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2 votes
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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: $$...
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} ...
  • 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 $$\mathcal E_T(\rho) = \int dU\, U^\dagger\mathcal E(U\rho U^\dagger)U,$$ ...
  • 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 ...

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