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Display information for equation id:math.2272.21 on revision:2272

* Page found: Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt (eq math.2272.21)

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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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MathML (25.567 KB / 2.909 KB) :

η ⁢ ( ρ ) = - k ⁢ Tr ⁡ ( ρ ⁢ ln ⁡ ρ ) = - k ⁢ ∑ m ⟨ r m | ρ ⁢ ln ⁡ ρ | r m ⟩ = - k ⁢ ∑ m r m ⁢ ln ⁡ r m 1 ≥ r m ≥ 0 ⇒ ln ⁡ r m ≤ 0 ⇒ η ⁢ ( ρ ) ≥ 0 missing-subexpression 𝜂 𝜌 𝑘 Tr 𝜌 𝜌 𝑘 subscript 𝑚 quantum-operator-product subscript 𝑟 𝑚 𝜌 𝜌 subscript 𝑟 𝑚 𝑘 subscript 𝑚 subscript 𝑟 𝑚 subscript 𝑟 𝑚 missing-subexpression 1 subscript 𝑟 𝑚 0 missing-subexpression ⇒ absent subscript 𝑟 𝑚 0 missing-subexpression ⇒ absent 𝜂 𝜌 0 {\displaystyle{\displaystyle\begin{aligned} &\displaystyle\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}}}\\ &\displaystyle 1\geq{{r}_{m}}\geq 0\\ &\displaystyle\Rightarrow\ln{{r}_{m}}\leq 0\\ &\displaystyle\Rightarrow\eta\left(\rho\right)\geq 0\\ \end{aligned}}}

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MathML (2.382 KB / 498 B) :

η(ρ)=−kTr⁡(ρln⁡ρ)=−k∑m⟨rm|ρln⁡ρ|rm⟩=−k∑mrmln⁡rm1≥rm≥0⇒ln⁡rm≤0⇒η(ρ)≥0

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Calculated based on the variables occurring on the entire Vorurteilsfreie Schätzung des statistischen Operators zu einem festen Zeitpunkt page

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  • η
  • ρ
  • k
  • ρ
  • ρ
  • k
  • m
  • rm
  • ρ
  • ρ
  • rm
  • k
  • m
  • rm
  • rm
  • rm
  • rm
  • η
  • ρ

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