Classical entropy of quasi-free states

17 April 1995 PHYSICS LETTEKS A

Physics Letters A 200 (1995) 91-94

ELSEVIER

Classical entropy of quasi-free states H. Scutaru 1 Department of Theoretical Physics, Institute of Atomic Physics, POB MG-6, Bucharest-Magurele, Romania

Received 7 October 1994; revised manuscript received 8 November 1994; accepted for publication 8 February 1995 Communicated by P.R. Holland

Abstract

We obtain a general formula for the classical entropy of a quasi-free state as a function of the correlation matrix. Using an earlier result of Balian, de Domincis and Itzykson concerning the structure of the Bogoliubov automorphisms we can separate the transformations which produce the squeezing from those which produce the correlations and we make precise their contribution to the classical entropy.

The notion of the quasi-free state has appeared and was developed in the framework of the C *-algebra approach to the canonical commutation relations (CCR) [1-10]. The quasi-free states are the natural ground states for Hamiltonians which are at most quadratic in the bosonic creation and annihilation operators. The essential property of the quasi-free states is that all their correlations are expressible in terms of the one- and two-point functions. The advantage of the C *-algebra approach to the theory of coherent and squeezed states (thermal or not) follows from the fact that the results are obtained in a representation independent form. We shall illustrate this assertion by giving a representation independent formula for the transition probability between a squeezed state and an arbitrary state of the C *-algebra of CCRs. When one of these states is a quasi-free state and the other is a coherent state labeled by a point of the

1E-mail: [email protected].

phase space, then the corresponding transition probability is a classical probability density on the phase space which is associated to the quasi-free state. The entropy of this probability density is called the classical entropy of the quasi-free state. This definition generalizes the corresponding notion introduced for Glauber coherent states in Refs. [11,12] and for the generalized coherent states in Ref. [13]. An explicit formula is given for the classical entropy of a quasi-free state as a function of the correlation matrix. From this formula it follows directly that the classical entropy has a positive lower bound if we take into account the restriction on the Gaussian generating function of a quasi-free state obtained in the C *-algebra description of these states [2-4,9]. This lower bound is attained by an ordinary coherent state. The phase space considered in the present paper is a finite dimensional symplectic space (E, o-). This is a real vector space endowed with a real, bilinear, antisymmetric form o'(.,.) which gives the symplectic structure on E. Then E is of even real dimension

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1t. Scutaru / Physics Letters A 200 (1995) 91-94

92

2n and there exist in E symplectic bases of vectors {e~, f~}j=l....... i.e. reference systems such that o'(ej, e k) = o'(fj, f k ) = 0 and o-(ej, f k ) = - o ' ( f k , e ) = 6jk, j, k = 1. . . . , n. The coordinates (~J, ~¢) of a vector u ~ E in a symplectic basis (u = Y'-~j=1 (~Jej + r/Jfj)) are called symplectic coordinates. The measure dm(u) = l-I"j = l d~:j dr/j is the same for all symplectic coordinates and is called the Liouville measure on (E,o'). There is a one-to-one correspondence between the symplectic bases and the linear operators J on E defined by Je k = - f k and Jft, = ek, k = 1. . . . . n. The essential properties of these operators are: tr(Ju, u) >~O, tr(Ju, v) + tr(u, Jr) = 0, u, v ~ E and j 2 = - I ( I denotes the identity operator on E). In the following we shall use the matrix notations with u ~ E as column vectors. Then tr(u, v) = uTJv and the scalar product is given by cr(Ju, v) = uTv, U, V ~ E. A linear operator S on E is called a symplectic operator if S T J S = J. When S is a symplectic operator then S T and S-1 are also symplectic operators. The group of all symplectic operators Sp(E, tr) is called the symplectic group of (E, tr). The Lie algebra of Sp(E, o') is denoted by sp(E, tr) and its elements are operators R on E with the property ( J R ) T = JR. The C *-algebra A of the canonical commutation relations (CCR) is obtained [1,2] by the completion of the *-algebra Span{8,: u ~ E}, the elements of which satisfy the Weyl relations,

0)(A,v) = 0)(A,O)OTv. Hence O)(A,v) is pure iff 0)(a,0) is pure and this is the case iff - ( J A ) 2 = I (this is equivalent with JA ~ Sp(E, o')) and particularly when A = I, i.e. when the state is a coherent state. The squeezed states are the quasi-flee states of the form 0)(t,,) o as = 0)(S.%sT,,)" The transition probability P(tol; 0)2) between two arbitrary states o 1 and 0)2 on A can be defined in many ways [14-16]. In the case when the state 0)1 is a squeezed state the transition probability to an arbitrary state 0)2 can be computed by the following representation independent formula,

6.6"=exp[-itr(u,

P ( w , ; 0)2)

v)/2]6u+~,

6.* = 6 _ ~ ,

u,v ~ E.

A Bogoliubov transformation of A is an *-automorphism of A defined for any S ~ Sp(E, o-) by Ols(6u)---6s, ,,

u~E.

(4)

The quasi-free automorphisms of A are those of the form L ° as where L, ~ E and S ~ Sp(E, o-). Let A be a symmetric ( A T = A ) and positive (uTAu >1 O, Vu ~ E) matrix. A quasi-free state 0)(a,,o on A is [3-10] a state whose characteristic function is given by the formula

0)(A,,,)(6,) = exp( --uTau/4 + iuTv)

(5)

with the restriction

- J A J > ~ A -1 .

(6)

Evidently we have (7)

O)(A,v ) o aS = O)(sTAs,STt, )

and

=

0),( a_o) 0):( 6.)

din(u),

(1)

(8)

A state on A is a positive linear functional 0) normalized by 0)(6 0) = 1. The complex valued function on the phase space 0)(6.) is called the characteristic or the generating function of the state 0). A continuous complex valued function f on E, with f(0) = 1, is the characteristic function of a state on A if and only if

where dm(u) i~ the Liouville measure defined above. A proof of this result can be obtained combining the construction of a projection as in l.emma 3 from Ref. [3] with the assertion (vi) of Section II from Ref. [17]. It is well known [15-17] that for any automorphism a of A we have

N

E

-5jak exp[itr(uj, u k ) / 2 ] f ( u k - u , )

>~0

(2)

j,k=l

for all a k ~ C , u k ~ E , k = l . . . . . N, and N ~ . For any v ~ E an automorphism % of A can be defined by ~',(x) = 6,.* x6,,.

(3)

n(wl

o o~; 0)2 ° a )

= n ( 0 ) , ; to2).

(9)

In particular, this property is a direct consequence of Eq. (7) when a is a quasi-flee automorphism. The function on E defined by (27r) -~ P(0); 0)(t,,:)) is a probability density on E associated with any state o9 of A. Indeed we have /(2~) "E

-" P ( to; to(t,,.)) dm(v) = 1.

(10)

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H. Scutaru / Physics Letters A 200 (1995) 91-94

The entropy of this probability density is called the classical entropy of the quantum state to, S¢l(tO ) = - d a r ( P ( o g ; ~o(t.)))/dr I r=l,

(11)

where Gr(P(w; o~(i,. ))) is the generating function of the entropy, defined by

G~(P( w;

the most general form of a positive definite matrix A,

M -1 oT DO

× 0 ( MO M-~0 ) 0 ' .

(20)

w(t ,. )))

= fE[(2~')"P( w; w(t,,.))] r dm(v).

(12)

From Eq. (8) it follows that

Sc, ( O.)(A,t)) -~- Scl( O.)(A,0)).

(13)

Hence it is sufficient to consider only the quasi-free state of the form tO(a,0). In this case we have Gr(tO(a.0)) = r n{det[(A + 1 ) / 2 ] } -(~-1)/2.

(14)

Various particular kinds of such matrices are obtained taking O, O', D or M to be equal or proportional to the corresponding identity operator. A pure squeezed state is obtained when D = I. If this condition is not satisfied, the state is a mixed state called thermal squeezed state [20]. When M = I there is no squeezing and the corresponding states are pure coherent states or thermal coherent states. All these states have correlations between the different modes produced by the orthogonal symplectic operators O and O'. Combining Eqs. (7) and (15) we obtain that

Then it follows that

Sd(tO(a.o)) = n + ln{det[(a + 1 ) / 2 ] } / 2 .

Scl ( O)a° O~S) : S c l (O.)a)

Evidently we have Sc,(og(t,0)) = n.

(16)

Moreover, from the well known inequality det[(A + B)/2] >/(det A det B) 1/2 with B = I and from the fact that, as a consequence of the inequality (6) we have det A >t 1, follows the entropic uncertainty relation Sc,(o)(a,o))

>in.

(17)

Because the correlation matrix A is positive definite it follows [18] that there exists a S ~ Sp(E, o') such that

where D is a diagonal n × n matrix. The most general real symplectic transformation S ~ Sp(E, tr) has [19] the following structure,

S= O

(o 0)

M - l O',

(21)

(15)

(19)

where O and O' are symplectic and orthogonal (OTO = I) operators and M is a diagonal n × n matrix. Combining these two assertions one obtains

iff STAS = A or iff sTs = I, i.e. iff S is orthogonal. The first situation appears for example in the case of a parametric amplifier [21] and this property explains the fact that the nonentanglement of states is preserved by such a transformation. Hence the parametric amplifier is an entropy preserving device. The second situation appears if there is no squeezing, i.e. M = I . As it was shown in Ref. [6] for any quasi-free state wa we have (-t)a = ( O.).l ~ (.1).2 ~ . . . ~ O.).n)O O~s,

(22)

where d~. . . . . d, are the diagonal elements of the matrix D, and where wa,, i = 1. . . . . n, are the one mode quasi-free states defined on the phase spaces E i generated by the vectors {ei, f/} (on which the operator A acts by Ae i = die i and Afi = dill) by the formula

tOd,(t~,,e,+o,L)=exp{-di[('i)e+(rli)2]/4}, (23) with d i ~> 1 as it follows from Eq. (6). From Eq. (15) it is clear that n Sc,(tOd, ® . . . . . ® Wd°) = Y'~Sc,( Wd,) . 1

(24)

H. Scutaru/ PhysicsLettersA 200 (1995)91-94

94

Combining Eq. (22) with Eq. (24) we obtain that the equality

Scl (O)A) : ~ S c l ((.Ddi)

(25)

l

is valid only for nonsqueezed states (i.e. when M = I). For pure squeezed states ( D = I ) we still have such a decomposition of the classical entropy as a sum of the classical entropies of the individual modes. Moreover, there is another class of quasi-free states, considered also in Ref. [22], defined by the condition O = I in Eq. (20), and for which such a decomposition is valid. The formula which describes these two situations is

Sc[ ((.OA) = n + ~'~_~ln[(dzi + a ) / g + ( N i ) ] 1

(26)

where

( N i) = ( d i - 1 ) / 2 + di[(m i+ m~-1)2/2 - 1] is the mean value of the number of particle observable for the mode i. In the one mode case in the situation in which either di = 1 or m~ --- 1 we recover the results from Refs. [23,24]. The thermal squeezed states considered in Ref. [25] are obtained when M = mI and D = d l . In this case the correlation matrix A takes the following form

A = o ' T ( m2dl 0

m-2dl 0 )O'.

(27)

We shall take into account the fact that the most general form of an orthogonal symplectic matrix is [181 O'=(

_yX

xY)'

(28)

where X and Y are n × n matrices which satisfy the conditions x Tx + Y T y = I and x T y = Y TX. When Y = 0 the corresponding quasi-free state is nonentangled and remains nonentangled in all symplectic frames obtained by such Bogoliubov transformations (i.e. orthogonal with Y = 0) [25]. The ideal (lossless) beam splitter effects a transformation of such kind and the nonentanglement is preserved only when all modes are at the same temperature and equally squeezed [21]. Because the corresponding

transformation is orthogonal it follows from the above discussion that the ideal beam splitter is an entropy preserving device. For the thermal coherent states (i.e. states with M = I and D = dI) the nonentanglement is preserved by all orthogonal symplectic matrices O' (i.e. when Y ~ 0) [25]. In both cases considered in Ref. [25] the classical entropies of the individual modes are all equal with the same value. When the individual modes are squeezed differently (i.e. m/~: mj for some values of i and j ) or at different temperatures (i.e. d i 4: dj for some values of i and j ) the orthogonal transformations produce entanglement but the classical entropy is preserved.

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