Questions tagged [kraus-representation]
For questions relating to the Kraus decomposition of quantum channels.
87
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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 $p_j\in[0,1]$ ...
- 457
2
votes
1
answer
62
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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)\...
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0
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Lindblad under time rescaling: Zero noise extrapolation
This question is related to an earlier post Zero noise extrapolation for error mitigation: Meaning of rescaled density matrix, specifically when there is no local hamiltonian evolution
If I have the ...
2
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1
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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 ...
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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 ...
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0
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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 ...
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1
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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 ...
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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 $d^4−d^2$ ...
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1
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178
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Kraus operators required for operations on $d$ dimensional Hilbert space
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,
$$
\mathcal{E}(\rho)=\sum_{k=1}^M ...
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4
votes
1
answer
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Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$
In Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
Given a trace-preserving quantum operation expressed in the ...
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1
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Why can any quantum channel be represented as a matrix?
In this PDF (page 43), it is argued that, given an arbitrary quantum channel with Kraus decomposition:
$$ E(\rho) = \sum_{j} K_j \rho K_j^{\dagger} $$
Such map can be represented with a matrix in $\...
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Show the linearity of $(\langle a_m|\otimes I_B\otimes I_C\otimes \langle d_q|) U(I_{A}\otimes I_B\otimes |0_{C}\rangle\otimes |0_{D}\rangle)$
Suppose a composite system $AB$ initially in an unknown quantum state $\rho$ is brought into contact with a composite system $CD$ initially in some standard state $|0\rangle$, and the two systems ...
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2
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Find an operator-sum representation for a depolarizing channel acting on 2qubit
In Nielsen and Chuang (page:379), it shows how to represent a 1 qubit depolarizing channel in operator-sum representation.
$$
\mathcal{E}_1(\rho)=pI/2+(1-p)\rho
=(1-3p/4)\rho+p/4(X\rho X+Y\...
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1
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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 ...
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1
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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\...
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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 ...
3
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1
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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:
...
glS♦
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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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146
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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 ...
- 562
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1
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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 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 $$U=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 &...
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Qutip choi_to_kraus and to_kraus functions are not returning list of Kraus representation
I am trying to find my kraus representation from my process matrix.
Suppose that, I have these process matrix:
...
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1
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408
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How to find the Kraus operators from the process matrix?
I am trying to find the Kraus operator from process matrix.
For instance, suppose that for single qubit identity gate, I have the following process matrix:
...
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1
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243
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How to write the Kraus representation for many-qubit states?
The most general formula of Kraus operator on density matrix is:
$$\rho\to \sum_k A_k^\dagger\rho A_k.$$
If I want to write this equation for one qubit, the most general way will be:
$\rho_f = (a^*I+b^...
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0
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What's a "natural" way to show that, for unital channels, $\Phi(X)=X$ iff $[X,A_a]=0$ for all Kraus operators $A_a$?
This is a statement proved in Watrous, Theorem 4.25, page 229 of the online version.
Let $\Phi\in\mathrm C(\mathcal X)$ be a unital channel with Kraus representation $\Phi(X)=\sum_a A_a X A_a^\dagger$....
glS♦
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Is there an upper-bound on the operator norm (max-singular value) of the matrix representation of a quantum channel?
Suppose $\Phi$ is a CPTP map with Kraus operators $\phi_n$, so that $\hat{\Phi} := Σ_n (\phi_n ⊗ \phi_n^*)$ is the matrix representation (here $*$ being entry-wise complex conjugate).
Is there an ...
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Is the composition of two extremal channels also extremal?
In this question, I follow the terminology and notation of the book of Watrous, most notably chapter two.
Extremal channels
An extremal channel $\Phi(X) \in C(\mathcal{X},\mathcal{Y})$ is a channel ...
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Implication of SWAP being not positive in terms of quantum channel
I am going over chapter 3 of Preskill's lecture notes regarding complete positivity. Specifically, on page 19, it is mentioned that since SWAP has eigenstates with eigenvalue -1, it is not positive, ...
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What is the root of the non-trace-preserving bit-flip map
I have a quantum channel defined by the Kraus operators:
$$
U_1 =
\begin{bmatrix}
p & 0 \\
0 & p
\end{bmatrix},\quad
U_2 =
\begin{bmatrix}
0 & p \\
p & 0
\end{bmatrix}
$$
i.e.
$$
U_1\...
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votes
1
answer
270
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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 ...
glS♦
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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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How does the invertibility of a quantum map reflect on its Kraus operators?
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 ...
glS♦
- 21.7k
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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 ...
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0
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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 $\Lambda_{A}^{*}$...
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2
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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 ...
- 2,782
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votes
1
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389
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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(...
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votes
2
answers
591
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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
$$
\mathcal{E}(\rho) = \sum_{i=1}^...
- 5,770
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1
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613
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What are examples of Kraus operators describing the process of control error?
Noise, such as photon loss or dephasing, is often described with Kraus operators.
Are there examples of Kraus operators describing the process of control error?
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2
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Can Kraus operators change a mixed state into a pure state?
It seems that Kraus operators cannot change a pure state into a mixed one (wrong). For any pure state can be written as $|\psi\rangle\langle\psi|$, so after the Kraus operators. It becomes $$\sum_l\...
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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:
$$\operatorname{Tr}_{\rm env}[\hat{U}(|\psi\rangle\otimes|0\rangle)(\langle\psi|\otimes\langle 0|)\hat{U}^\dagger].
$$...
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