On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures
<p>(<b>a</b>) the domain <math display="inline"> <semantics> <msub> <mi mathvariant="script">D</mi> <mi>p</mi> </msub> </semantics> </math> for a given <span class="html-italic">p</span> is represented by the gray area (here <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>></mo> <mn>2</mn> </mrow> </semantics> </math>). The thick line belongs to <math display="inline"> <semantics> <msub> <mi mathvariant="script">D</mi> <mi>p</mi> </msub> </semantics> </math>. The dashed line represents <math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mi>p</mi> </msub> </semantics> </math>, corresponding to the Lutwak situation of <a href="#sec2dot3dot1-entropy-19-00493" class="html-sec">Section 2.3.1</a>, where the relation holds and the minimizers are explicitly known (stretched deformed Gaussian distributions), whereas <math display="inline"> <semantics> <msub> <mover> <mi mathvariant="script">L</mi> <mo stretchy="false">¯</mo> </mover> <mi>p</mi> </msub> </semantics> </math> corresponds to <a href="#sec2dot3dot2-entropy-19-00493" class="html-sec">Section 2.3.2</a> (<math display="inline"> <semantics> <msub> <mi mathvariant="script">B</mi> <mi>p</mi> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mover> <mi mathvariant="script">B</mi> <mo stretchy="false">¯</mo> </mover> <mi>p</mi> </msub> </semantics> </math> obtained by the Gagliardo–Nirenberg inequality are their restrictions to <math display="inline"> <semantics> <msub> <mi mathvariant="script">D</mi> <mi>p</mi> </msub> </semantics> </math>); (<b>b</b>) same situation for <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics> </math>, with the domains <math display="inline"> <semantics> <msub> <mi mathvariant="script">A</mi> <mn>2</mn> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mover> <mi mathvariant="script">A</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> </semantics> </math> (dashed lines) that correspond to the situations of <a href="#sec2dot3dot3-entropy-19-00493" class="html-sec">Section 2.3.3</a> and <a href="#sec2dot3dot4-entropy-19-00493" class="html-sec">Section 2.3.4</a>, respectively, (<math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mn>2</mn> </msub> </semantics> </math> and <math display="inline"> <semantics> <msub> <mover> <mi mathvariant="script">L</mi> <mo stretchy="false">¯</mo> </mover> <mn>2</mn> </msub> </semantics> </math> are not represented for the clarity of the figure).</p> "> Figure 2
<p>Given a <span class="html-italic">p</span>, the domain in gray represents <math display="inline"> <semantics> <msub> <mover accent="true"> <mi mathvariant="script">D</mi> <mo stretchy="false">˜</mo> </mover> <mi>p</mi> </msub> </semantics> </math>, where we know that the <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics> </math>-Fisher–Rényi complexity is optimally lower bounded and where the minimizers can be deduced from proposition 2. (<b>a</b>) the domain in dark gray represents <math display="inline"> <semantics> <msub> <mi mathvariant="script">D</mi> <mi>p</mi> </msub> </semantics> </math>, which is obviously included in <math display="inline"> <semantics> <msub> <mover accent="true"> <mi mathvariant="script">D</mi> <mo stretchy="false">˜</mo> </mover> <mi>p</mi> </msub> </semantics> </math>; the dot is a particular point <math display="inline"> <semantics> <mrow> <mrow> <mo>(</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> <mo>∈</mo> <msub> <mi mathvariant="script">D</mi> <mi>p</mi> </msub> </mrow> </semantics> </math> and the dotted line represents its transform by <math display="inline"> <semantics> <mi mathvariant="fraktur">A</mi> </semantics> </math>; (<b>b</b>) the domain in dark gray represents <math display="inline"> <semantics> <mrow> <mi mathvariant="fraktur">A</mi> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">L</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> <mo>⊂</mo> <msub> <mover accent="true"> <mi mathvariant="script">D</mi> <mo stretchy="false">˜</mo> </mover> <mi>p</mi> </msub> </mrow> </semantics> </math>, which obviously contains <math display="inline"> <semantics> <msub> <mi mathvariant="script">L</mi> <mi>p</mi> </msub> </semantics> </math> represented by the dashed line; (<b>c</b>) same as (<b>b</b>) with <math display="inline"> <semantics> <msub> <mover> <mi mathvariant="script">L</mi> <mo stretchy="false">¯</mo> </mover> <mi>p</mi> </msub> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mi mathvariant="fraktur">A</mi> <mrow> <mo>(</mo> <msub> <mover> <mi mathvariant="script">L</mi> <mo stretchy="false">¯</mo> </mover> <mi>p</mi> </msub> <mo>)</mo> </mrow> <mo>⊂</mo> <msub> <mover accent="true"> <mi mathvariant="script">D</mi> <mo stretchy="false">˜</mo> </mover> <mi>p</mi> </msub> </mrow> </semantics> </math>. This illustrates that <math display="inline"> <semantics> <mrow> <msub> <mover accent="true"> <mi mathvariant="script">D</mi> <mo stretchy="false">˜</mo> </mover> <mi>p</mi> </msub> <mo>=</mo> <mi mathvariant="fraktur">A</mi> <mrow> <mo>(</mo> <msub> <mi mathvariant="script">L</mi> <mi>p</mi> </msub> <mo>)</mo> </mrow> <mo>∪</mo> <mi mathvariant="fraktur">A</mi> <mrow> <mo>(</mo> <msub> <mover> <mi mathvariant="script">L</mi> <mo stretchy="false">¯</mo> </mover> <mi>p</mi> </msub> <mo>)</mo> </mrow> </mrow> </semantics> </math>.</p> "> Figure 3
<p>Fisher information <math display="inline"> <semantics> <msub> <mi>F</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </semantics> </math> (left graph), Rényi entropy power <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>λ</mi> </msub> </semantics> </math> (center graph), and <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics> </math>-Fisher–Rényi complexity <math display="inline"> <semantics> <msub> <mi>C</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> </mrow> </msub> </semantics> </math> (right graph) of the radial hydrogenic distribution in position space with dimensions <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>(</mo> <mo>∘</mo> <mo>)</mo> <mo>,</mo> <mspace width="0.222222em"/> <mn>12</mn> <mo>(</mo> <mo>*</mo> <mo>)</mo> </mrow> </semantics> </math> versus the quantum numbers <span class="html-italic">n</span> and <span class="html-italic">l</span>. The complexity parameters are <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>λ</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>.</p> "> Figure 4
<p><math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics> </math>-Fisher–Rényi complexity (normalized to its lower bound), <math display="inline"> <semantics> <msub> <mi>C</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> </mrow> </msub> </semantics> </math>, with <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>β</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0.8</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>,</mo> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> </mrow> </semantics> </math> for the radial hydrogenic distribution in the position space with dimensions <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>(</mo> <mo>∘</mo> <mo>)</mo> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mn>12</mn> <mo>(</mo> <mo>*</mo> <mo>)</mo> </mrow> </semantics> </math>.</p> "> Figure 5
<p>Fisher information <math display="inline"> <semantics> <msub> <mi>F</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </semantics> </math> (left graph), Rényi entropy power <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>λ</mi> </msub> </semantics> </math> (center graph), and <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics> </math>-Fisher–Rényi complexity <math display="inline"> <semantics> <msub> <mi>C</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> </mrow> </msub> </semantics> </math> (right graph) of the radial hydrogenic distribution in momentum space with dimensions <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>(</mo> <mo>∘</mo> <mo>)</mo> <mo>,</mo> <mspace width="0.222222em"/> <mn>12</mn> <mo>(</mo> <mo>*</mo> <mo>)</mo> </mrow> </semantics> </math> versus the quantum numbers <span class="html-italic">n</span> and <span class="html-italic">l</span>. The complexity parameters are <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>λ</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>.</p> "> Figure 6
<p><math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics> </math>-Fisher–Rényi complexity (normalized to its lower bound), <math display="inline"> <semantics> <msub> <mi>C</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> </mrow> </msub> </semantics> </math>, with <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>β</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0.8</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.222222em"/> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.222222em"/> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> </mrow> </semantics> </math> for the radial hydrogenic distribution in the momentum space with dimensions <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>(</mo> <mo>∘</mo> <mo>)</mo> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <mn>12</mn> <mo>(</mo> <mo>*</mo> <mo>)</mo> </mrow> </semantics> </math>.</p> "> Figure 7
<p>Fisher information <math display="inline"> <semantics> <msub> <mi>F</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> </mrow> </msub> </semantics> </math> (left graph), Rényi entropy power <math display="inline"> <semantics> <msub> <mi>N</mi> <mi>λ</mi> </msub> </semantics> </math> (center graph), and <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics> </math>-Fisher–Rényi complexity <math display="inline"> <semantics> <msub> <mi>C</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> </mrow> </msub> </semantics> </math> (right graph) versus <span class="html-italic">n</span> and <span class="html-italic">l</span> for the radial harmonic system in position space with dimensions <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>(</mo> <mo>∘</mo> <mo>)</mo> <mo>,</mo> <mspace width="4pt"/> <mn>12</mn> <mo>(</mo> <mo>*</mo> <mo>)</mo> </mrow> </semantics> </math>. The informational parameters are <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>β</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mi>λ</mi> <mo>=</mo> <mn>7</mn> </mrow> </semantics> </math>.</p> "> Figure 8
<p><math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> <mo>)</mo> </mrow> </semantics> </math>-Fisher–Rényi complexity (normalized to its lower bound) <math display="inline"> <semantics> <msub> <mi>C</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>λ</mi> </mrow> </msub> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>λ</mi> <mo>,</mo> <mi>β</mi> <mo>)</mo> <mo>=</mo> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0.8</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.166667em"/> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> <mo>,</mo> <mspace width="0.166667em"/> <mo>(</mo> <mn>5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>7</mn> <mo>)</mo> </mrow> </semantics> </math> for the oscillator system in the position space with dimensions <math display="inline"> <semantics> <mrow> <mi>d</mi> <mo>=</mo> <mn>3</mn> <mo>(</mo> <mo>∘</mo> <mo>)</mo> <mo>,</mo> <mn>12</mn> <mo>(</mo> <mo>*</mo> <mo>)</mo> </mrow> </semantics> </math>.</p> ">
Abstract
:1. Introduction
2. -Fisher–Rényi Complexity and the Extended Stam Inequality
2.1. Rényi Entropy, Extended Fisher Information and Rényi–Fisher Complexity
2.2. Shift and Scale Invariance, Bounding from Below and Minimizing Distributions
2.3. Some Explicitly Known Minimizing Distributions
2.3.1. The Case
2.3.2. Stretched Deformed Gaussian: The Symmetric Case
2.3.3. Dealing with the Usual Fisher Information
2.3.4. The Symmetrical of the Usual Fisher Information
3. Extended Optimal Stam Inequality: A Step Further
3.1. Differential-Escort Distribution: A Brief Overview
3.2. Enlarging the Validity Domain of the Extended Stam Inequality
- Consider a point and find an index such that , which is a point of the intersection between and the line joining and .
- Apply Proposition 2 for the point , leading to the minimizing distribution and its corresponding bound.
- Then, remarking that , the minimizer of the extended complexity writes and the corresponding bound can be computed from this minimizer or noting that .
4. Applications to Quantum Physics
4.1. Brief Review on the Quantum Systems with Radial Potential
4.2. -Fisher–Rényi Complexity and the Hydrogenic System
4.3. -Fisher–Rényi Complexity and the Harmonic System
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A. Proof of Proposition 2
Appendix A.1. The Case λ ≠ 1
Appendix A.1.1. The Sub-Case λ < 1
Appendix A.1.2. The Sub-Case λ > 1
Appendix A.2. The Case λ = 1
Appendix B. Proof of Proposition 3
Appendix C. Proof of Proposition 5
Appendix C.1. The (p,β,λ)-Fisher–Rényi Complexity is Lowerbounded over
Appendix C.2. Explicit Expression for the Minimizers.
Appendix C.2.1. The Case 1 − p*β < λ < 1
Appendix C.2.2. The Case λ > 1
Appendix C.2.3. The Case λ = 1
Appendix C.3. Symmetry through the Involution .
Appendix C.4. Explicit Expression of the Lower Bound.
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Zozor, S.; Puertas-Centeno, D.; Dehesa, J.S. On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures. Entropy 2017, 19, 493. https://doi.org/10.3390/e19090493
Zozor S, Puertas-Centeno D, Dehesa JS. On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures. Entropy. 2017; 19(9):493. https://doi.org/10.3390/e19090493
Chicago/Turabian StyleZozor, Steeve, David Puertas-Centeno, and Jesús S. Dehesa. 2017. "On Generalized Stam Inequalities and Fisher–Rényi Complexity Measures" Entropy 19, no. 9: 493. https://doi.org/10.3390/e19090493