states pi and P2, with the density matrices belong-
ing to the set 3 defined by Eq. (11). The main idea is to build
up auxiliary states pg) and p(2), equivalent to pi and P2,
respectively, as far as their entanglement and entropy are
concerned, but fulfilling the condition of being connected the
one to the other by one of the S transformations described in
Sec. III C. This makes these states compatible with earlier
prescriptions, and thus with earlier results. The states p(l)
and p(2) are defined as follows. Let P(), P' ), P21) , and P(31)
denote the eigenvalues of the density matrix Pi, while
p( 2) ,P(22), and P32) denote the eigenvalues of the den-
sity matrix P2; we define the auxiliary states pg ) and p2) by
the expressionsp= Pl em)(8mf Pyej
j=1(P1),p(2)(p(PB)) P( )em)(eml +~ P()e)(e,
j=1
(31)
where the S evolutions of Pm(t) and Pi(e) are given byand
Pm(s)= p + (p(2) _ P())
P () _ 1))(32)
(33)with j running from 1 to 3, respectively, and S belonging to
the interval [0,1]. The transformation E has the required
properties: (a) it keeps the state E B[p~),p2)](PB)) within
the set 3 for every value of S belonging to the interval [0,1],
(b) ~o[p r ( P2)](p(B)) p(BI) , and (c) S1[pB ) 2,P)]p(l))
p(2) . Note that the functions Pm( ), PI(), P2(e), and
P3( ) are eigenvalues of the quantum states S [p(,
p(z)](p( )), and are defined in the interval [0,1]. They fulfill
the properties of Eq. (13), the parameter conditions of Sec.
IIIB, the relation dPm / d>0 in the case P ()< P() , and
the relation dPm/d < 0 in the case Ps()> P) . This makes
it possible for us to use Q*(1, 2) of Eq. (23) and
Q**(1 ,2) of Eq. (24), and the relations on which these
quantities rest as well, to derive Q(p1 ,p2). This is done as
follows. We write the explicit forms that Q*(S1,S2) and
Q**( 1, 2) gain when 1 = 0 and 2 1. Applying the trans-
formations of Eqs. (32) and (33) to the prescriptions of Eqs.
(23)-(28) , we obtain the following expressions:
Q*(0,1) sup max{1,6(1 0),l ((),l l()}. (34)
e [0,1]
Here the function /3.(), with j= 1, 2, and 3, is defined as
follows. If the constraints
:2)(s)>1:)(s) and P)+S()-_ 1))>O, (35)
with j= 1,3, hold true, we set(36)
(29)
032310-5
and
PHYSICAL REVIEW A 64 032310
p() _ p(2)
In 3
p(2) _ p(1)
m m
In (j) (2) _ P(1))
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Giraldi, Filippo & Grigolini, Paolo. Quantum Entanglement and Entropy, article, August 20, 2001; [College Park, Maryland]. (digital.library.unt.edu/ark:/67531/metadc67627/m1/5/: accessed November 24, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.