Questions tagged [kraus-representation]
For questions relating to the Kraus decomposition of quantum channels.
48
questions
1
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3answers
162 views
Quantum Operation - non trace increasing
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....
5
votes
2answers
228 views
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
$$
\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}
$$...
7
votes
1answer
433 views
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 ...
2
votes
1answer
98 views
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 $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
2
votes
1answer
88 views
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 ...
7
votes
0answers
156 views
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 ...
4
votes
1answer
115 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
0answers
41 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
2answers
109 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
1answer
49 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
2answers
184 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
$$
\mathcal{E}(\rho) = \sum_{i=1}^...
3
votes
1answer
97 views
What are examples of Kraus operators describing the process of control error?
Noise, such as photon loss or dephasing, is often described with Kraus operators.
Are there examples of Kraus operators describing the process of control error?
3
votes
2answers
151 views
Can Kraus operators change a mixed state into a pure state?
It seems that Kraus operators cannot change a pure state into a mixed one (wrong). For any pure state can be written as $|\psi\rangle\langle\psi|$, so after the Kraus operators. It becomes $$\sum_l\...
4
votes
1answer
93 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
1answer
203 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
1answer
47 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
2answers
96 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\...
2
votes
3answers
140 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 ...
2
votes
1answer
65 views
Why does $\sum_n \langle n|M_m\rho M_m^\dagger|n\rangle$ simplify to $\langle \psi|M_m^\dagger M_m|\psi\rangle$?
I was trying to derive the formula for $p(m)$ in exercise 8.2 on page 357 in Nielsen & Chuang. But I am wondering what rule I can apply to simplify this
$$\mathrm{tr}(\mathcal{E}_m(\rho) )= \...
0
votes
1answer
88 views
Derivation of Equation $8.7$ in Nielsen Chuang [duplicate]
Equation \eqref{eq:sp1} represents the reduced state of the system after tracing over environment.(Page number 358)
$$\mathcal{E}(\rho) = \mathrm{tr}_{env}(\lbrack U(\rho \otimes \rho_{env} )U^{\...
4
votes
1answer
81 views
Unitary over bipartite states that can turn a non-product state into a product state
Consider a bipartite quantum state $\rho_{AB}$ over a product of finite-dimensional Hilbert spaces $\mathcal{H}_A \otimes \mathcal{H}_B$. Does there exists a unitary $U$ over $\mathcal{H}_A \otimes \...
1
vote
1answer
79 views
Prove that different Kraus decompositions are related through a unitary, using the Choi isomorphism
I consider a process $\mathcal{E}$ that is at least CP and hermitian preserving.
I know that the Choi matrix then has the form:
$$ M = \sum_k |M_k \rangle \rangle \langle \langle M_k | $$
Where $|M_k \...
1
vote
1answer
96 views
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 ...
1
vote
0answers
47 views
Qiskit function phase_amplitude_damping_error
I am reading about the noise models that can be simulated in Qiskit and I found out the phase_amplitude_damping_error function. I read about it and it seems to be a function that simulates a combined ...
4
votes
1answer
159 views
How to find the unitary operation of a depolarizing channel?
Suppose we have a depolarizing channel operation
$$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$
acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
2
votes
1answer
80 views
Finding the irreducible representation of Kraus operators of a dephasing channel
I would like to understand an example of finding a noiseless subsystem of a quantum channel from the irreducible representation of its Kraus operators.
Assume we have $2$ dephasing channels acting on ...
2
votes
1answer
100 views
Can any channel be written as $\Phi(X)=\operatorname{Tr}_{\mathcal Z}[U(X\otimes \sigma)U^\dagger]$ for any state $\sigma$?
We know that every CPTP map $\Phi:\mathcal X\to\mathcal Y$ can be represented via an isometry $U:\mathcal X\otimes\mathcal Z\to\mathcal Y\otimes\mathcal Z$, as
$$\Phi(X) = \operatorname{Tr}_{\mathcal ...
0
votes
1answer
70 views
What is the Kraus representation of quantum-to-classical channels?
As discussed in Watrous' book, quantum-to-classical channels are CPTP maps whose output is always fully depolarised. These can always be written as
$$\Phi_\mu(X) = \sum_a \langle X,\mu(a)\rangle E_{a,...
2
votes
4answers
178 views
Can a Kraus representation act as the identity on any operator?
In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A,
$$\...
2
votes
2answers
225 views
Find the Kraus operators of a combined amplitude and phase damping channel
I am going through the paper Surface code with decoherence: An analysis of three superconducting architectures and I have a doubt about how the authors get what they refer to as the combined channel ...
2
votes
2answers
293 views
Prove that the depolarizing channel is completely positive
In two dimensions, for a density operator $\rho$ and probability $\lambda$, a depolarizing channel can be written as:
$$\mathcal{E}(\rho) = (1-\lambda) \frac{\mathbb{I}}{2} + \lambda\rho$$
In ...
1
vote
2answers
61 views
Prove that the partial trace is a quantum operation, finding its Kraus representation
I am referring to Nielsen and Chuang Quantum Computation and Quantum Information 10th Anniversary Edition Textbook, Chapter 8.3.
A linear operator $E_i:H_{QR}\longrightarrow H_Q $ is defined by:
$$...
3
votes
1answer
65 views
Implementing sum on Boolean with Grover algorithm
We are trying to implement a "sum over 4 booleans = k" in the spirit of Grover search. First, we have 4 qubits, one for each boolean q00, q01, q02, q03,; then 4 ...
1
vote
0answers
145 views
Affine Map of the Bloch sphere
I am referring to Equation (8.89) to (8.92) in Chapter 8 of "Quantum Computing and Information 10th Anniversary Edition" by Nielsen and Chuang. This section deals with the geometric picture of single ...
1
vote
1answer
214 views
How does qiskit finally implement a noise model?
I have been reading qiskit documentation for hours and I still don't get how does it implements noise in the circuit. I have understood that it works with a objects of the class QuantumError which ...
4
votes
1answer
200 views
What does it mean "less than identity" in the operator sum representation?
In Quantum Computation and Quantum Information by Isaac Chuang and Michael Nielsen, section 8.2.3, $\mathcal{E}=\sum_{k}E_k\rho E_k^{\dagger}$ gives the operator-sum representation. In general, it ...
1
vote
1answer
56 views
Kraus decomposition for non trace preserving operation: shouldn't we have $0 \leq \sum_k E_k^{\dagger} E_k \leq I$
In N&Chuang, on page 368 is written the following theorem:
The map $\mathcal{E}$ satisfies axioms A1,A2,A3 if and only if
$$\mathcal{E}(\rho)=\sum_k E_k \rho E_k^{\dagger}$$
Where $\sum_k ...
5
votes
3answers
513 views
How is the partial trace related to the operator sum representation? [duplicate]
In Quantum Computation and Quantum Information by Nielsen and Chuang, the authors introduce operator sum representation in Section 8.2.3. They denote the evolution of a density matrix, when given an ...
4
votes
2answers
637 views
Do the Kraus operators of a CPTP channel need to be orthogonal?
Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map.
Any such channel admits a Kraus decomposition of the form
$$\Phi(X)=\sum_a A_a X A_a^\dagger,$$
for a set of operators $A_a\in\mathrm{Lin}(\...
6
votes
1answer
442 views
Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?
If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that
$$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$
in the ...
5
votes
2answers
310 views
Find the quantum operation corresponding to a given unitary evolution and projective measurement
I'm trying to (understand and) solve this problem from Nielsen and Chuang's Quantum Computation and Quantum Information.
I know the definition of Operation Elements: $\sum_{k} E_k \rho E_k^†$ with $...
3
votes
1answer
183 views
Direct derivation of the Kraus representation from the natural representation, using SVD
$\newcommand{\Y}{\mathcal{Y}}\newcommand{\X}{\mathcal{X}}\newcommand{\rmL}{\mathrm{L}}$As explained for example in Watrous' book (chapter 2, p. 79), given an arbitrary linear map $\Phi\in\rmL(\rmL(
\X)...
5
votes
1answer
676 views
How does the vectorization map relate to the Choi and Kraus representations of a channel?
I know that the Choi operator is a useful tool to construct the Kraus representation of a given map, and that the vectorization map plays an important role in such construction.
How exactly does the ...
5
votes
2answers
312 views
How does $\mathcal E(\rho)=\mathrm{Tr}_{env}[U(\rho\otimes\rho_{env})U^\dagger]$ turn into $P_0\rho P_0+P_1\rho P_1$?
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation:
$$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$...
4
votes
2answers
2k views
How to find the operator sum representation of the depolarizing channel?
In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity ...
5
votes
1answer
1k views
Deduce the Kraus operators of the dephasing channel using the Choi
I'm trying to deduce the Kraus representation of the dephasing channel using the Choi operator (I know the Kraus operators can be guessed in this case, I want to understand the general case).
The ...
4
votes
1answer
389 views
Tensor product properties used to obtain Kraus operator decomposition of a channel
I work on a Quantum Information Science II: Quantum states, noise and error correction MOOC by Prof. Aram Harrow, and I do not understand which property of tensor products is used in one of the ...
8
votes
2answers
569 views
Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?
I understand that there are two ways to think about 'general quantum operators'.
Way 1
We can think of them as trace-preserving completely positive operators. These can be written in the form
$$\...
