# Questions tagged [kraus-representation]

For questions relating to the Kraus decomposition of quantum channels.

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### Does the unitary freedom in choice of Kraus operators come from the freedom in the choice of purifications?

Does the unitary freedom in the choice of Kraus operators for a given quantum channel just come from the unitary freedom in choice of purification of a quantum state? Here's what I'm thinking. If I ...
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### What is the adjoint of the complementary channel?

Given a channel $\phi$ with the set of kraus operators; $(K_1, K_2,...,K_n)$, I know the complementary channel is; $\phi^c(A)=\sum_{i,j}tr(K^*_jK_iA)E_{ij}$ what will be the adjoint of this ...
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### 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 $\{I\}$...
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### Qiskit: Sum of Kraus operators are not equal to identity matrix

I know that the one of the property of the Kraus Operator is: So in qiskit, I first converted my array to super operator and then I found my kraus operators. However the sum of kraus operator is not ...
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### What is the Stinespring dilation of $T\otimes I$ for some CPTP map $T$?

Let $T: \mathcal{H}_A \rightarrow \mathcal{H}_B$ be a CPTP map with Stinespring extension $U: \mathcal{H}_{A} \rightarrow \mathcal{H}_{B} \otimes \mathcal{H}_E$. That is $U$ is an isometry such that ...
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### 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 ...
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### Special properties of a channel whose Kraus decomposition contains Identity

I would like to know if there are any special properties of channels that permit a Kraus representation that includes an identity? That is, if I am given a Kraus representation of a CPTP map $\Phi$ ...
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### Can every unitary on $\mathcal{H}\otimes \mathcal{K}$ be modelled by quantum operations on $\mathcal{H}$?

In section 8.2.3 of Nielsen and Chuang, they discuss how unitary dynamics of a system and environment arise from quantum operations (i.e. Kraus operators $E_k$ such that $\sum_k E_k^*E_k=I$). ...
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### 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....
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### 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 $$\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}$$...
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### 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 ...
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### How does the Kraus decomposition imply the Stinespring representation?

To show that the Kraus decomposition $\Phi(\rho)=\sum_{k=1}^D M_k\rho_S M_k^\dagger$ implies the Stinespring form $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
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### Finding Kraus operators from the output density matrix

I have a question regarding Kraus operators. Any quantum channel can be written in terms of Kraus operators as $E(\rho)= \sum_{i=0}^n K_i \rho K_i^{\dagger}$ where $\rho$ is the initial density ...
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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 $\mathcal ... 4 votes 1 answer 162 views ### Is a quantum channel reversible if all Kraus operators are proportional to unitaries? In preskill's online lecture p.13, he stated that if a channel is reversible, i.e.,$\varepsilon^{-1}\circ\varepsilon(\rho)=\rho$for any$\rho$, then the kraus operator of the quantum channel must be ... 1 vote 0 answers 44 views ### How is$I(\rho^{QC})=I_{CC}(\rho^{QC})$On page 3 of this paper, for the proof of theorem 1, it states that, using Lemma 2 from the previous page, that if $$I(\Lambda_{A}\otimes\Gamma_{B})[\rho]=I(\rho))$$ then there exists$\Lambda_{A}^{*}$... 4 votes 2 answers 175 views ### What properties do Kraus operators of Markovian processes have? It is well-known that the Kraus operator can describe more kinds of processes than master equations. For example, the master equation cannot describe non-markovian processes while the Kraus operator ... 3 votes 1 answer 148 views ### Equivalent statement of the unitary freedom of Kraus operator? There is a well-known form of the unitary freedom of Kraus operators, which can be found in Nielsen's book, stating that two sets of Kraus operators describe the same physical process of the system(... 4 votes 2 answers 310 views ### Determining whether there exists an equivalent set of unitary Kraus operators I have a CPTP quantum channel$\mathcal{E}$that I've characterized by an operator sum representation$\{E_i\}$for$i=1, \dots, m$which acts on an input state like $$\mathcal{E}(\rho) = \sum_{i=1}^... 3 votes 1 answer 211 views ### What are examples of Kraus operators describing the process of control error? Noise, such as photon loss or dephasing, is often described with Kraus operators. Are there examples of Kraus operators describing the process of control error? 4 votes 2 answers 198 views ### Can Kraus operators change a mixed state into a pure state? It seems that Kraus operators cannot change a pure state into a mixed one (wrong). For any pure state can be written as |\psi\rangle\langle\psi|, so after the Kraus operators. It becomes$$\sum_l\... 5 votes 1 answer 131 views ### How to use the Kraus operators to represent the total density matrix instead of the reduced one? In Nielsen's book, the Kraus operator can be attained by trace out the enviroment: $$\operatorname{Tr}_{\rm env}[\hat{U}(|\psi\rangle\otimes|0\rangle)(\langle\psi|\otimes\langle 0|)\hat{U}^\dagger].$$... 5 votes 1 answer 230 views ### Quantum capacity for serial composition of quantum channels Recently, I have been working with quantum channel capacity for quantum-quantum channels and I was wondering if there exist some results for channel compositions. Specifically, I have been looking for ... 3 votes 1 answer 64 views ### Kraus representation of a convex combination of CPT maps Let$\Phi_1,\Phi_2$be CPT maps with Kraus decomposition \begin{equation} \Phi_1=\sum_{k=1}^{d_1}M_k\rho M_k^\dagger, \quad \Phi_2=\sum_{k=1}^{d_2} N_k\rho N_k^\dagger, \quad \text{s.t.} \quad \sum_{k=... 1 vote 2 answers 175 views ### How do I derive Stinespring and Kraus representations of a map such that$\Lambda(\rho)=|0\rangle\langle0|$for all$\rho$? Can't find any info on Stinespring dilation so I thought I could post here. If I have a qubit complete positive map$\Lambda$, that maps all inputs to the output$|0\rangle$,$\Lambda(\rho)=|0\rangle\... 257 views

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

This matrix $$c_{\lambda} = \lambda |\phi^+\rangle \langle\phi^+| + (1-\lambda )|\phi^-\rangle \langle\phi^-|$$ is the Choi–Jamiołkowski matrix of a quantum channel for any $\lambda \in [0,1]$. The ...
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### Choi matrix in QETLAB

I am using QETLAB, a package for working with quantum information theory in Matlab and I have some doubts. I am trying to calculate diamond norms using such for some quantum channels. However, when ...
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