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Friday, 24 January 2014

Some more facts concerning the von Neumann entropy

Let $\reg{X}$, $\reg{Y}$, and $\reg{Z}$ be registers, assume that the classical state set of $\reg{X}$ is $\Sigma$, and let $n = \abs{\Sigma}$.

Theorem:
For every state $\rho\in\Density(\X\otimes\Y\otimes\Z)$ of $(\reg{X},\reg{Y},\reg{Z})$ it holds that
\[
      S(\reg{X},\reg{Y} : \reg{Z})
      \leq S(\reg{Y}:\reg{X},\reg{Z}) + 2\log(n).
\]
     

Proof:
 
    From the result proved in a previous post, it holds that for every choice of registers $\reg{X}$ and $\reg{Z}$, and for any state of $\Density(\X\otimes\Z)$, $S(\reg{Z})\leq S(\reg{X})+S(\reg{X}, \reg{Z})$, or equivalently that
\[
     0\leq S(\reg{X})+S(\reg{X}, \reg{Z})-S(\reg{Z}).
\]
 Also, by sub-additivity $S(\reg{X},\reg{Y})\leq S(\reg{X})+S(\reg{Y})$, or equivalently
\[
 0\leq S(\reg{X})+S(\reg{Y})-S(\reg{X},\reg{Y}).
\]
 Then by adding these two inequalities, it must also hold that
\[
  0\leq S(\reg{X})+S(\reg{X}, \reg{Z})-S(\reg{Z})+S(\reg{X})+S(\reg{Y})-S(\reg{X},\reg{Y}),
\]
and since in general $S(\reg{X})\leq \log(n)$ or $2S(\reg{X})\leq 2\log(n)$,
\[
   0\leq S(\reg{X}, \reg{Z})-S(\reg{Z})+S(\reg{Y})-S(\reg{X},\reg{Y})+2\log(n).
\]
 Therefore,
\[
S(\reg{Z})+S(\reg{X},\reg{Y})\leq S(\reg{X}, \reg{Z})+S(\reg{Y})+2\log(n),
\]
 Adding $-S(\reg{X},\reg{Y}, \reg{Z})$ to both sides of this inequality yields
\[
S(\reg{Z})+S(\reg{X},\reg{Y})-S(\reg{X},\reg{Y}, \reg{Z})\leq S(\reg{X}, \reg{Z})+S(\reg{Y})-S(\reg{X},\reg{Y}, \reg{Z})+2\log(n),
\]
 or equivalently
\[
 S(\reg{X},\reg{Y} : \reg{Z})\leq S(\reg{Y}:\reg{X},\reg{Z}) + 2\log(n).
\]



Here is an example, for $\Sigma = \{0,1\}$, of a state $\rho$ for which this inequality becomes an equality.

Consider the three qubit pure state
\[
\left|\psi\right>_{\reg{X},\reg{Y},\reg{Z}}=\frac{1}{\sqrt{2}}(\left|0\right>_{\reg{X}}\left|0\right>_{\reg{Y}}\left|0\right>_{\reg{Z}}+\left|1\right>_{\reg{X}}\left|0\right>_{\reg{Y}}\left|1\right>_{\reg{Z}}).
\]
Then the states of  the following particular subystems are also pure :
\[\begin{align*}
\left|\psi\right>_{\reg{Y}}&=\left|0\right>_{\reg{Y}}\\
\left|\psi\right>_{\reg{X},\reg{Z}}&=\frac{1}{\sqrt{2}}(\left|0\right>_{\reg{X}}\left|0\right>_{\reg{Z}}+\left|1\right>_{\reg{X}}\left|1\right>_{\reg{Z}}).
\end{align*}\]

However, the following subsystems are in the maximally mixed state:
\[\begin{align*}
\rho_{\reg{X}}=\frac{1}{2}(\left|0\right>\left<0\right|+\left|1\right>\left<1\right|) \\
\rho_{\reg{Z}}=\frac{1}{2}(\left|0\right>\left<0\right|+\left|1\right>\left<1\right|).
\end{align*}\]
Moreover, the state of the subsystem $\reg{X},\reg{Y}$ is in the tensor product state
\[\begin{align*}
\rho_{\reg{X},\reg{Y}}&=\rho_{\reg{X}}\otimes \rho_{\reg{Y}}\\
&=\frac{1}{2}\bigl(\left|0\right>\left<0\right|+\left|1\right>\left<1\right|\bigr)\otimes \left|0\right>\left<0\right|
\end{align*}\]

The entropy of a pure state is zero and the entropy of a maximally entangled state in this case is $\log(n)=\log(2)$. Then the entropies of the states listed above are
\[\begin{align*}
S(\reg{Y})=S(\reg{X},\reg{Z})&=0 \\
S(\reg{X})=S(\reg{Z})&=\log(2) \\
S(\reg{X},\reg{Y})=S(\reg{X})+S(\reg{Y})&=\log(2).
\end{align*} \]

Therefore,
\[\begin{align*}
S(\reg{X},\reg{Y} : \reg{Z}) - S(\reg{Y}:\reg{X},\reg{Z})&=S(\reg{X},\reg{Y})-S(\reg{Y})-S(\reg{X},\reg{Z})+S(\reg{Z})+\left(S(\reg{X},\reg{Y},\reg{Z})-S(\reg{X},\reg{Y},\reg{Z})\right) \\
&=S(\reg{X},\reg{Y})-S(\reg{Y})-S(\reg{X},\reg{Z})+S(\reg{Z}) \\
&=S(\reg{X}+S(\reg{Y})-S(\reg{Y})-S(\reg{X},\reg{Z})+S(\reg{Z}) \\
&=S(\reg{X}-S(\reg{X},\reg{Z})+S(\reg{Z}) \\
&=\log(2)-0+\log(2) \\
&=2\log(2),
\end{align*}\]
which implies that
\[
S(\reg{X},\reg{Y} : \reg{Z}) = S(\reg{Y}:\reg{X},\reg{Z})+2\log(2).
\]

Theorem:

Let $p\in\P(\Sigma)$ be a probability vector, let $\{\sigma_a\,:\,a\in\Sigma\} \subset \Density(\Y\otimes\Z)$ be a collection of density operators, and let
\[
      \rho = \sum_{a\in \Sigma} p(a) E_{a,a}\otimes \sigma_a.
\]
In other words, $\rho$ is a state of $(\reg{X},\reg{Y},\reg{Z})$ in which we view $\reg{X}$ as a classical register. With respect to the state $\rho$, it holds that
\[
      S(\reg{X},\reg{Y} : \reg{Z})
      \leq S(\reg{Y}:\reg{X},\reg{Z}) + \log(n).
\]
   

Proof:

First, observe that
\[\begin{align*}
S(\reg{X}|\reg{Y})-S(\reg{X}|\reg{Z})&=S(\reg{X},\reg{Y})-S(\reg{Y})-S(\reg{X},\reg{Z})+S(\reg{Z})\\
&=S(\reg{X},\reg{Y})-S(\reg{Y})-S(\reg{X},\reg{Z})+S(\reg{Z})+\left(S(\reg{X},\reg{Y},\reg{Z})-S(\reg{X},\reg{Y},\reg{Z})\right) \\
&=S(\reg{X},\reg{Y} : \reg{Z}) - S(\reg{Y}:\reg{X},\reg{Z}).
\end{align*}\]

Now consider the individual bounds on the quantities $S(\reg{X}|\reg{Y})$ and $S(\reg{X}|\reg{Z})$ in order to infer a bound on the difference $S(\reg{X}|\reg{Y})-S(\reg{X}|\reg{Z})$. In this case, since the state of register $\reg{X}$ is classical the conditional entropies  are at most $S(\reg{X}|\reg{Y})\leq \log(n)$ and likewise $S(\reg{X}|\reg{Z})\leq \log(n)$. On the contrary, it could be the case that $S(\reg{X}|\reg{Y})\leq 0$ or $S(\reg{X}|\reg{Z})\leq 0$ in the presence of stronger entanglement correlations in which case $S(\reg{Y})\leq S(\reg{X},\reg{Y})$ or $S(\reg{Z})\leq S(\reg{X},\reg{Z})$. Therefore, the largest the difference of the two could be is when $S(\reg{X}|\reg{Y})=\log(n)$ and $S(\reg{X}|\reg{Z})=0$. Hence, $S(\reg{X}|\reg{Y})-S(\reg{X}|\reg{Z})\leq \log(n)$, or equivalently $S(\reg{X},\reg{Y} : \reg{Z})-S(\reg{Y}:\reg{X},\reg{Z}) \leq \log(n)$ implying that  $S(\reg{X},\reg{Y} : \reg{Z}) \leq S(\reg{Y}:\reg{X},\reg{Z}) + \log(n)$.

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