# Questions tagged [kraus-representation]

For questions relating to the Kraus decomposition of quantum channels.

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### Get matrix for an X gate for a given fidelity p

Wanted to check on how to mathematically obtain the matrix of an X gate which has fidelity/probability $p$? (i.e. it acts as an $X$ gate with probability $p$)
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### Unital channel which is not mixed unitary

How to prove that for a multi-qubit system a unital channel is not necessarily mixed unitary? This is Problem 8.3 in Nielsen and Chuang. Here's a snippet of the text: Shall I need to take two ...
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### Interconversion between different representations of quantum channels

I was reading TQI-notes by Watrous where they introduce different representations for quantum channels and wondering how to go from one to the other. I have: \begin{align} &|\Phi(\rho)\rangle\!\...
1 vote
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### What is the relation between the Choi matrix and the Liouville space (superoperator) representations of a channel?

A.S. Fletcher, P. W. Shor, and M. Z. Win Phys. Rev. A 75, 012338 (2007) says the Choi matrix for the operation $\mathcal{A}$ is given by $X_A \equiv \sum_k |A_k\rangle\!\rangle\langle\!\langle A_k|$, ...
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### Can any channel be represented as $A\rho A^\dagger$ for some $A$?

Consider an arbitrary quantum operation defined by a series of Kraus operators $\sum_j K_j\rho K_j^\dagger$ over the density matrix of the system $\rho$. The operation might or might not be unitary, ...
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### Are $n$-qubit Pauli channels $\mathcal E(\rho)=\sum_j p_j P_j \rho P_j$ invertible?

An $n$-qubit Pauli channel $\mathcal E$ acting on a quantum state $\rho$ is of the form \begin{equation}\label{PauliChannel} \mathcal E(\rho)=\sum_jp_jP_j\rho P_j \end{equation} where $p_j\in[0,1]$ ...
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### How to recover the original density matrix from the output of amplitude damping channel?

For amplitude damping, we have the below expression $$\xi(\rho)=E_0\rho{E_0}^\dagger + E_1\rho{E_1}^\dagger.$$ How can I perform a matrix inverse operation on $\xi(\rho)$ at the receiver to get back ...
1 vote
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### Kraus Operators for Weak Measurments

I have system which has states two spin states and an optically excited states. Everytime optically excited states relaxes, a photon is emitted, which I can detect a fraction of. If I simulatenously ...
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### Simulation of environment noise in tensor networks

I want to simulate a noisy version of a tensor network efficiently. I do not want to bring noisy gates into the picture and make the "environment" my only noise source. I know the ...
1 vote
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### Why is dual-rail encoding called an error-detecting code for amplitude damping?

Exercise 8.23 : Suppose that a single qubit state is represented by using two qubits, as $|\psi\rangle=a|01\rangle+b|10\rangle$. Show that $\mathcal{E}_{AD}\otimes\mathcal{E}_{AD}$ applied to this ...
1 vote
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### How to derive the number of independent parameters in the $\chi$ matrix from the Choi matrix?

In the section on Quantum process tomography, Page 391, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang. it is given that In general, $\chi$ will contain $d^4−d^2$ ...
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### Why does the $\chi$ matrix have $d^4-d^2$ independent parameters?

In the section on Quantum process tomography, Page 391, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang. it is given that In general, $\chi$ will contain $d^4−d^2$ ...
1 vote
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### Why Kraus operator is not a number?

Quantum operations can be represented in an elegant form known as the operator-sum representationn, namely $\mathcal{E}(\rho)=\sum_kE_k\rho E_k^\dagger$, where $E_k=\langle e_k|U|e_0\rangle$, the ...
1 vote
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### Check that a channel implements a particular unitary

Consider a channel $C$ with Kraus operators $\{K_k\}$ and a unitary U. How can I check that $C$ implements $U$ ? One can write that their Choi matrices are equal i.e: \begin{equation} \sum_{i,j}|i\...
1 vote
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### Is there an inverse for Stinespring dilation?

Given a set of Kraus operators we can find a unitary that does the equivalent map on an extended space including the environment using Stinespring dilation. My question is how do we go about doing the ...
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### What is the most general way to describe post-measurement states?

Background Generally speaking, the description of post-measurement states associated with a POVM seems to always pass through, in some form or another, the formalism of Kraus operators. For example: ...
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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 ...
1 vote
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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 ...
To show that the Kraus decomposition $\Phi(\rho)=\sum_{k=1}^D M_k\rho_S M_k^\dagger$ implies the Stinespring form $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
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 ...