110 lines
4.0 KiB
Markdown
110 lines
4.0 KiB
Markdown
# Math 401 Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)
|
|
|
|
[Concentration of measure effects in quantum information](https://www.ams.org/books/psapm/068/2762144)
|
|
|
|
A more comprehensive version of this paper is in [Aspect of generic entanglement](https://arxiv.org/pdf/quant-ph/0407049).
|
|
|
|
## 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 (hopefully very soon).
|
|
|
|
---
|
|
|
|
### 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 as side note 3](Math401_P1_3.md)
|
|
|
|
### 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.
|
|
|
|
$$
|
|
\mathbb{E}[H(\psi_A)] \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
|
|
$$
|
|
|
|
[Decomposing the statement in detail as side note 2](Math401_P1_2.md)
|
|
|
|
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 $d_B\geq d_A\geq 3$.
|
|
|
|
> Noted in [Aspect of generic entanglement](https://arxiv.org/pdf/quant-ph/0407049) $C_3=(8\pi^2\ln(2))^{-1}$.
|
|
|
|
#### 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
|
|
|
|
> The role of the paper in Physics can be found in (15.86) on book Geometry of Quantum states. |