22
votes
Accepted
How does the vectorization map relate to the Choi and Kraus representations of a channel?
One way to understand the relationship between the Choi representation of a channel and its possible Kraus representations is to use the vectorization map.
Suppose that we have two finite-dimensional ...
- 5,158
12
votes
Accepted
What is the "complementary map" of a channel with given Kraus decomposition?
Let's start by finding a complementary channel for any channel given by a Kraus representation
$$
\Phi(X) = \sum_{k=1}^n A_k X A_k^{\dagger}.
$$
To make the necessary equations clear, let us assume ...
- 5,158
11
votes
Accepted
What does it mean "less than identity" in the operator sum representation?
Matrix inequalities of the form $A\ge B$ should be read as
$$
A-B\ge 0\ ,
$$
which in turn means that all eigenvalues of $A-B$ are larger or equal than zero.
In the given case, $M\le I$ means that ...
- 5,647
11
votes
Accepted
Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?
This question is posed, and answered positively, in Nielsen & Chuang in a subsection of chapter 8 entitled "System-environment models for and operator-sum representation". In my version, it can be ...
- 55.7k
11
votes
Accepted
Do the Kraus operators of a CPTP channel need to be orthogonal?
There is an ambiguity in the choice of Kraus operators: If $\{E_a\}$ is a set of Kraus operators for a channel $\mathcal E$, so is $\{F_b\}$ with $F_b=\sum_a v_{ab} E_a$, with $(v_{ab})$ an isometry.
...
- 5,647
10
votes
Accepted
Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?
You cannot always find such a Kraus decomposition. Notice that any CPTP map $\mathcal E$ which does have a decomposition as a probabilistic mixture unitaries is unital, which is to say that it maps ...
- 11.8k
10
votes
Accepted
What is the rank of a quantum channel?
Every quantum channel has many Kraus representations that may differ in the number of Kraus operators. For example, for any positive integer $n$ and numbers $p_i$ with $i=1,\dots,n$ and $\sum_{i=1}^...
- 20.7k
8
votes
Accepted
How to find the operator sum representation of the depolarizing channel?
This really depends where you want to start from. For instance, you can construct the Choi state of $\mathcal E$, i.e.,
$$
\sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ ,
$$
...
- 5,647
7
votes
Accepted
Can any channel be written as $\Phi(X)=\operatorname{Tr}_{\mathcal Z}[U(X\otimes \sigma)U^\dagger]$ for any state $\sigma$?
No, this is not always possible.
A counterexample is given by $\sigma=I/d'$ and $\Phi(X)=\mathrm{tr}(X)|0\rangle\langle0|$.
To see this, note that for $X=I/d$,
\begin{align}
2(1-1/d) & =
\|\,|0\...
- 5,647
6
votes
Accepted
Direct derivation of the Kraus representation from the natural representation, using SVD
As matrices, the natural representation and Choi representation of a map $\Phi$ have exactly the same entries, but arranged into matrices in different ways. One way to express this is like this:
$$
\...
- 5,158
6
votes
Accepted
How do I derive Stinespring and Kraus representations of a map such that $\Lambda(\rho)=|0\rangle\langle0|$ for all $\rho$?
Stinespring dilation can be thought of as a way of representing an arbitrary completely positive trace preserving map $\Lambda$ on a system $A$ as a composition of two simpler maps: a unitary ...
- 20.7k
6
votes
Accepted
How does the spectral decomposition of the Choi operator relate to Kraus operators?
Choi operator of a linear map $\mathcal{E}$ is defined as
$$
J(\mathcal{E}) = \sum_{ij} \mathcal{E}(|i\rangle\langle j|)\otimes |i\rangle\langle j|.\tag1
$$
Substituting $\mathcal{E}(\rho)=\sum_k E_k\...
- 20.7k
6
votes
Accepted
Why does a quantum operation being trace-preserving imply that $\sum_k E_k^\dagger E_k=I$?
Suppose that
$$
\mathrm{tr}\left(\sum_k E_k\rho E_k^\dagger\right) = \mathrm{tr}(\rho)
$$
for all $\rho$. Then
$$
\mathrm{tr}\left(\sum_k E_k^\dagger E_k\rho\right) = \mathrm{tr}(I\rho)
$$
for all $\...
- 20.7k
6
votes
Accepted
What is the root of the non-trace-preserving bit-flip map
Assuming w.l.o.g. that $p\in\mathbb{R}$, the linear map in the question may be rewritten as
$$
\mathcal{E}(\rho) = p^2\rho+p^2X\rho X = 2p^2\left(\frac12\rho + \frac12 X\rho X\right)
$$
where $X$ is ...
- 20.7k
6
votes
Accepted
Is there an upper-bound on the operator norm (max-singular value) of the matrix representation of a quantum channel?
Summary
Below, we prove that
$$
\|\hat\Phi\|_{\rm op}=\sup_{\rho\in D(\mathcal{X})}\sqrt\frac{\gamma(\Phi(\rho))}{\gamma(\rho)}
$$
where $D(\mathcal{X})$ denotes the set of density matrices on the ...
- 20.7k
5
votes
Accepted
Find the quantum operation corresponding to a given unitary evolution and projective measurement
So, let the system be $\rho$, and the environment $|0\rangle \langle 0|$.
The given operation (which you can check is unitary, and incidentally happens to be the CNOT operation), is applied on $\rho \...
- 1,621
5
votes
Accepted
Find the Kraus operators of a combined amplitude and phase damping channel
You can obtain the Kraus operators of the combined channel by taking products of the Kraus operators of the individual channels (using the notation from the paper you linked):
Amplitude damping:
$E^{...
- 1,356
5
votes
How to find the operator sum representation of the depolarizing channel?
While the procedure in the existing answer, based on channel-state duality, applies to general channels, there's a more direct way to obtain Kraus operators for this particular case of the ...
- 181
5
votes
Accepted
How to find the unitary operation of a depolarizing channel?
From N&C: Assuming the environment is in some pure state we recall that Kraus representations comes from the unitary evolution $$\sum_{k}E_k\rho E_k^*=\sum_k \langle e_k |U\left(\rho\otimes|e_0\...
- 1,893
5
votes
Accepted
Choi matrix in QETLAB
QETLAB usually deals with channels as Choi operators. You can convert your Kraus operators to the Choi matrix by providing the Kraus operators as a cell array. Example with the amplitude damping ...
- 126
5
votes
Accepted
Derivation of Equation $8.7$ in Nielsen Chuang
Just plug in all of the relevant stuff you state in the question, i.e.
$$
U = |0\rangle \langle 0 | \otimes I + |1 \rangle \langle 1 | \otimes X
$$
and
$$
\rho_{\mathrm{env}} = |0\rangle \langle 0 |.
$...
- 5,345
5
votes
Quantum capacity for serial composition of quantum channels
TL;DR
Quantum capacity of $\mathcal{N}_2\circ\mathcal{N}_1$ can be anywhere between zero and the minimum of the quantum capacities of $\mathcal{N}_1$ and $\mathcal{N}_2$.
Background
Quantum capacity ...
- 20.7k
5
votes
Accepted
How to use the Kraus operators to represent the total density matrix instead of the reduced one?
You have
$$\newcommand{\ket}[1]{\lvert #1\rangle}U(\ket\psi\otimes\ket0)
= \bigg(I\otimes \underbrace{\sum_\ell \ket\ell\!\langle\ell|}_{\equiv I}\bigg) U (\ket\psi\otimes\ket0) \\
= \sum_\ell (I\...
glS♦
- 23.3k
5
votes
What do commuting quantum channels look like?
Examples
The condition $[A_a, B_b]=0$ is sufficient, but not necessary for $\Phi$ and $\Psi$ to commute. Indeed, the standard Kraus representations of many commuting pairs of channels have non-...
- 20.7k
5
votes
Accepted
Implication of SWAP being not positive in terms of quantum channel
Quoting from the linked source: "thus SWAP has negative
eigenvalues, which means that $T\otimes I$ is not positive and therefore $T$ is not
completely positive", where $T$ is the transpose. ...
glS♦
- 23.3k
5
votes
What are the possible Kraus operators of the identity channel?
Short version
Another approach: observe that finding the Kraus operators for a channel $\Phi$ is equivalent to finding a decomposition for the Choi $J(\Phi)$ in terms of positive-semidefinite unit-...
glS♦
- 23.3k
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:
$$\operatorname{vec}_c(\rho)=\left(\begin{array}{c}
\...
- 2,912
5
votes
How to recover the original density matrix from the output of amplitude damping channel?
Inverting a quantum channel
Kraus representation does not make it convenient to find the inverse. However, quantum channels are linear maps, so we can represent them as matrices which can be inverted ...
- 20.7k
5
votes
Accepted
Can any channel be represented as $A\rho A^\dagger$ for some $A$?
No, this is not possible in general.
To see it, consider for example what happens taking the trace of that expression. You'd get:
$$\operatorname{tr}(A^\dagger A\rho)=\operatorname{tr}\left(\sum_j K_j^...
glS♦
- 23.3k
4
votes
Accepted
Deduce the Kraus operators of the dephasing channel using the Choi
Acting with the dephasing channel on the possible states of a single qubit:
\begin{align}D\left(\left|0\rangle\langle0\right|\right) &= \left|0\rangle\langle0\right| \\ D\left(\left|0\rangle\...
- 3,667
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