AREA LAW FOR MEASUREMENT ENTROPY IN QUANTUM
MECHANICS
R. Englman
Soreq NRC,Yavne 81800,Israel
Phone, fax:972 2 6430546
e-mail: englman@vms.huji.ac.il
May 28, 2009
Abstract
A tentative, approximate and preliminary derivation is given for an expression which
equates the measurement entropy (of Balian) with an area in parameter space. This
might be a non-thermodynamical generalization of the entropy-horizon area law for
black holes. The derivation is speci_cally for time dependent, cyclically, slowly
varying quantum mechanical systems and includes as theoretical requisites (time-)
analytic properties of the wave function, Hilbert transforms and geometrical aspects
of the Berry phase.
1 Introduction
Starting with the recognition that black holes behave as thermodynamic entities, and
therefore are de_ned by an entropy SBH and a temperature, Bekenstein derived a
proportionality between SBH and the area Ah of the black hole horizon [1]. A
derivation by Hawking has further established the proportionality, leading to the
relation [2]
in which LP is the four-dimensional Planck length. (In view of the genesis of this
expression the subscripts in SBH could equally be taken to stand for "black-holes" and
"Bekenstein-Hawking".) For reviews of the subject and more recent speculations,
including also logarithmic corrections, one can turn to [3]-[5]. Arising from the above
equality is the "holographic 1 principle", which states that the description of a volume
of space should be thought as encoded in a surface-like part associated with the
volume, in the black hole case the area of the horizon.
To my knowledge the entropic-area law and the holographic principle have so far
been deduced only on the basis of thermodynamic and information theoretical
considerations, but a purely quantum mechanical derivation is lacking, and so is also
an extension of the applicability of these to situations outside black holes and
gravitation. The very word "principle" cries out for a wider relevance than these! I
attempt to supply here a broad extension of the area law (and the associated
"principle") for a very general quantum mechanical situation, in fact for any time
dependent system subject to non-abrupt cyclic (time-periodic) forces.
2 Previous Studies
The motivation for this arose on the one hand from the famous result for the Berry (or
topological, or geometric) phase [6], which quanti_ed this phase as the area of the
spherical cap on a unit sphere, which is centered at a degeneracy point. Another
motivation arose from relationships derived by this author, during his collaboration
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with A. Yahalom and (partly) with M. Baer in several publications, between the log
amplitude of a time (t)
- dependent wave function ln j (r; t)j and the corresponding wave-function argument
or phase arg (r; t) [7]-[9]. These relationships were based on certain analytic
properties possessed by the wave function in the Argandplane of complex time (t) and
took the form of Hilbert transforms in a time plane (similar to the well known
Kramers-Kronig or dispersion relations in a complex frequency plane). A noteworthy
stimulant was the expression for the quantum mechanical expectation value of the
complex function ln (r; t) as two terms: an entropy-like real term and a Berry phaselike imaginary term in [10] (Equations (7) and (17) there). Hilbert transforms
connecting these two parts would be expected to generate some relationship between
the entropy and the cap-form (=an area) of the Berry-phase, of the form envisaged by
the entropy-area law.
A parallel stimulant came from the works of Alonso Botero ([11]) which derived
generalized Cauchy-Riemann conditions connecting real and imaginary parts of the
logarithm of a time dependent wave function, valid under conditions of locally
holomorphic (analytic) behavior of the wave function. This contrasts with the
requirement of the requirement of globally analytic behavior, for the application of
the Hilbert transform method. The 2 Cauchy-Riemann connection was privately
brought to my notice earlier by A. Yahalom. As is well known, the formal expression
of this connection for a function W(z) of the complex variable z = x+iy, separated into
real and imaginary parts as W(z) = U(x; y) + iV (x; y) reads
The extension by [11] essentially allows for the presence of a gauge _eld. The second
member in our suggested formulation of an entropy-area relationship is the
measurement entropy, SM, introduced by Balian [12]. This di_ers from both the
Gibbs-entropy and the von Neumann (state-) entropy and is de_ned by
Here ρ is the density operator, ΠAi is the projection operator (on the set of eigenstates)
belonging to the i'th eigenvalue of the operator A^. Alternatively,
which equation is the specialization of the previous one for the (rather more common)
case when the operator A^ refers to experimentally establishing the values of the
coordinates; then, of course, (r; t) stands for a solution of the time-dependent
Schr•odinger equation in the coordinate representation. While the von Neumann
entropy quanti_es the uncertainty in (not) having the knowledge of a state, the
measurement entropy ScoordM (t) gives the uncertainty in the positions of the particles
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(while, possibly, having the full possession of the knowledge of the state) that is
relieved when the coordinate measurements are made. Balian [12] points out that
either of the two entropies can be bigger than the other, depending on the case. In the
sequel we shall use the second version of the measurement entropy, denoting it as SM.
3 Algebraic derivation
The basic relation that is used is one of the two reciprocal relations ["Hilbert
transforms", Eqs. (9) and (10) in [9]] involving real and imaginary parts of
and valid for for functions of the complex variable t that satisfy the analyticity
conditions laid out in [13] (of which more, later). Additional constraints that we
impose on the functions are that the Hamiltonian (of which they are solutions) is
periodic in time with period T and that the time variation is adiabatic (slow, compared
to the inverse of the magnitude of the Hamiltonian). The relation in question is
where P stands for the principal part of the integral. We wish to derive an
(approximate) "Area Law" for the time average of the measurement entropy < SM >
over the period T. Referring to the measurement entropy, as de_ned in equation (4) ,
we have the Hilbert transform expression
Noting that for Z = X + iY , arg Z _ arctan(Y=X) and rewriting the last factor ("arg")
on the right hand side as the inde_nite integral of the di_erential of arg
and further noting that
, we obtain
11
where in going from equation (8) to equation (9) we have interchanged the order of
the t,r and t0 integrations and transferred the principal integral symbol to before the
singular integral. We now assume that for a periodic and adiabatically varying j (r;
t)j2, this quantity does not vary much throughout the period, which enables us to let
the wave function factor in the t-integrand of equation (9) cancel the same factor in
the divisor of the t"-integrand, giving us
In the last line we have changed the position of the coordinate integration.
The principal integral
can now be evaluated. It takes the values
The values in both lines range between
and
, but the first line shows a
smoother variation, whereas the second is sharper, exhibiting an anti-resonance like
behavior. We wish to argue that the opposing signs persisting for a long range will
lead to a cancellation in the t'-integration, which will not be the case for the value in
equation (13) . We therefore consider only the latter case, which means taking (0; T)
for the range of the t' integration.
For t' in this range the integral in over t" equation (11) has a well established meaning.
It is the phase acquired by the wave function during a not fully completed periodic
path. It is called the open path phase [14] and will be denoted here by
. Clearly
is the (gauge invariant) Berry phase [6]. As function of t',
is monotonic and in a
general case may be assumed to be a linear function of its argument, so that
It is now elementary to evaluate the measurement entropy as
12
Famously, the geometric phase is
the solid angle element [15]. Therefore, in the
last line we have introduced the area measure A(R) on a solid sphere of 5 radius R
centered in the parameter-space singularity, which area the contour circumnavigates
during a period T. This is the final result of this paper, giving a new area law for the
measurement entropy, which is to be compared to the black hole area law in equation
(1) .
4 Applicability. Analytic conditions
The conditions for the Hilbert transform equation (6) to be applicable is that the
complex function ψ(r; t), being a solution of the time dependent Schr•odinger
equation, satisfy as a function of the complex variable t and for given values of r the
following conditions: It be without singularities and zeros in the lower half of the
complex t-plane and that its logarithm tend to a constant (possibly zero) suitably fast
in the far regions of the real axis and that it vanish for large enough values of
[13]. Some expanding wave packets, "frozen Gaussians" satisfy these conditions [8].
The physical rationale of the analytic requirements is the lower-boundedness of
energies for bound states ([16]- [17]). When not all of these requirements are
satisfied, e.g. the asymptotics are not as required or there are zeros with negative
imaginary values, it is possible to neutralize these defects by defining an associated
function which has some added factors. One can then make the Hilbert transform on
this and finally correct the obtained phase modulus relations by deducting
algebraically the part due to the added factors. However, my results will be subject to
modifications, which remain to be given.
5 Conclusion
Intrigued by the fundamental importance in Black Hole Physics of the BekensteinHawking entropy-area law, I have attempted to obtain a derivation of another (in
some sense, more general) measurement entropy-area law, rooted in pure quantum
mechanics and stimulated by Berry's area law for his geometrical phase. Several (and
severe) approximations have been made, indicating that the proposed law in equation
(18) is only approximative. Applications and eventual verifications in particular cases
remain to be made. Also, any logical (as opposed to the demonstrated, formal)
connection between the above result and the black hole entropy law needs to be
explored.
6 References
[1] J.D. Bekenstein, Lett. Nuovo Cimento 4, 737 (1972); Phys. Rev. D 7
2333, 9 3292 (1974), 12 3077 (1975)
[2] S.W. Hawking, Nature 248,30 (1974); Commun. Math. Phys. 43 189
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(1973)
[3] D.J. Raine and E.G. Thomas, Black Holes, An Introduction (Imperial
College Press, London, 2003)
[4] S. Das and S. Shankaranaryanan, Class. Quant. Gravity 24, 5249 (2007)
[5] S. Das, S. Shankaranaryanan and S. Sur,"Black Hole Entropy from Entanglement
and Area Law", ArXiv:086.0402 [gr-qc] 2 June 2008
[6] M.V. Berry, Proc Roy Soc. London, Ser. A 392 45 (1984)
[7] R. Englman, A.Yahalom and M. Baer, J. Chem. Phys. 109 6550 (1998);
Phys. Letters A 251 223 (1999);Europ. Phys. Journal D 1 (2000)
[8] R. Englman and A. Yahalom, Phys. Rev. A 60 1802 (1999); Phys. Rev.
B 61 2716 (2000); J. Phys. Chem. A 107 7170 (2003)
[9] R. Englman and A. Yahalom, Adv. Chem. Phys. 124 197 (2002)
[10] R. Englman and A. Yahalom, Phys. Rev. A67 054103 (2003)
[11] A. Botero, J. Math. Phys. 41 5279 (2003);"Geometric phase and modulus
relations for SU(n) matrix elements in the de_ning representation"
arXiv:math-ph/0310065 18 Nov 2003
[12] R. Balian, Europ. J. Phys. 10 208 (1989)
[13] E.C. Titchmarsh Introduction to the Theory of Fourier Integrals
(Clarendon Press, Oxford,1948) Chapter V; also C. Caratheodory Theory
of Functions (Chelsea, New York, 1958) Vol.I, Chapter 3.
[14] M. Mukunda and R. Simon, Ann. Phys. (N.Y.) 228, 205 (1993); A.K.
Pati, Phys. Rev. A 52, 2576 (1995); S.R. Jain and A.K. Pati, Phys. Rev.
Lett. 80, 650 (1998)
[15] M.S. Child, Adv. Chem. Phys. 124 1 (2002) [Eq. (49)]
[16] L.A. Khal_n, Sov. Phys. JETP 8 1053 (1958)
[17] M.E. Perel'man and R. Englman, Modern Phys. Lett. B 14 907 (2000)
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