Skip to main content

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.

Filter by
Sorted by
Tagged with
1 vote
0 answers
27 views

Generators of positive quantum evolution

It is known that a generator of completely positive evolution, a Lindbladian $\mathcal{L}$, can always be represented in the following form: \begin{equation*} \mathcal{L}(\rho) = -i [H,\rho] + \Phi(...
trurl's user avatar
  • 141
-1 votes
0 answers
9 views

Is there any method to obtain the post projection state using Qiskit?

As shown in image, I have two qubit state and a projection state. I want to obtain the normalized state after the projection operation using Qiskit. There is a way to do that using ancilliary qubit ...
Piyush Verma 19349's user avatar
3 votes
2 answers
44 views

Does $N(U\rho U^\dagger)=U' N(\rho)U'^\dagger$ for unitaries $U,U'$ and a channel $N$ imply $UK_i=K_i U'$?

Let $H_A, H_B$ be Hilbert spaces and let a channel $N_{A\rightarrow B}$ be a CPTP map between them. If there exist that unitaries $U\in H_A$ and $U'\in H_B$ such that for all $\rho\in H_A$ $$N(U\rho U^...
user1936752's user avatar
  • 3,075
1 vote
1 answer
22 views

Are peripheral eigenvalues of a completely positive map always semisimple?

It is known that all peripheral eigenvalues (i.e. all eigenvalues $\lambda\in\mathbb C$ such that $|\lambda|$ equals the spectral radius) of positive trace-preserving or positive unital maps are ...
Frederik vom Ende's user avatar
4 votes
1 answer
71 views

What is the domain of the dual map of a quantum channel?

Possibly a naive question...if the dual map of a quantum channel gives the evolution of the system in the Heisenberg picture by acting on observables, and observables are self-adjoint operators on the ...
Mara Jade's user avatar
1 vote
1 answer
27 views

Finding a succinct representation of a CPTP map

Consider a single qubit CPTP map $\mathcal{N}$ such that $$\mathcal{N}(I) = I + pZ,~~~~~~\mathcal{N}(Z) = (1-p)Z,$$ where $I$ and $Z$ are Pauli operators. For an $n$ qubit Pauli operator $P$, made ...
BlackHat18's user avatar
  • 1,363
1 vote
1 answer
33 views

Are Stinespring unitaries that give rise to the same channel locally unitarily equivalent?

It is well known that for if any two linear maps $V_1,V_2:\mathbb C^n\to\mathbb C^k\otimes\mathbb C^m$ (isometry or not) satisfy $$ {\rm tr}_{\mathbb C^m}(V_1(\cdot)V_1^\dagger)={\rm tr}_{\mathbb C^m}(...
Frederik vom Ende's user avatar
3 votes
1 answer
145 views

Do all Hermiticity-preserving maps generate completely positive maps?

I am confused about what kinds of maps are valid infinitesimal generators of completely positive maps. I know that any Markovian completely positive map can be written in the form $e^{t \mathcal{L}}$, ...
nlupugla's user avatar
0 votes
1 answer
28 views

Given $\Psi$ completely positive when do there exist $K_1,K_2$ such that $K_2\Psi(K_1^\dagger(\cdot)K_1)K_2^\dagger$ is also trace preserving?

In quantum information it occasionally happens that one ends up with a completely positive but not yet trace-preserving map $\Psi$ which one wants to make trace preserving somehow; this often comes up ...
Frederik vom Ende's user avatar
1 vote
2 answers
142 views

Definition of a quantum gate

A quantum gate is usually defined as a unitary transformation, like the definition found in "Mathematics of Quantum Mechanics" by Scherer. According to this definition, can we consider a ...
Josh's user avatar
  • 407
2 votes
3 answers
179 views

Why do we need/have the operator sum representation (Kraus representation)?

I am reading through Nielsen & Chuang, and I am on the section about operator sum representation. They performed this derivation. Why is it important and useful for us to bundle together the ...
researcher101's user avatar
0 votes
1 answer
39 views

Resource for geometric representation of quantum channels

I was wondering if anyone knows about any good resources on representing unital/quantum channels by using rotations/pauli matrices. It is mentioned in Nielsen&Chuang on p774, but i feel it is ...
Pink Elephants's user avatar
2 votes
1 answer
38 views

Is the adjoint of a strictly positive channel again strictly positive?

Building on the concept of positive definite operators${}^1$—denoted $A>0$—a linear map $\Phi:\mathbb C^{n\times n}\to\mathbb C^{k\times k}$ is called strictly positive if $\Phi(A)>0$ for all $A&...
Frederik vom Ende's user avatar
1 vote
0 answers
11 views

Solving KnapSack problem on D Wave hybrid CQM

I am solving 01 KnapSack problem for 500k items with the help of hybrid CQM solver of D Wave. And for comparison I solved same problem with CPLEX. Solution quality of CPLEX solver is better than D-...
Maths_hawk's user avatar
3 votes
3 answers
253 views

How to mathematically describe the action of CNOT on the control qubit alone?

Basically the title. From what I know, starting from the $|{+0}\rangle$ state where the reduced density matrix of the first qubit $|+\rangle$ is $\frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{...
Jack Nathan's user avatar
0 votes
0 answers
37 views

How to interpret result for an OR problem solved in Qiskit

I solved "01 KnapSack problem" in Qiskit optimization module. Though the iterations doesn't gets converged but I got an answer (may be infeasible). But my question is that how to interpret ...
Maths_hawk's user avatar
3 votes
1 answer
86 views

What are kraus operators of a qubit interacting a thermal environment?

Suppose a qubit that interacting a thermal environment. The thermal environment can be a thermal field for example. What is the kraus operators for this case?
reza's user avatar
  • 733
1 vote
1 answer
67 views

Can the spectral radius of a completely positive map exceed the spectral radius of its transition matrix?

Recalling the spectral radius $r(T):=\max_{\lambda\in\sigma(T)}|\lambda|$ of a linear map $T$ (where $\sigma(T)$ refers to the spectrum of $T$), it is known that every quantum channel $\Phi:\mathbb C^{...
Frederik vom Ende's user avatar
4 votes
0 answers
41 views

What are examples of channels whose Holevo capacity can be computed explicitly?

Given a channel $\Phi:\operatorname{Lin}(\mathbb{C}^n)\to\operatorname{Lin}(\mathbb{C}^m)$, we define its Holevo capacity as $$\chi(\Phi) = \sup_\eta \chi(\Phi(\eta)),$$ with the sup taken with ...
glS's user avatar
  • 25.6k
2 votes
2 answers
90 views

Kraus decomposition of merging in lattice surgery

I am reading about lattice surgery from this paper. I am interested in the merge operation which takes 2 qubits to 1 qubit. I want to understand the logical-level Kraus operation that the merge does. ...
qubit's user avatar
  • 23
0 votes
0 answers
22 views

Transforming spin operators into fermionic operators and finding their anticommutation relations

The Jordan-Wigner transformation (JWT) is a method used in quantum mechanics to map spin operators, which are typically associated with spin-1/2 particles, to fermionic operators, which describe ...
amirhoseyn Asghari's user avatar
3 votes
1 answer
157 views

For how many different times do I have to know that $e^{tL}$ is a quantum channel to conclude that $L$ is of Lindblad form?

As first shown by Gorini, Kossakowski, Sudarshan and Lindblad given some linear map $\mathcal L:\mathbb C^{n\times n}\to\mathbb C^{n\times n}$, $e^{t\mathcal L}$ is a quantum channel for all $t\geq 0$ ...
Frederik vom Ende's user avatar
1 vote
1 answer
36 views

Is every quantum channel covariant with respect to some non-trivial Hamiltonian?

When asking whether every channel is covariant with respect to some non-trivial unitary channel I mean the following: Does there for every CPTP map $\Phi:\mathbb C^{n\times n}\to\mathbb C^{n\times n}$...
Frederik vom Ende's user avatar
1 vote
1 answer
52 views

To what extent is the normal form of the Pauli transfer matrix unique?

In order to properly state the question let me be precise about the object at the core of this question's title. First, given any orthonormal basis of $G:=\{G_j\}_{j=1}^{n^2}$ of $\mathbb C^{n\times n}...
Frederik vom Ende's user avatar
0 votes
1 answer
38 views

Solving optimization problems on real quantum hardware

I want to know how to solve a 01 Knapsack problem (or any optimization problem) on real Quantum Hardware only. I don't want to use an Application class or any classical simulation technique. If anyone ...
Maths_hawk's user avatar
1 vote
1 answer
49 views

IBM quantum computer backend cycle time and real gate duration

I am new to dynamical decoupling and is trying to study this from qiskit: https://docs.quantum.ibm.com/api/qiskit/qiskit.transpiler.passes.PadDynamicalDecoupling . There, they are specifying the ...
Tianqi Chen's user avatar
5 votes
1 answer
185 views

Is the trace of a positive map always positive?

Obviously, positive semi-definite operators always admit a positive trace as ${\rm tr}(A)=\|A\|_1\geq 0$ whenever $A\geq 0$. This motivates the following "lifted" question: Given any ...
Frederik vom Ende's user avatar
2 votes
1 answer
42 views

If states are close together does there always exist a channel close to the identity mapping one to the other?

Question: Given states $\rho,\omega\in\mathbb C^{n\times n}$ and $\varepsilon>0$ such that $\rho$ and $\omega$ are $\varepsilon$-close in trace norm does there exist a channel $\Phi$ with $\Phi(\...
Frederik vom Ende's user avatar
2 votes
0 answers
58 views

How to systematically find the kernel of a channel from its Kraus operators?

A quantum channel is a completely positive trace-preserving map. Given a quantum state $\rho$ and channel $N$, let the output be $N(\rho)$. Given the Kraus operators of the channel, how can one find ...
user1936752's user avatar
  • 3,075
0 votes
0 answers
17 views

Quantum Channel with least disturbance for any input and output dimensions

Let $n$ and $m$ be two arbitrary dimensions of the input Hilbert space and output Hilbert space respectively. What is the quantum channel that preserves information as much as possible (i.e. with the ...
Shadumu's user avatar
  • 321
1 vote
1 answer
106 views

How to view operator norms on open-system representation of quantum channels

I know how the operator norms $\| X\|_{1}$,$\| X\|_{2}$, and $\| X\|_{\infty}$ are defined for any operator $X\in B(\mathcal{H})$. My question is about how to view$\| T(X)\|_{1}$,$\| T(X)\|_{2}$, and ...
Pink Elephants's user avatar
0 votes
1 answer
28 views

Can any separable $\rho=\sum_i p_i\sigma_i\otimes\tau_i$ be written as $\rho=(I\otimes T)(\sum_ip_i\sigma_i\otimes|i⟩\!⟨i|)$ for some channel $T$?

I am struggling with the following exercise, and was wondering if anybody had any good tips on how to attack the problem/where to begin: Given a separable quantum state $$\rho_{AB'}=\sum_{i=1}^{k}p_{i}...
Pink Elephants's user avatar
5 votes
1 answer
63 views

Does monotonicity of diamond distance hold for intermediate channels?

It is well known that $\|\mathcal{E} \circ \mathcal{F} - \mathcal{E}\|_\lozenge \leq \|\mathcal{F} - \mathcal{I}\|_\lozenge$. What if I have $\|\mathcal{A} \circ \mathcal{E} \circ \mathcal{F} - \...
usermm's user avatar
  • 101
5 votes
1 answer
114 views

What is the meaning of $\sum_i K_iK_i^\dagger$ for a quantum channel with Kraus operators $K_i$?

Let a channel $N$ be given in terms of its Kraus operators $K_i$ as $$N(\rho) = \sum_i K_i\rho K_i^\dagger.$$ Is the sum $\sum_i K_iK_i^\dagger$ a meaningful quantity? I know that $\sum_i K_i K_i^\...
Pluto's user avatar
  • 53
1 vote
1 answer
32 views

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
2 votes
1 answer
323 views

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
20 votes
6 answers
1k views

Counterexamples in quantum information theory

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
1 vote
1 answer
98 views

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
  • 21
2 votes
2 answers
119 views

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
  • 31
2 votes
0 answers
33 views

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
8 votes
2 answers
106 views

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
3 votes
1 answer
169 views

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
103 views

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
1 vote
0 answers
29 views

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
1 vote
0 answers
49 views

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
1 vote
1 answer
61 views

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
4 votes
1 answer
95 views

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
4 votes
2 answers
167 views

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
6 votes
3 answers
430 views

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 ...
Rell's user avatar
  • 61
0 votes
0 answers
15 views

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

1
2 3 4 5
11