Entropic uncertainty relations for nonextensive quantum scattering

28 October 1999

Physics Letters B 466 Ž1999. 27–32

Entropic uncertainty relations for nonextensive quantum scattering D.B. Ion, M.L.D. Ion National Institute for Physics and Nuclear Engineering Horia Hulubei, NIPNE-HH, P. O. Box MG-6, Bucharest, Romania Received 31 July 1999; accepted 7 September 1999 Editor: R. Gatto

Abstract In this paper the angle-angular momentum entropic uncertainty relations are obtained for Tsallis-like entropies for nonextensive quantum scattering of spinless particles. The number-phase entropic uncertainty relations are also proved for nonextensive quantum scattering. Numerical results on the experimental tests of these entropic uncertainty relations, for the nonextensive Ž q / 1. statistics case are obtained by calculations of Tsallis-like scattering entropies from the 48 experimental sets of the pion-nucleus phase shifts. q 1999 Published by Elsevier Science B.V. All rights reserved. PACS: 03.65.Ca; 25.80.Dj

1. Introduction The entropy of a quantum state Žsee e.g. Ref. w1x. describing a physical system is a quantity expressing not only the uncertainty or randomness of the system but also can be regarded as the amount of information carried by its state. Recently the investigation of the quantum entropy was substantially extended not only by proving new entropic uncertainty relations ŽEUR. Žsee e.g. Refs. w1–4x. for the standard additive systems but also by the generalization of such results to the nonextensive statistics w5–9x. In Ref. w4x the state independent angle-angular momentum entropic lower bounds are proved by using Tsallis-like entropies and the Riesz theorem w10x for the quantum scattering of the spinless particles. Moreover, the optimal entropic upper bounds for each Tsallis-like scattering entropy w4x Su Ž q ., SLŽ q ., Su LŽ q ., were recently w11x proved in terms of optimal entropies

derived from the principle of minimum distance in the space of states w12x. In this way, the entropic uncertainty band as a new concept in quantum physics is introduced. However, one of the basic ingredients of the entropic bands, the entropic uncertainty relation, is rigorously proved in Refs. w4,5x only for the extensive case q s 1. The generalized entropy as a measure of quantum uncertainty was also discussed in Ref. w13x in connection with thenumber-phase entropic uncertainty measure for the coherent states within the Pegg-Barnet quantum theory w14x. In this paper the entropic uncertainty relations ŽEUR. are proved for Tsallis-like entropies for nonextensive quantum scattering of spinless particles. The results of the consistent experimental tests of these EUR in the nonextensive Žq / 1. statistics cases, obtained by calculations of the Tsallis-like scattering entropies from the 48 experimental sets of pion-nucleus phase shifts w15–19x, are presented.

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 9 . 0 1 0 5 2 - 7

D.B. Ion, M.L.D. Ion r Physics Letters B 466 (1999) 27–32

28

Extension of these results in the case of the numberphase entropic uncertainty relation for nonextensive quantum scattering are also obtained.

tivity entropy rule Eq. Ž3.. In the limit t s 1, we get the following quantum scattering entropies: 1

Su s lim Su Ž q . s y

Hy1dxP Ž x . ln P Ž x . ,

q™1

2. Entropic uncertainty relations for the nonextensive quantum scattering Tsallis-like entropies for the quantum scattering of spinless particles are defined as follows: 1

Su Ž q . s

qy1

½

1y

1

Hy1dx

q

PŽ x.

5

,

qgR ,

Ž 1.

L

SL s lim SLŽ q . s y q™1

SLŽ q . s

qy1

½

L

1y

Ý

Ž 2 l q 1. P w pl x

q

lso

5

qgR ,

Ž 2.

1 s qy1

½

L

1y

Ý Ž 2 l q 1. ls0

p lq

1

Hy1dx

PŽ x.

q

5

s Su Ž q . q SLŽ q . q Ž 1 y q . Su Ž q . SLŽ q . , qgR ,

Ž 3.

w 1 y 2 1y t x F Su LŽ t . , for any t g R .

PŽ x. s

2p

p l s 4p P

sel

P

dV

N fl N 2

sel

Pu l s P Ž x . . p l ,

Ž x. ,

1

Hy1P Ž x . dx s 1 ,

Ž 4.

L

,

Ý Ž 2 l q 1. P pl s 1 ,

Ž 5.

ls0 1

Ž 11 .

The equality sign holds in (10), (11) for certain, i.e., when p l s d l 0 and P(x) s 1 r 2. Proof. Here the proof of the result (10), (11) will be presented for two distinctiÕe cases characterized by the entropic inequalities (see Ref. [4]):

ry1

ds

Ž 10 .

In the limit t s 1, we get the usual entropic uncertainty relation (see Eq. (17) from Ref. [4]):

1

where

Ž 9.

EUR-1. The entropic uncertainty relations for the Tsallis-like entropies for nonextensiÕe quantum scattering of spinless particles are as follows:

ln2 F Su q SL .

Su LŽ q .

Ž 8.

lso

q™1

ty1

,

Ý Ž 2 l q 1. P pl ln pl ,

Su L s lim Su LŽ q . s Su q SL .

1 1

Ž 7.

w 1 y 2 1y r x F Su Ž r . , and 0 F SLŽ r . ,

for t s r g Ž y`,0 . , Su Ž t . F

1 ty1

Ž 12 .

w 1 y 2 1y t x , and 0 F SLŽ t . ,

for t g Ž 0,q ` . .

Ž 13 .

L

Hy1 Ý Ž 2 l q 1. P l Ž x ,l . s 1 , u

ls0

Ž 6. where ddVs Ž x . s < f Ž x .< 2 and sel are the differential and integrated cross sections of the two-body elastic scattering of spinless particles, f l are partial amplitudes from the usual phase shift analysis Žsee Ref. w4x for details.. The index q / 1 controls the degree of entropy nonextensivity reflected in the pseudoaddi-

The left inequalities (12), (13) are proÕed Õia Lagrange multipliers by solÕing the optimization problem: find an extremum of the Su (q)-entropy subject to the normalization constraint (4). So, from the first deriÕatiÕe we obtain the following equations: E £ r E P s q(q y 1)y 1P q y 1 y l s 0, where l is the Lagrange multiplier. The solution of this problem is P(x) s 12 , and, according to the sign of the second deriÕatiÕe E 2 £ r E P 2 s qP q y 2 we get the left inequalities (12), (13).

D.B. Ion, M.L.D. Ion r Physics Letters B 466 (1999) 27–32

29

Ži. Case t s r g (y`,0): The proof of (10) in the case (i) can be directly obtained from the entropic bounds (12) which are equiÕalent to

Now, from (18), (20) we obtain the following inequalities:

2 1y r F 1 q Ž 1 y r . Su Ž r . ,

ty1

1

w 1 y 2 1y t x F Su LŽ t . F2 1ytSLŽ t . 1

and 1 F 1 q Ž 1 y r . SLŽ r . ,

q

respectively, for t s r g Ž y`,0 . .

ty1

Ž 14 .

Ž 1 y 2 1y t . ,

for any t g Ž 0,q ` . .

Ž 21 .

So, combining these inequalities, we get 1 ry1

w 1 y 2 1y r x F Su LŽ r . , for any r g Ž y`,0 . Ž 15 .

since 2 1y r F 1 q Ž 1 y r . SLŽ r .

1 q Ž 1 y r . Su Ž r .

s 1 q Ž 1 y r . Su LŽ r . ,

Ž 16 .

where Su L(t), t g (y`,q `) is giÕen by Eq. (3). Žii. Case t g (0,q `): In this case, from the inequalities (13), we get

Next, using only Ž16., Ž18. and Ž20., we can prove the following generalization of the entropic inequalities. Entropic Inequality-1. The Tsallis-like scattering entropies Su L(t), corresponding to the nonextensiÕity indices t s u g (y`,1) and t s q g (1,q `) satisfy the following generalized inequalities: 1

1 q Ž 1 y q . Su LŽ q . F exp

1 q Ž 1 y s . Su Ž s . F 2 1y s , 1 F 1 q Ž 1 y s . SLŽ s . ,

for any t s s g Ž 0,1 . , Ž 17 .

2q

½

1

1 y

2q

2u

1

ln2

5

1 q Ž 1 y u . Su LŽ u .

2u

Ž 22 .

or

Proof. The proof of (22) is obtained by using the following inequalities (see (16), (18) and (20))

2 1y s F 1 q Ž 1 y s . Su LŽ s .

2 1y r F 1 q Ž 1 y r . Su LŽ r . ,

for t s r g Ž y`,0 . ,

2 1y s F 1 q Ž 1 y s . Su LŽ s . ,

for t s s g Ž 0,1 . ,

1 q Ž 1 y q . Su LŽ q . F 2 1y q ,

for q g Ž 1,q ` .

F 2 1y s 1 q Ž 1 y s . SLŽ s . ,

s g Ž 0,1 .

Ž 18 .

and

Ž 23 .

2 1y q F 1 q Ž 1 y q . Su Ž q . ,

to obtain

1 q Ž 1 y q . SLŽ q . F 1 , for any t s q g Ž 1,q ` . ,

Ž 19 .

2

1yu

1

2u

2q

1 q Ž 1 y q . Su LŽ q .

or F2

2 1y q 1 q Ž 1 y q . SLŽ q . F 1 q Ž 1 y q . Su LŽ q . F 2 1y q , t s q g Ž 1,q ` . .

1y q

1

2q

2u

1 q Ž 1 y u . Su LŽ u .

u g Ž y`,1 . ,

Ž 20 .

q g Ž 1,q ` .

,

Ž 24 .

D.B. Ion, M.L.D. Ion r Physics Letters B 466 (1999) 27–32

30

Table 1 A brief summary of the state independent properties of the Tsallis-like entropies SLŽ t ., Su Ž t ., Su LŽ t . and also for SN Ž t ., Sw Ž t ., Sw N Ž t . SLŽ t .

Su Ž t .

Su LŽ t .

1.

0 F SLŽ t ., t g R

1 ty 1

2.

SLŽ t . s 0 for  pl 4 s  d l 0 4

3. 4.

0 F lim t ™ 1 SLŽ t . s SL SLŽ0. s LŽ L q 1.

Ž1 y 2 1y t . F Su Ž t ., t g Ry Su Ž t . F t y1 1 Ž1 y 2 1y t ., t g Rq Su Ž t . s t y1 1 Ž1 y 2 1y t . for P Ž x . s 12 or  pl 4 s  d l 0 4 lim t ™ 1 Su Ž t . s Su F ln2 Su Ž0. s 1

Su LŽ t . s t y1 1 Ž1 y 2 1yt ., t g R, for P Ž x . s 12 or  pl 4 s  d l 0 4 ln2 F lim t ™ 1 Su LŽ t . s Su q SL Su LŽ0. s 2 LŽ L q 1. q 1

SN Ž t .

Sw Ž t .

Sw N Ž t .

0 F SN Ž t ., t g R

1 ty 1

1.

1 ty 1

1 y Ž 2p .

1yt

Sw Ž t . F t y1 1 1 y Ž 2p .

1y t

Sw N Ž t . s t y1 1 1 y Ž 2p .

1 2p

3. 4.

In particular, for 12 - p F 1 and q defined by q 21p s 1, the result Ž22. includes in a general and exact form the inequalities 1 2p

1 q Ž 1 y q . Su LŽ q .

then, in this case, the Tsallis-like entropies can be defined as follows:

2q 1

p

2p

1 q Ž 1 y p . Su LŽ p .

1 ty1

½

1y

2p

H0

dw P Ž w .

t

5

,

tgR ,

Ž 27 .

py1

F2

, t g R,

for P Ž x . s or  pn 4 s  dn0 4 lnŽ2p . F lim t ™ 1 Sw N Ž t . s Sw q SN Sw N Ž0. s N q 1

Sw Ž t . s

1

1yt

1 2p

for P Ž x . s or  pn 4 s  dn0 4 lim t ™ 1 Sw Ž t . s Sw F lnŽ2p . Sw Ž0. s 1

0 F lim t ™ 1 SN Ž t . s SN SN Ž0. s N

w1 y Ž2p .1y t x F Sw N Ž t ., t g R

, t g Rq

1y t Sw Ž t . s t y1 1 1 y Ž 2p .

SN Ž t . s 0 for  pn 4 s  dn0 4

2.

1 ty 1

F Sw Ž t ., t g Ry

Ž1 y 2 1yt . F Su LŽ t ., t g R

,

Ž 25 .

which can be obtained by combining Eqs. Ž19. with Ž20. from Ref. w4x. A brief summary of the state independent properties of the Tsallis-like entropies SLŽ t . Su Ž t ., Su LŽ t . for the quantum scattering of spinless particles is presented in Table 1.

SN Ž q . s Sw N Ž t . s

1 ty1 1 ty1

½1 y Ý p 5 , t n

tgR ,

Ž 28 .

n

½

1 y Ý pnt n

2p

H0

dw P Ž w .

t

5

s Sw Ž t . q SN Ž t . q Ž 1 y t . Sw Ž t . SN Ž t . , tgR

Ž 29 .

and 3. Number-phase entropic uncertainty relations for nonextensive scattering

P Ž w . s < f Ž w . < 2r 2p

Now, suppose that scattering amplitude f Ž w . g L2 Ž0,2p . and let the Fourier coefficients c n be defined by cn s

1

2p

H 2p 0

H0

< f Ž w . < 2dw ,

P Ž w . dw s 1 ,

pn sN c n N 2r Ý N c n N 2 ,

Ž 30 .

Ý pn s 1 ,

n

P Ž w ,n . s P Ž w . P pn ,

f Ž w . exp  yin w 4 d w ,

Ž n s 0," 1," 2, . . . . ,

2p

H0

2p

Ý pn H n

Ž 26 .

Ž 31 .

n

P Ž w . dw s 1 .

0

Ž 32 .

D.B. Ion, M.L.D. Ion r Physics Letters B 466 (1999) 27–32

31

Fig. 1. The experimental tests of the generalized entropic uncertainty relations Ž10. with the Tsallis-like entropy Su LŽ t . as a function of nonextensivity parameter, obtained by using the 48-sets of the p 0-nucleus phase shifts w15–19x.

Entropic Inequality-2. Let 12 - p F 1, and suppose that f (t) g L p(0,2p ) and let the Fourier coefficients c n be defined by Eq. (26). Then, we haÕe

and 1

1

F exp

½

py1 2p

1 q Ž 1 y q . Sw N Ž q .

2q

1 q Ž 1 y q . Sw Ž q .

ln Ž 2p .

5

F exp

½

py1 p

2q

ln Ž 2p .

5

1

= 1 q Ž 1 y p . SN Ž q .

2p

1

,

Ž 33 .

1

1 q Ž 1 y q . SN Ž q . F exp

½

py1 2p

= 1 q Ž 1 y p . Sw N Ž q .

2p

.

Ž 35 .

2q

ln Ž 2p .

5 1

= 1 q Ž 1 y p . Sw Ž q .

2p

,

Ž 34 .

Proof. The proof of (33) – (35) is similar to that of the [u ,L] results from Ref. [4] and can be obtained using the Hausdorff-Young theorem 2.3 (Ref. [9], p. 101) (with p ™ 2p and pX ™ 2q, so that, p y 1 q q y 1 s 2) for f (w ) g L p(0,2p ).

D.B. Ion, M.L.D. Ion r Physics Letters B 466 (1999) 27–32

32

EUR-2. If the scattering amplitude f g L p(0,2p ) the generalized phase-number uncertainty relations are giÕen by 1 ty1

1 y Ž 2p .

1y t

F Sw N Ž t . ,

t g Ž y`.q ` . .

Ž 36 . In the limit t s 1, we haÕe the usual EUR ln Ž 2p . F Sw q SN .

Ž 37 .

ized entropic formulations of the Uncertainty Principle with the aid of e Tsallis-like entropies Žsee e.g., Ref. w7x. or with more general nonextensive entropies Žsee Refs. w8,9x.. Moreover, due to the general results Ž22. – Ž25. proved here, extensions of the Riesz ŽTheorem 2.8, Ref. w10x, p.102. as well as of the Hausdorff-Young ŽTheorem 2.3, Ref. w10x, p.101. results, towards lower indices p g Žy`, 12 ., are expected to be proved.

The equality sign holds in (36) and (37) for certain, i.e., when pn s dn 0 and P(w ) s 21p . Proof. The proof of the results (36) and (37) can be obtained just as in the aboÕe [u ,L] case and is not giÕen here. A brief summary of the state independent properties of the Tsallis-like entropies SN Ž t ., Sw Ž t ., Sw N Ž t . for the quantum scattering of spinless particles is also presented in Table 1.

4. Experimental tests We proved the state independent entropic uncertainty relations Ž10. for Tsallis-like entropies for all t g Žy`,q `.. For an experimental test of these important w u , L x-entropic uncertainty relations it is interesting to calculate the entropies Ž1. – Ž3. by reconstruction of the pion-nucleus scattering amplitudes using all the available experimental phase-shifts w15–19x for p 0 – 4 He, p 0 – 12 C and p 0 – 16 O, p 0 – 40 Ca scattering. The results obtained in this way on the first experimental test of the entropic uncertainty relations Ž10. are presented in Fig. 1 where the joint entropy Su LŽ q . are represented as functions of the nonextensivity parameter t. From Fig. 1 we see that the state independent w u , L x-entropic uncertainty relations Ž10. are clearly experimentally verified with high accuracy for 0 F t F 2. The methods and results presented in this paper can also be extended to other quantitative general-

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