# Tag Info

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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 ...
• 4,568
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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 ...
• 4,568
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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 ...
• 5,007
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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 ...
• 47.5k
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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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### 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., $$\sigma = (\mathcal E \otimes \mathbb I)(|\Omega\rangle\langle\Omega|)\ ,$$ ...
• 5,007
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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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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 $\Phi:\mathrm{Lin}(\mathcal ... • 19k 4 votes ### Quantum capacity for serial composition of quantum channels TL;DR Quantum capacity of$\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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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 $$\Phi_Y(X) = \operatorname{Tr}(X) Y.$$ ...