Entropic uncertainty relations

Volume 103A, number 5

PHYSICS LETTERS

9 July 1984

ENTROPIC UNCERTAINTY RELATIONS Iwo BIALYNICKI-BIRULA

Instttute for TheorettcalPhystcs,PohshAcademy of Sciences, Lotmkow 32/46, 02-668 Warsaw,Poland Received 16 April 1984

New entroplc uncertainty relations for angle-angular momentum and posmon-momentum, derived recently by Partovl are related to older relations of a simdar type, which were proved by Blalymckl-Blrula and Myclelskl Significantly improved lower bounds are obtained m both cases

In two recent papers, Deutsch [1] and Partovi [2] have introduced new uncertainty relations in quantum mechanics, expressed in terms of the information ent r o p y The purpose o f this note is to significantly sharpen their results in the case of canonically conjugate observables and to remark, also, on the history o f this subject The entroplc uncertainty relation for two observables, A and B, has the form [2]

S A +sB>~b

So+SLz=-~

f

dOl¢(O)i 2

t AOt

×in(s

A0 t

where ~(~) is the angular wave function and cm are ItS expansion coefficients into the elgenfunctlons of Lz,

(1)

The entropies S A and S B are defined by the standard formula sA =_~pA

In the first case, the l e • h a n d side of (1) is

lnpA,

(2)

ff(q~) = (2rr)-1/2 ~ Cm e'mO

(4)

m

Since x in x iS a convex function, the following inequality holds* 1 [3]

t

where pA is the probablhty to find the value of the observable A in the tth interval of the spectrum of A, or in the tth bin, in the terminology of ref [2] Deutsch studied the simpler case of observables with purely discrete spectra Partovl extended these methods to the general case o f discrete and continuous spectra and applied them to the most Important pairs o f observables angle-angular m o m e n t u m and posmonm o m e n t u m In these two cases, he obtained certain expressions for the lower bounds b, but his results are not optimal In ~ hat follows, I shall derive the optimal bound for the angle-angular m o m e n t u m pair and unprove the lower bound for the position-momentum pair 0 375-9601/84/$ 03 00 © Elsevier Science Pubhshers B V (North-Holland Physics Pubhshmg Division)

1 fd~iffl2 A¢ A¢

lnl~kl2

1 r A~

(5) A¢

I shall assume that all the lntegratlox~ intervals in (3) are of equal length and next I wall use (5) to obtain the Inequality

,1 The value of a convex function at the mean value of the argument does not exceed the mean value of the function 253

Volume 103A, number 5

PHYSICS LETTERS

27r

Se~ + s L z ~

- ~ m

- f 0

d4>lffl21n[ffl2-1n/Xq 5

[cm 121nlCm 12

(6)

At th:s point I can use, for the sum of the two entropies appearing on the right-hand side of (6), the following mequahty proved by MyclelsM and myself some t:me ago [4] ,2 27r

d ~ b l f f i 2 1 n l f f ] 2 - ~ ICm121nicm12~ln27r m

-f 0

(7) This is the crumal step m m y derlvatmn It leads to the opt:mal entropm uncertainty relation,

S ~ + S Lz >~ --ln(A¢/2rr),

(8)

for the angle-angular momentum pair The sum of the two entropms attains :ts lower bound on the mgenstates of L z In a similar fashmn, I can improve the lower bound for the posmon-momentum pair To this end, I shall use another entropm uncertainty relatmn, -fdxlffl21nlff[ 2-

/dpl~121nl~12h~>l+lnn,

(9) where = (2rch)- 1/2 f

dx exp(-lpx/ll)t~(x)

(10)

This mequahty has an interesting history It has been ,2 Notice a difference m the normahzatlon of the wave functions

254

9 July 1984

conjectured (but not proven) by Everett m 1957 m an extended version of his famous Ph D thesis [5], In which he Introduced the many-worlds lnterpretatlon of quantum mechamcs and also independently, at the same time, by Hlrschman [6] The first proof of this lnequahty was given by Blalymck:-Blrula and Myclelskl [4] and independently by Beckner [7] It was also used to obtain the lower energy energy bound for the logarlthmm Schrodmger equation [8] Following the same hne of reasoning, as m the derwatlon of (8), one can obtain with the help of (9)

S x +S p ~> 1 - In 2 - In 7 ,

(11)

where 7 = zkxz2xp/h This is not the opt:mal bound However, It becomes one m the hmlt, when 7 tends to zero For example, for 7 = 0 01, the relative difference between the left- and the r:ght-hand s:de of (11) is only 3 5%, for gauss:an wave functmns, which saturate (9) Thus for small bm s:zes, my lower bound s:gmficantly improves the result 2 ln[2/(1 + x/-~)], obtained :n ref

[2] References [1] D Deutsch, Phys Rev Lett 50 (1983)631 [2] M H Partovl, Phys Rev Lett 50 (1983) 1883 [3] E F Beckenbach and R Bellman, Inequahtles (Springer, Berhn, 1961)§ 16 [4] I Blalymcka-Blrulaand J Mymelsk&Commun Math Phys 44 (1975) 129 [5] H Everett III, m The many-worlds interpretation of quantum mechamcs, eds B S DeWltt and N Graham (Princeton Umv Press, Princeton, 1973)p 52 [6] I I Hlrschman, Am J Math 79 (1957)152 [7] W Beckner, Ann Math 102 (1975) 159 [8] I Blalymcka-Blrulaand J Myclelska, Ann Phys (NY)100 (1976) 62