DESIGUALDADES DE HEISENBERG GENERALES

Mariela Portesi (IFLP & DF, La Plata)In collaboration with: Steeve Zozor (Grenoble) Christophe Vignat (Paris) Jesús S. Dehesa (Granada) Pablo Sánchez-Moreno (Granada)

VI WORKSHOP MECÁNICA ESTADÍSTICA Y TEORÍA DE LA INFORMACIÓN

Hotel Iruña, Mar del Plata 26 - 28 Mayo 2010

MotivationsMeasure of uncertainty / information :

measures of the dispersion or randomness of a statemeasure of the “complexity" of a signal, of a time series, of atomic organization...in (tele)communications: measure of the information that can be transmitted through a channelin time-frequency representations: measure of uncertainty of a signal in the “dual" time and frequency spaces; search for the time-frequency “atoms" that minimize the “joint“ uncertainty

Quantum Mechanics’ Uncertainty Principle: The precision with which incompatible physical observables can be prepared, is limited by an upper bound. There exists an irreducible lower bound for the uncertainty in the result of a simultaneous mesaurement of non-commuting observables.

( , ; ) ( , )U A B A Bψ ≥B

U: VARIANCES / HIGHER ORDER MOMENTSHeisenberg inequality 1927 Schrödinger – Robertson generalized relation 1929Sánchez-Moreno, González-Férez & S.Dehesa 2006Zozor, MP, Sánchez-Moreno & S.Dehesa 2010

U: ENTROPIES (Shannon / Rényi / Tsallis)Deutsch 1983Maassen & Uffink 1988; Uffink thesis 1990 Bialynicki-Birula 1975; 2007MP & Plastino 1995Vignat & Zozor 2007A. Luis 2007Zozor, MP & Vignat 2008

Outline of the talk

Variance formulation of Uncertainty Principle

Entropic formulation of Uncertainty Principle:

• Shannon entropy

• Havrda&Charvat-Daroczy-Tsallis entropy

• Rényi entropy; Rényi entropy power

( Both discrete and continuous cases )

Beyond the variance formulation: higher order moments

Final comments

Variance formulation of UPUse product of variances as measures of dispersion

Variance of observable C when the system is in state ψ:

gives a measure of uncertainty in the value of C; measures spread or localization for simple probability distributions (one hump)

In random vector point of view, C corresponds to a random variable X distributed according to |ψ(x)|2. Or in time-frequency domain, signal x(t) can be normalized so that |x(t)|2 sums to 1

2( ) ( )C C CψΔ = ⟨ − ⟨ ⟩ ⟩ | |C Cψ ψ⟨ ⟩ = ⟨ ⟩

Variance formulation of UPGENERALIZED HEISENBERG(-ROBERTSON) INEQUALITY:

has sense provided both variances existdoes not provide a universal lower bound for uncertainty (unless [A,B] is a c-number: Heisenberg case)involves only second moments of A and B

--------------------ALTERNATIVE FORMULATIONS: entropies, Fisher info OTHER ALTERNATIVES: higher order moments

Heisenberg, Z.Phys. (1927); Robertson, Phys.Rev. 34 (1929)

1 [ , ]2

A B A BΔ Δ ≥ ⟨ ⟩

Entropic formulation of UP: ShannonUse sum of Shannon entropies as measure of uncertainty

Shannon entropy for prob.distribution obtained by projection of ψ on discrete n-dimensional eigenbasis of C, gives missing or lack of information

SHANNON ENTROPIC UNCERTAINTY INEQUALITY:

provides informative universal lower bound, independent of A and Beigenvalues; more stringent than Heisenberg relation.Search for infimum (or bounds) for

Deutsch, Phys.Rev.Lett. 50 (1983)

, ,1

( ; ) lnn

C i C ii

S C p pψ=

= −∑ 2, |C i ip c ψ= ⟨ ⟩

( ; ) ( ; ) ( , )S A S B A Bψ ψ+ ≥B

( ) ( ) ( )1 , U A B S A S B= +

Bounds for Shannon entropy UPLet c be the maximum overlap between A and B eigenstates.

Then

Particularly, for complementary observables (corresponding to ψand its discrete Fourier transform), one has c=1/√n .Then

Other alternative measures for missing information: extensions of Shannon entropy

Kraus, Phys.Rev.D (1987); Maassen & Uffink, Phys.Rev.Lett. 60 (1988)

11 2( , ) 2 ln 2 ln

1U A B

c c≥ ≥

+

1( , ) lnB nU A ≥

Entropic formulation of UP: HCDTUse q-generalized sum of q-entropies as measure of uncertainty

Havrda&Charvat-Daroczy-Tsallis q-entropy (q>0) for probability distribution pA,i ; same for B with identical value of index q

HCDT q-ENTROPIC UNCERTAINTY INEQUALITY:

provides informative universal lower bound, indep.of A,B eigenvalues

Search for infimum (or bounds) for

MP, Thesis UNLP (1995); MP & Plastino, Physica A 225 (1996)

,1

1( ; ) 11

nT qq A i

iS A p

qψ

=

⎛ ⎞⎜ ⎟⎝

= −− ⎠

∑

( ) ( ) (1 ) ( ) ( ) ( , )T T T T Tq q q q qS A S B q S A S B A B+ + − ≥B

( ) ( ) ( ), T T Tq q q qU A B S A S B= ⊕

Bounds for HCDT entropy UPLet c be the maximum overlap between A and B eigenstates.

Then

where

Particularly, for complementary observables: c=1/√n . Then

Other alternative measures for (missing) information

MP, Thesis UNLP (1995); MP & Plastino, Physica A 225 (1996)

2

2

1 2 1( , ) ln ln , 11 2

Tq q qU A B q

c c⎛ ⎞≥ ≥ ≤ ≤⎜ ⎟+⎝ ⎠

11ln ( )1

q

qxx

q

−−≡

−

lnTq qU n≥

UP in the continuous caseUse (generalized) sum of (q-)entropies, for the probability densities

associated to observable A and its conjugate Ã, e.g.:X and P in d dimensions, or angular position and ang. momentum

Heisenberg product saturates for CS (d=1 Gaussian wf of width δ):

Shannon entropic ineq. is stronger than Heisenberg’s; saturates for CS or HO wf:

HCDT q-entropic uncertainty for Coherent States is δ-independent:

Bialynicki-Birula et al., Comm.Math.Phys. 44 (1975), Phys.Lett.A 108 (1985), Phys.Rev.A 74 (2006)

( )2 2( ) | ( ) | and ( ) | ( ) | ( )x x p p= = =ρ ψ γ ψ ψ ψF

( ) ( )2 2

X G P Gδ δδ

δΔ Δ =

1( , ; ) (1 ln )U X P dψ π≥ +

1( , ; ) (1 ln )q qU X P Gqδ π= +

Shannon entropic UP implies Heisenberg inequality

is equivalent to

for the Shannon entropy power

Maximizer of N(ρ) under variance constraint ( <r2> fixed ) is a Gaussian, and the entropy power is <r2> / d . Same for N(γ).

Then

from which

( ) ( ) (1 ln )S S d+ ≥ +ρ γ π ( ) ( ) 1/ 4N N ≥ρ γ

1 2( ) exp ( )2

N Se d

⎛ ⎞= ⎜ ⎟⎝ ⎠

ρ ρπ

2 21 ( ) ( ) ( ) ( )4 G G

r pN N N N

d d= ρ γ ≤ ρ γ =

22 2 2 ( ) ( )

4dr p d N N≥ ρ γ ≥

Entropic formulation of UP: RényiUse sum of Rényi entropies (or product of Rényi entropy powers) as measure of uncertainty

Rényi α-entropy (α>0, approaches S when α 1) forprobab.distribution pA,i ; same for B with arbitrary β

Rényi α-entropy power :(N α =M α +1 : M&U certainty; N2: Larsen’s purity)

RÉNYI ENTROPIC UNCERTAINTY INEQUALITY:

RÉNYI ENTROPY POWER (EP)UNCERTAINTY INEQUALITY:

Zozor & Vignat, Physica A 375 (2007); A.Luis, Phys.Rev.A 75 (2007); B-B, Found.Prob.& Phys.889 (2007)

, 21

1 2( ; ) ln ln1 1

nR

A ii

S A p αα α

αψ ψα α=

⎛ ⎞= =⎜ ⎟− −⎝ ⎠

∑1( ) exp ( )RN A S Adα α

⎛ ⎞= ⎜ ⎟⎝ ⎠

,( ; ) ( ; ) ( , )R R RS A S B A Bα β α βψ ψ+ ≥B

,( ; ) ( ; ) ( , )N A N B A Bα β α βψ ψ ≥C

Rényi EP uncertainty prod. for A,ÃAssume A and B=Ã are complementary (or conjugate) observables

(e.g., X and P), with associated probability densities and

, i.e., the corresponding wavefunctions are linked by a

Fourier transform .

We consider continuous and discrete state space.

Search for non-trivial infimum (or lower bounds), if there exists, for the Renyi entropy power uncertainty product

2| ( ) |pψ

2| ( ) |xψ

( ), ( , ) ( ) ??U A Ã N A N Ãα β α β= ≥ …

1.Particular situation: conjugated indices

Property:

Universal lower bounds (i.e. independent of the state of the system)In C-C case saturates iff ψ=Gaussian; in D-D case saturates iff ψ=Kronecker indicator or

constantOnly possibility with same uncertainty quantifier : α=β=1 (Shannon)

Bialynicki-Birula, Phys.Rev.A 74 (2006); Zozor & Vignat, Physica A 375 (2007)

1 1 1, 12 2p q p

p qα β≡ ≡ + = ≥and with

1 1( , ) : 1,2 2

α βα β+ =For there exists a Rényi EP UP of the form

/2 /21 1

( 2) ( 2)

( ) ( ) 2

2

p q

p q

nN A N A

p q

π

π − −

⎧⎪⎪≥ ⎨⎪⎪⎩

discrete-discrete case

discrete- continuous case

continuous- cont. case

C-C case: hints for the proof

p-norm : Fourier transform :

Babenco – Beckner inequality :

then

finally use p= 2α and q=2β , and take log to obtain entropy power

1/

( )d

pp d

px d x

⎛ ⎞ψ = ψ⎜ ⎟

⎝ ⎠∫

( )

( )( )

/21/

1/

/ 2 1 1sup , 1/ 2d

p

dpq

qL p

pp qqψ∈

⎡ ⎤ψ π= + =⎢ ⎥

ψ π⎢ ⎥⎣ ⎦

( )( )

( )/21

2tix p d

dp x e d x−ψ = ψπ ∫

( ),

d

p qq pCψ ≤ ψ

2.General situation: arbitrary indices

Gives flexibility to choose information measures in position and in momentum space, as αand β are unrelated. Also same quantifiers can be taken, other than Shannon

Define

and

On hyperbola C (in D), 2α and 2β are conjugated. Shannon case (α ,β) = (1,1) is on C.

1 1 12 2α β

+ ≠

1/ 2 2 22 1αα α α αα

= ≥−

with (then and are conjugated)

0

20

2

1( , ) : and 2

1( , ) : ,2

\10 ;2

[ )

α β α β α

α β α β α

+

⎧ ⎧ ⎫= > >⎨ ⎬⎪ ⎩ ⎭⎪⎪ ⎧ ⎫= ≥ =⎪ ⎨ ⎬⎨ ⎩ ⎭⎪ =⎪⎪

=⎪⎩

D

C

D D

S

arbitrary indices (cont.)Discrete-discrete case:

Property D-D:

Saturates for Kronecker or constant wf

Discrete-continuous case:

Property D-C:

Sharp lower bound, saturates for Kronecker indicator. Not solved yet in D0.

Continuous-continuous case:

E.g. α = β = 2: N2(A) N2(Ã) is not saturated forGaussians !!Indeed, A. Luis found d=1 exponential states with U2,2(E) < 2π = U2,2(G)

M & U, Phys.Rev.Lett. 60 (1988); A.Luis, Phys.Rev.A 75 (2007); Zozor, MP & Vignat, Phys.A 387(2008)

2

2//2

/ 2

For any ( , ) , there exists a Rényi EP UP of the form

2( ) ( )1

dd

d

nN A N Anα β

α β +∈

⎛ ⎞≥ ⎜ ⎟+⎝ ⎠

For any ( , ) , there exists a Rényi EP UP of the form

( ( ) 2)N A N Aα β

α β

π

∈

≥

D

Property C-C1:

This relation is not sharp and/or not saturated for Gaussians .

Property C-C2:

Example: the 2-parameters Student-t wf gives arbitrarily small Rényi EP uncertaintyproduct for given values of those parameters.

The case (α, β) = (2,2) corresponds to a point in D0, what explains Luis results !!

Zozor, MP & Vignat, Physica A 387 (2008)

arbitrary indices (cont.)

0For any ( , ) , no Rényi EP UP exists since

( ) ( ) can be arbitrarily small (and is positive).N A N Aα β

α β ∈D

1 12( 1) 2( 1)

For any ( , ) , there exists a Rényi EP UP of the form( ) in \ with

( ) ( ) ( ) in \ with 2 in

where ( ) withx x

BN A N A B

B x x x

α β

α βα α ββ β απ

π − −

∈

≥⎧ ⎫⎪ ⎪≥ ≥⎨ ⎬⎪ ⎪⎩ ⎭

=

DDD

SS

S

/ (2 1) x x x= −

Higher order moments as UPUse moments higher than variance as general measure of dispersion

Then Heisenberg-like relations:

for any positive a, b. Hint: , ...

2/( )

aaa X rΔ =ψ

2/( )

bbbP pΔ =ψ

2/ 2/...??

a ba br p ≥

/( ) ( , )

d aar N g aα≥ ρ × α

The case a=b=2 improved for V(r)Heisenberg inequality in d dimensions:

improved for d-dim central potentials: ( l = hyperangular q.number)

Reduces to Heisenberg for s states, but improves for l>0 states (growing as l2)

Saturates for nodeless isotropic HO wavefunctions (g.s.):

For then( nr = No.nodes )

For Coulomb potential V(r) = -1/r : ( L=l+(d-3)/2 , η=n+(d-3)/2 )Uncertainty product is larger than bound.

22 2

4dr p ≥

22 2

2dr p l⎛ ⎞≥ +⎜ ⎟

⎝ ⎠

2 2( ) / 2V r r= ω2

2 2

2rdr p n l⎛ ⎞= + +⎜ ⎟

⎝ ⎠

( )2 2 21 5 3 ( 1) 12

r p L L= η − + +

Sánchez-Moreno et al., NJP 8 (2006)

Final commentsWe consider the mathematical formulation of the Uncertainty Principle for two conjugate

observables A and à (in state space and Fourier transformed space, with discrete orcontinuous spectra), in terms of products of Rényi entropy powers:

Previous known results refer mostly to inequalities for pairsof conjugated indices: (α,β) located on curve CWe extend use of products of entropy powers to general situation of arbitrary, positive non-conjugated indices:

Continuos-Continuous case :we show that entropic uncertainty relations exist for (α,β) in D, with different bounds. Conversely, in D0

the positive product Nα(A) Nβ(Ã) can be arbitrarily small (e.g. (2,2) ).

NEXT STEPS: deeper study of absence of restriction in D0, search for the best bounds for entropy-products in D, search for states that minimize entropy-product in D.Analisis of Heisenberg-like relations of higher order.

1 1 12 2α β

+ ≠

,( ) ( ) ( , )N A N A A Aα β α β≥C 1( ) exp ( )RN A S Adα α

⎛ ⎞= ⎜ ⎟⎝ ⎠

GRACIAS !!

Steeve Zozor, Mariela Portesi, Christophe VignatSOME ENTROPIC EXTENSIONS OF THE UNCERTAINTY PRINCIPLEPhysica A 387 (18-19), 4800-4808 (2008)arXiv:0709.3011 [math-PR]

Steeve Zozor, Mariela Portesi, Pablo Sánchez-Moreno, Jesús S. DehesaON MOMENTS-BASED HEISENBERG INEQUALITIES(2010)