Questions tagged [kraus-representation]
For questions relating to the Kraus decomposition of quantum channels.
96
questions
2
votes
1
answer
31
views
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, ...
- 295
2
votes
0
answers
23
views
Kraus decomposition in the infinite-dimensional case
The action of a quantum channel $\Lambda$ on a generic state $\rho_S$ coupled to a pure environment state $|0\rangle_E$ can be written in the Stinespring representation as
$\Lambda(\rho_S) = \text{tr}...
- 21
0
votes
1
answer
56
views
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 $p$)
- 69
0
votes
1
answer
162
views
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 ...
2
votes
2
answers
44
views
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. $\...
- 237
1
vote
2
answers
145
views
Action of a CPTP map on Identity
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^+ =\...
- 153
0
votes
0
answers
47
views
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
1
answer
99
views
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 $X_A \equiv \sum_k |A_k\rangle\!\rangle\langle\!\langle A_k|$, ...
4
votes
1
answer
65
views
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, ...
- 199
2
votes
1
answer
164
views
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 $p_j\in[0,1]$ ...
- 657
2
votes
1
answer
99
views
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)\...
- 157
0
votes
0
answers
26
views
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
votes
1
answer
107
views
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 ...
- 157
3
votes
1
answer
277
views
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 ...
- 51
1
vote
0
answers
48
views
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 ...
- 614
0
votes
0
answers
56
views
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 ...
- 179
1
vote
1
answer
130
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 ...
- 791
1
vote
1
answer
203
views
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 $d^4−d^2$ ...
- 791
5
votes
2
answers
302
views
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 $d^4−d^2$ ...
- 791
1
vote
1
answer
256
views
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,
$$
\mathcal{E}(\rho)=\sum_{k=1}^M ...
- 791
4
votes
1
answer
309
views
Show that $E_k=(I\otimes\langle e_k|)U(I\otimes|e_0\rangle)$ implies $U=\begin{bmatrix}[E_1]&\cdots\\ [E_2]&\cdots\\\vdots&\ddots\end{bmatrix}$
In Page 365, Operator-sum representation, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that
Given a trace-preserving quantum operation expressed in the ...
- 791
2
votes
1
answer
205
views
Why can any quantum channel be represented as a matrix?
In this PDF (page 43), it is argued that, given an arbitrary quantum channel with Kraus decomposition:
$$ E(\rho) = \sum_{j} K_j \rho K_j^{\dagger} $$
Such map can be represented with a matrix in $\...
- 337
1
vote
1
answer
128
views
Show the linearity of $(\langle a_m|\otimes I_B\otimes I_C\otimes \langle d_q|) U(I_{A}\otimes I_B\otimes |0_{C}\rangle\otimes |0_{D}\rangle)$
Suppose a composite system $AB$ initially in an unknown quantum state $\rho$ is brought into contact with a composite system $CD$ initially in some standard state $|0\rangle$, and the two systems ...
- 791
2
votes
2
answers
177
views
Find an operator-sum representation for a depolarizing channel acting on 2qubit
In Nielsen and Chuang (page:379), it shows how to represent a 1 qubit depolarizing channel in operator-sum representation.
$$
\mathcal{E}_1(\rho)=pI/2+(1-p)\rho
=(1-3p/4)\rho+p/4(X\rho X+Y\...
- 29
1
vote
1
answer
342
views
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 ...
- 657
1
vote
1
answer
47
views
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\...
- 360
1
vote
1
answer
118
views
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
votes
1
answer
509
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♦
- 23.4k
2
votes
2
answers
262
views
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 ...
- 485
1
vote
1
answer
129
views
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 ...
8
votes
3
answers
418
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 $\{I\}$...
- 279
0
votes
0
answers
181
views
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 ...
- 614
5
votes
1
answer
112
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 ...
- 469
4
votes
1
answer
193
views
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 $$U=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 &...
- 197
2
votes
1
answer
150
views
Qutip choi_to_kraus and to_kraus functions are not returning list of Kraus representation
I am trying to find my kraus representation from my process matrix.
Suppose that, I have these process matrix:
...
- 614
2
votes
1
answer
569
views
How to find the Kraus operators from the process matrix?
I am trying to find the Kraus operator from process matrix.
For instance, suppose that for single qubit identity gate, I have the following process matrix:
...
- 614
0
votes
1
answer
314
views
How to write the Kraus representation for many-qubit states?
The most general formula of Kraus operator on density matrix is:
$$\rho\to \sum_k A_k^\dagger\rho A_k.$$
If I want to write this equation for one qubit, the most general way will be:
$\rho_f = (a^*I+b^...
- 614
4
votes
0
answers
276
views
What's a "natural" way to show that, for unital channels, $\Phi(X)=X$ iff $[X,A_a]=0$ for all Kraus operators $A_a$?
This is a statement proved in Watrous, Theorem 4.25, page 229 of the online version.
Let $\Phi\in\mathrm C(\mathcal X)$ be a unital channel with Kraus representation $\Phi(X)=\sum_a A_a X A_a^\dagger$....
glS♦
- 23.4k
3
votes
4
answers
743
views
Is there an upper-bound on the operator norm (max-singular value) of the matrix representation of a quantum channel?
Suppose $\Phi$ is a CPTP map with Kraus operators $\phi_n$, so that $\hat{\Phi} := Σ_n (\phi_n ⊗ \phi_n^*)$ is the matrix representation (here $*$ being entry-wise complex conjugate).
Is there an ...
- 53
5
votes
1
answer
179
views
Is the composition of two extremal channels also extremal?
In this question, I follow the terminology and notation of the book of Watrous, most notably chapter two.
Extremal channels
An extremal channel $\Phi(X) \in C(\mathcal{X},\mathcal{Y})$ is a channel ...
- 5,359
3
votes
1
answer
335
views
Implication of SWAP being not positive in terms of quantum channel
I am going over chapter 3 of Preskill's lecture notes regarding complete positivity. Specifically, on page 19, it is mentioned that since SWAP has eigenstates with eigenvalue -1, it is not positive, ...
- 73
5
votes
1
answer
288
views
What is the root of the non-trace-preserving bit-flip map
I have a quantum channel defined by the Kraus operators:
$$
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\...
- 157
6
votes
1
answer
341
views
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♦
- 23.4k
3
votes
1
answer
125
views
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 $\Phi$ ...
- 6,123
2
votes
1
answer
77
views
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 $\sum_k E_k^*E_k=I$). ...
- 1,893
6
votes
3
answers
377
views
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....
- 109
8
votes
3
answers
859
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
$$
\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}
$$...
- 613
7
votes
1
answer
1k
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 ...
- 613
3
votes
1
answer
1k
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 $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
- 165
2
votes
1
answer
2k
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 ...
- 58