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Display information for equation id:math.2258.21 on revision:2258
* Page found: Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt (eq math.2258.21)
(force rerendering)Occurrences on the following pages:
Hash: b3eac053b8292be4dc0a6d2c3c9d8434
TeX (original user input):
\begin{align}
& \eta \left( \rho \right)=-k\operatorname{Tr}(\rho \ln \rho )=-k\sum\limits_{m}^{{}}{\left\langle {{r}_{m}} \right|\rho \ln \rho \left| {{r}_{m}} \right\rangle }=-k\sum\limits_{m}^{{}}{{{r}_{m}}\ln {{r}_{m}}} \\
& 1\ge {{r}_{m}}\ge 0 \\
& \Rightarrow \ln {{r}_{m}}\le 0 \\
& \Rightarrow \eta \left( \rho \right)\ge 0 \\
\end{align}
TeX (checked):
{\begin{aligned}&\eta \left(\rho \right)=-k\operatorname {Tr} (\rho \ln \rho )=-k\sum \limits _{m}^{}{\left\langle {{r}_{m}}\right|\rho \ln \rho \left|{{r}_{m}}\right\rangle }=-k\sum \limits _{m}^{}{{{r}_{m}}\ln {{r}_{m}}}\\&1\geq {{r}_{m}}\geq 0\\&\Rightarrow \ln {{r}_{m}}\leq 0\\&\Rightarrow \eta \left(\rho \right)\geq 0\\\end{aligned}}
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<math class="mwe-math-element" xmlns="http://www.w3.org/1998/Math/MathML"><mrow data-mjx-texclass="ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow data-mjx-texclass="ORD"><mtable columnalign="right left right left right left right left right left right left" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true" rowspacing="3pt"><mtr><mtd></mtd><mtd><mi>η</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ρ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>=</mo><mo>−</mo><mi>k</mi><mi data-mjx-texclass="OP" mathvariant="normal">Tr</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>ρ</mi><mi>ln</mi><mo>⁡</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mi>k</mi><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="ORD"><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">⟨</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo data-mjx-texclass="CLOSE">|</mo></mrow><mi>ρ</mi><mi>ln</mi><mo>⁡</mo><mi>ρ</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">|</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo data-mjx-texclass="CLOSE">⟩</mo></mrow></mrow><mo>=</mo><mo>−</mo><mi>k</mi><munderover><mo form="prefix" texclass="OP">∑</mo><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow><mrow data-mjx-texclass="ORD"></mrow></munderover><mrow data-mjx-texclass="ORD"><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mi>ln</mi><mo>⁡</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd></mtd><mtd><mn>1</mn><mo>≥</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>≥</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>ln</mi><mo>⁡</mo><msub><mi>r</mi><mrow data-mjx-texclass="ORD"><mi>m</mi></mrow></msub><mo>≤</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd><mtd><mo>⇒</mo><mi>η</mi><mrow data-mjx-texclass="INNER"><mo data-mjx-texclass="OPEN">(</mo><mi>ρ</mi><mo data-mjx-texclass="CLOSE">)</mo></mrow><mo>≥</mo><mn>0</mn></mtd></mtr><mtr><mtd></mtd></mtr></mtable></mrow></mstyle></mrow></math>
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