The renyi entropy and the uncertainty relations in quantum mechanics
- 1. The R´nyi entropy and e the uncertainty relations in quantum mechanics Iwo Bialynicki-Birula Center for Theoretical Physics Warsaw, Poland www.cft.edu.pl/˜ birula November 2008
- 2. Heisenberg uncertainty relation The standard formulation of the uncertainty relations for the measurements of the position and momentum has the following form h ¯ ∆x∆p ≥ 2 ∆x and ∆p are the mean standard deviations ∆a = (a − a )2 2 2 (∆x) = dxρ(x)(x − x )2 (∆p) = dp˜(p)(p − p )2 ρ This form of the uncertainty is simple to prove The proof requires only the Schwartz inequality and is usually included in the first course on quantum mechanics
- 3. Critique of the standard uncertainty relations The mean standard deviation ∆a (dispersion) is used a lot in the statistical analysis of experiments that measure a It is a reasonable measure of the spread when (as in measurements) the distribution in question is of a simple ”hump” type For example, it is a very good characteristic for a Gaussian distribution since it determines the half-width of this distribution However, when the distribution of the values has more than one hump the standard deviation loses some of its usefulness especially in connection with the notion of uncertainty Examples of two states of a particle in one dimension First state describes a particle with a uniformly distributed probability in a box of total length L Second state describes a particle localized with equal probabilities in two smaller boxes each of length L/4
- 4. The probability distributions corresponding to these two states F=flat and C=clustered, are graphically represented as follows: I II III IV State F II IV I III State C 1 inside the box, State F ρ= L 0 outside the box, 2 State C ρ= L inside the box, 0 outside the box. In which case, F or C, is the uncertainty in the position greater?
- 5. According to our intuition, the uncertainty is greater in the case F In the case C we know more about the position; we know that the particle is not in the regions II and III. However, when we calculate the standard deviation ∆x we obtain the opposite result: Case F L ∆xF = √ 12 Case C 7 L ∆xC = √ 4 12
- 6. The second, somewhat more dramatic example of the situation where the standard deviation does not give a sensible measure of uncertainty is provided by the following distribution of probability. State D L(1-1/N) ————— Distance NL —————– L/N The probability density is constant in two regions I and II separated by a large distance N L (N is a large number) The region I has the size L(1 − 1/N ) and the distant region II has the size L/N
- 7. 1 L in region I, State D ρ= 1 in region II, L 0 elsewhere. For large N the standard deviation ∆x is approximately equal to: Case D √ L ∆xD ∼ 1 + 12N √ 12 This expression tends to infinity with N in spite of the fact that the probability of finding the particle in the region I tends to 1 This example shows clearly what is wrong with the standard deviation It gets very high contributions from distant regions because these contribution enter with a large weight: namely, the distance from the mean value.
- 8. Information entropy as a measure of uncertainty The best measure of information was introduced in 1948 by Shannon in his paper “The mathematical theory of communication” Shannon entropy closely resembles the physical entropy H=− pi ln pi i where pi is the probability of the occurrence of the i-th event or the a priori probability of the i-th message In information theory one uses the logarithms to the base of 2 H is then measured in bits but a change of the base results only in a change of the scale The information entropy may serve as a precise measure of uncertainty It has even been described already by Shannon as a measure of “information, choice or uncertainty”
- 9. H may indeed serve at the same time as a measure of information and as a measure of uncertainty, because the two notions are directly related The information gained in the measurement removes the uncertainty about its outcome The information uncertainty is in many ways a much better measure of uncertainty than the standard deviation The Shannon entropy has a simple intuitive interpretation The average number of questions needed to discover ”the truth” (that is to remove uncertainty) which is hidden in one of the boxes with the probability distribution pi is bounded from below by H
- 11. To apply the Shannon measure of uncertainty to quantum mechanics we must discretize the distributions of the position and momentum
- 12. Histograms The probability density to find a quantum particle at x is given by the modulus squared of the wave function ρ(x) = |ψ(x)|2 The probability density to find the momentum p is given by ˜ the modulus squared of the Fourier transform ρ(p) = |ψ(p)|2 ˜ In order to have a discrete set of probabilities we introduce the finite resolutions (bins) for the measurements of x and p To distinguish them from the statistical dispersions ∆x and ∆p they will be denoted by δx and δp The probabilities ql and k to find the values of x and p in the l-th and the k-th bin are xl +δx pk +δp ql = dx ρ(x) pk = dp ρ(p) ˜ xl pk The values of ql and pk represent a histogram of measurements
- 13. Following D. Deutsch (1983) and H. M. Partovi (1983) we can formulate the uncertainty relations between the positions and momenta that restrict the available information about x and p measured by the Shannon entropies H (x) = − ql ln ql and H (p) = − pk ln pk The best known (but not sharp) bound expressing the uncertainty relations (I. Bialynicki-Birula 1984) is (x) (p) δxδp H +H ≥ 1 − ln(2) − ln h RHS 3 2 1 ∆x∆p 0.5 1. 1.5 2. h
- 14. The R´nyi entropy e Is there an uncertainty relation expressed in term of the R´nyi entropy? e R´nyi entropy is a one parameter extension of the Shannon entropy e 1 Hα = ln( pα ) k 1−α The limit α → 1 produces the Shannon entropy ( pk = 1) d ln( pα ) = k pk ln pk dα R´nyi entropy has many applications in information theory (coding and e cryptography), physics (particle production, diffusion), signal analysis, nonlinear dynamics, fractals, etc. (150 000 hits in Google)
- 15. R´nyi entropy shares many properties with Shannon entropy e • Hα is nonnegative • Hα vanishes when the uncertainty is the smallest (only one pk = 0) • Hα takes on the maximal value ln n when all n probabilities are equal • Hα is extensive — satisfies the following condition If pij is a product of two probability distributions pij = pi pj then 1 1 α α Hα = ln( pα ) = ij ln( pi p j) = Hα + Hα 1−α ij 1−α i j R´nyi entropies for the position and momentum are easily defined using e the same probabilities pk and ql as for the Shannon entropy (x) α (p) Hα = ln ql /(1 − α) Hα = ln pα /(1 − α) k (x) (p) What are the limitations on the values of Hα and Hα ?
- 16. Brief mathematical interlude The (q, q )-norm of T is the smallest k(q, q ) such that Tψ q ≤ k(q, q ) ψ q 1/q + 1/q = 1 q≥q 1 q ψ q = dx|ψ(x)|q K. I. Babenko (1961) and W. Beckner (1975) found k(q, q ) for the Fourier transformation ˜ Choosing the physical normalization of ψ we obtain ˜ 1 ψ(p) = √ dx e−ipx/¯ ψ(x) h 2π¯ h 1 1 − 2q ˜ 1 1 q−2 q 2q q ψ q ≤¯ h ψ q (I) 2π 2π This inequality is saturated for all Gaussian wave functions
- 17. It is convenient to change q → 2α and q → 2β 1 1 α β + =2 = α > 1 =⇒ β < 1 α β α−1 1−β and rewrite the norms in the inequality (I) as follows 1 1 2α 2β ˜ ψ 2α = dp|˜(p)|α ρ ψ 2β = dx|ρ(x)|β ˜ ρ(x) = |ψ(x)|2 and ρ(p) = |ψ(p)|2 ˜ are quantum mechanical probability densities After squaring both sides of the inequality (I) we obtain (for α > β) 1 1 1 α −1 α 2α β 2β ρ ˜ α ≤ n(α, β) ρ β n(α, β) = ¯ h (II) π π
- 18. To make use of this mathematical inequality in the proof of the physical uncertainty relations we have to introduce into this analysis the probabilities pk and ql This is done by splitting the integrals into bins xk +δp dp(˜(p))α = ρ dp (˜(p))α ρ xk and using the famous property of convex functions: The value of the function at the average point cannot exceed the average value of the function In the present case I need a somewhat more general version of this statement in the form of an integral Jensen inequality Φ dx h(x)w(x) ≤ dx Φ (h(x)) w(x) D D which is valid for a convex function Φ, an arbitrary function h(x) and a nonnegative weight function w(x) normalized to 1 ( D dx w(x) = 1)
- 19. The power xα is a convex function for α > 1 hence xk +δp α xk +δp pk α xk dp ρ(p) ˜ xk dp (˜(p))α ρ = ≤ δp δp δp α δp1−α pα ≤ k dp (˜(p))α = ( ρ α ) ρ ˜ (A) For concave functions we have: The average value of the function cannot exceed the value of the function at the average point β β ( ρ β) = dx (ρ(x))β ≤ δx1−β ql (B) Raising (A) to the power 1/α and (B) to the power of 1/β and we the use of the basic inequality (II) we obtain 1 1 1−α α 1−β β β δp α pα k ≤ n(α, β)δx β ql 1−α 1 1−β 1 β ln(δp) + ln pα k ≤ ln(δx) + ln n(α, β) + ln ql α α β β
- 20. Final form of the uncertainty relation From the last inequality with the use of α/(α − 1) = β/(1 − β) after some work we obtain finally (x) (p) 1 ln α ln β δxδp Hβ + Hα ≥ − + − ln(2) − ln 2 1−α 1−β h In the limit when α → 1 also β → 1 With the help of the l’Hˆpital rule one obtains o ln α lim = −1 α→1 1 − α The inequality for R´nyi entropies reduces to the uncertainty relation e for the Shannon entropies proved by me twenty years ago (x) (p) δxδp H +H ≥ 1 − ln(2) − ln h
- 21. Uncertainty relations for N -level systems For quantum systems described by vectors in the N -dimensional Hilbert space the analog of the uncertainty relation for the R´nyi entropies is e N N 1 1 ln ρα ˜k + ln ρβ l ≥ ln N, 1−α 1−β k=1 l=1 where ρk = |˜k |2 , ρl = |al |2 and the amplitudes ak and al are ˜ a ˜ connected by the discrete Fourier transformation N 1 ak = √ ˜ exp(2πik l/N ) al . N l=1 This uncertainty relation is saturated for the states that are either localized in “position space” exactly one of the amplitudes al is different from zero or are localized in “momentum space” exactly one of the amplitudes ak is different from zero ˜ The bound does not depend on α and β
- 22. Uncertainty relations for mixed states The uncertainty relations for the R´nyi entropies also e hold for all mixed states This result is not obvious because the R´nyi entropy is e not a convex function of the probability distributions for all values of α The terms on the left hand side of the uncertainty relation may decrease as a result of mixing In order to extend our results to mixed states one has to go back to the original inequalities that were used in the proof and show that they still hold
- 23. Uncertainty relations for continuous distributions There also exist purely mathematical versions of the uncertainty relations that do not involve the experimental resolutions δx and δp of the measuring devices ∞ ∞ 1 1 ln dp (˜(p))α + ρ ln dx (ρ(x))β 1−α −∞ 1−β −∞ 1 α 1 β ≥− ln − ln 2(1 − α) π 2(1 − β) π On the left hand side of this inequality we have, what might be called, the continuous or integral versions of the R´nyi entropies e In order to derive this purely mathematical inequality that includes no reference to the finite resolution of the physical measurements I have dropped ¯ in the definition of the Fourier transform h This inequality has been independently proven by Zozor and Vignat Analogous relations for the continuous Tsallis entropies for x and p were obtained by Rajagopal
- 24. Uncertainty relations for continuous Shannon entropies Assuming that the distributions ρ(x) and ρ(p) are normalized ˜ in the limit, when α → 1, β → 1 from the formula for the entropic uncertainty relation in terms of the R´nyi entropies e we obtain the classic uncertainty relation for the continuous Shannon entropies ∞ ∞ − dp ρ(p) ln ρ(p) − ˜ ˜ dx ρ(x) ln ρ(x) ≥ ln(eπ) −∞ −∞ This inequality had been conjectured by Hirschman (1957) and later proved by Bialynicki-Birula and Mycielski (1975) and by Beckner (1975)
- 25. References 1. I. Bialynicki-Birula, “Renyi entropy and the uncertainty rela- tions”, Foundations of Probability and Physics, Ed. G. Adenier, C. A. Fuchs, and A. Yu. Khrennikov, American Institute of Physics, Melville, 2007. 2. I. Bialynicki-Birula, “Formulation of the uncertainty relations in terms of the Renyi entropies” Phys. Rev. A 74, 052101 (2006). Both references can be found on my home page: www.cft.edu.pl/˜ birula
- 26. Summary I have shown that the R´nyi entropies for canonically conjugate e physical variables obey the inequalities that have the interpretation of the uncertainty relations In order to use the R´nyi entropies in their standard form e (i.e. using a discrete set of probabilities), I had to introduce the finite resolutions δx and δp of the measuring devices The resulting new uncertainty relations for the position and momentum acquire an additional meaning not found in their standard version The more accurately we wish to measure x and p the stronger the limitations imposed by the uncertainty relation 1 ln α ln β δxδp The lower limit − + − ln increases 2 1−α 1−β π¯ h