90 lines
3.4 KiB
Markdown
90 lines
3.4 KiB
Markdown
# Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration
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## Typicality
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> The idea of typicality in high-dimensions is very important topic in understanding this paper and taking it to the next level of detail under language of mathematics. I'm trying to comprehend these material and write down my understanding in this note.
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Let $X$ be the alphabet of our source of information.
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Let $x^n=x_1,x_2,\cdots,x_n$ be a sequence with $x_i\in X$.
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We say that $x^n$ is $\epsilon$-typical with respect to $p(x)$ if
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- For all $a\in X$ with $p(a)>0$, we have
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$$
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\|\frac{1}{n}N(a|x^n)-p(a)|\leq \frac{\epsilon}{\|X\|}
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$$
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- For all $a\in X$ with $p(a)=0$, we have
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$$
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N(a|x^n)=0
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$$
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Here $N(a|x^n)$ is the number of times $a$ appears in $x^n$. That's basically saying that:
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1. The difference between **the probability of $a$ appearing in $x^n$** and the **probability of $a$ appearing in the source of information $p(a)$** should be within $\epsilon$ divided by the size of the alphabet $X$ of the source of information.
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2. The probability of $a$ not appearing in $x^n$ should be 0.
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Here are two easy propositions that can be proved:
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For $\epsilon>0$, the probability of a sequence being $\epsilon$-typical goes to 1 as $n$ goes to infinity.
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If $x^n$ is $\epsilon$-typical, then the probability of $x^n$ is produced is $2^{-n[H(X)+\epsilon]}\leq p(x^n)\leq 2^{-n[H(X)-\epsilon]}$.
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The number of $\epsilon$-typical sequences is at least $2^{n[H(X)+\epsilon]}$.
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Recall that $H(X)=-\sum_{a\in X}p(a)\log_2 p(a)$ is the entropy of the source of information.
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## Shannon theory in Quantum information theory
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Shannon theory provides a way to quantify the amount of information in a message.
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Practically speaking:
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- A holy grail for error-correcting codes
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- Conceptually speaking:
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- An operationally-motivated way of thinking about correlations
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- What’s missing (for a quantum mechanic)?
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- Features from linear structure:
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- Entanglement and non-orthogonality
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## Partial trace and purification
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### Partial trace
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Recall that the bipartite state of a quantum system is a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
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#### Definition of partial trace
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Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
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An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as (by the definition of [tensor product of linear operators](https://notenextra.trance-0.com/Math401/Math401_T2#tensor-products-of-linear-operators))
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$$
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T=\sum_{i=1}^n a_i A_i\otimes B_i
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$$
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where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$.
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The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by
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$$
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\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
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$$
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### Purification
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Let $\rho$ be any [state](https://notenextra.trance-0.com/Math401/Math401_T6#pure-states) (may not be pure) on the finite dimensional Hilbert space $\mathscr{H}$. then there exists a unit vector $w\in \mathscr{H}\otimes \mathscr{H}$ such that $\rho=\operatorname{Tr}+2(|w\rangle\langle w|)$ is a pure state.
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<details>
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<summary>Proof</summary>
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</details>
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## MM space
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