For questions relating to the Kraus decomposition of quantum channels.
Questions tagged [kraus-representation]
126 questions
16
votes
1 answer
How does the vectorization map relate to the Choi and Kraus representations of a channel?
I know that the Choi operator is a useful tool to construct the Kraus representation of a given map, and that the vectorization map plays an important role in such construction.
How exactly does the vectorization map work in this context, and how…
Tobias Fritzn
- 721
- 4
- 11
12
votes
1 answer
How does the invertibility of a quantum map reflect on its Kraus operators?
Consider a quantum map $\Phi\in\mathrm T(\mathcal X)$, that is, a linear operator $\Phi:\mathrm{Lin}(\mathcal X)\to\mathrm{Lin}(\mathcal X)$ for some finite-dimensional complex vector spaces $\mathcal X$.
In the specific case in which $\Phi$ is also…
glS
- 27,510
- 7
- 37
- 125
11
votes
2 answers
How to find the operator sum representation of the depolarizing channel?
In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity matrix with
$$\frac{\mathbb{I}}{2} = \frac{\rho +…
user1936752
- 3,311
- 1
- 9
- 24
10
votes
1 answer
What does it mean "less than identity" in the operator sum representation?
In Quantum Computation and Quantum Information by Nielsen and Chuang, Section 8.2.3, $\mathcal{E}=\sum_{k}E_k\rho E_k^{\dagger}$ gives the operator-sum representation. In general, it requires $\sum_k E_k E_k^{\dagger}\leq I$. But, what does it mean…
czwang
- 949
- 1
- 6
- 17
10
votes
2 answers
How does the spectral decomposition of the Choi operator relate to Kraus operators?
In Nielsen and Chuang's QCQI, there is a proof states that
Theorem 8.1: The map $\mathcal{E}$ satisfies axioms A1, A2 and A3 if and only if
$$
\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}
$$
for some set of operators…
Sherlock
- 745
- 3
- 15
10
votes
2 answers
Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?
I understand that there are two ways to think about 'general quantum operators'.
Way 1
We can think of them as trace-preserving completely positive operators. These can be written in the form
$$\rho'=\sum_k A_k \rho A_k^\dagger \tag{1}$$
where…
Quantum spaghettification
- 1,532
- 11
- 29
9
votes
1 answer
Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?
If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that
$$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$
in the same spirit as any density matrix $\rho$ can be…
user2723984
- 1,156
- 8
- 16
9
votes
3 answers
What are the possible Kraus operators of the identity channel?
Consider a Kraus representation $\{A_a\}_a$ of the identity channel $\mathcal{I}$ that maps any state to itself. Of course, $\{A_a\}_a$ are not the simplest Kraus operators, which would just be $\{I\}$, and they need not to be orthogonal. Is there a…
Shadumu
- 373
- 1
- 5
8
votes
2 answers
Do the Kraus operators of a CPTP channel need to be orthogonal?
Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map.
Any such channel admits a Kraus decomposition of the form
$$\Phi(X)=\sum_a A_a X A_a^\dagger,$$
for a set of operators $A_a\in\mathrm{Lin}(\mathcal X,\mathcal Y)$ satisfying $\sum_a…
glS
- 27,510
- 7
- 37
- 125
8
votes
2 answers
Deduce the Kraus operators of the dephasing channel using the Choi
I'm trying to deduce the Kraus representation of the dephasing channel using the Choi operator (I know the Kraus operators can be guessed in this case, I want to understand the general case).
The dephasing channel maps a density operator $\rho$…
user2723984
- 1,156
- 8
- 16
8
votes
3 answers
Why does a quantum operation being trace-preserving imply that $\sum_k E_k^\dagger E_k=I$?
I am reading Nielsen Chuang Chapter 8. They say that if a quantum operation is trace-preserving, then
\begin{equation}
Tr\left(\sum_k E_k^{\dagger}E_k \rho\right) = 1
\end{equation}
which I understand. They however then claim that as this is true…
alpha
- 149
- 5
8
votes
1 answer
What is the rank of a quantum channel?
I read the following sentence in a paper:
We consider a quantum channel $\mathcal{E}_{\omega}(\rho)=\sum_{i=1}^{r} K_{i} \rho K_{i}^{\dagger}$ where $r$ is the rank of the channel.
I didn't find the definition of the rank of the quantum channel…
Sherlock
- 745
- 3
- 15
7
votes
2 answers
Find the quantum operation corresponding to a given unitary evolution and projective measurement
I'm trying to (understand and) solve this problem from Nielsen and Chuang's Quantum Computation and Quantum Information.
Exercise 8.4: (Measurement) Suppose we have a single qubit principal system, interacting with a single qubit environment…
Bashir
- 179
- 8
7
votes
2 answers
How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation:
$$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$
They show an example with
$\rho_{env} =…
Mahathi Vempati
- 1,731
- 10
- 21
7
votes
1 answer
What do commuting quantum channels look like?
Consider two channels, $\Phi,\Psi\in\mathrm C(\mathcal X)$ acting on some space $\mathcal X$, and suppose they commute, that is,
$$\Phi(\Psi(\rho))=\Psi(\Phi(\rho))$$
for all states $\rho$. Can anything be said about the structure, e.g. in terms of…
glS
- 27,510
- 7
- 37
- 125