On a conjecture regarding Fisher information

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.

A particular instance of great relevance is that of translational families [1]. These are distribution functions whose form remains invariant under displacements of a shift parameter θ. Thus, they are shift invariant distributions (in a Mach sense, there is no absolute origin for θ). The measure exhibits Galilean invariance [1]. Given a probability density f (x, θ), with x ∈ R D and θ = (θ i ) 1≤i≤n a family of parameters, the concomitant Fisher matrix is [26] where dx = D k=1 dx k is the volume element in R D . In particular, for θ ∈ R D , one defines translational fam- where ∂ i represents the partial derivative with respect to the coordinate x i . The trace of this matrix, given by , is a good uncertainty indicator for probability distributions associated to quantum wave functions [25]. If ψ(x) is a normalized wave function in coordinate space (D-dimensions) andψ(p) = (2π) −D/2 e −ix·p ψ(x) dx is its momentumcounterpart, the corresponding probability densities are, respectively, ρ(x) = |ψ(x)| 2 andρ(p) = |ψ(p)| 2 , with associated Fisher measures allow one to study uncertainty relations via the product I x I p [25]. For instance, one can demonstrate that, if ψ(x) (or ψ(p)) is real, then I x I p ≥ 4D 2 [5], with equality for coherent states of the harmonic oscillator (HO) [25]. For general, mixed, states it is clear that the product I x I p does not possess a non trivial lower bound. (For example, one can use thermal HO states, represented by 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 I x I p was an open question. Hall conjectured that the relation I x I p ≥ 4 might hold in general for pure states in one dimension [25, p. 3]. We will next present a couple of counterexamples that show this conjecture to be incorrect. Our examples give rise to a new conjecture: for a bounded wave function one has I x I p → 0 when t → ∞.

II. COUNTEREXAMPLES
Our first example is taken from the considerations (in a different context) of reference [5]. A free-particle's one dimensional wave packet ψ(x, t) (unit mass) evolves according to Schrödinger's equation Setting the initial conditions with A 0 = ∆ −1/2 π −1/4 ,Ã 0 = ∆ 1/2 π −1/4 and ∆ > 0, that correspond to a Gaussian packet, one finds the solution where The product We pass now to another free-particle solution, given by the first partial derivative of ψ(x, t) with respect to x: i ∂ ∂t The new solution is with A (1) (t) = −2 1/2 π 1/4 ∆ 3/2 (∆ 2 + it) −3/2 . The two corresponding densities are The product is I x I (1) x I p → 0 when t → ∞.
In general, one can show that the whole family of solutions of Eq. (4) given by successive derivatives of ψ(x, t), i.e., the set {ψ (n) (x, t)|ψ (n) (x, t) = N n ∂ n x ψ(x, t), n = 0, 1, 2...}, verifies that I x I p → 0 when t → ∞, with N n the pertinent normalization constants. Thus, the family {ψ (n) (x, t)} n∈N0 yields infinite counterexamples to Hall's conjecture. To see this, one needs first to rewrite the Fisher measure in wave function's terms, so that Eq. (2) becomes equivalent to or, in one dimension, I x = (∂ x |ψ|) 2 dx. Further, |ψ| = ψ * ψ. Thus, I x can be expressed in terms of ψ and ψ (1) . In one dimension one has In general, for ψ (k) , the Fisher's measure associated to the distribution |ψ (k) | 2 becomes We show now that the integrand tends to zero for t → ∞. Thus, I (k) x → 0 in such a limit. Actually, we will show that t) is proportional to the k-th derivative of a Gaussian distribution, given by where c(t) 2 = (2(∆ 2 + it)) −1 and H k (y) is the Hermite polynomial of degree k in the variable y. The timedependent quantities c(t), ψ (0) (x, t), and A(t) vanish for t → ∞. What is the behavior of N k ? Let us see what happens withψ (k) (p, t), the k-th solution in momentum space, corresponding to the Fourier transform of ψ (k) (x, t). We havẽ Demanding normalization leads to Thus, and |N k (t)| 2 is time-independent. We can conclude that with A(t) = π −1/4 √ 2∆c(t), and One finds the following limits for the absolute values of the wave functions: Eq. (20) indicates that all functions ψ (k) (x, t) vanish for t → ∞. Accordingly, from Eq. (13) we find I (k) x → 0 for t → ∞ for all k = 0, 1, 2.... Further, Eq. (21) shows that ψ (k) (p, t) does not vanish in this limit. In fact, |ψ (k) (p, t)| does not depend on t.
So as to understand what happens with I (k) p let us see an expression analogous to Eq. (13) in momentum space: Expanding the integrand using Eq. (15) we havẽ Introducing this into Eq. (22), and remembering that both N k and c(t) are independent of p, we find Since |N k (t)| 2 ∼ 1 (see Eq. (17)), one has |N k (t)N k+1 | 2 ∼ 1 and thus I

III. CONCLUSIONS
We conclude by reiterating that we have found an infinite number of counterexamples to the conjecture I x I p ≥ 4, for pure states, put forward in [25]. On the basis of these results, we conjecture that, for any normalizable wave function ψ(x, 0), the corresponding time-dependent solution ψ(x, t) of the free-particle Schrödinger equation, satisfies I x I p → 0 for t → ∞.