Jump to content

The accessible information for a mixed state with 4 dimension


guo-jyun

Recommended Posts

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.

Link to comment
Share on other sites

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 by guo-jyun
Link to comment
Share on other sites

  • 3 weeks later...

Create an account or sign in to comment

You need to be a member in order to leave a comment

Create an account

Sign up for a new account in our community. It's easy!

Register a new account

Sign in

Already have an account? Sign in here.

Sign In Now
×
×
  • Create New...

Important Information

We have placed cookies on your device to help make this website better. You can adjust your cookie settings, otherwise we'll assume you're okay to continue.