Von Neumann and Shannon-Wehrl Entropy for Squeezed States And

Home , Wehrl entropy

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VON NEUMANN AND SHANNON-WEHRL ENTROPY

FOR SQUEEZED STATES

AND COSMOLOGICAL PARTICLE PRODUCTION

M. Gasp erini and M. Giovannini

Dipartimento di FisicaTeorica, UniversitadiTorino,

Via P.Giuria 1, 10125 Turin, Italy

ABSTRACT

We show that the e ective coarse graining of a two-mo de squeezed density

matrix, implicit in the Wehrl approaches to a semiclassical phase-space

distribution, leads to results in agreement with previous di erent de -

nitions of entropy for the pro cess of pair pro duction from the vacuum.

We also present, in this context, a p ossible interpretation of the entropy

growth as an ampli cation (due to the squeezing) of our lackofknowledge

ab out the initial conditions, which gives rise to an e ective decoherence

of the squeezed density matrix.

Some years ago it was suggested that the in quantum pro cess of pair pro duction

from the vacuum, induced by the action of a time-varying classical background,

the growth of the average numb er of quanta of a given eld should b e naturally

asso ciated to the growth of entropy of that eld [1,2]. More recently, suchan

entropy growth was quanti ed on the grounds of a squeezed state approach to the

pro cess of particle pro duction [3-7]. With this approach one can easily compute, in

particular, the total entropy S stored in the cosmological p erturbations ampli ed

by in ation, and one obtains [4,6]

3=2

S ' S (1)

cmb

where S is the thermal black-body entropy of the 3K photon microwave back-

cmb

ground, H the nal curvature scale of in ation, and M the Planck mass.

f p

The use of the squeezed state formalism to compute entropy pro duction might

seem to give rise to a puzzle, since the squeezed evolution is unitary. The solu-

tion of the puzzle is that, of course, there is no loss of information in principle

asso ciated to the evolution of the initial state into a nal squeezed quantum state 1

(and so no p ossible information paradox, like in the case of black-hole dynamics).

The loss of information o ccurs however in practice, as a consequence of the pro cess

of measure of the observables characterizing the nal squeezed state. There are

indeed variables (related to the so-called \sup er uctuant" op erators [8]) which are

ampli ed by the squeezed evolution and b ecome thus available for macroscopic ob-

servations, while other variables (the \sub uctuant" ones [8]) are corresp ondingly

\squeezed" and b ecome unobservable.

A simple and intuitiveway to illustrate this situation is to consider a semi-

classical description of the squeezing pro cess as a parametric ampli cation of the

wave function, \hitting" the e ective p otential barrier of a Schro edinger-like equa-

tion (with time-likevariable) [9,10]. The nal amplitude of the wave scattered by

the barrier may b e larger or smaller than the initial amplitude, dep ending on the

phase with which the waveenters the barrier. What is macroscopically detected,

however, is the average ampli cation of the wave, out of an initial random distri-

bution of phases. This p ointwas clearly stressed already in the rst pap er on the

cosmological ampli cation of tensor p erturbations [11]. A macroscopic observa-

tion of the quantum uctuations of the metric, ampli ed by the cosmic evolution,

traces out over phases.

In order to compute the entropy asso ciated to a squeezing pro cess, this macro-

scopic loss of information has to b e taken into account in the form of a suitable

\coarse graining" of the density matrix of the nal (pure) squeezed state. Sucha

coarse graining was originally p erformed in a Fock basis, i.e. with resp ect to the

eigenstates of the numb er op erator, byaveraging over phases [3-5]. The number

op erator, however, is only a particular example of sup er uctuantvariable which

grows in the case of a pair pro duction pro cess, while the conjugate variable (the

phase) is corresp ondingly squeezed. This suggests the use, in general, of the eigen-

states of the sup er uctuant op erator as the natural basis for the coarse graining of

a squeezed density matrix [6,7]. A p ossible di erent basis, that of coherent states

with real amplitude, has b een suggested however in [12].

It is imp ortant to stress that all these di erent approaches [3-7,12] to the

entropy of a squeezed state exactly lead to the same result when the entropy

growth S , for eachmodek,isevaluated in the large squeezing regime, namely

S ' 2r ; r >> 1 (2)

k k k

where r is the mo dulus of the (complex) squeezing parameter [8]. In the previ-

ously quoted pap ers, however, the decoherence pro cess which leads to a reduced 2

density matrix (from which to compute the quantum Von Neumann entropy)

red

is imp osed \by hand", by setting the o -diagonal terms of the matrix to zero in

some chosen basis; at b est, like in the case of the numb er basis, it is justi ed in a

semiclassical way using a sto chastic approach and a sort of random-phase approx-

imation [3,4]. The problem is that it is formally imp ossible in the numb er basis,

as well as in the more general sup er uctuant basis, to compute the quantum von

) obtained Neumann entropy(Tr ln ) from a reduced matrix ( 6=

red red red

red

by tracing out phases, or another sub uctuantvariable. Indeed, a sub uctuant

op erator do es not commute with the conjugate sup er uctuant one, so that b oth

op erators cannot b e simultaneously included in the complete set of observables

characterizing a pure squeezed state.

In view of these preliminary remarks, we can say that the aim of this pap er is

twofold. First of all wewant to compute the semiclassical Shannon-Wehrl entropy

[13,14] for a two-mo de squeezed vacuum state, in order to show that also in that

approach one recovers the result (2) for the large squeezing limit. This con rms

that the entropy of a highly squeezed state is very robust to the particular scheme

of coarse graining implemented (as discussed also in [15,16]).

In the second place wewant to present a p ossible justi cation to the reduction

of the density matrix in a sup er uctuant basis [6,7], byinterpreting such a reduc-

tion as the consequence of our lackofknowledge ab out the initial con guration

which is the starting p oint of the squeezed evolution. To this purp ose we shall

represent the initial state, in general, as a statistical mixture of coherent states.

This representation automatically de nes a quantum von Neumann entropy asso-

ciated to the initial mixture. This entropyhowever is constant, as the squeezed

evolution is unitary, and then it do es not contribute to the di erence S between

nal and initial entropy.However, if the statistical distribution satis es (at least

approximately) a principle of equipartition of the probabilities among the states

of the mixture, then the o -diagonal terms of the density matrix turn out to b e

suppressed. An e ective decoherence is thus pro duced, whose asso ciated entropy

in the sup er uctuant basis is eventually ampli ed by the squeezed evolution.

The arguments presented in this pap er are intended to apply to any pro cess of

pair pro duction, describ ed by a Bogoliub ov transformation connecting the particle

y y

~ ~

fb; b g and anti-particle fb; b g annihilation and creation jini op erators to the jouti

y y

ones, fa; a ; a;~ a~ g . Such a transformation reads, for eachmodek,

a =c (k)b +c (k)b

k + k

y y

~a =c (k)b +c (k)b (3)

k k 3

2 2

where the Bogoliub ov co ecients c (k ) satisfy jc j jc j = 1. By putting

c (k ) = cosh r ; c (k )=e sinh r (4)

+ k k

the transformation (3) can b e rewritten as a unitary transformation,

y y y y

a = b ; a~ = b (5)

k k k k

generated by the two-mo de squeezing op erator [8]

y y

~ ~

= exp z b b z b b ; z = r e (6)

k k k k k k

k k

Our discussion can thus b e applied in general to any dynamic situation in which the

initial vacuum state j0i evolves towards a nal two-mo de squeezed vacuum state,

jz i = j0i,characterized by a non-vanishing value of the squeezing parameter

r = jz j (related to the exp ectation value of the numb er of pro duced particles by

k k

y y

hn i = hz jb b jz i = h0ja a j0i = jc (k )j = sinh r ).

k k k k k k

k k

It is imp ortant to note that in the context of such a dynamic evolution one can

always de ne two op erators x andx ~, called sup er uctuant, whose variances (x) ,

(x ~) are ampli ed with resp ect to their vacuum value (x) ,(~x) , namely [8]

z 0 0

r r

(x) ! (x) =(x) e ; (x ~ ) ! (x ~) =(~x) e (7)

0 z 0 0 z 0

2 2 2

where (x) = hz jx jz i(hzjxjzi) (and the same forx ~). The variance of

the canonically conjugate op erators y; y~ is corresp ondingly squeezed, (y ) =

(y ) e =(~y) . In terms of the sup er uctuantvariables the op erators b and

0 z

b have the di erential representation [6,7,8]

e (x ix~ + @ i@ ) b =

x x~

b = e (x + ix~ + @ + i@ ) (8)

x x~

(henceforth the mo de index k is to b e understo o d, if not explicitly written, and

any correlation among mo des with di erent jk j will b e neglected, following the

coarse graining approach of [2]). The normalized wave function for the two-

mo de squeezed vacuum (such that a =0 = ~ ) b ecomes, in the (x; x~)-space

z z

representation [6,7],

1=2

2 2

(x +~x )

(x; x~)=hxx~jj0ihxx~jzi= e ;

= e ; hz jz i =1 (9)

The squeezed evolution j0i!jzi=j0i maythus represented, in the sup er uc-

p p

tuant basis, as a scaling transformation x ! x,~x! x~, with real p ositive

parameter 1, related to the squeezing parameter r according to eq.(9).

In order to compute the Shannon-Wehrl entropy for a squeezed state we recall

that the classical entropy is de ned, for any given phase-space distribution f (q; p),

S (f )= dq dpf (q; p)ln f(q; p) (10)

while the quantum (or von Neumann) entropy is de ned in terms of the density

op erator as

S ()=Tr ln (11)

For a pure state, in particular, = and S = 0. Supp ose now that wewantto

compute, for a given quantum state represented by , its entropy in the \semiclas-

sical" limit. It is well known that a p ossible pro cedure to obtain the semiclassical

limit of a quantum observable is to compute the exp ectation value of that observ-

able in the basis of the coherent states j i. According to Wehrl [14], a natural

semiclassical phase-space distribution asso ciated to is thus given by

h jj i (12) f ( )=

(the so-called Glaub er distribution function [17]), which leads to de ne the semi-

classical (Wehrl) entropy S as

S = S ( f ( )) (13)

W cl

Foratwo-mo de squeezed state jz i (see [18,19] for the one-mo de case) wehave,

in particular,

h ~jzihz j ~ i (14) f ( ; ~ )=

where the coherent states j ~ i satisfy

bj ~ i = j ~ i; bj ~ i =~ j ~i (15)

(the op erators b; b are de ned in eq.(8); their corresp onding eigenvalues are com-

plex numb ers, = + i ,~ =~ +i ~ ). By expanding in the sup er uctuant

1 2 1 2

basis jxx~ i one easily obtains

1 1 1 1

2 2 2 2

~ ~

hxx~ j ~ i = exp (

2 2 2

h ~ j ~ i =1 (16)

where

i i

k = ie ( +~ ); k=e (~ ) (17)

Moreover

0 0 0 0 0 0

f ( ; ~ )= dxdxdx~ dx~ (xx~ ) (x x~ )h ~ jxx~ihx x~ j ~ i =

2i 2i 2 2

= exp tanh r ( e~ + ~ e ) j j j ~j (18)

cosh r

With this Glaub er distribution, the semiclassical Wehrl entropy (13) for the two-

mo de squeezed vacuum is then

2 2

S = d d f~ ( ; ~ )lnf( ; ~ ) = 2+2ln + 2 ln cosh r (19)

The same result holds for a squeezed-coherent state jz ~ i =j ~ i, obtained

0 0 0 0

by applying a squeezing transformation to the \displaced" vacuum j ~ i =

0 0

D ( )D (~ )j0i, where D ( ) = exp( b b) is the Glaub er displacement op er-

0 0

ator.

It is imp ortant to note that for r>>1 this semiclassical entropy repro duces

the large squeezing b ehavior of eq.(2), previously obtained [1,3-7,12] with di erent

approaches to the entropy of a particle pro duction pro cess. We also note that the

entropy (19) is exactly the sum of two \marginal" entropies,

W W

S = S + S (20)

~ ~

1 1 2 2

obtained by decomp osing the probability distribution f ( ; ~ ) in the complex ; ~

planes, and by tracing out one of the two cartesian variables,

Z Z

= d d ~ f ( ; ~ )lnf( ; ~ ); f( ; ~ )= d d ~ f ( ; ~ ) S

1 1 1 1 1 1 1 1 2 2

1 1

Z Z

= d d ~ f ( ; ~ )lnf( ; ~ ); f( ; ~ )= d d ~ f ( ; ~ ) (21) S

2 2 2 2 2 2 2 2 1 1

2 2

This is a non-trivial result, as it implies that the information-theoretical Araki-Lieb

inequality [20], S S + S , is maximized b oth by the two-mo de squeezed

~ ~

1 1 2 2

vacuum and by a squeezed-coherent state, in agreement with the prop erties of

one-mo de squeezed states [18]. 6

According to eq.(19), the semiclassical entropy asso ciated to a pure squeezed

state jz i is thus non-vanishing, in spite of the fact that = jz ihz j = and S ()=

0. This o ccurs b ecause an e ective reduction of the quantum density matrix is

implicit in the semiclassical limit, as even a pure state is seen, classically,asa

disordered one. However, the coarse graining asso ciated to the semiclassical limit

cannot give us explicit information ab out the physical origin of the decoherence

mechanism, which can only b e describ ed in the context of some dynamic mo del

for the quantum to classical transition (see also [21]). For a squeezed state such

a mo del must account, in particular, for the fact that some degree of freedom

of the system is made dynamically less relevant than others for what concerns

macroscopic observations.

Let us consider now the p ossibility of de ning also a von Neumann entropy

for the squeezing pro cess, in terms of a prop erly reduced density op erator. In the

approach of Refs.[5-7] the reduction was p erformed by neglecting the o -diagonal

elements of the density matrix in a sup er uctuant representation. The use of

such a representation is justi ed b ecause the squeezed density op erator can only

b e reconstructed, observationally,by measuring sup er uctuantvariables and their

momenta. But is the reduction a formally justi ed pro cedure?

The answer is certainly not for the pure squeezed vacuum state (9), whose

density matrix in the sup er uctuant representation

0 0 0 0 0 0

(x; x ; x;~ x~ )=hxx~jzihz jx x~ i = (x; x~) (x ; x~ )=

z z

h i

2 02 2 02

= exp (x + x +~x +~x ) (22)

0 0

is p erfectly symmetric in the (x; x ) and (x; ~ x~ ) planes. The same answer holds

for a pure squeezed numb er and squeezed-coherent state. On the contrary, the

reduction would b e naturally justi ed if the o -diagonal matrix elements would b e

suppressed, j (x; x)j >> j (x; x 6= x)j (and the same forx ~), for any given value

z z

of the squeezing parameter. It is thus interesting to note that such requirement

can b e satis ed when the initial state is not pure, but is represented by a mixture,

whose statistical weights tend to b e equally distributed among all the states of the

mixture.

Supp ose to start, in fact, from an initial con guration more general than the

vacuum, represented by an ensemble of states which, like the vacuum, minimize

however the quantum uctuations x,~x( and their conjugate y ,~y). This

corresp onds to an initial density op erator which can b e expressed in terms of the 7

coherent states (15), (16) as

2 2

= d d P~ ( ; ~ )j ~ ih ~ j (23)

where the statistical weights P 0 satisfy the normalization condition

2 2

d d P~ ( ; ~ )=1 (24)

The squeezed evolution leads then to a nal density op erator, = , which

f i

in the sup er uctuant basis is explicitly represented by

0 0 2 2 0 0

(x; x ; x;~ x~ )= d d P~ ( ; ~ ) (x; x~) (x ; x~ ) (25)

f z ~

z ~

Here (x; x~) is the squeezed-coherentwave function, representing in the (x; x~)

z ~

space the eigenfunctions of fa; a~ g with eigenvalues f ; ~ g, namely

(x; x~)=hxx~jz ~i = hxx~jj ~i =

z ~

1 1

2 2 2 2

~ ~

= exp (

2 2 2

(k; k are de ned in eq.(17)). Note that, for 6= , there is a non-vanishing

2 2

quantum entropy, S = d d P~ ( ; ~ )lnP( ~), naturally asso ciated to the

initial mixture. Suchanentropy,however, is constant throughout the whole pro-

cess of squeezing ! (which describ es a unitary evolution), so that it cannot

i f

contribute to the overall entropy di erence S = S = S .

f i

The choice of the vacuum as initial condition corresp onds to the assumption

2 2

P ( ; ~ )= ( ) (~ ) in eq.(25). We shall consider here an initial distribution

which, instead of b eing in nitely p eaked up on the vacuum, tends to b e spread

rather uniformly over a wide range of states, so as to re ect our lack of knowledge

ab out the initial conditions. We shall assume, however, that such a distribution is

always centered around the vacuum, and we shall conveniently translate the (x; x~)

frame in sucha way that hz ~ jxjz ~ i =0=hz ~ jx~jz ~i (which implies

a normalized square-like distribution, in which the probability of a state j ~ i is

constant and equal to 1=L for j j; j ~ j L=2,

namely

2 2

d d P~ ( ; ~ )=d d d ~ d ~ ( ~ ) ( +~ )

1 2 1 2 1 1 2 2

[ ( + L=2) ( L=2)] [ ( + L=2) ( L=2)] (27)

1 1 2 2

Here is the Heaviside step function, and the two delta distributions have b een

inserted to implement the conditions hxi =0=hx~i;wehave also rescaled the

variables ; ~ so as to absorb the overall phase e which do es not contribute to

the integration over the whole complex plane.

With this probability distribution the squeezed matrix (25) b ecomes

p p

0 0

2 02 2 02

sin [L (x x)] sin [L (~x x~ )]

(x +x +~x +~x ) 0 0

p p

(28) (x; x ; x;~ x~ )= e

0 0

L (x x) L (~x x~ )

and it is immediately evident that, for large enough L, the o -diagonal terms

0 0

x 6= x ,~x6=~x are suppressed with resp ect to the diagonal ones. Such a suppression

grows with L, namely with the numb er of states which are considered equiprobable

in the initial distribution, i.e. with the growth of our \ignorance" ab out the initial

con guration. For large L wemaythus approximate the squeezed matrix with

the diagonal elements only, and in particular we can formally de ne in this basis

a (normalized) reduced density matrix, in the limit L !1 (at nite ), as

0 0 2 0 0

(x; x ; x;~ x~ ) = lim fL (x; x ; x;~ x~ )g =

red f

L!1

2 2

(x ++x ~ ) 0 0

= e (x x ) (~x x~ ) (29)

This is exactly the representation in the (x; x~) basis of the reduced op erator

= dxdx~j (x; x~)j jxx~ ihxx~ j (30)

red z

previously used in Refs.[6,7].

Note that Tr = 1, but 6= . This reduction leads then to a quan-

red red

red

tum entropy S ( ) which, unlike the entropy of the mixture S ( ), dep ends on

q red q f

, and in particular grows with the degree of squeezing. By taking the di erence

between nal and initial entropywe nd indeed the result of eq.(2),

S =(Tr ln ) (Tr ln ) =2r (31)

red red red red

The accuracy of this estimate in our context b ecomes larger and larger as the

initial probability distribution b ecomes atter, so as to share probabilities among

a larger numb er of states. It is also imp ortant to note that if, for a given L, the

reduced matrix represent a go o d approximation to the exact matrix (28) at large 9

squeezing (>>1), then the approximation is even b etter as we go back in time

towards the initial con guration, where =1.

Wemaythus conclude that the e ective reduction of the squeezed density

op erator, intro duced \ad ho c" in previous pap ers [6,7], can b e formally justi ed

through a limiting pro cedure based on an appropriate mixture of coherent states,

assumed to represent the initial con guration. The corresp onding entropy growth

maythus physically interpreted as an ampli cation, due to the squeezed evolu-

tion, of our ignorance ab out the initial conditions, in a context in whichwehave

no motivation to single out some preferred initial state, and we cannot reconstruct

the initial state through nal measurements of sup er uctuantvariables only. Such

an interpretation seems to b e particularly appropriate in the context of cosmolog-

ical p erturbation theory, where the ampli cation of the background uctuations is

computed starting from an initial state which is left unsp eci ed except for the re-

quirement of minimizing the quantum uctuations, and which can thus b e prop erly

represented as a general mixture of coherent states.

Acknowledgments

One of us (M. Gasp erini) wishes to thank D. Boyanovsky for rising stimulating

questions that motivated in part the work rep orted here. We are also grateful to

G. Veneziano for many helpful discussions.

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