Update Math401_P1.md

content/Math401/Math401_P1.md

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 Encoding channel and decoding channel
 
+That is basically two maps that encode and decode the qbits. You can think of them as a quantum channel.
+
 #### Quantum capacity for a quantum channel
 
+The quantum capacity of a quantum channel is governed by the HSW noisy coding theorem, which is the counterpart for the Shannon's noisy coding theorem in quantum information settings.
+
 #### Lloyd-Shor-Devetak theorem
 
+Note, the model of the noisy channel in quantum settings is a map $\eta$: that maps a state $\rho$ to another state $\eta(\rho)$. This should be a CPTP map.
+
+Let $A'\cong A$ and $|\psi\rangle\in A'\otimes A$.
+
+Then $Q(\mathcal{N})\geq H(B)_\sigma-H(A'B)_\sigma$.
+
+where $\sigma=(I_{A'}\otimes \mathcal{N})\circ|\psi\rangle\langle\psi|$.
+
+(above is the official statement in the paper of Patrick Hayden)
+
+That should means that in the limit of many uses, the optimal rate at which A can reliably sent qbits to $B$ ($1/n\log d$) through $\eta$ is given by the regularization of the formula
+
+$$
+Q(\eta)=\max_{\phi_{AB}}[-H(B|A)_\sigma]
+$$
+
+where $H(B|A)_\sigma$ is the conditional entropy of $B$ given $A$ under the state $\sigma$.
+
+$\phi_{AB}=(I_{A'}\otimes \eta)\circ\omega_{AB}$
+
+(above formula is from the presentation of Patrick Hayden.)
+
+For now we ignore this part if we don't consider the application of the following sections. The detailed explanation will be added later.
+
 ### Surprise in high-dimensional quantum systems
 
 #### Levy's lemma
 
+Given an $\eta$-Lipschitz function $f:S^n\to \mathbb{R}$ with median $M$, the probability that a random $x\in_R S^n$ is further than $\epsilon$ from $M$ is bounded above by $\exp(-\frac{C(n-1)\epsilon^2}{\eta^2})$, for some constant $C>0$.
+
+$$
+\operatorname{Pr}[|f(x)-M|>\epsilon]\leq \exp(-\frac{C(n-1)\epsilon^2}{\eta^2})
+$$
+
+Decomposing the statement in detail,
+
+#### $\eta$-Lipschitz function
+
+Let $(X,\operatorname{dist}_X)$ and $(Y,\operatorname{dist}_Y)$ be two metric spaces. A function $f:X\to Y$ is said to be $\eta$-Lipschitz if there exists a constant $L\in \mathbb{R}$ such that
+
+$$
+\operatorname{dist}_Y(f(x),f(y))\leq L\operatorname{dist}_X(x,y)
+$$
+
+for all $x,y\in X$. And $\eta=\|f\|_{\operatorname{Lip}}=\inf_{L\in \mathbb{R}}L$.
+
+That basically means that the function $f$ should not change the distance between any two pairs of points in $X$ by more than a factor of $L$.
+
+> This theorem is exactly the 5.1.4 on the _High-dimensional probability_ by Roman Vershynin.
+
+#### Isoperimetric inequality on $\mathbb{R}^n$
+
+Among all subsets $A\subset \mathbb{R}^n$ with a given volume, the Euclidean ball has the minimal area.
+
+That is, for any $\epsilon>0$, Euclidean balls minimize the volume of the $\epsilon$-neighborhood of $A$.
+
+Where the volume of the $\epsilon$-neighborhood of $A$ is defined as
+
+$$
+A_\epsilon(A)\coloneqq \{x\in \mathbb{R}^n: \exists y\in A, \|x-y\|_2\leq \epsilon\}=A+\epsilon B_2^n
+$$
+
+Here the $\|\cdot\|_2$ is the Euclidean norm. (The theorem holds for both geodesic metric on sphere and Euclidean metric on $\mathbb{R}^n$.)
+
+#### Isoperimetric inequality on the sphere
+
+Let $\sigma_n(A)$ denotes the normalized area of $A$ on $n$ dimensional sphere $S^n$. That is $\sigma_n(A)\coloneqq\frac{\operatorname{Area}(A)}{\operatorname{Area}(S^n)}$.
+
+Let $\epsilon>0$. Then for any subset $A\subset S^n$, given the area $\sigma_n(A)$, the spherical caps minimize the volume of the $\epsilon$-neighborhood of $A$.
+
+The above two inequalities is not proved in the Book _High-dimensional probability_.
+
+To continue prove the theorem, we use sub-Gaussian concentration of sphere $\sqrt{n}S^n$. 
+
+This will leads to some constant $C>0$ such that
+
+
+
+> The Levy's lemma can also be found in _Metric Structures for Riemannian and Non-Riemannian Spaces_ by M. Gromov. $3\frac{1}{2}.19$ The Levy concentration theory.
+
+#### Theorem $3\frac{1}{2}.19$ Levy concentration theorem:
+
+An arbitrary 1-Lipschitz function $f:S^n\to \mathbb{R}$ concentrates near a single value $a_0\in \mathbb{R}$ as strongly as the distance function does.
+
+That is
+
+$$
+\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
+$$
+
+where 
+
+$$
+\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
+$$
+
+Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
+
 ### Random states and random subspaces
 
+Choose a random pure state $\sigma=|\psi\rangle\langle\psi|$ from $A'\otimes A$.
+
+The expected value of the entropy of entanglement is kown and satisfies a concentration inequality.
+
+$$
+\mathbb{E}[H(\psi_A)] \leq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
+$$
+
+From the Levy's lemma, we have
+
+If we define $\beta=\frac{d_A}{\log_2(d_B)}$, then we have
+
+$$
+\operatorname{Pr}[H(\psi_A) \geq \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{(d_Ad_B-1)C\alpha^2}{(\log_2(d_A))^2}\right)
+$$
+where $C$ is a small constatnt and $d_B\geq d_A\geq 3$.
+
+
 #### ebits and qbits
 
 ### Superdense coding of quantum states