| Summary: | It is known that Kaniadakis entropy, a generalization of the Shannon–Boltzmann–Gibbs entropic form, is always super-additive for any bipartite statistically independent distributions. In this paper, we show that when imposing a suitable constraint, there exist classes of maximal entropy distributions labeled by a positive real number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℵ</mo><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> that makes Kaniadakis entropy multi-additive, i.e., <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mrow><mi mathvariant="normal">A</mi><mo>∪</mo><mi mathvariant="normal">B</mi></mrow></msup><mo>]</mo></mrow><mo>=</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mo>ℵ</mo><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mfenced separators="" open="(" close=")"><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mo>]</mo></mrow><mo>+</mo><msub><mi>S</mi><mi>κ</mi></msub><mrow><mo>[</mo><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mo>]</mo></mrow></mfenced></mrow></semantics></math></inline-formula>, under the composition of two statistically independent and identically distributed distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mrow><mi mathvariant="normal">A</mi><mo>∪</mo><mi mathvariant="normal">B</mi></mrow></msup><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mspace width="0.166667em"></mspace><mi>y</mi><mo>)</mo></mrow><mo>=</mo><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, with reduced distributions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mi mathvariant="normal">A</mi></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mi>p</mi><mi mathvariant="normal">B</mi></msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> belonging to the same class.
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