Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration

Typicality

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.

Let X be the alphabet of our source of information.

Let x^n=x_1,x_2,\cdots,x_n be a sequence with x_i\in X.

We say that x^n is $\epsilon$-typical with respect to p(x) if

For all a\in X with p(a)>0, we have


\|\frac{1}{n}N(a|x^n)-p(a)|\leq \frac{\epsilon}{\|X\|}

For all a\in X with p(a)=0, we have


N(a|x^n)=0

Here N(a|x^n) is the number of times a appears in x^n. That's basically saying that:

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.
The probability of a not appearing in x^n should be 0.

Here are two easy propositions that can be proved:

For \epsilon>0, the probability of a sequence being $\epsilon$-typical goes to 1 as n goes to infinity.

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]}.

The number of $\epsilon$-typical sequences is at least 2^{n[H(X)+\epsilon]}.

Recall that H(X)=-\sum_{a\in X}p(a)\log_2 p(a) is the entropy of the source of information.

Shannon theory in Quantum information theory

Shannon theory provides a way to quantify the amount of information in a message.

Practically speaking:

A holy grail for error-correcting codes
Conceptually speaking:
An operationally-motivated way of thinking about correlations
What’s missing (for a quantum mechanic)?
- Features from linear structure:
  - Entanglement and non-orthogonality

Partial trace and purification

Partial trace

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.

Definition of partial trace

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.

An operator T on \mathscr{H}=\mathscr{A}\otimes \mathscr{B} can be written as (by the definition of tensor product of linear operators)


T=\sum_{i=1}^n a_i A_i\otimes B_i

where A_i is a linear operator on \mathscr{A} and B_i is a linear operator on \mathscr{B}.

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


\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i

Purification

Let \rho be any state (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.

Proof

3.4 KiB Raw Blame History Unescape Escape