- Email: [email protected]
Quantum fisher metric and uncertainty relations
Volume 126, number 4
PHYSICS LETTERS A
4 January 1988
QUANTUM FISHER METRIC AND UNCERTAINTY RELATIONS
‘~
E.R. CAIANIELLO and W. GUZ’ Department of TheoreticalPhysics, University ofSalerno, 84100 Salerno, Italy Received 24 August 1987; accepted for publication 27 October 1987 Communicated by J.P. Vigier
The connection between the recently discovered quantum-mechanical analogue of the Fisher information metric and the uncertainty relations is further investigated , with special emphasis on the role of the quantum counterpart of the Cramér—Rao inequality.
Recently, the Fisher information metric [1] has
g(O) =4J0,[Pm(X)1
been shown to possess a quantum-mechanical counterpart, leading to the quantum analogue of the Cramér—Rao inequality [2—41,the latter implying in turn the quantum-mechanical uncertainty relations.In this not we develop some aspects ofthe above mentioned approach, which might be of interest for quantum physics. In order to establish the notation, we begin with a few basic definitions. Let (~‘ 1, ~u)be a measure space, and M a real n-dimensional differentiable manifold such that with every point m of M there is associated a probability measure Fm on (~‘ E), which is assumed to be absolutely continuous with respect to u. The Radon—Nikodym derivative OfPm with respect to /~will be denoted by p,~,i.e., dPrn=prn d/1. Let (U, 0) be any local coordinate chart such that mEU. The Fisher information is defined [1] as the covariant tensor field on M, whose components in a given local coordinate frame (U, 0) are given by g,~(0)
=
$
0, log pm(X) O~log Pm (x) Pm(dX),
(1)
(2)
markable chain of Cramér—Rao-like inequalities varmFy(X, X) ~var~FFg(X, X)
8100k, xe~.Clearly, g can equivalently be expressed as [2—4] Work supported by the Italian Ministero della Pubblica Istruzione, fondo 60%, at-i. 65 D.P.R. 382/80. On leave of absence from the Institute of Theoretical Physics and Astrophysics, University of Gdañsk, 80-952 Gdai%sk, Poland.
9u(dx)
and the appearence of the square root of the probability density in (2) suggested defining the quantum counterpart of g,, as follows [2,31 8jV’,n), (3) Y~(O)=(ôiWrn, where yi~ is a parametric family, labelled by points m of M, of normalized Hilbert space vectors V/mE it” with ( , ) denoting the inner product in J~°,giving therefore a complex extension of the Fisher information metric g,~.Using the spectral theorem for a (fixed) self-adjoint operator F on J~°in the “multiplication operator form”, which states that there exists an isometric isomorphismj onto L2((~,Z, au), where (~,~, ~z)is some measure space, such that [J(F~)](x) =P(x)[j(y.’)](x), V/E.)f’, x~, for some measurable function P: ~—P (in the sequel, j will simply be omitted), we arrive at the following re-
where 0k =
*
t”2a~[p~(x)1 112
[41
1< df X> I (4) where, by definition, ft m)X)=y = (V/rn, Fy.im), ‘I’m, 2~rn)”2,y(X, t forvarrnF= any (real) [F—f(m)] 0X’X vector Xtangent at meM, and similarly for ~g, where ~g denotes the Fisher metric corresponding to the parametric Pm(X)
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
family
of
probability
densities
I Wm(X) 12, and finally, denotes the 223
Volume 126, number 4
PHYSICS LETTERS A
value on X of the differential df at the point me M. In a given local coordinate chart (U, 0), =>~1(o1f)X7. It has been shown in ref. [4} that the standard quantum-mechanical uncertainty relations (e.g., position—momentum and time—energy) can be easily obtained as consequencesof (4). The expression (3) is, however, by no means the only one possible quantum-mechanical extension of the Fisher (1). It is even more natural to replace (3) by metric the following complex hermitian metric tensor [4]
4 January 1988
where X, Y are two (real) vectors tangent at me M and, according to (5), Im y,~(O) = —2i[(a~~,~, 8,’I/,~) (8jwm~ôjV/rn)]
,
—
(10)
to the parametric family of Hilbert space vectors 1I’ m =exp[(i/h)(q.P+pQ]V/, 2”, (11) mm(q,p)eEP where ~i’ is an arbitrary norm-one Hubert space vector, and
= (ôiV/,n, 0 3V/,n) 8iV/pn,V/m)(V/m,0jV/,n)
(5)
.
Re(0)=2(OIp~2,OJp~2)HS,
this
oj-~, =
since its real part can be reexpressed as
where p~=
[Q1,Q~,]= [P1, F1] =0, [Q,,F3] =ihö~1
In
(
(6)
I V/rn> <‘I’
m ~and ( , ) HS stands for the customary inner product inin the.W, Banach of the Hubert—Schmidt operators so thatspace the correspondence between the classical and quantum Fisher metrics becomesnow more transparent. Using ~, one
case,
(12)
Ojyim=(i/h)(—~pj+Pj)V/,,, and ~i~qj+Qj)V/m, where O,=0/Oq’, O~-=O/O,j,
f=j+n, so that
~ 0) = 4~—2 cov,~ covrn(Q,, 2 (P,, P~) Qj)
5~,-~-(0)=4h~‘~1(0) = 4h —2 COVm ( Q~’P,) 2ih ‘(5 ~( 0) = 4/~—2 COVm( Qi, P~)+ 2ih —‘~5 —
-
Ii,
(13)
Ii,
can easily prove [4] that var,~Fy(X,X) ~var,~F2(X, X)
where, by definition, cov,~(A,B) = (V/m,~(AB+BA)V/m)
?~var,~FFg(X,X)~II
(7) (V/,n,AV/,n)(V/,n,BV/m).
and again the quantum-mechanical uncertainty relations can readily be obtained [4] as consequences of (7). The most remarkable fact, however, which deserves particular attention is the intermediate inequality varmF Fg(X, X) ~ 1< df X> 2 , (8) which is strongly reminiscent of the classical-like Uncertaintyrelations derived recently within the framework of Nelson’s stochastic mechanics (see, e.g., refs. [5,6] and references quoted therein). There is no doubt that connections between these two approaches must exist and are to be established in the near future. The main aim of this note is to show, first of all, that also the imaginary part of the metric tensor ~i can be used to derive the usual quantummechanical uncertainty relations. To show this, we shall apply the inequality IIm~(X,Y)I2~(X,X)~(Y,Y), (9) 224
Inserting in (9) in place of X and Y, respectively, the orthonormal vectors E~and E 1 defined by E~= a, b = 1, 2, 2n, we immediately get the position— momentum uncertainty relations (var,~P,) X (var~Q~) ? ~ Clearly, the same result can be obtained by using the real part of the metric tensor ~ and the quantum analogue of the Cramér—Rao inequality (cf. (7)): var,~F $~(X, X) ~ 1< dJ X> 2 (14) ...,
Indeed, for F= Q, one has O~f(m)= (v/rn, Q1PjV/m), ~9i-f(m) = 0, and by inserting X= E~in (14) one gets varrnQi varmPj ~ ~ as before. Putting, however, F= Q~and X= E,+ tE~t real, one easily gets from (14) var~1Q, var~,P~,~ 2 2varmQj] (15) —varmQj[2tcovrn(Qj, F~)+t
Volume 126, number 4
PHYSICS LETTERS A
and looking for the maximum of the r.h.s. of the inequality above we easily find that it attains its maximum
varrnFj varmQj~1h2ô +varmP, var;’P~cov~(Q~, P~),
varmQ, var;’ Q~cov~(Q~, ~+
which also reduces to (17), when n=2.
~
at the value 1= —covm(Qj, F~)var;’Q~,so that the equality (15) then becomes varmQ, varmFj ~ ~ +varmQ, var;’ Q~cov~(Q~, Pa.)
.
(16) If n = 2, the inequality above reduces to the well known uncertainty relation between Q and F in the Schrodinger form
4January 1988
(18)
References [1] R.A. Fisher, Proc. Cambridge Philos. Soc. 122 (1925) 700. [2] E.R. Caianiello, Lett. Nuovo Cimento 38 (1983) 539. [3] E.R. Caianiello, in: Frontiers of non-equilibrium statistical physics, eds. G.T. Moore and M.O. Scully, (Plenum, New York, 1986). [4] E.R. Caianiello, D. de Falco and W. Guz, A comparison be-
(17)
tween classical and quantum information geometry, University of Salerno preprint (1987). [5] D. de Falco, S. Dc Martino and S. De Siena, Phys. Rev. Lett. 49 (1982) 181.
1 and X= tE~+E1, t real, we find in the same manner that
[6] 5. Golin, J. Math. Phys. 26 (1985) 2781. [71 E. Schrodinger, Sitzungsber. Preuss. Akad. Wiss. Phys. Math.
varm Q varmP ~ h2 + cov~,( Q, F) We
note, finally,
that
.
putting
F=P
KI. (1930) 296.
225
PHYSICS LETTERS A
4 January 1988
QUANTUM FISHER METRIC AND UNCERTAINTY RELATIONS
‘~
E.R. CAIANIELLO and W. GUZ’ Department of TheoreticalPhysics, University ofSalerno, 84100 Salerno, Italy Received 24 August 1987; accepted for publication 27 October 1987 Communicated by J.P. Vigier
The connection between the recently discovered quantum-mechanical analogue of the Fisher information metric and the uncertainty relations is further investigated , with special emphasis on the role of the quantum counterpart of the Cramér—Rao inequality.
Recently, the Fisher information metric [1] has
g(O) =4J0,[Pm(X)1
been shown to possess a quantum-mechanical counterpart, leading to the quantum analogue of the Cramér—Rao inequality [2—41,the latter implying in turn the quantum-mechanical uncertainty relations.In this not we develop some aspects ofthe above mentioned approach, which might be of interest for quantum physics. In order to establish the notation, we begin with a few basic definitions. Let (~‘ 1, ~u)be a measure space, and M a real n-dimensional differentiable manifold such that with every point m of M there is associated a probability measure Fm on (~‘ E), which is assumed to be absolutely continuous with respect to u. The Radon—Nikodym derivative OfPm with respect to /~will be denoted by p,~,i.e., dPrn=prn d/1. Let (U, 0) be any local coordinate chart such that mEU. The Fisher information is defined [1] as the covariant tensor field on M, whose components in a given local coordinate frame (U, 0) are given by g,~(0)
=
$
0, log pm(X) O~log Pm (x) Pm(dX),
(1)
(2)
markable chain of Cramér—Rao-like inequalities varmFy(X, X) ~var~FFg(X, X)
8100k, xe~.Clearly, g can equivalently be expressed as [2—4] Work supported by the Italian Ministero della Pubblica Istruzione, fondo 60%, at-i. 65 D.P.R. 382/80. On leave of absence from the Institute of Theoretical Physics and Astrophysics, University of Gdañsk, 80-952 Gdai%sk, Poland.
9u(dx)
and the appearence of the square root of the probability density in (2) suggested defining the quantum counterpart of g,, as follows [2,31 8jV’,n), (3) Y~(O)=(ôiWrn, where yi~ is a parametric family, labelled by points m of M, of normalized Hilbert space vectors V/mE it” with ( , ) denoting the inner product in J~°,giving therefore a complex extension of the Fisher information metric g,~.Using the spectral theorem for a (fixed) self-adjoint operator F on J~°in the “multiplication operator form”, which states that there exists an isometric isomorphismj onto L2((~,Z, au), where (~,~, ~z)is some measure space, such that [J(F~)](x) =P(x)[j(y.’)](x), V/E.)f’, x~, for some measurable function P: ~—P (in the sequel, j will simply be omitted), we arrive at the following re-
where 0k =
*
t”2a~[p~(x)1 112
[41
1< df X> I (4) where, by definition, ft m)X)=y = (V/rn, Fy.im), ‘I’m, 2~rn)”2,y(X, t forvarrnF= any (real) [F—f(m)] 0X’X vector Xtangent at meM, and similarly for ~g, where ~g denotes the Fisher metric corresponding to the parametric Pm(X)
0375-9601/88/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
family
of
probability
densities
I Wm(X) 12, and finally,
Volume 126, number 4
PHYSICS LETTERS A
value on X of the differential df at the point me M. In a given local coordinate chart (U, 0),
4 January 1988
where X, Y are two (real) vectors tangent at me M and, according to (5), Im y,~(O) = —2i[(a~~,~, 8,’I/,~) (8jwm~ôjV/rn)]
,
—
(10)
to the parametric family of Hilbert space vectors 1I’ m =exp[(i/h)(q.P+pQ]V/, 2”, (11) mm(q,p)eEP where ~i’ is an arbitrary norm-one Hubert space vector, and
= (ôiV/,n, 0 3V/,n) 8iV/pn,V/m)(V/m,0jV/,n)
(5)
.
Re(0)=2(OIp~2,OJp~2)HS,
this
oj-~, =
since its real part can be reexpressed as
where p~=
[Q1,Q~,]= [P1, F1] =0, [Q,,F3] =ihö~1
In
(
(6)
I V/rn> <‘I’
m ~and ( , ) HS stands for the customary inner product inin the.W, Banach of the Hubert—Schmidt operators so thatspace the correspondence between the classical and quantum Fisher metrics becomesnow more transparent. Using ~, one
case,
(12)
Ojyim=(i/h)(—~pj+Pj)V/,,, and ~i~qj+Qj)V/m, where O,=0/Oq’, O~-=O/O,j,
f=j+n, so that
~ 0) = 4~—2 cov,~ covrn(Q,, 2 (P,, P~) Qj)
5~,-~-(0)=4h~‘~1(0) = 4h —2 COVm ( Q~’P,) 2ih ‘(5 ~( 0) = 4/~—2 COVm( Qi, P~)+ 2ih —‘~5 —
-
Ii,
(13)
Ii,
can easily prove [4] that var,~Fy(X,X) ~var,~F2(X, X)
where, by definition, cov,~(A,B) = (V/m,~(AB+BA)V/m)
?~var,~FFg(X,X)~I
(7) (V/,n,AV/,n)(V/,n,BV/m).
and again the quantum-mechanical uncertainty relations can readily be obtained [4] as consequences of (7). The most remarkable fact, however, which deserves particular attention is the intermediate inequality varmF Fg(X, X) ~ 1< df X> 2 , (8) which is strongly reminiscent of the classical-like Uncertaintyrelations derived recently within the framework of Nelson’s stochastic mechanics (see, e.g., refs. [5,6] and references quoted therein). There is no doubt that connections between these two approaches must exist and are to be established in the near future. The main aim of this note is to show, first of all, that also the imaginary part of the metric tensor ~i can be used to derive the usual quantummechanical uncertainty relations. To show this, we shall apply the inequality IIm~(X,Y)I2~(X,X)~(Y,Y), (9) 224
Inserting in (9) in place of X and Y, respectively, the orthonormal vectors E~and E 1 defined by E~= a, b = 1, 2, 2n, we immediately get the position— momentum uncertainty relations (var,~P,) X (var~Q~) ? ~ Clearly, the same result can be obtained by using the real part of the metric tensor ~ and the quantum analogue of the Cramér—Rao inequality (cf. (7)): var,~F $~(X, X) ~ 1< dJ X> 2 (14) ...,
Indeed, for F= Q, one has O~f(m)= (v/rn, Q1PjV/m), ~9i-f(m) = 0, and by inserting X= E~in (14) one gets varrnQi varmPj ~ ~ as before. Putting, however, F= Q~and X= E,+ tE~t real, one easily gets from (14) var~1Q, var~,P~,~ 2 2varmQj] (15) —varmQj[2tcovrn(Qj, F~)+t
Volume 126, number 4
PHYSICS LETTERS A
and looking for the maximum of the r.h.s. of the inequality above we easily find that it attains its maximum
varrnFj varmQj~1h2ô +varmP, var;’P~cov~(Q~, P~),
varmQ, var;’ Q~cov~(Q~, ~+
which also reduces to (17), when n=2.
~
at the value 1= —covm(Qj, F~)var;’Q~,so that the equality (15) then becomes varmQ, varmFj ~ ~ +varmQ, var;’ Q~cov~(Q~, Pa.)
.
(16) If n = 2, the inequality above reduces to the well known uncertainty relation between Q and F in the Schrodinger form
4January 1988
(18)
References [1] R.A. Fisher, Proc. Cambridge Philos. Soc. 122 (1925) 700. [2] E.R. Caianiello, Lett. Nuovo Cimento 38 (1983) 539. [3] E.R. Caianiello, in: Frontiers of non-equilibrium statistical physics, eds. G.T. Moore and M.O. Scully, (Plenum, New York, 1986). [4] E.R. Caianiello, D. de Falco and W. Guz, A comparison be-
(17)
tween classical and quantum information geometry, University of Salerno preprint (1987). [5] D. de Falco, S. Dc Martino and S. De Siena, Phys. Rev. Lett. 49 (1982) 181.
1 and X= tE~+E1, t real, we find in the same manner that
[6] 5. Golin, J. Math. Phys. 26 (1985) 2781. [71 E. Schrodinger, Sitzungsber. Preuss. Akad. Wiss. Phys. Math.
varm Q varmP ~ h2 + cov~,( Q, F) We
note, finally,
that
.
putting
F=P
KI. (1930) 296.
225