# 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 February 26, 2017), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT College of Arts and Sciences.