The accessible information for a mixed state with 4 dimension

[guo-jyun]

If I have four states in

\ket{S_1}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{+1})_{AB}
\ket{S_2}=\frac{1}{\sqrt{2}}(\ket{-1}-\ket{10})_{AB}
\ket{S_3}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{+1})_{AB}
\ket{S_4}=\frac{1}{\sqrt{2}}(\ket{00}+\ket{-1})_{AB},

and its density matrix is

\rho=frac{1}{2}(\ket{S_1}\bra{S_1}+\ket{S_2}\bra{S_2}+\ket{S_3}\bra{S_3}+\ket{S_4}\bra{S_4}).

Using the Holevo's theorem the bound of mutual information can be calculated as

I(X;Y)\leqslant S(\rho)-\frac{1}{4}(S(\ket{S_1}\bra{S_1})+S(\ket{S_2}\bra{S_2})+S(\ket{S_3}\bra{S_3})+S(\ket{S_4}\bra{S_4}))=1.60087603669285.

How to calculate the best measurement probability p from the accessible information?

For example to simple explain my question, a density matrix in mixed state

\rho=\frac{1}{2}(\ket{0}\bra{0}+\ket{+}\ket{+}),

where 

\ket{0}=\begin{pmatrix}1 \\ 0\end{pmatrix} and \ket{+}=\frac{1}{\sqrt{2}}\begin{pmatrix}1 \\ 1\end{pmatrix}.

We can then calculate the mutual information and its accessible information.

After that, we can estimate the best p from Shannon entropy as

-p\log_2(p)-(1-p)\log_2(1-p)=S(\rho)-\frac{1}{2}(S(\ket{0}\bra{0})+S(\ket{+}\bra{+}))=0.600876036692856,

and thus 

p=0.85355.

I am a beginner. If the concept is wrong, please correct me.

[studiot]

Is this homework?

[Sensei]

Your LaTex codes don't work on this forum..

 

Edited August 25, 2018 by Sensei

[guo-jyun]

2 hours ago, studiot said:

Is this homework?

No, that's what's bothering me right now.

I can't use my existing literature knowledge to calculate him.

If you know the answer, please tell me.

2 hours ago, Sensei said:

Your LaTex codes don't work on this forum..

I know, but I choose the latex type....

Please try this pdf https://www.papeeria.com/d/file/15def8a2-7af6-4670-a776-7c9455d90206/15def8a2-7af6-4670-a776-7c9455d90206.pdf/Demo - main.pdf

Edited August 25, 2018 by guo-jyun

[guo-jyun]

The pdf link is disabled.

Therefore, I upload the pdf file.

question.pdf