19 votes
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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 ...
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12 votes
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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 ...
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11 votes
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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 ...
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10 votes
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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 ...
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9 votes
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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 ...
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8 votes
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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., $$ \sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ , $$ ...
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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. ...
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8 votes
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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}^...
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7 votes
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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\...
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6 votes
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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: $$ \...
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6 votes
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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 ...
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6 votes
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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 $\...
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6 votes
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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 ...
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6 votes
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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 ...
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5 votes
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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 \...
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5 votes
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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\...
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5 votes
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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\...
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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\...
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5 votes
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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. ...
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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-...
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4 votes
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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\...
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  • 3,477
4 votes
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How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?

Let's start with a general state $$ \rho\otimes\rho_0=\sum_{x,y\in\{0,1\}}\langle x|\rho|y\rangle|x\rangle\langle y|\otimes |0\rangle\langle 0|. $$ If we apply the controlled-not, we have $$ \...
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4 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 ...
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  • 171
4 votes

Why for $\Phi(\rho)=\sum_j F_j^\dagger\rho F_j$ to be trace preserving we need the condition $\sum_j F_j F_j^\dagger=I$?

Recall that the trace is both linear and invariant under cyclic permutation of the operators $$ \mathrm{Tr}(\Phi(\rho))=\mathrm{Tr}\left(\sum_j F_j^\dagger \rho F_j\right)=\sum_j\mathrm{Tr}\left( F_j^...
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4 votes
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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^{...
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4 votes
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How is the partial trace related to the operator sum representation?

I think it helps here to write things explicitly. Suppose $\mathcal E(\rho)=\operatorname{Tr}_E[U(\rho\otimes|\mathbf e_0\rangle\!\langle\mathbf e_0|)U^\dagger]$. Pick a basis for the environment in ...
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4 votes
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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 |. $...
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  • 4,202
4 votes

What is the Kraus representation of the quantum channel with Choi $\lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$?

Another way is to observe that Choi $J(\Phi)\in\mathrm{Lin}(\mathcal Y\otimes\mathcal X)$ and Kraus operators $\{A_a\}_a\subset\mathrm{Lin}(\mathcal X,\mathcal Y)$ of a map $\Phi:\mathrm{Lin}(\mathcal ...
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4 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 ...
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4 votes
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Can Kraus operators change a mixed state into a pure state?

More generally, given any two states, you can always find some channel sending one into the other. Consider for example replacement maps, which have the form $$\Phi_Y(X) = \operatorname{Tr}(X) Y.$$ ...
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