Questions tagged [quantum-operation]
For questions about completely positive (CP) linear maps between quantum states. Can also be used for trace-preserving CP maps (quantum channels). For questions about unitary operations, please use quantum-gate instead.
372
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Applications of quantum channel's different representations in photonic quantum computing?
We know that quantum channel has different representations, for example, Kraus representation and Stinespring representation. Do these representations have applications in photonic quantum computing ...
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Do the operators $\Phi(\sigma_k)$ form a basis if $\sigma_k$ do?
Assume we have a quantum channel $\Phi$. The single qubit Pauli basis is $\sigma_0, \sigma
_1, \sigma_2, \sigma_3$. Now we apply $\Phi$ to Pauli basis and get $\gamma_0=\Phi(\sigma_0), \gamma_1 = \Phi(...
- 337
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How to Implement Measure and Prepare Channel?
In arXiv:2207.14734v2 the authors introduce the following measure and prepare channel (Equ. 2):
$$ \Psi_0(X) = E_U \Bigl[ \sum_{j=1}^d \langle j | U^\dagger X U |j\rangle U |j \rangle \langle j | U^\...
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1
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Do quantum channels satisfy $\Phi(0)=0$?
Assume $\Phi$ is a quantum channel, which is a CPTP map. Does the following equality holds and why?
\begin{align}
\Phi(0) = 0.
\end{align}
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Why is the quantum capacity quantified by the coherent information?
Most types channel capacities associated to a given quantum channel are quantified using mutual informations (sometimes classical, sometimes quantum, sometimes regularised), which is not surprising ...
glS♦
- 21.3k
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Prove that the twirling operation on a channel gives a decomposition $\int dU\, U^\dagger\mathcal E(U\rho U^\dagger)U=\alpha P+\beta Q$
The twirled operation of a quantum channel $\mathcal E$ is defined as
\begin{align}
\mathcal E_T(\rho)
&= \int dU U^\dagger \mathcal E(U \rho U^\dagger)U,
\end{align}
where the integral is over ...
- 337
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1
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Prove the invariance upon change of variables in the definition of twirled channel
The twirled operation of a quantum channel $\varepsilon$ is defined as
\begin{align}
\varepsilon_T(\rho)
&= \int dU U^\dagger \varepsilon(U \rho U^\dagger)U,
\end{align}
where the integral is over ...
- 337
2
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1
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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)\...
- 117
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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 ...
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Zero noise extrapolation for error mitigation: Meaning of rescaled density matrix, specifically when there is no local hamiltonian evolution
I have a few questions regarding dynamics rescaling for zero noise extrapolation. In the paper Error mitigation for short-depth quantum circuits, in equation (30), they write
We redefine $T \...
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1
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What are examples of quantum maps with complex eigenvalues?
Chapter 6 of Michael Wolf's notes (MichaelWolf/QChannelLecture.pdf) discuss the structure of the spectrum of quantum maps and channels. However, it seems like the only explicit example given in the ...
glS♦
- 21.3k
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Is it possible to obtain a closed-form expression of the diamond distance between two single-qubit channels?
I would like to compute the diamond norm of the difference of two single-qubit channels $\Phi_1$ and $\Phi_2$. This difference is equal to, for any $2\times2$ complex matrix $\rho$:
$$\...
- 4,278
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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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Can two measurements be represented as a single measurement when they are acted upon sequentially?
Let two different POVM measurements represent as $\mathcal{M}_1=\{\Pi_i\}_{i=1}^k$ where $\Pi_i$ is element of the $\mathcal{M}_1$ measurement and $\mathcal{M}_2=\{E_j\}_{j=1}^n$ where $E_j$ is the ...
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Why is the operation in Nielsen and Chuang's Section 8.5 not a quantum operation?
In Section 8.5 of the 10th anniversary edition, Nielsen and Chuang discuss the limitations of the quantum operations framework. They give an example of a qubit prepared in an unknown state $\rho$, ...
- 171
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Deriving the choi matrix definition of the quantum depolarizing channel
I was reading up on the quantum depolarizing channel (Preskill's Notes) (stack exchaange explanation), and I'm failing to see how the form
\begin{align}
\sigma &= (\mathcal E \otimes \mathbb I)(|\...
- 31
5
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2
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363
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What is the difference between quantum gates and quantum channels?
I'm not sure if this is a dumb question, since they seem to be very basic building blocks of quantum information theory; however, I can't seem to wrap my head around the difference between the two. As ...
- 51
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1
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33
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time evolution of Hamiltonian to generate the Bell pair
Consider two different Hamiltonians: $H_1(t) = ZZ + \alpha(t)X_1 + \beta(t)X_2$ and $H_2(t) = XX + \alpha(t)Z_1 + \beta(t)Z_2$, where $\alpha(t)$ and $\beta(t)$ are time-dependent functions. Starting ...
- 413
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58
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Why Qiskit's circuit denotes the top bit as the least significant bit?
The descriptions can be found here:Qiskit Circuit. Where it says:
It really confuses me, because it's very inconvenient to run quantum circuits designed in a classical way in the physics textbook and ...
- 79
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0
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28
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Understanding adversarial Channels
The paper here (Definition 1) defines adversarial channels as $N(\rho)= \sum_i A_i \rho A_i$ with the mention that the $A_i$ is chosen only after a communication strategy is decide. This gives the ...
- 449
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1
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Is tomography of the Choi state sufficient for channel tomography?
Given that there is an isomorphism between quantum states and quantum channels (the Choi-Jamiolkowski isomorphism) and given that state tomography is well-researched, why is quantum process or quantum ...
- 2,447
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0
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87
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How comes the definition of Pauli transfer matrix?
In addition to the Kraus operator and Choi matrix, the Pauli transfer matrix (PTM) is another useful representation of a quantum map, its matrix entries are
$(R)_{ij}=\frac{1}{d}tr\{P_i\Lambda(P_j)\}$,...
- 437
4
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1
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119
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How to splice Hamiltonians corresponding to channels $\Phi_1$ and $\Phi_2$ so as to obtain a Hamiltonian corresponding to $\Phi_2\circ\Phi_1$?
Suppose I have two quantum channels $\Phi_1:B(\mathcal{H}_1)\rightarrow B(\mathcal{H}_2)$ and $\Phi_2:B(\mathcal{H}_2)\rightarrow B(\mathcal{H}_3)$, and let $\Phi=\Phi_2\circ \Phi_1$.
Stinespring ...
- 1,802
2
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1
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Does $\Phi(A^\dagger) = \Phi^\dagger(A)$?
Define linear map $\Phi \in T(X)$, linear operator $A \in L(X)$.
Is it true that
\begin{equation}
\Phi(A^\dagger) = \Phi^\dagger(A)?
\tag{1}
\end{equation}
What are the conditions that will let ...
- 337
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votes
1
answer
41
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An inequality involving quantum channels
Consider two quantum circuits $\mathsf{C}$ and $\mathsf{D}$ applied to $|0^n\rangle$. Then, measure in the standard basis and, for $x \in \{0, 1\}^n$, consider two probabilities:
\begin{equation}
p_{x,...
- 1,043
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1
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Quick conditions to see that $I - E_1 -E_2$ is positive for $E_1, E_2$ positives (example from Nielsen and Chuang)
In the Nielsen and Chuang ("Quantum Computation and Quantum Information"), section 2.2.6 POVM measurements, they define these three operators:
$E_1 = \frac{\sqrt{2}}{1+\sqrt{2}} |1\rangle \...
- 81
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2
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Converting $T_1$ and $T_2$ decay rates to noise supported by stim
Stim only supports Pauli noises like DEPOLARIZE1, DEPOLARIZE2, X_ERROR, ...
- 55
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1
answer
49
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Question regarding the monotonicity of Mutual Information of a tripartite state under multiple non-local commuting unitaries
Given a system $\rho_{AB}\otimes\rho_{C}$, and a unitary interaction $U_{BC}$, due to the monotonicity of the relative entropy under the actions of the partial trace map, $$I(A:B)=I(A:U_{BC}(B,C))\ge ...
- 1,087
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Why is error correction very different for circuits compared to channels?
Background
Suppose one wishes to communicate information using a noisy channel $N$ instead of an ideal channel $I$. The general framework to communicate reliably is
Take $n$ copies of $N$.
For some ...
- 2,447
3
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1
answer
181
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How to recover the original density matrix from the output of amplitude damping channel?
For amplitude damping, we have the below expression
$$\xi(\rho)=E_0\rho{E_0}^\dagger + E_1\rho{E_1}^\dagger.$$
How can I perform a matrix inverse operation on $\xi(\rho)$ at the receiver to get back ...
- 31
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1
answer
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How does the number of copies affect the diamond distance?
Suppose we are given two maps $\Phi$ and $\Psi$ such that
$$\|\Phi-\Psi\|_{\diamond}\leqslant\varepsilon.$$
What can we say about $\left\|\Phi^{\otimes t}-\Psi^{\otimes t}\right\|_{\diamond}$? Is it ...
- 4,278
1
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1
answer
26
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Creating a gate which have multiple register as an input
I'm trying to create a new gate in qiskit which the input is multiple register but the program output an error that says:
...
- 21
4
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1
answer
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What is the adjoint of the depolarizing channel?
Consider the single qubit depolarizing noise channel given by
$$\Phi(\rho) = \frac{\lambda}{d} \mathbb{I} + (1- \lambda) \rho.$$
What might be the adjoint $\Phi^{*}(\cdot)$ of this channel? In ...
- 315
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1
answer
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Complementary channel of binary sum channel
This isn't strictly a quantum question but the idea of complementary channels is the following: Take any channel $N_{A\rightarrow B}$. Take it's Stinespring dilation (which is an isometry) $V_{A\...
- 2,447
3
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1
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Does any quantum channel satisfy ${\rm Tr}(\mathcal E^\dagger \mathcal E) \in[0, d^2]$?
I am reading the paper "Direct Fidelity Estimation from Few Pauli Measurements".
According to the paper, the entanglement fidelity between the a unitary channel $\mathcal U$ and a quantum ...
- 337
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0
answers
49
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Given a POVM, what's the channel that optimally preserves coherence in the post-measurement outcomes?
It is well-known that a POVM $\boldsymbol\mu\equiv (\mu_a)_{a\in\Sigma}$ describes outcome probabilities, but not post-measurement outcomes, which in many scenarios exist and are of interest.
To ...
glS♦
- 21.3k
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1
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Is LOCC equivalence the same as LU equivalence?
I'm currently learning on LOCC transformations. In the Dur, 2000 paper, there is a statement that
(...) two pure states $|\psi\rangle$ and $|\phi\rangle$ can be obtained with certainty from each ...
- 163
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1
answer
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Why is it safe to ignore the phase factor when working with unitary operations? (and potentially elsewhere?)
After not understanding the explanation of the no-cloning theorem proof in my lecture notes I turned to Wikipedia, this explanation made more sense to me however it had an extra phase factor that is ...
- 131
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0
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noise and classical bits (measurements ) in qiskit
I am using qiskit and LocalNoisePass to add custom noise to the gates. But there is a problem when there are classical bits in the circuits. The localNoisePass give this error
...
- 111
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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 ...
- 667
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2
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Expectation value of Pauli Z for locally rotated Bell state
Suppose we have a Bell state $\frac{\lvert 00 \rangle + \lvert 11 \rangle}{\sqrt{2}}$. The expectation value of the Bell state with respect to $Z \otimes I $ is $\langle Bell|Z_1|Bell\rangle = 0$. Now,...
- 307
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Is there a CPTP map that takes $\rho_{AB}$ to $\rho_A\otimes\rho_B$?
Given some joint state $\rho_{AB}$, one can find either the marginal state $\rho_A$ or the marginal state $\rho_B$ through a CPTP map. The proof being that partial tracing is indeed CPTP.
Is a CPTP ...
- 2,447
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142
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Prove $\beta=\Lambda\otimes\Lambda$, where $\Lambda=\dfrac{1}{2}\begin{bmatrix}I&X\\X&-I\end{bmatrix}$ for single qubit tomography
In the Section on single qubit quantum process tomography, Box 8.5, Page 393, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, and in Prescription for experimental ...
- 667
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votes
1
answer
112
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Nielsen and Chuang: Solving equation of motion for amplitude damping
I would like to know how to obtain a solution to the equation of motion given in Section 8.4.1 Master equations of Nielsen and Chuang, 10th edition.
The equation of motion that allows getting the ...
- 2,041
1
vote
1
answer
55
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Why is $\chi$ not uniquely determined by $\sum_{mn}\beta_{jk}^{mn}\chi_{mn}=\lambda_{jk}$?
The mathematical construct of the Quantum process tomography is given in Page 391, 392, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, as follows
Let a fixed set of ...
- 667
2
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0
answers
57
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Can channels be generalized to linear maps from $\mathbb{C}^{n^k}$ to $\mathbb{C}^{n^k}$?
First let's set some terminology. Recall that a quantum channel is in particular a linear map $\Phi : \text{L} ( \mathcal{X}) \rightarrow \text{L} ( \mathcal{Y})$ where $\mathcal{X}$ and $\mathcal{Y}$ ...
- 51
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0
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117
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Matrix Representation of Quantum Channels
I am working on a project and I expect to have expressions of a bunch of quantum channels of interest. The quantum channels will be in matrix form. For example for a 2 qubit system, the quantum ...
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6
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0
answers
71
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Why is combined amplitude and phase damping considered sufficient for noise modeling?
In QECC literature, I often come across the "combined amplitude and phase damping channel" as being representative of a realistic noise model which makes sense (as amplitude damping and de-...
- 167
1
vote
1
answer
35
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How to create Parametrized Operator
How to transform the U3 gate of parameters (theta, phi, lambda) to an Operator. (in qiskit)
The following code don't work
...
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vote
1
answer
79
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Number of independent parameters in $\chi$ 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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