# 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 $$p\_i$$ $$|i \rangle \longrightarrow p\_i \buildrel Bob \over \longrightarrow | b\_j\rangle$$ ### $$\def\myHearts#1#2{\color{#1}{\heartsuit}\kern-0pt\color{#2}{\heartsuit}} \myHearts{red} {blue}$$ $$| 0\rangle$$ ### 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] ``` $$\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]$$ ### Function definition ``` f(z) = \left\{ \begin{array}{rcl} \overline{\overline{z^2}+\cos z} & \mbox{for} & |z|<3 \\ 0 & \mbox{for} & 3\leq|z|\leq5 \\ \sin\overline{z} & \mbox{for} & |z|>5 \end{array}\right. ``` $$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}$$ ### Equations align ``` \eqalign{ (a+b)^2 &= (a+b)(a+b) \\ &= a^2 + ab + ba + b^2 \\ &= a^2 + 2ab + b^2 } ``` $$\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}$$ ### Matrices #### Type 1 ``` \begin{matrix} \hline x_{11} & x_{12} \\ x_{21} & x_{22} \strut \\ \hdashline x_{31} & x_{32} \strut \end{matrix} ``` $$\begin{matrix} \hline x\_{11} & x\_{12} \ x\_{21} & x\_{22} \strut \ \hdashline x\_{31} & x\_{32} \strut \end{matrix}$$ #### Type 2 ``` A = \pmatrix{ a_{11} & a_{12} & \ldots & a_{1n} \cr a_{21} & a_{22} & \ldots & a_{2n} \cr \vdots & \vdots & \ddots & \vdots \cr a_{m1} & a_{m2} & \ldots & a_{mn} \cr } ``` $$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$$ ### References A complete list is available at ## Quantum operations In general, what are the legal transformations $$\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$$ Why A3b? consider $$\varepsilon: \matrix{ a & b \cr c & d } \longrightarrow \pmatrix{ a & c \cr b & d }$$ $$\sum\_{jk} C\_{jk} | j k | \buildrel \varepsilon \over \rightarrow \sum\_{jk} C\_{jk} | k j |$$ --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://i.janardhanpulivarthi.com/tools/quantum-holevos-theorem.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.