Questions tagged [quantum-operation]

For questions about quantum channels or more generally quantum maps and the related formalism. For questions about unitary operations, please use quantum-gate instead.

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relationship between helstrom operators acting on different pairs of quantum states

Let $\rho_1, \rho_2, \rho_3, \rho_4$ be arbitrary single-qubit density matrices. Let $A$ be an Hermitian operator and its spectral decomposition as $A = \sum_i \lambda_i \lvert i \rangle \langle i \...
user185671631's user avatar
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How can the depolarizing channel be a quantum operation?

In Quantum Computing: From Linear Algebra to Physical Realizations it states that A quantum operation maps a density matrix to another density matrix linearly But let $\rho\in M_2$ be a density ...
John Hippisley's user avatar
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Is there a comprehensive list of counterexamples in quantum information?

As was already asked about in this phys.SE question many years ago---which, sadly, got closed and never received an answer---is there a collection of counterexamples in quantum information theory, &...
Frederik vom Ende's user avatar
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If $\text{tr}_B \rho \in A$, then $\rho \in A \otimes B$?

Let our Hilbert space be $H = (A \otimes B) \oplus (A \otimes B)^{\perp}$. If $\rho \in A \otimes B$, then we have $\text{tr}_B \rho \in A$. Is the converse true: if $\text{tr}_B \rho \in A$, then $\...
karavan's user avatar
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Resources for understanding non-unitary channels and operators

I need some resources to understand non-unitary channels and operators in depth in order to simulate non-unitary channels instead of unitary ones in some problems. I would appreciate any guidance or ...
Titan78's user avatar
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What is the rank of a superoperator of the form $\Xi (\cdot) = \sum_i^n U_i^\dagger {\cdot}\, U_i$?

Given a superoperator $\Xi$ as $\Xi (\cdot) = \sum_i^n U_i^\dagger \cdot U_i $ where $U_i$ are unitary. What can I say about the image of this map or about the rank of $\Xi$? Also, do you have some ...
relativeentropy's user avatar
7 votes
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What is the definition of physical gate error rate?

The fidelity of two quantum states $\rho$ and $\sigma$ is a well-defined (up to discussions about a square): $$ F(\rho, \sigma) = \text{Tr}\left( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2. $$ ...
Frederik Ravn Klausen's user avatar
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Why is the coefficient-squared the probability, and not just the coefficient itself?

Context: I have decided not to accept the postulates of quantum mechanics blindly as gospel. There must be a way someone arrived at those postulates, and I want to know the basic reasoning behind them,...
Abhay Agarwal's user avatar
4 votes
2 answers
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Show that all extensions of $\rho$ can be obtained as a channel applied to its purification

I am struggling with this exercise here: Let $H:A, H_E$ and $H_{E′}$ denote complex Euclidean spaces. Consider a purification $|ψ_{AE}⟩⟨ψ_{AE}| ∈ D(H_A ⊗ H_E)$ of a quantum state $ρ_A ∈ D(H_A)$ and a ...
Pink Elephants's user avatar
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qml.StronglyEntanglingLayers custom CNOT placement

The qml.StronglyEntanglingLayers function works great for what I need. However, I'd like to modify so that for each layer, only the first qubit is the control and the rest are targets of the control ...
TuktukTaxi's user avatar
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Trouble in Depolarizing Error Simulation with Qiskit

I'm currently attempting to simulate depolarizing errors using Qiskit, but I'm encountering an issue where it appears that no errors are being introduced into my simulation. After running the ...
Byeongyong Park's user avatar
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What is the smallest environment size that allows to represent every quantum channel in fixed dimensions?

From the Stinespring dilation, we have that the dual or complementary channel can be observed in/expressed with the environment. Can we reconstruct any channel for environments with $\text{dim}>1$ ...
Pink Elephants's user avatar
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How to get the Kraus decomposition of the amplitude damping channel from its Choi?

I found going from the Choi-matrix of a quantum channel to the Choi-Kraus decomposition a bit difficult. I know that it follows from the eigen-decomposition of the Choi-matrix. But I struggle with ...
Pink Elephants's user avatar
3 votes
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What can we say about the eigendecomposition of quantum channels?

It is known that quantum channels, being CPTP maps, map density operators to density operators. And thus, they can be seen as superoperators. Similar to operators, where eigenstates and eigenvalues ...
ironmanaudi's user avatar
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Can a CPTP map increase the purity of a state?

I am wondering if there exist CPTP maps $T$ such that the purity of a quantum state $\rho$ can increase, i.e. $$ \text{tr} ( T ( \rho )^2 ) \geq \text{tr} ( \rho ^2). $$ If so, what are the conditions ...
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Multimode unitary channel in terms of action on characteristic function

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
Phil K.'s user avatar
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Liouville superoperators of multi-qubit quantum channels : Need for Numerics

As greatly described here, a density matrix can be vectorized in column-major order such that a (unitary) channel can be written $$ \rho \rightarrow |\rho\rangle = vec(\rho) \\ \rho' = U \rho U^{\...
Onur Danaci's user avatar
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How do I prove the following maps are completely positive?

I am trying to prove that the following superoperators are quantum channels, that is completely positive and trace-perserving linear maps 1 $\Psi[M]=WMW^\dagger$ where $W$ is an isometry 2 $\Psi[M_A]=...
darkside's user avatar
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How to prove there's no quantum channel that clones all classical states?

Considering a qubit $\scr H =\Bbb C^2$ I have seen a proof of the no-cloning theorem for pure states. I wonder how do you prove it for a classical state? 1)That is, how do I prove that there is no ...
darkside's user avatar
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Physical Realizability of Outer product

I have a quantum state $|\Psi\rangle$ of $n$ qubits whose exact coefficients/probabilities I do not know. I want to operate another unknown quantum state $|\Phi\rangle$ with $U = a|\Psi\rangle \langle ...
Balaji sb's user avatar
1 vote
2 answers
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Find the Kraus operators for the amplitude damping channel from its isometric representation

I am currently learning about quantum channels and am sadly stuck at a rudimentary problem, where I don't understand how to find the Kraus matrices of a quantum channel. The amplitude damping channel ...
Alex1111's user avatar
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Show that the Choi of a tensor product is the tensor product of the Chois

I have the following problem. Let $N:L(H_A)\rightarrow L(H_A)$ be a quantum superoperator. The quantum state corresponding to this operator (via Choi-Jamiolkowski Isomorphism) is $\Gamma_A^{N}=id\...
Piotr Masajada's user avatar
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5 answers
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Does a quantum channel always preserve the identity matrix?

Does a quantum channel (a completely positive trace-preserving map) always map the identity to the identity? In other words, suppose that $ \mathcal{E}: \mathbb{C}^{N \times N} \to \mathbb{C}^{N \...
Ian Gershon Teixeira's user avatar
2 votes
1 answer
130 views

How many free real parameters in a general CPTP map?

The question is how many free real parameters a general CPTP map can maximally have. Let's assume the CPTP map $\Phi:L(\mathcal{H}_A) \rightarrow L(\mathcal{H}_B)$ is given in the Kraus representation ...
Tobias95's user avatar
4 votes
2 answers
242 views

Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

This problem is from a "passing remark" in this lecture notes. With the help of some colleagues I managed to find a way for this supposedly elementary fact, but I would like to see if there ...
Evangeline A. K. McDowell's user avatar
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1 answer
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Two-photon N00N state through Mach-Zehnder interferometer

I am interested in modelling a two-photon N00N state sent through a Mach-Zehnder interferometer, which consists of a beam-splitter (50:50), a phase shift operator on the first mode, a phase shift ...
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Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

There is another question asked on this on stack exchange but I did not find any answers there that fully answered the question. In Gottesman's paper "The Heisenberg Representation of Quantum ...
am567's user avatar
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Is the "unitary twirling operation" physically realizable?

In this neat answer by Markus Heinrich, it is shown that twirling an arbitrary quantum channel $\Lambda$ over the unitary group $U(d)$ yields a depolarizing channel $\tilde{\Lambda}$ given by $$ \...
Eric Kubischta's user avatar
1 vote
1 answer
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Affine transformation of the Bloch sphere to Kraus representation of qubit channels

It is known that qubit channels can be written in the form: $$ \begin{align} \Phi(\rho) = \frac{1}{2}\left(I+(T\vec{r}+\vec{t})\cdot\sigma\right)\ \end{align} $$ where $\vec{r}$ is the Bloch vector ...
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What is the meaning of complex expected values?

Introduction I am using BQSKIT to compile an approximation of a Toffoli gate (for testing purposes) and output to a QISKIT QuantumCircuit. I want to find the expected value of this approximation to ...
Shadow43375's user avatar
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Partial decoherence of a general one-photon state

Let $\rho_1$ be the pure one-photon state described by the ket $$|\psi_1\rangle = \int dk\ A(k)a^\dagger(k)|0\rangle$$ for a complex amplitude function $A(x)$ and an empty ket $|0\rangle$. This state ...
Bentanglement's user avatar
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Dimension reduction to subsystem in Pauli-Liouville basis: How to implement partial trace of Pauli-Transfer Matrices?

I have a complementary question to my previous (thankfully answered, but I can't verify for larger systems) question: Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-...
Onur Danaci's user avatar
1 vote
1 answer
185 views

Process matrix of CNOT gate

The fig below is the process matrix of the CNOT gate from this paper: where the legend explains that red corresponds to $\frac14$, green to $-\frac14$ and white to zero. I know the $U_{CNOT} = \frac{...
karry's user avatar
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1 answer
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Multi-qubit quantum channels in Pauli-Liouville basis: Tensor product of Pauli-Transfer Matrices?

I would like to verify something, need a sanity check. Are the quantum channels for different qubits in the Pauli-Liouville basis (Pauli Transfer Matrices) also given by a tensor product? The Kraus ...
Onur Danaci's user avatar
3 votes
2 answers
182 views

Equal partial traces

Given an arbitrary state $\rho_{AB}$, is it always possible to construct an extension $\rho_{ABC}$ such that $$Tr_B(\rho_{ABC}) := \rho_{AC} = \rho_{AB} := Tr_C(\rho_{ABC})$$ If yes, does there exist ...
user1936752's user avatar
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How do you work out the matrix for controlled-U operations?

I see this equations all over for controlled-U operations: $$ \left|{0}\right>\left<{0}\right| \otimes \mathbf{1} + \left|{1}\right>\left<{1}\right|\otimes U = \begin{pmatrix} \mathbb{1} &...
grepgrok's user avatar
4 votes
1 answer
271 views

Qiskit reverse_bits is not equivalent to swapping qubits

For three qubits, I thought swapping the first and third qubit is equivalent to vertically flipping the circuit. However, performing one and then the other in Qiskit returns two different operators, ...
Dylan VanAllen's user avatar
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1 answer
45 views

Obtaining the reduced density matrices for both subsystems of a bipartite system [duplicate]

If we have a single copy of a bipartite quantum system with density matrix $\rho$, is it possible to extract the reduced density matrices of the constituent subsystems separately, i.e. to achieve the ...
Bard's user avatar
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3 votes
1 answer
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Asymptotic purity from the spectrum of the Choi matrix?

I have a completely positive map $T$ and a sequence of $d\times d$ states $S_1,S_2,\ldots$ obtained by applying $T$ repeatedly to the identity matrix. I'm interested in quantifying what happens to ...
Yaroslav Bulatov's user avatar
2 votes
0 answers
36 views

Fisher information of parametric channel

Suppose $\Phi_\theta$ is a quantum channel whose action can be written for any state $\rho\in \mathcal S(\mathcal H_S)$ in the Stinespring representation as $\Phi_\theta(\rho)= \text{Tr}_E(U_\theta (\...
Quantastic's user avatar
2 votes
0 answers
24 views

Specifying the image of a set of states under the action of a channel

I have a generic channel $\mathcal{N}$ acting on a subspace of states defined on a $d$-dimensional Hilbert space $\mathcal{H}$. I am trying to make a statement about the dimension of the image of that ...
forky40's user avatar
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4 votes
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A measure of entanglement created by a unitary operation

Let $U$ be a unitary matrix acting on a 3-qubit system. If there is no correlation among any pairs of the three qubits, the unitary operation can be represented as $U = U_1 \otimes U_2 \otimes U_3$, ...
user185671631's user avatar
2 votes
1 answer
78 views

Is a process matrix of rank $1$ unique?

It is said that when an unknown process is unitary, its $\chi$ matrix is rank-$1$ and possesses only one positive eigenvalue. See eg https://arxiv.org/abs/2306.07867. So when the process matrix has ...
karry's user avatar
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4 votes
1 answer
237 views

How to characterize the extreme points of the set of CPTP maps?

The set of CPTP maps is convex, therefore, it is enough to perform the needed optimizations over the set of extreme points. Is there any way of characterizing the said extreme points that would lend ...
Cain's user avatar
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1 answer
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Question about Nielson & Chuang Problem 9.2

I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang. Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum ...
DJD's user avatar
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4 votes
1 answer
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What is known about the size of the spectral gap of unital quantum channels?

I am interested in the spectrum of unital quantum channels $\Phi$ (which act on finite dimensional spaces). The literature is extremely vast on such objects so I hope some experts can point me along ...
nervxxx's user avatar
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1 answer
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Derivation of Choi-Jamiolkowski isomorphism

I'm following the course Mathematical Methods of Quantum Information Theory by Reinhard Werner. In lecture 6, he gave a derivation of Choi-Jamiolkowski isomorphism, and I'm struggling to understand ...
Manuel E's user avatar
2 votes
1 answer
44 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, ...
am567's user avatar
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1 vote
1 answer
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What are the singular values of a quantum channel?

I have tried to find the explicit definition of them but was not able to. My guess is that they are eigenvalues of the superoperator $\Phi^{\ast}(\Phi)$, where $\Phi$ is the channel and $\Phi^{\ast}$ ...
trurl's user avatar
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3 votes
0 answers
40 views

Infidelity as distance measure

Let $\mathcal{X} \in {\rm CP}(\mathcal{H}, \mathcal{K})$ and unital (compositive positive and unital maps). Let $\mathcal{Y} \in {\rm CPT}(\mathcal{H}, \mathcal{K})$(complete positive and trace ...
Michael.Andy's user avatar

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