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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What does the inequality $\sum_k A_k^* A_k \leq I$ represent in Kraus' theorem?
Nielsen and Chuang's statement of Kraus' Theorem includes the inequality
$$ \sum_k A_k^* A_k \leq I.$$
What does this inequality represent? What quantities are being compared by this symbol?
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Finding solutions to commutation relation
In a Hilbert space of dimension $d$, a positive definite operator $A>0$ is given. I would be interested in the set of operators $B>0$ such that there exists some $C>0$ that satisfies:
$$i[A,B]...
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Reset channels for two-qubit space
In the paper "Unbiased Simulation of Near-Clifford Quantum Circuits" by Ryan S. Bennink et al. (arXiv) the authors define a Pauli reset ("Pauli measurements followed by Clifford ...
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Improving the Bound on $\|\Phi^k[\mathbf{I}_d] - \Phi^{k+1}[\mathbf{I}_d]\|$ for a CP Map
I am analyzing a completely positive (CP) linear map $\Phi : \mathbb{R}^{d \times d} \to \mathbb{R}^{d \times d}$ with the following properties:
$\Phi[\mathbf{I}_d] \preccurlyeq \mathbf{I}_d$,
All ...
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How to obtain $U$, representing the action of the amplitude dampening channel?
How to obtain $U$ for the amplitude dampening channel?
The behaviour of the amplitude damping channel on the system can be represented by $U$, a unitary matrix which has the following effect:
$U|00\...
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Quantum process tomography of a non-CPTP map
Let's say I have a theoretical/ computational process that inverts the outputs of a quantum channel and returns noise-free input density matrices. I perform quantum process tomography on my process ...
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Fidelity bound under composed channels?
Suppose $\mathcal{E}_1, \mathcal{E}_2$ are CPTP maps. Is the following bound true?
$$F(\mathcal{E}_1 \circ \mathcal{E}_2 (\rho), \rho) \stackrel{?}{\geq} F(\mathcal{E}_1(\rho), \rho) + F(\mathcal{E}_2(...
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How to switch between the definitions of Choi as $(\mathcal{N}_A \otimes id_B)(\phi^+_{AB})$ and $(id_A \otimes \mathcal{N}_B)(\phi^+_{AB})$?
For the channel $\mathcal{N}$ the Choi state can be either defined as $(\mathcal{N}_A \otimes id_B)(\phi^+_{AB})$ or $(id_A \otimes \mathcal{N}_B)(\phi^+_{AB}) $, where $\phi^+_{AB}$ is the maximally ...
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Why do completely positive maps satisfy ${\rm Tr}[\Psi(\rho)_++\Psi(-\rho)_+]\leq{\rm Tr}[\Psi(\rho_+)]+{\rm Tr}[\Psi((-\rho)_+)]$?
I am studying a paper of M. Plenio, "Logarithmic Negativity: A Full Entanglement Monotone That is not Convex", PRL 2005 [arXiv:quant-ph/0505071].
In the paper, I do not fully understand the ...
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Basic questions about google superconducting qubit system model
In this paper about Google superconducting qubit system, I want to understand the origin of equation (1), and the origin of the $\nu/2 Z$ term in equation (2). Regarding the first point, I want to ...
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Prove a linear map $\mathcal{N}$ is completely positive if its Choi operator is positive semi-definite
I'm doing exercise 4.4.1 in Quantum information theory by Wilde. The exercise asks to prove that a linear map $\mathcal{N}_{A\to B}$ is completely positive if its Choi operator is a positive semi-...
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Bloch representation of a quantum channel acting on a 2-qubit density-matrix
A previous answer nicely show the relationship between the Pauli transfer matrix (PTM) and the Bloch representation of a quantum channel that acts on single-qubit density matrix.
In short, given a ...
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Finding a density matrix for a distribution of pure states
Let $\theta$ be a Gaussian variable with mean 0 and variance 1. Then for $t>0$, the variable $\theta \sqrt{t}$ is also Gaussian with mean $0$ and variance $t$. Let $|\psi_0\rangle$ be an arbitrary ...
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What would a master equation describe for a Hamiltonian $H=\alpha 1 + \beta \sigma_x$?
Consider the following master equation:
$$
\partial_{t} \rho = -i [H, \rho ] + \gamma (\sigma_{-} \rho \sigma_{+} - \frac{1}{2} \sigma_{+} \sigma_{-} \rho - \frac{1}{2} \rho \sigma_{+} \...
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When performing a projective measurement on a subsystem X entangled with another system Y, can the evolution of Y be unitary?
I am interested in the evolution of a subsystem Y entangled with another subsystem X.
X and Y are initially in a pure product state. They undergo some global joint evolution E (not necessarily unitary)...
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Invalid coefficients in Quantum Process Tomography of the Hadamard gate
I am trying to perform QPT for the Hadamard operation as outlined in this document. The basis they use is $$ \{ \rho_{j} \} = \{|0\rangle \langle 0|, |1\rangle \langle 1|, |+\rangle \langle +|, |+i\...
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Mathematical properties used to derive Kraus operators
In this answer, it was very well explained why Kraus operators are not numbers as it might seem when reading Nielsen and Chuang for the first time. I have a minor, purely technical and probably simple ...
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What is the operator-sum representation of the two-qubit depolarizing channel?
I want to get the operator-sum representation of the two-qubit depolarizing channel
$$ \mathcal{E}(\rho) = (1-\lambda)\rho + \frac{\lambda I}{4}$$
Using $\frac{I}{2} = \frac{\rho +X\rho X +Y\rho Y +Z\...
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Can this minus operation be implemented via a quantum channel?
For an any given $\rho_1$ and $\rho_2=1-\rho_1$, the initial density matrix $\varrho_0$ is $\varrho_0=$Diag$[\rho_1,\rho_2,0]$.
(Assume $\rho_1>\rho_2$, or swap them)
Can we extract $\rho_2/2$ from ...
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Do POVM and generalized measurements really describe all possible measurements we can do on a quantum system (open dynamics)?
My understanding in the motivation of POVMs and generalized measurements is that they allow to describe more general measurements than projective ones, if we don't describe the environment around the ...
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What are "Nice" characterizations of every $n$-qubit CPTP
I would like to know if there is a "nice" way to parametrize the set of $n$-qubit CPTP map?
I guess I could parametrize it by using the channel state duality (to every CPTP I can attribute a ...
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Trouble understanding operator sum representation [duplicate]
I am having lot of trouble trying to understand the operator sum representation in Nielsen and Chuang:
I get the very first equation in above but how does that translate to 2 and how does the 3rd ...
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Tests for unitarity of a quantum channel that don't require the Choi state
I am aware that the rank of the Choi state is a test for whether or not a channel is unitary.
Are there any other equivalent tests that we can do to test if a channel is unitary without involving a ...
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In the Choi of the state preparation map, what is the maximally entangled state with the trivial space $\mathbb{C}$?
I am following Section 4.6.1 of Mark Wilde's book. The preparation map goes from a trivial input Hilbert space $\mathbb{C}$ to some output Hilbert space $\mathcal{H}_A$. Let it prepare the state $\...
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Is the extension of a pure operation also a pure operation?
Given some quantum channel—or, more generally, some positive linear map—$\Phi:\mathbb C^{n\times n}\to\mathbb C^{n\times n}$ one usually calls $\Phi$ a pure operation if for all $\rho\geq 0$ pure (...
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What are use cases for learning about unknown quantum states and processes?
I've seen numerous research papers about learning unknown quantum states or unitary operations from multiple copies of them. This includes fields such as quantum machine learning and tomography.
I’m ...
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Can a quantum operation inflate the Bloch sphere?
The depolarizing noise channel uniformly deflates the Bloch sphere to a single point, which is $\mathbf{n}= (0,0,0)$ or in terms of quantum qubit states, we get a maximally mixed state $\rho = \frac{1}...
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Could this map be a quantum channel?
A quantum channel is defined as a completely positive, trace-preserving (CPTP) linear map between spaces of operators.
Suppose we have the following operation
$$\mathcal{E}(\rho) = \sum_{j=1}^n K_j \...
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Where does the spectral decomposition $\mathcal M=\sum_k \lambda_k u_k\otimes v_k$ for a quantum channel come from?
I'm currently watching a lecture where the lecturer talks about unistochastic maps $\mathcal{M}$ that map $d\times d $ dimensional operators to other $d\times d$ dimensional operators.
He then goes on ...
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Heisenberg picture for a quantum measurement where the outcome is recorded
I asked a related question here. From the answer by Mateus, we see that one cannot describe a quantum measurement (where the measurement outcome is recorded) as a quantum map. That is because the ...
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What is the quantum map that describes a measurement where the measurement outcome is known?
My question is the other case of this question.
Suppose I do a measurement in the $Z$ basis with outcomes $\lambda_0, \lambda_1$ but I don't record the outcome. In this case, I can describe what ...
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Eigenvalues of Pauli Gate and connection to measurement
Suppose I measure a qubit in the $Z$ basis. If I measure and obtain the outcome $+1$, I get the post-measurement state $\vert 0\rangle\langle 0\vert$ and if I measure and obtain the outcome $-1$, I ...
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In exercise 8.23 of Nielsen and Chuang why is the quantum operation no longer trace-preserving?
My question is about this exercise from Nielsen & Chuang:
For context, $\mathcal{E}_{AD}$ is the quantum operation for amplitude damping on a single physical qubit, with operation elements given ...
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Unital qubit channels as a convex combination of entanglement-breaking and unitary channel
I am trying to show that for $T:B(\mathbb{C}^{2})\rightarrow B(\mathbb{C}^{2})$ a unital qubit channel, that T is a convex combination $T=pB+(1-p)Ad_{V}$, where B is a Entanglement-Breaking(EB) ...
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What does it mean for a gate to "commute with a measurement"?
I think I understand intuitively what this means. For example, I can apply an $X$ gate and then perform a measurement in the $X$ basis and I will get the same post-measurement states as if I measured ...
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Single-qubit quantum channel from the CNOT gate
I am studying quantum noise, chapter $8$ in Nielsen and Chuang. Section $8.2.2$ introduces an example for the definition of quantum operations, in particular the CX gate is introduced as an example. I ...
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unitary that transform $\sigma^x \pm \alpha \sigma^z$ into $\sigma^x$ and $\sigma^z$
Consider $\sigma^x \pm \sigma^z$, where $\sigma^x$ and $\sigma^z$ are Pauli $X$ and $Z$ matrices. Let unitary $U$ be a $\pi/4$ rotation matrix around $Y$-axis. Then,
\begin{equation}
U(\sigma^x + \...
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Nontrivial channel that is identity only on separable states?
Is there a channel that acts like the identity channel on any separable state but acts non-trivially on at least one entangled state? If not, is there a proof that such a channel cannot exist?
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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(...
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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^...
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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 ...
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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 ...
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Finding a succinct representation for the CPTP map ${\cal N}^{\otimes n}$ such that ${\cal N}(I)=I+pZ$ and ${\cal N}(Z)=(1-p)Z$
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 ...
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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}(...
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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 $e^{t \mathcal{L}}$, ...
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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 ...
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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 ...
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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.
\begin{align}
\mathcal{E}(\rho) &= \sum_k \langle e_k | U \...
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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 ...
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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&...
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