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Questions tagged [kraus-representation]

For questions relating to the Kraus decomposition of quantum channels.

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Are peripheral eigenvalues of a completely positive map always semisimple?

It is known that all peripheral eigenvalues (i.e. all eigenvalues $\lambda\in\mathbb C$ such that $|\lambda|$ equals the spectral radius) of positive trace-preserving or positive unital maps are ...
Frederik vom Ende's user avatar
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Are Stinespring unitaries that give rise to the same channel locally unitarily equivalent?

It is well known that for if any two linear maps $V_1,V_2:\mathbb C^n\to\mathbb C^k\otimes\mathbb C^m$ (isometry or not) satisfy $$ {\rm tr}_{\mathbb C^m}(V_1(\cdot)V_1^\dagger)={\rm tr}_{\mathbb C^m}(...
Frederik vom Ende's user avatar
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Given $\Psi$ completely positive when do there exist $K_1,K_2$ such that $K_2\Psi(K_1^\dagger(\cdot)K_1)K_2^\dagger$ is also trace preserving?

In quantum information it occasionally happens that one ends up with a completely positive but not yet trace-preserving map $\Psi$ which one wants to make trace preserving somehow; this often comes up ...
Frederik vom Ende's user avatar
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How does measuring a density matrix give Kraus operators?

I am trying to complete this exercise regarding noisy channels. I need to measure a density matrix to get the Kraus operators. However, if I measure, I only get scalars. Can someone please explain how ...
researcher101's user avatar
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3 answers
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Why do we need/have the operator sum representation (Kraus representation)?

I am reading through Nielsen & Chuang, and I am on the section about operator sum representation. They performed this derivation. Why is it important and useful for us to bundle together the ...
researcher101's user avatar
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Why is the Choi matrix different from the analytic form for a depolarizing channel?

I'm currently trying to implement the depolarizing channel on qiskit. But, as I see in my calculation it doesn't match with the qiskit aer_noise. So, for the Depolarizing Channel we got : $$ \mathcal{...
felipechoy's user avatar
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What are kraus operators of a qubit interacting a thermal environment?

Suppose a qubit that interacting a thermal environment. The thermal environment can be a thermal field for example. What is the kraus operators for this case?
reza's user avatar
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Is the trace of a positive map always positive?

Obviously, positive semi-definite operators always admit a positive trace as ${\rm tr}(A)=\|A\|_1\geq 0$ whenever $A\geq 0$. This motivates the following "lifted" question: Given any ...
Frederik vom Ende's user avatar
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What is the meaning of $\sum_i K_iK_i^\dagger$ for a quantum channel with Kraus operators $K_i$?

Let a channel $N$ be given in terms of its Kraus operators $K_i$ as $$N(\rho) = \sum_i K_i\rho K_i^\dagger.$$ Is the sum $\sum_i K_iK_i^\dagger$ a meaningful quantity? I know that $\sum_i K_i K_i^\...
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Counterexamples in quantum information theory

As was already asked about in this phys.SE question many years ago—which, sadly, got closed and never received an answer—is there a collection of counterexamples in quantum information theory, "...
Frederik vom Ende's user avatar
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How to get the Kraus decomposition of the amplitude damping channel from its Choi?

I found going from the Choi-matrix of a quantum channel to the Choi-Kraus decomposition a bit difficult. I know that it follows from the eigen-decomposition of the Choi-matrix. But I struggle with ...
Pink Elephants's user avatar
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How to model the IBM quantum errors?

I'm conducting a study of the quantum channel errors, and a question has arisen: Is it possible to model the errors of an IBM quantum computer with Kraus operators?
Leandro Matheus Morais da Silv's user avatar
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2 answers
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Find the Kraus operators for the amplitude damping channel from its isometric representation

I am currently learning about quantum channels and am sadly stuck at a rudimentary problem, where I don't understand how to find the Kraus matrices of a quantum channel. The amplitude damping channel ...
Alex1111's user avatar
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How many free real parameters in a general CPTP map?

The question is how many free real parameters a general CPTP map can maximally have. Let's assume the CPTP map $\Phi:L(\mathcal{H}_A) \rightarrow L(\mathcal{H}_B)$ is given in the Kraus representation ...
Tobias95's user avatar
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Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

This problem is from a "passing remark" in this lecture notes. With the help of some colleagues I managed to find a way for this supposedly elementary fact, but I would like to see if there ...
Evangeline A. K. McDowell's user avatar
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Affine transformation of the Bloch sphere to Kraus representation of qubit channels

It is known that qubit channels can be written in the form: $$ \begin{align} \Phi(\rho) = \frac{1}{2}\left(I+(T\vec{r}+\vec{t})\cdot\sigma\right)\ \end{align} $$ where $\vec{r}$ is the Bloch vector ...
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Completeness relation of kraus operators not satisfied in Fake backends

With the following code ...
stopper's user avatar
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How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?

I am confused as to how we get $M_{0} = \sqrt{1-p} I$ and the following $M_{1}, M_{2}, M_{3}$. The above notes say that we should partially trace over the environment in the $|{0}\rangle, |{1}\rangle, ...
am567's user avatar
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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 $p$)
codeit's user avatar
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Can a unital channel not be 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 ...
Sudhir Kumar's user avatar
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2 answers
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Proof that two sets of quantum maps are equivalent only when they are related by a unitary transformation

I am trying to show that the two different quantum maps $\rho'=\sum_{\alpha} K_{\alpha} \rho K_{\alpha}^{\dagger}$ and $\rho''=\sum_{\beta} L_{\beta} \rho L_{\beta}^{\dagger}$ are equivalent i.e. $\...
Anindita Sarkar's user avatar
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What can we say about $\sum_i K_i K_i^\dagger$ for non-unital CPTP maps?

Suppose we have a CPTP map $\Phi(\rho)=\sum_i K_i \rho K_i^+$, such that, $\sum_i K_i^+K_i=\mathbb{I}$. In case the map preserves Identity, is unital, then we immediately have $\sum_i K_i K_i^+ =\...
Cain's user avatar
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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\!\...
Saurabh Shringarpure's user avatar
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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 $X_A \equiv \sum_k |A_k\rangle\!\rangle\langle\!\langle A_k|$, ...
Saurabh Shringarpure's user avatar
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1 answer
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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, ...
Zarathustra's user avatar
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1 answer
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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 $p_j\in[0,1]$ ...
karry's user avatar
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How to extract probabilities from Kraus representation?

Consider a quantum operation described by Kraus operators $K_1, ..., K_n$. As I understand the effect of this operation on a density matrix $\rho$ can be described as $ \mathcal{E}(\rho)= \sum_{i}p(i)\...
EugeneB's user avatar
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What is the meaning of $\langle e_k|U|e_0\rangle$ when $U$ acts on a larger Hilbert space than that in which $|e_0\rangle$ and $|e_k\rangle$ live?

In Nielsen and Chuang, 10th Anniversary Edition, there is a definition of the operator sum representation of a quantum operation: $\mathcal{E}(\rho)=\sum_{k}\langle e_k|U[\rho\otimes|e_0\rangle\langle ...
EugeneB's user avatar
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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 ...
joy Jaganath's user avatar
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2 answers
239 views

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 ...
Sooraj S's user avatar
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1 vote
1 answer
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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 $d^4−d^2$ ...
Sooraj S's user avatar
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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 $d^4−d^2$ ...
Sooraj S's user avatar
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1 vote
1 answer
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Prove that at most $d^2$ Kraus operators are sufficient to represent any quantum operation

All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $d$ can be generated by an operator-sum representation containing at most $d^2$ elements, $$ \mathcal{E}(\rho)=\sum_{k=1}^M ...
Sooraj S's user avatar
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Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$

In Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that Given a trace-preserving quantum operation expressed in the ...
Sooraj S's user avatar
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2 votes
1 answer
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Why can any quantum channel be represented as a matrix?

In this PDF (page 43), it is argued that, given an arbitrary quantum channel with Kraus decomposition: $$ E(\rho) = \sum_{j} K_j \rho K_j^{\dagger} $$ Such map can be represented with a matrix in $\...
NYG's user avatar
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1 answer
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Show the linearity of $(\langle a_m|\otimes I_B\otimes I_C\otimes \langle d_q|) U(I_{A}\otimes I_B\otimes |0_{C}\rangle\otimes |0_{D}\rangle)$

Suppose a composite system $AB$ initially in an unknown quantum state $\rho$ is brought into contact with a composite system $CD$ initially in some standard state $|0\rangle$, and the two systems ...
Sooraj S's user avatar
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2 answers
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Find an operator-sum representation for a depolarizing channel acting on 2qubit [duplicate]

In Nielsen and Chuang (page:379), it shows how to represent a 1 qubit depolarizing channel in operator-sum representation. $$ \mathcal{E}_1(\rho)=pI/2+(1-p)\rho =(1-3p/4)\rho+p/4(X\rho X+Y\...
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1 answer
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Why are Kraus operators $E_k=\langle e_k|U|e_0\rangle$ not just numbers?

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 ...
karry's user avatar
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1 answer
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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\...
Nichola's user avatar
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1 answer
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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 ...
Discord Warrior's user avatar
3 votes
1 answer
767 views

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: ...
glS's user avatar
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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 ...
SescoMath's user avatar
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1 answer
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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 ...
Charity Adu Amankwaa's user avatar
8 votes
3 answers
555 views

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 $\{I\}$...
Shadumu's user avatar
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Qiskit: Sum of Kraus operators are not equal to identity matrix

I know that one of the properties of the Kraus Operator is: $$\sum_k A_k^\dagger A_k=I\,.$$ So, in qiskit, I first converted my array to a super operator, and then I found my Kraus operators. However, ...
quest's user avatar
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6 votes
1 answer
133 views

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 ...
Root's user avatar
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4 votes
1 answer
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What is the technique for calculating $\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$?

I am stuck on calculating $\mathcal{E}(\rho)=\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$. For example, in the case when $U$ is the CNOT matrix $$U=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 &...
zdm's user avatar
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1 answer
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Qutip choi_to_kraus and to_kraus functions are not returning list of Kraus representation

I am trying to find my kraus representation from my process matrix. Suppose that, I have these process matrix: ...
quest's user avatar
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2 votes
1 answer
777 views

How to find the Kraus operators from the process matrix?

I am trying to find the Kraus operator from process matrix. For instance, suppose that for single qubit identity gate, I have the following process matrix: ...
quest's user avatar
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1 answer
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How to write the Kraus representation for many-qubit states?

The most general formula of Kraus operator on density matrix is: $$\rho\to \sum_k A_k^\dagger\rho A_k.$$ If I want to write this equation for one qubit, the most general way will be: $\rho_f = (a^*I+b^...
quest's user avatar
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