Questions tagged [kraus-representation]

For questions relating to the Kraus decomposition of quantum channels.

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31 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(...
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2answers
147 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}^...
2
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1answer
51 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?
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141 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
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1answer
80 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: $$\operatorname{Tr}_{\rm env}[\hat{U}(|\psi\rangle\otimes|0\rangle)(\langle\psi|\otimes\langle 0|)\hat{U}^\dagger]. $$...
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1answer
197 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
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1answer
39 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=...
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2answers
84 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
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3answers
109 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
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1answer
60 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) )= \...
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1answer
79 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^{\...
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1answer
79 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 \...
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1answer
59 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 \...
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1answer
89 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 ...
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0answers
44 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
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1answer
141 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\...
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1answer
71 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
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1answer
86 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 ...
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1answer
67 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,...
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4answers
155 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, $$\...
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2answers
195 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 ...
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2answers
239 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 ...
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2answers
57 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: $$...
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1answer
63 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 ...
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0answers
134 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 ...
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1answer
188 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 ...
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1answer
186 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 ...
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1answer
53 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 ...
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3answers
463 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 ...
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2answers
541 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}(\...
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1answer
425 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
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2answers
307 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 $...
2
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1answer
159 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
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1answer
606 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
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2answers
287 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: $$\mathcal E(\rho) = \mathrm{Tr}_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$...
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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
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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
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1answer
380 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 ...
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537 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 $$\...