Quantum - Holevo's theorem
$\chi (p_i, |v_i \rangle) = H( \sum_i p_i | v_i \rangle \langle v_i|$
Allowed to send
$p_1, p_2, p_3, ..., p_k.$
Capacity is
$\frac{max}{p_i, p_i} \chi = \frac{max}{p_i, p_i} \chi - \sum_i p_i H(p_i)$
block length: n
capacity: C
pick
$2^{n(c - \epsilon)}$
random codewords with letter chosen with probability maximizing
$H(B) - H(B | A)$
.
Steps:
• Alice sends Bob codeword
• Bob gets codeword with noise
• Find the codeword most likely to have been the input words as
$n \rightarrow \infty , \in \rightarrow 0$
.

### How about quantum case?

Upper bound:
• Alice sends
$p_i$
to Bob
Will show that for single state decoding, Shannon information provided by any measurement of Bob's
$\lt$
Holevo's information
$\chi$
.
Alice's record of
$p_i$

## ​

$\def\myHearts#1#2{\color{#1}{\heartsuit}\kern-0pt\color{#2}{\heartsuit}} \myHearts{red} {blue}$
$| 0\rangle$

## Brackets

\left\langle \frac{1}{2} \middle| 1 \right\rangle
$\left\langle \frac{1}{2} \middle| 1 \right\rangle$

## Fractions and brackets

\left( \begin{array}{cc} 2\tau & 7\phi-frac5{12} \\
3\psi & \frac{\pi}8 \end{array} \right)
\left( \begin{array}{c} x \\ y \end{array} \right)
\mbox{~and~} \left[ \begin{array}{cc|r}
3 & 4 & 5 \\ 1 & 3 & 729 \end{array} \right]

## Function definition

f(z) = \left\{ \begin{array}{rcl}
\overline{\overline{z^2}+\cos z} & \mbox{for}
& |z|<3 \\ 0 & \mbox{for} & 3\leq|z|\leq5 \\
\sin\overline{z} & \mbox{for} & |z|>5
\end{array}\right.

## Diagrams

\begin{matrix}
&& \Bbb Q(\sqrt{2},i) & \\
&\huge\diagup & \huge| & \huge\diagdown \\
\Bbb Q(\sqrt{2}) & & \Bbb Q(i\sqrt{2})& & \Bbb Q(i)\\
&\huge\diagdown & \huge| & \huge\diagup \\
&&\Bbb Q
\end{matrix}
$\begin{matrix} && \Bbb Q(\sqrt{2},i) & \\ &\huge\diagup & \huge| & \huge\diagdown \\ \Bbb Q(\sqrt{2}) & & \Bbb Q(i\sqrt{2})& & \Bbb Q(i)\\ &\huge\diagdown & \huge| & \huge\diagup \\ &&\Bbb Q \end{matrix}$

## Equations align

\eqalign{
(a+b)^2 &= (a+b)(a+b) \\
&= a^2 + ab + ba + b^2 \\
&= a^2 + 2ab + b^2
}

## Hat

\hat\imath
\hat\jmath
\hat ab
\hat{ab}
$\hat\imath \\ \hat\jmath \\ \hat ab \\ \hat{ab}$

## Matrices

### Type 1

\begin{matrix}
\hline
x_{11} & x_{12} \\
x_{21} & x_{22} \strut \\
\hdashline
x_{31} & x_{32} \strut
\end{matrix}

### Type 2

A = \pmatrix{
a_{11} & a_{12} & \ldots & a_{1n} \cr
a_{21} & a_{22} & \ldots & a_{2n} \cr
\vdots & \vdots & \ddots & \vdots \cr
a_{m1} & a_{m2} & \ldots & a_{mn} \cr
}

## Limits

\sum_{k=1}^n a_k
$\sum_{k=1}^n a_k$

# Quantum operations

In general, what are the legal transformations
$\rho \rightarrow \varepsilon (\rho)$
?
Def: Operation
$\varepsilon$
is a valid quantum op iff
A1
$Tr(\varepsilon(\rho)) = 1$
A2
$\varepsilon$
is convex and linear.
$\varepsilon (\sum_k P_k \rho_k) = \sum_k P_k \varepsilon (\rho_k)$
A3
$\varepsilon$
is completely positive.
a. if
$\rho \geq 0$
then
$\varepsilon(\rho) \geq 0$
b.
$(I_R \varepsilon_Q) (\rho_{RQ}) \geq 0 \forall \rho_{AB} \geq 0$
Why A3b?
consider