201 lines
7.0 KiB
Markdown
201 lines
7.0 KiB
Markdown
# Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)
|
|
|
|
[PDF](https://www.ams.org/books/psapm/068/2762144)
|
|
|
|
## Quantum codes
|
|
|
|
### Preliminaries
|
|
|
|
#### Daniel Gottesman's mathematics of quantum error correction
|
|
|
|
##### Quantum channels
|
|
|
|
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 *(Chapter 3 of _High-dimensional probability_ by Roman Vershynin)* of sphere $\sqrt{n}S^n$.
|
|
|
|
This will leads to some constant $C>0$ such that the following lemma holds:
|
|
|
|
#### The "Blow-up" lemma
|
|
|
|
Let $A$ be a subset of sphere $\sqrt{n}S^n$, and $\sigma$ denotes the normalized area of $A$. Then if $\sigma\geq \frac{1}{2}$, then for every $t\geq 0$,
|
|
|
|
$$
|
|
\sigma(A_t)\geq 1-2\exp(-ct^2)
|
|
$$
|
|
|
|
where $A_t=\{x\in S^n: \operatorname{dist}(x,A)\leq t\}$ and $c$ is some positive constant.
|
|
|
|
#### Proof of the Levy's concentration theorem
|
|
|
|
Proof:
|
|
|
|
Without loss of generality, we can assume that $\eta=1$. Let $M$ denotes the median of $f(X)$.
|
|
|
|
So $\operatorname{Pr}[|f(X)\leq M|]\geq \frac{1}{2}$, and $\operatorname{Pr}[|f(X)\geq M|]\geq \frac{1}{2}$.
|
|
|
|
Consider the sub-level set $A\coloneqq \{x\in \sqrt{n}S^n: |f(x)|\leq M\}$.
|
|
|
|
Since $\operatorname{Pr}[X\in A]\geq \frac{1}{2}$, by the blow-up lemma, we have
|
|
|
|
$$
|
|
\operatorname{Pr}[X\in A_t]\geq 1-2\exp(-ct^2)
|
|
$$
|
|
|
|
And since
|
|
|
|
$$
|
|
\operatorname{Pr}[X\in A_t]\leq \operatorname{Pr}[f(X)\leq M+t]
|
|
$$
|
|
|
|
Combining the above two inequalities, we have
|
|
|
|
$$
|
|
\operatorname{Pr}[f(X)\leq M+t]\geq 1-2\exp(-ct^2)
|
|
$$
|
|
|
|
> 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 known and satisfies a concentration inequality.
|
|
|
|
$$
|
|
\mathbb{E}[H(\psi_A)] \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
|
|
$$
|
|
|
|
From the Levy's lemma, we have
|
|
|
|
If we define $\beta=\frac{d_A}{\log_2(d_B)}$, then we have
|
|
|
|
$$
|
|
\operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{(d_Ad_B-1)C\alpha^2}{(\log_2(d_A))^2}\right)
|
|
$$
|
|
where $C$ is a small constant and $d_B\geq d_A\geq 3$.
|
|
|
|
#### ebits and qbits
|
|
|
|
### Superdense coding of quantum states
|
|
|
|
It is a procedure defined as follows:
|
|
|
|
Suppose $A$ and $B$ share a Bell state $|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, where $A$ holds the first part and $B$ holds the second part.
|
|
|
|
$A$ wish to send 2 classical bits to $B$.
|
|
|
|
$A$ performs one of four Pauli unitaries on the combined state of entangled qubits $\otimes$ one qubit. Then $A$ sends the resulting one qubit to $B$.
|
|
|
|
This operation extends the initial one entangled qubit to a system of one of four orthogonal Bell states.
|
|
|
|
$B$ performs a measurement on the combined state of the one qubit and the entangled qubits he holds.
|
|
|
|
$B$ decodes the result and obtains the 2 classical bits sent by $A$.
|
|
|
|
|
|
### Consequences for mixed state entanglement measures
|
|
|
|
#### Quantum mutual information
|
|
|
|
### Multipartite entanglement
|
|
|