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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 ...
John Watrous's user avatar
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13 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 ...
John Watrous's user avatar
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12 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 ...
DaftWullie's user avatar
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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 ...
Norbert Schuch's user avatar
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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. ...
Norbert Schuch's user avatar
10 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 ...
Niel de Beaudrap's user avatar
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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}^...
Adam Zalcman's user avatar
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Counterexamples in quantum information theory

Quantum Channels Quantum channels: general properties Not every positive map is completely positive. One may argue that this is the mother of all counterexamples in quantum information: the ...
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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., $$ \sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ , $$ ...
Norbert Schuch's user avatar
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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 ...
Adam Zalcman's user avatar
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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\...
Norbert Schuch's user avatar
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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 $$\Phi_Y(X) = \operatorname{Tr}(X) Y.$$ ...
glS's user avatar
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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: $$ \...
John Watrous's user avatar
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6 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 ...
kianna's user avatar
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6 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\...
Condo's user avatar
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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 ...
Adam Zalcman's user avatar
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6 votes
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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\...
Adam Zalcman's user avatar
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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 $\...
Adam Zalcman's user avatar
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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 ...
Adam Zalcman's user avatar
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6 votes

Counterexamples in quantum information theory

Quantum Computing / Quantum Complexity Theory Requirements for exponential speedup Clifford circuits can (1) create superposition such as with Hadamard gates, (2) create entanglement such as with ...
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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 \...
Mahathi Vempati's user avatar
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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^{...
chrysaor4's user avatar
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5 votes
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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 ...
user13507's user avatar
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5 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 |. $...
Rammus's user avatar
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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 ...
Adam Zalcman's user avatar
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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\...
glS's user avatar
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5 votes
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Is a quantum channel reversible if all Kraus operators are proportional to unitaries?

No. The key thing about what Preskill is saying is that all the Kraus operators must be proportional to the same unitary. Your two Kraus operators are proportional to different Kraus operators.
DaftWullie's user avatar
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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-...
Adam Zalcman's user avatar
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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. ...
glS's user avatar
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5 votes
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What is the Stinespring dilation of $T\otimes I$ for some CPTP map $T$?

Yes. Note that in general ${\rm Tr}_1(\rho_{12} \otimes \rho_3) = {\rm Tr}_1(\rho_{12}) \otimes \rho_3$. It's easy to verify your equality for $\rho_{AC} = \rho_A \otimes \rho_C$ : $$ {\rm Tr}_{E} \...
Danylo Y's user avatar
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