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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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 ...
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What does it mean "less than identity" in the operator sum representation?
In Quantum Computation and Quantum Information by Nielsen and Chuang, Section 8.2.3, $\mathcal{E}=\sum_{k}E_k\rho E_k^{\dagger}$ gives the operator-sum representation. In general, it requires $\sum_k ...
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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:
$$i[A,B]...
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Reset channels for two-qubit space
In the paper "Unbiased Simulation of Near-Clifford Quantum Circuits" by Ryan S. Bennink et al. (arXiv) the authors define a Pauli reset ("Pauli measurements followed by Clifford ...
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How does the CPTP constraint reflect on the matrix representation of a qubit channel in the Pauli basis?
Let us write the possible states of a qubit in the Bloch representation as
$$\newcommand{\bs}[1]{{\boldsymbol{#1}}}\rho_{\bs r}\equiv \frac{I+\bs r\cdot\bs \sigma}{2},$$
where $\bs\sigma=(\sigma_1,\...
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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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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, "...
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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♦
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What is the "Stinespring Dilation"?
I've consulted Nielsen and Chuang to understand the Stinespring Dilation, but wasn't able to find anything useful. How does this operation relate to partial trace, Kraus operators, and purification?
glS♦
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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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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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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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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) = \left(\text{Tr} \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2.
$$
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Infidelity as distance measure
Let $\mathcal{X} \in {\rm CP}(\mathcal{H}, \mathcal{K})$ and unital (compositive positive and unital maps). Let $\mathcal{Y} \in {\rm CPT}(\mathcal{H}, \mathcal{K})$(complete positive and trace ...
CommunityBot
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Fidelity bound under composed channels?
Suppose $\mathcal{E}_1, \mathcal{E}_2$ are CPTP maps. Is the following bound true?
$$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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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 $\rho$, ...
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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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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 ...
glS♦
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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-...
glS♦
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What would a master equation describe for a Hamiltonian $H=\alpha 1 + \beta \sigma_x$?
Consider the following master equation:
$$
\partial_{t} \rho = -i [H, \rho ] + \gamma (\sigma_{-} \rho \sigma_{+} - \frac{1}{2} \sigma_{+} \sigma_{-} \rho - \frac{1}{2} \rho \sigma_{+} \...
glS♦
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Bloch representation of a quantum channel acting on a 2-qubit density-matrix
A previous answer nicely show the relationship between the Pauli transfer matrix (PTM) and the Bloch representation of a quantum channel that acts on single-qubit density matrix.
In short, given a ...
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When performing a projective measurement on a subsystem X entangled with another system Y, can the evolution of Y be unitary?
I am interested in the evolution of a subsystem Y entangled with another subsystem X.
X and Y are initially in a pure product state. They undergo some global joint evolution E (not necessarily unitary)...
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Verification of local unitary equivalence between two pure states
This might be a non-trivial and hard problem. I've been thinking about this for days but couldn't find a good answer, so I hope any of you could give me a good answer/intuition for me to move forward.
...
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Invalid coefficients in Quantum Process Tomography of the Hadamard gate
I am trying to perform QPT for the Hadamard operation as outlined in this document. The basis they use is $$ \{ \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 ...
glS♦
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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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How can the depolarizing channel, defined as $\mathcal E(\rho) = (1-p)\rho + p\frac{I}{2}$, be a linear quantum operation? [duplicate]
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 ...
glS♦
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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\...
glS♦
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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 ...
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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 ...
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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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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♦
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Why are entanglement breaking channels, defined as $\Phi(\rho)=\sum_a \operatorname{Tr}(\mu(a)\rho)\sigma_a$, entanglement breaking?
Define an entanglement breaking channel $\Phi$ as a channel (CPTP map) of the form
$$\Phi(\rho) = \sum_a \operatorname{Tr}(\mu(a)\rho) \sigma_a\tag A$$
for some POVM $\{\mu(a)\}_a$ and states $\...
glS♦
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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 ...
glS♦
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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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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 ...
glS♦
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time evolution of Hamiltonian to generate the Bell pair
Consider two different Hamiltonians: $$H_1(t) = ZZ + \alpha(t)X_1 + \beta(t)X_2\quad\text{ and }H_2(t) = XX + \alpha(t)Z_1 + \beta(t)Z_2\,,$$ where $\alpha(t)$ and $\beta(t)$ are time-dependent ...
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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 $\...
glS♦
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Is the extension of a pure operation also a pure operation?
Given some quantum channel—or, more generally, some positive linear map—$\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 ...
glS♦
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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}...
glS♦
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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
$$\mathcal{E}(\rho) = \sum_{j=1}^n K_j \...
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Twirling Quantum Channels: Pauli and Clifford Twirling
I am currently working through some papers related with approximations of more general quantum channels such as amplitude and phase damping channels to Pauli channels. The reason to do so is so that ...
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Affine transformation of the Bloch sphere to Kraus representation of qubit channels
It is known that qubit channels can be written in the form:
$$
\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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Find the Kraus operators of a combined amplitude and phase damping channel
I am going through the paper Surface code with decoherence: An analysis of three superconducting architectures and I have a doubt about how the authors get what they refer to as the combined channel ...
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