Fisher's information measure plays a very important role in diverse areas of
theoretical physics. The associated measures as functionals of quantum
probability distributions defined in, respectively, coordinate and momentum
spaces, are the protagonists of our present considerations. The product of them
has been conjectured to exhibit a non trivial lower bound in [Phys. Rev. A
(2000) 62 012107]. We show here that such is not the case. This is illustrated,
in particular, for pure states that are solutions to the free-particle
Schr\"odinger equation. In fact, we construct a family of counterexamples to
the conjecture, corresponding to time-dependent solutions of the free-particle
Schr\"odinger equation. We also give a new conjecture regarding any
normalizable time-dependent solution of this equation.Comment: 4 pages; revised equations, results unchange

Bellomo, Guido

Plastino, Angel R.

Plastino, Angelo

Advances in Mathematical Physics

English

arXiv

Fisher’s information measure I plays a very important role in diverse areas of theoretical physics. The associated measures Ix and Ip, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The product IxIp has been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimension IxIp≥4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifies IxIp→0 for t→∞

Angelo Plastino

Guido Bellomo

Angel Ricardo Plastino

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On a Conjecture regarding Fisher Information

Fisher’s information measureIplays a very important role in diverse areas of theoretical physics. The associated measuresIxandIp, as functionals of quantum probability distributions defined in, respectively, coordinate and momentum spaces, are the protagonists of our present considerations. The productIxIphas been conjectured to exhibit a nontrivial lower bound in Hall (2000). More explicitly, this conjecture says that for any pure state of a particle in one dimensionIxIp≥4. We show here that such is not the case. This is illustrated, in particular, for pure states that are solutions to the free-particle Schrödinger equation. In fact, we construct a family of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schrödinger equation. We also conjecture that any normalizable time-dependent solution of this equation verifiesIxIp→0fort→∞.</jats:p

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Research ArticleOn a Conjecture regarding Fisher InformationAngelo Plastino,1 Guido Bellomo,1 and Angel Ricardo Plastino21 Instituto de Fı´sica La Plata (IFLP-CONICET) and Departamento de Fı´sica, Facultad de Ciencias Exactas,Universidad Nacional de La Plata, No. 115 and No. 49, C.C. 67, 1900 La Plata, Argentina2CeBio y Secretaria de Investigacion, Universidad Nacional del Noroeste de la Provincia de Buenos Aires-UNNOBA and CONICET,Roque Saenz Pena 456, Junin, ArgentinaCorrespondence should be addressed to Guido Bellomo; gbellomo@fisica.unlp.edu.arReceived 10 December 2014; Revised 23 January 2015; Accepted 23 January 2015Academic Editor: Xiao-Jun YangCopyright © 2015 Angelo Plastino et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Fisher’s informationmeasure 𝐼 plays a very important role in diverse areas of theoretical physics.The associatedmeasures 𝐼𝑥and 𝐼𝑝,as functionals of quantum probability distributions defined in, respectively, coordinate andmomentum spaces, are the protagonistsof our present considerations. The product 𝐼𝑥𝐼𝑝has been conjectured to exhibit a nontrivial lower bound in Hall (2000). Moreexplicitly, this conjecture says that for any pure state of a particle in one dimension 𝐼𝑥𝐼𝑝≥ 4. We show here that such is not the case.This is illustrated, in particular, for pure states that are solutions to the free-particle Schro¨dinger equation. In fact, we construct afamily of counterexamples to the conjecture, corresponding to time-dependent solutions of the free-particle Schro¨dinger equation.We also conjecture that any normalizable time-dependent solution of this equation verifies 𝐼𝑥𝐼𝑝→ 0 for 𝑡 → ∞.1. IntroductionA very important informationmeasure, withmanifold physi-cal applications, was conceived by R. A. Fisher in the 1920s—for detailed discussions see [1–4]. Recent developments showthat Fisher’s information has a fundamental role in quantummechanics [5–19]. In particular, it allows for the formulationof new quantum uncertainty principles [20–24]. It is usuallyabbreviated as 𝐼 and can be thought of as a measure of theexpected error in a measurement [1].A particular instance of great relevance is that of trans-lational families [1]. These are distribution functions whoseform remains invariant under displacements of a shift param-eter 𝜃. Thus, they are shift invariant distributions (in a Machsense, there is no absolute origin for 𝜃). The measure exhibitsGalilean invariance [1]. Given a probability density 𝑓(x, 𝜃),with x ∈ R𝐷 and 𝜃 = (𝜃𝑖)1≤𝑖≤𝑛a family of parameters, theconcomitant Fisher matrix is [25]𝐼𝑗𝑘:= ∫1𝑓 (x, 𝜃)(𝜕𝑓𝜕𝜃𝑗)(𝜕𝑓𝜕𝜃𝑘) dx, (1)where dx = ∏𝐷𝑘=1d𝑥𝑘is the volume element in R𝐷. Inparticular, for 𝜃 ∈ R𝐷, one defines translational fami-lies 𝑓(x − 𝜃), with elements 𝐼𝑗𝑘= ∫(1/𝑓)(𝜕𝑗𝑓)(𝜕𝑘𝑓)dx,where 𝜕𝑖represents the partial derivative with respect tothe coordinate 𝑥𝑖. The trace of this matrix, given by 𝐼 =∫(1/𝑓)[∑𝐷𝑘=1(𝜕𝑘𝑓)2]dx, is a good uncertainty indicator forprobability distributions associatedwith quantumwave func-tions [26]. If𝜓(x) is a normalizedwave function in coordinatespace (𝐷-dimensions) and ?̃?(p) = (2𝜋)−𝐷/2 ∫ 𝑒−ix⋅p𝜓(x)dxis its momentum-counterpart, the corresponding probabilitydensities are, respectively, 𝜌(x) = |𝜓(x)|2 and 𝜌(p) = |?̃?(p)|2,with associated Fisher measures𝐼x = ∫1𝜌[∇x𝜌]2 dx, (2)𝐼p = ∫1𝜌[∇p𝜌]2dp, (3)which allowone to study uncertainty relations via the product𝐼x𝐼p [26].Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015, Article ID 120698, 4 pageshttp://dx.doi.org/10.1155/2015/1206982 Advances in Mathematical PhysicsFor instance, one can demonstrate that if𝜓(x) (or ?̃?(p)) isreal, then 𝐼x𝐼p ≥ 4𝐷2 [5], with equality for coherent states ofthe harmonic oscillator (HO) [26]. For general, mixed statesit is clear that the product 𝐼x𝐼p does not possess a nontriviallower bound (e.g., one can use thermalHO states, representedby Gaussian distributions in both coordinates and momenta,in the high temperature limit). In the case of pure states,though, the existence of such a lower bound for 𝐼x𝐼p was anopen question. Hall conjectured that the relation 𝐼𝑥𝐼𝑝≥ 4might hold in general for pure states in one dimension [26,page 3].Wewill next present a couple of counterexamples thatshow this conjecture to be incorrect. Our examples give riseto a new conjecture: for a bounded wave function one has𝐼x𝐼p → 0 when 𝑡 → ∞.2. CounterexamplesOur first example is taken from the considerations (in a dif-ferent context) of [5]. A free-particle’s one-dimensional wavepacket 𝜓(𝑥, 𝑡) (unit mass) evolves according to Schro¨dinger’sequationi𝜕𝜓𝜕𝑡= −12𝜕2𝜓𝜕𝑥2. (4)Setting the initial conditions𝜓 (𝑥, 0) = 𝐴0exp[− 𝑥22Δ2] ,?̃? (𝑝, 0) = 𝐴0exp[−Δ2𝑝22] ,(5)with 𝐴0= Δ−1/2𝜋−1/4, 𝐴0= Δ1/2𝜋−1/4, and Δ > 0, thatcorrespond to a Gaussian packet, one finds the solution𝜓 (𝑥, 𝑡) = 𝐴 (𝑡) exp[− 𝑥22Δ2 (1 + i𝑡/Δ2)] , (6)where 𝐴(𝑡) = 𝐴0(1 + i𝑡/Δ2)−1/2. The associated probabilitydensities are𝜌 (𝑥, 𝑡) =Δ√𝜋 (Δ4 + 𝑡2)exp[− Δ2𝑥2Δ4 + 𝑡2] ,𝜌 (𝑝, 𝑡) =Δ√𝜋exp [−Δ2𝑝2] .(7)The product 𝐼𝑥𝐼𝑝= 4Δ4(Δ4+𝑡2)−1 obeys the relation 𝐼𝑥𝐼𝑝< 4for 𝑡 > 0. Also, one has 𝐼𝑥𝐼𝑝→ 0 for 𝑡 → ∞.We pass now to another free-particle solution, givenby the first partial derivative of 𝜓(𝑥, 𝑡) with respect to 𝑥:𝜓(1)(𝑥, 𝑡) ∝ 𝜕𝑥𝜓(𝑥, 𝑡) (see (4)), correctly normalized. Itis easy to see that 𝜓(1)(𝑥, 𝑡) is a solution by deriving bothmembers of (4); that is,i 𝜕𝜕𝑡𝜕𝜓𝜕𝑥= −12𝜕2𝜕𝑥2𝜕𝜓𝜕𝑥. (8)The new solution is𝜓(1)(𝑥, 𝑡) = 𝐴(1)(𝑡) exp[− 𝑥22Δ2 (1 + i𝑡/Δ2)] , (9)with𝐴(1)(𝑡) = −21/2𝜋1/4Δ3/2(Δ2+i𝑡)−3/2.The two correspond-ing densities are𝜌(1)(𝑥, 𝑡) =2Δ3√𝜋 (Δ4 + 𝑡2)3𝑥2 exp[− Δ2𝑥2Δ4 + 𝑡2] ,𝜌(1)(𝑝, 𝑡) =2Δ3√𝜋𝑝2 exp [−Δ2𝑝2] .(10)The product is 𝐼(1)𝑥𝐼(1)𝑝= 36Δ4(Δ4+𝑡2)−1, verifying 𝐼(1)𝑥𝐼(1)𝑝< 4for 𝑡 > 2√2Δ2, and 𝐼(1)𝑥𝐼(1)𝑝→ 0 when 𝑡 → ∞.In general, one can show that the whole family of solu-tions of (4) given by successive derivatives of 𝜓(𝑥, 𝑡), that is,the set {𝜓(𝑛)(𝑥, 𝑡) | 𝜓(𝑛)(𝑥, 𝑡) = 𝑁𝑛𝜕𝑛𝑥𝜓(𝑥, 𝑡), 𝑛 = 0, 1, 2, . . .},verifies that 𝐼𝑥𝐼𝑝→ 0 when 𝑡 → ∞, with𝑁𝑛the pertinentnormalization constants. Thus, the family {𝜓(𝑛)(𝑥, 𝑡)}𝑛∈N0yields infinite counterexamples to Hall’s conjecture. To seethis, one needs first to rewrite the Fisher measure in wavefunction’s terms, so that (2) becomes equivalent to𝐼x = 4∫ (∇x󵄨󵄨󵄨󵄨𝜓󵄨󵄨󵄨󵄨)2 dx, (11)or, in one dimension, 𝐼𝑥= ∫(𝜕𝑥|𝜓|)2d𝑥. Further, |𝜓| = 𝜓∗𝜓.Thus, 𝐼𝑥can be expressed in terms of 𝜓 and 𝜓(1). In onedimension one has𝐼𝑥= 4∫ (𝜓(1)∗𝜓 + 𝜓∗𝜓(1))2d𝑥. (12)In general, for 𝜓(𝑘), the Fisher’s measure associated with thedistribution |𝜓(𝑘)|2 becomes𝐼(𝑘)𝑥= 4∫ (𝜓(𝑘+1)∗𝜓(𝑘)+ 𝜓(𝑘)∗𝜓(𝑘+1))2d𝑥. (13)We shownow that the integrand tends to zero for 𝑡 → ∞.Thus, 𝐼(𝑘)𝑥→ 0 in such a limit. Actually, we will show that𝜓(𝑘)(𝑥, 𝑡) → 0 for 𝑡 → ∞. The 𝑘th derivative of 𝜓(𝑥, 𝑡) ≡𝜓(0)(𝑥, 𝑡) is proportional to the 𝑘th derivative of a Gaussiandistribution, given by𝜓(𝑘)(𝑥, 𝑡) = 𝑁𝑘(𝑡)𝜕𝑘𝜕𝑥𝑘𝜓(0)(𝑥, 𝑡)= 𝑁𝑘(𝑡)𝜕𝑘𝜕𝑥𝑘(𝐴 (𝑡) 𝑒−𝑐(𝑡)2𝑥2)= 𝑁𝑘(𝑡) 𝐴 (𝑡) (−1)𝑘𝑐 (𝑡)2𝐻𝑘(𝑐 (𝑡) 𝑥) 𝑒−𝑐(𝑡)2𝑥2= 𝑁𝑘(𝑡) (−1)𝑘𝑐 (𝑡)2𝐻𝑘(𝑐 (𝑡) 𝑥) 𝜓(0)(𝑥, 𝑡) ,(14)where 𝑐(𝑡)2 = (2(Δ2 + i𝑡))−1 and 𝐻𝑘(𝑦) is the Hermitepolynomial of degree 𝑘 in the variable 𝑦.The time-dependentAdvances in Mathematical Physics 3parameters 𝑐(𝑡), 𝜓(0)(𝑥, 𝑡), and𝐴(𝑡) vanish for 𝑡 → ∞. Whatis the behavior of𝑁𝑘? Let us see what happens with ?̃?(𝑘)(𝑝, 𝑡),the 𝑘th solution in momentum space, corresponding to theFourier transform of 𝜓(𝑘)(𝑥, 𝑡). We have?̃?(𝑘)(𝑝, 𝑡) =1√2𝜋∫ 𝑒−i𝑥𝑝𝜓(𝑘)(𝑥, 𝑡) d𝑥= 𝑁𝑘(𝑡) 𝐴 (𝑡) ∫ 𝑒−i𝑥𝑝 𝜕𝑘𝜕𝑥𝑘𝑒−𝑐(𝑡)2𝑥2d𝑥= 𝑁𝑘(𝑡) 𝐴 (𝑡) (i𝑝)𝑘 1√2𝑐 (𝑡)exp[−𝑝24𝑐 (𝑡)2]= 𝑁𝑘(𝑡) √𝑐 (𝑡) (i𝑝)𝑘 ?̃?(0) (𝑝, 𝑡) .(15)Demanding normalization leads to1 = ∫ ?̃?(𝑘)(𝑝, 𝑡) ?̃?(𝑘)∗(𝑝, 𝑡) d𝑝=󵄨󵄨󵄨󵄨𝑁𝑘 (𝑡)󵄨󵄨󵄨󵄨2|𝑐 (𝑡)| ∫ 𝑝2𝑘󵄨󵄨󵄨󵄨󵄨?̃?(0)󵄨󵄨󵄨󵄨󵄨2d𝑝=󵄨󵄨󵄨󵄨𝑁𝑘 (𝑡)󵄨󵄨󵄨󵄨2|𝑐 (𝑡)|Δ√𝜋∫𝑝2𝑘𝑒−Δ2𝑝2d𝑝=󵄨󵄨󵄨󵄨𝑁𝑘 (𝑡)󵄨󵄨󵄨󵄨2|𝑐 (𝑡)|Δ√𝜋Γ(𝑘 +12)Δ−2𝑘−1.(16)Thus,󵄨󵄨󵄨󵄨𝑁𝑘 (𝑡)󵄨󵄨󵄨󵄨2=√𝜋Δ2𝑘Γ (𝑘 + 1/2)2√Δ4 + 𝑡2. (17)Remember that𝜓(0)(𝑥, 𝑡) = 𝐴 (𝑡) 𝑒−𝑐(𝑡)2𝑥2, (18)with 𝐴(𝑡) = 𝜋−1/4√2Δ𝑐(𝑡), and?̃?(0)(𝑝, 𝑡) =√Δ𝜋1/4exp[−𝑝24𝑐 (𝑡)2] . (19)One finds the following limits for the absolute values of thewave functions:󵄨󵄨󵄨󵄨󵄨𝜓(𝑘)(𝑥, 𝑡)󵄨󵄨󵄨󵄨󵄨2=󵄨󵄨󵄨󵄨󵄨𝑁𝑘(𝑡) 𝑐 (𝑡)2𝐻𝑘(𝑐 (𝑡) 𝑥) 𝜓(0)(𝑥, 𝑡)󵄨󵄨󵄨󵄨󵄨2∼ |𝑐 (𝑡)|−1|𝑐 (𝑡)|4|𝑐 (𝑡)|2𝑘|𝑐 (𝑡)|2𝑒−2R(𝑐(𝑡)2)𝑥2∼ |𝑐 (𝑡)|2𝑘+5 exp[− Δ2𝑥2Δ4 + 𝑡2] 󳨀→𝑡→∞0,(20)󵄨󵄨󵄨󵄨󵄨?̃?(𝑘)(𝑝, 𝑡)󵄨󵄨󵄨󵄨󵄨2=󵄨󵄨󵄨󵄨󵄨󵄨𝑁𝑘(𝑡) √𝑐 (𝑡) (i𝑝)𝑘 ?̃?(0) (𝑝, 𝑡)󵄨󵄨󵄨󵄨󵄨󵄨2∼ |𝑐 (𝑡)|−1|𝑐 (𝑡)|󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨exp[−𝑝24𝑐 (𝑡)2]󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨2= 𝑒−Δ2𝑝2.(21)Equation (20) indicates that all functions 𝜓(𝑘)(𝑥, 𝑡) vanish for𝑡 → ∞. Accordingly, from (13) we find 𝐼(𝑘)𝑥→ 0 for 𝑡 → ∞for all 𝑘 = 0, 1, 2, . . .. Further, (21) shows that ?̃?(𝑘)(𝑝, 𝑡) doesnot vanish in this limit. In fact, |?̃?(𝑘)(𝑝, 𝑡)| does not dependon 𝑡.So as to understand what happens with 𝐼(𝑘)𝑝let us see anexpression analogous to (13) in momentum space:𝐼(𝑘)𝑝= 4∫ (?̃?(𝑘+1)∗?̃?(𝑘)+ ?̃?(𝑘)∗?̃?(𝑘+1))2d𝑥. (22)Expanding the integrand using (15) we have?̃?(𝑘+1)∗?̃?(𝑘)+ ?̃?(𝑘)∗?̃?(𝑘+1)= 2I (𝑁𝑘𝑁∗𝑘+1) |𝑐 (𝑡)| 𝑝2𝑘+1󵄨󵄨󵄨󵄨󵄨?̃?(0)(𝑝, 𝑡)󵄨󵄨󵄨󵄨󵄨2.(23)Introducing this into (22), and remembering that both 𝑁𝑘and 𝑐(𝑡) are independent of 𝑝, we find𝐼(𝑘)𝑝= 16I (𝑁𝑘𝑁∗𝑘+1)2|𝑐 (𝑡)|2Δ2𝜋∫𝑝4𝑘+2𝑒−2Δ2𝑝2= 16I (𝑁𝑘𝑁∗𝑘+1)2|𝑐 (𝑡)|2Δ2𝜋Γ (2𝑘 + 3/2)(2Δ2)2𝑘+3/2.(24)Since |𝑁𝑘(𝑡)|2∼ |𝑐(𝑡)|−1 (see (16)), one has|𝑁𝑘(𝑡)𝑁𝑘+1|2|𝑐(𝑡)|2∼ 1 and thus 𝐼(𝑘)𝑝becomes bounded.Thus, there exists 𝐼(𝑘)𝑝,max ∈ R>0 such that 𝐼(𝑘)𝑝≤ 𝐼(𝑘)𝑝,max for all𝑘 = 0, 1, 2, . . .. We conclude that 𝐼(𝑘)𝑥𝐼(𝑘)𝑝→ 0 for 𝑡 → ∞ forthe whole family of solutions {𝜓(𝑘)(𝑥, 𝑡)}𝑘∈N0.3. ConclusionsWe conclude by reiterating that we have found an infinitenumber of counterexamples to the conjecture 𝐼𝑥𝐼𝑝≥ 4, forpure states, put forward in [26]. On the basis of these results,we conjecture that, for any normalizable wave function𝜓(𝑥, 0), the corresponding time-dependent solution 𝜓(𝑥, 𝑡)of the free-particle Schro¨dinger equation satisfies 𝐼𝑥𝐼𝑝→ 0for 𝑡 → ∞.Conflict of InterestsThe authors declare that there is no conflict of interestsregarding the publication of this paper.References[1] B. R. Frieden, Physics from Fisher Information, CambridgeUniversity Press, Cambridge, UK, 1998.[2] B. R. Frieden, Science from Fisher Information, CambridgeUniversity Press, Cambridge, UK, 2004.[3] R. Carroll,On the EmergenceTheme of Physics, World Scientific,Singapore, 2010.[4] F. Pennini and A. Plastino, “Heisenberg-Fisher thermal uncer-tainty measure,” Physical Review E, vol. 69, no. 5, Article ID057101, 2004.4 Advances in Mathematical Physics[5] P. Sa´nchez-Moreno, A. R. Plastino, and J. 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On a conjecture regarding Fisher information

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