Area Law for Measurement Entropy in Quantum

AREA LAW FOR MEASUREMENT ENTROPY IN QUANTUM

MECHANICS

R. Englman

Soreq NRC,Yavne81800,Israel

Phone, fax:972 2 6430546

e-mail:

May 28, 2009

Abstract

A tentative, approximate and preliminary derivation is given for anexpression which equates the measurement entropy (of Balian) withan area in parameter space. This might be a non-thermodynamicalgeneralization of the entropy-horizon area law for black holes. Thederivation is speci_cally for time dependent, cyclically, slowly varyingquantum mechanical systems and includes as theoretical requisites(time-) analytic properties of the wave function, Hilbert transformsand geometrical aspects of the Berry phase.

1 Introduction

Starting with the recognition that black holes behave as thermodynamicentities, and therefore are de_ned by an entropy SBH and a temperature,Bekenstein derived a proportionality between SBH and the area Ah of theblack hole horizon [1]. A derivation by Hawking has further established theproportionality, leading to the relation [2]

in which LP is the four-dimensional Planck length. (In view of the genesisof this expression the subscripts in SBH could equally be taken to standfor "black-holes" and "Bekenstein-Hawking".) For reviews of the subjectand more recent speculations, including also logarithmic corrections, onecan turn to [3]-[5]. Arising from the aboveequality is the "holographic1principle", which states that the description of a volume of space should bethought as encoded in a surface-like part associated with the volume, in theblack hole case the area of the horizon.

To my knowledge the entropic-area law and the holographic principlehave so far been deduced only on the basis of thermodynamic and informationtheoretical considerations, but a purely quantum mechanical derivationis lacking, and so is also an extension of the applicability of these to situationsoutside black holes and gravitation. The very word "principle" criesout for a wider relevance than these! I attempt to supply here a broadextension of the area law (and the associated "principle") for a very generalquantum mechanical situation, in fact for any time dependent systemsubject to non-abrupt cyclic (time-periodic) forces.

2 Previous Studies

The motivation for this arose on the one hand from the famous result for theBerry (or topological, or geometric) phase [6], which quanti_ed this phaseas the area of the spherical cap on a unit sphere, which is centered at adegeneracy point. Another motivation arose from relationships derived bythis author, during his collaboration with A. Yahalom and (partly) withM. Baer in several publications, between the log amplitude of a time (t)

- dependent wave function ln j (r; t)j and the corresponding wave-functionargument or phase arg (r; t) [7]-[9]. These relationships were based oncertain analytic properties possessed by the wave function in the Argandplaneof complex time (t) and took the form of Hilbert transforms in a timeplane (similar to the well known Kramers-Kronig or dispersion relations ina complex frequency plane). A noteworthy stimulant was the expressionfor the quantum mechanical expectation value of the complex functionln (r; t) as two terms: an entropy-like real term and a Berry phase-likeimaginary term in [10] (Equations (7) and (17) there). Hilbert transformsconnecting these two parts would be expected to generate some relationshipbetween the entropy and the cap-form (=an area) of the Berry-phase, of theform envisaged by the entropy-area law.

A parallel stimulant came from the works of Alonso Botero ([11])whichderived generalized Cauchy-Riemann conditions connecting real and imaginaryparts of the logarithm of a time dependent wave function, valid underconditions of locally holomorphic (analytic) behavior of the wave function.This contrasts with the requirement of the requirement of globally analyticbehavior, for the application of the Hilbert transform method. The2Cauchy-Riemann connection was privately brought to my notice earlier byA. Yahalom. As is well known, the formal expression of this connection fora function W(z) of the complex variable z = x+iy, separated into real andimaginary 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-arearelationship 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, ΠAiis the projection operator (on the set ofeigenstates) belonging to the i'th eigenvalue of the operator A^. Alternatively,

which equation is the specialization of the previous one for the (rather morecommon) case when the operator A^ refers to experimentally establishingthe values of the coordinates; then, of course, (r; t) stands for a solution ofthe time-dependent Schr odinger equation in the coordinate representation.While the von Neumann entropy quanti_es the uncertainty in (not) havingthe knowledge of a state, the measurement entropy ScoordM (t) gives the uncertaintyin the positions of the particles (while, possibly, having the fullpossession of the knowledge of the state) that is relieved when the coordinatemeasurements are made. Balian [12] points out that either of the twoentropies can be bigger than the other, depending on the case. In the sequelwe shall use the second version of the measurement entropy, denoting it asSM.

3 Algebraic derivation

The basic relation that is used is one of the two reciprocal relations ["Hilberttransforms", 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 analyticityconditions laid out in [13] (of which more, later). Additional constraintsthat we impose on the functions are that the Hamiltonian (of which theyare solutions) is periodic in time with period T and that the time variationis adiabatic (slow, compared to the inverse of the magnitude of theHamiltonian). 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_nedin equation (4) , we have the Hilbert transform expression

Noting that for Z = X + iY , arg Z _ arctan(Y=X) and rewriting the lastfactor ("arg") on the right hand side as the inde_nite integral of the di_erentialof arg and further noting that, we obtain

where in going from equation (8) to equation (9) we have interchangedthe order of the t,r and t0 integrations and transferred the principal integralsymbol to before the singular integral. We now assume that for a periodicand adiabatically varying j (r; t)j2, this quantity does not vary muchthroughout the period, which enables us to let the wave function factor inthe t-integrand of equation (9) cancel the same factor in the divisor of thet"-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 lineshows a smoother variation, whereas the second is sharper, exhibiting ananti-resonance like behavior. We wish to argue that the opposing signspersisting 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 thereforeconsider only the latter case, which means taking (0; T) for the range of thet'integration.

For t'in this range the integral in over t" equation (11) has a wellestablished meaning. It is the phase acquired by the wave function during anot fully completed periodic path. It is called the open path phase [14] andwill be denoted here by . Clearly

is the (gauge invariant) Berry phase [6]. As function of t', is monotonicand 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

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 of5radius R centered in the parameter-space singularity, which area the contourcircumnavigates 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 comparedto the black hole area law in equation (1) .

4 Applicability. Analytic conditions

The conditions for the Hilbert transform equation (6) to be applicable isthat the complex function ψ(r; t), being a solution of the time dependentSchr odinger equation, satisfy as a function of the complex variable t andfor given values of r the following conditions: It be without singularitiesand zeros in the lower half of the complex t-plane and that its logarithmtend to a constant (possibly zero) suitably fast in the far regions of thereal axis and that it vanish for large enough values of [13]. Someexpanding wave packets, "frozen Gaussians" satisfy these conditions [8]. Thephysical rationale of the analytic requirements is the lower-boundedness ofenergies for bound states ([16]- [17]). When not all of these requirementsare satisfied, e.g. the asymptotics are not as required or there are zeroswith negative imaginary values, it is possible to neutralize these defects bydefining an associated function which has some added factors. One can thenmake the Hilbert transform on this and finally correct the obtained phasemodulusrelations by deducting algebraically the part due to the addedfactors. However, my results will be subject to modifications, which remainto be given.

5 Conclusion

Intrigued by the fundamental importance in Black Hole Physics of theBekenstein-Hawking entropy-area law, I have attempted to obtain a derivationof another (in some sense, more general) measurement entropy-arealaw, rooted in pure quantum mechanics and stimulated by Berry's area lawfor his geometrical phase. Several (and severe) approximations have beenmade, indicating that the proposed law in equation (18) is only approximative.Applications and eventual verifications in particular cases remainto be made. Also, any logical (as opposed to the demonstrated, formal)connection between the above result and the black hole entropy law needsto be explored.

6References

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[16] L.A. Khal_n, Sov. Phys. JETP 8 1053 (1958)

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