New answers tagged quantum-operation
2
votes
Accepted
Action of a channel on an "unphysical" state
A quantum channel sends states to states, which are positive semidefinite matrices. But it is a linear map. There's no problem in extending its action to all matrices by linearity, since every matrix ...
- 6,013
3
votes
Accepted
How to obtain the unitary operator to get specific partial trace?
I think the unitary is $\sqrt{\frac{1}{2}}I+\sqrt{\frac{1}{2}}iS$ where $S$ is swap operator such that $S|i\rangle\otimes|j\rangle=|j\rangle\otimes|i\rangle$, and have matrix form $S=\sum_{ij}{|ij\...
- 2,607
6
votes
How can I represent the completely mixed state as $\frac I2=\frac14(\rho+X\rho X+Y\rho Y+Z\rho Z)$?
There are two ways to see it: mathematically, and "intuitively".
Mathematically
$\rho$ can be written as:
$$\rho=\begin{pmatrix}a&b\\\overline{b}&1-a\end{pmatrix}$$
Thus, we have:
$$...
- 2,850
5
votes
Accepted
Why can any quantum channel be represented as a matrix?
Mind that $E\left( \cdot \right) $ is a linear map, and can be written as matrix act on a vector. If we write matrix in vector form as follows:
$$\operatorname{vec}_c(\rho)=\left(\begin{array}{c}
\...
- 2,607
1
vote
Why does the twirl of a quantum channel give a depolarizing channel?
I will give an extended explanation of Nielsen's proof, i.e. your first ref link.
The idea is that, $\rho=\sum_ip_i|i\rangle\langle i|$, we can prove it's depolarizing channel for each $|i\rangle\...
- 2,607
2
votes
Accepted
How to calculate the action of a channel on part of a quantum state?
you can absolutely do the calculation in this comment. That is, you can just compute
$$(I\otimes \mathcal E)\rho = \sum_{ij} (I\otimes \mathcal E)(|i\rangle\!\langle j|\otimes \sigma_{ij})
= \sum_{ij} ...
glS♦
- 19.6k
2
votes
Are continuous probability distributions over quantum channels possible?
Sure. A standard example of this is the use of "twirling operations": given a channel $\mathcal E$, one can define
$$\mathcal E_T(\rho) = \int dU\, U^\dagger\mathcal E(U\rho U^\dagger)U,$$
...
glS♦
- 19.6k
3
votes
Are continuous probability distributions over quantum channels possible?
Quantum channels describe stochastic events, so it should be completely legit to consider continuous sets of events. Integrating over a continuous set of quantum channels with some distribution give ...
- 51
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