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.

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1answer
44 views

What is the certain error rate in a quantum channel?

Quantum error correction is a fundamental aspect of quantum computation. I have read some material about "Quantum Channel" and "Quantum error correction". I have known the formula ...
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1answer
37 views

Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)

This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
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2answers
32 views

Calculate probability of a state after depolarization

Let's say I have a particle in the quantum state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, represented as a density operator (1st matrix) that went through a depolarizing chanel (2nd ...
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1answer
85 views

Prove that $A\preceq B$ implies $A=\Psi(B)$ for some channel $\Psi$

Define $\newcommand{\PP}{\mathbb{P}}\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\tr}{\operatorname{tr}}\newcommand{\ketbra}[1]{\lvert #1\rangle\!\langle #1\rvert}\PP_\psi\equiv\ketbra\psi$, and ...
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2answers
76 views

Why do quantum operations need to be reversible?

Why do quantum operations need to be reversible? Is it because of the physical implementation of the qubits or operators neesd to be unitary so that the resultant states fit in the Bloch sphere?
2
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1answer
29 views

Given a channel $\Phi(X)=\sum_k c_k(X)\sigma_k$, are there always $F_k\ge0$ such that $\Phi(X)=\sum_k \operatorname{tr}(F_k X)\sigma_k$?

Fix a finite number of states $\sigma_k$, and consider a channel of the form $$\Phi(X)=\sum_k c_{k}(X)\sigma_k.$$ For $\Phi$ to be linear and trace-preserving we must have: $$c_k(X+X') = c_k(X) + c_k(...
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1answer
33 views

Can any channel be written as $\Phi(X)=\operatorname{Tr}_{\mathcal Z}[U(X\otimes \sigma)U^\dagger]$ for any state $\sigma$?

We know that every CPTP map $\Phi:\mathcal X\to\mathcal Y$ can be represented via an isometry $U:\mathcal X\otimes\mathcal Z\to\mathcal Y\otimes\mathcal Z$, as $$\Phi(X) = \operatorname{Tr}_{\mathcal ...
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1answer
39 views

What is the Kraus representation of quantum-to-classical channels?

As discussed in Watrous' book, quantum-to-classical channels are CPTP maps whose output is always fully depolarised. These can always be written as $$\Phi_\mu(X) = \sum_a \langle X,\mu(a)\rangle E_{a,...
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3answers
93 views

What does the adjoint of a channel represent physically?

Given a quantum channel (CPTP map) $\Phi:\mathcal X\to\mathcal Y$, its adjoint is the CPTP map $\Phi^\dagger:\mathcal Y\to\mathcal X$ such that, for all $X\in\mathcal X$ and $Y\in\mathcal Y$, $$\...
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0answers
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Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$?

Given the state of a system as $\rho_s$ and that of the ancilla (pointer) as $\rho_a$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective ...
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3answers
91 views

Can a Kraus representation act as the identity on any operator?

In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A, $$\...
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1answer
42 views

Confused regarding explanation of Schumachers compression in N&C

On page 547 of N&C, for $|\psi_{0}\rangle=|0\rangle$ and $|\psi_{1}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and for $|\tilde{0}\rangle=\cos(\pi/8)|0\rangle+\sin(\pi/8)|1\rangle$ and $|\tilde{1}\...
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2answers
60 views

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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55 views

Coherent Information and Entanglement Breaking channels

The book by John Watrous, "The Theory of Quantum Information" is an exciting read for anyone wanting to research quantum information theory. The following question presumes some background ...
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0answers
57 views

How to compute the capacity of a quantum channel from its Kraus operators?

Is there a working rule to compute the capacity of a quantum channel described by a set of Kraus operators $\{K_i\}$?
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2answers
100 views

Prove that the depolarizing channel is completely positive

In two dimensions, for a density operator $\rho$ and probability $\lambda$, a depolarizing channel can be written as: $$\mathcal{E}(\rho) = (1-\lambda) \frac{\mathbb{I}}{2} + \lambda\rho$$ In ...
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1answer
28 views

Confusion over HSW theorem depicted in Nielsen and Chuang

On page 560, it states that $$C^{(1)} \geq S(\frac{\varepsilon(|{\psi}\rangle\langle{\psi}|) +\varepsilon(|{\varphi}\rangle\langle{\varphi}|)}{2} - \frac{1}{2}\varepsilon(|{\psi}\rangle\langle{\psi}|)-...
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1answer
45 views

Mutual information of Choi state=0, what would that imply about the quantum channel?

Classically, if the mutual information between the input and output of some channel or circuit $= 0$, it means the output is independent of the input, and the circuit is in a way 'useless'. For the ...
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1answer
30 views

Quantum operation to get rid of small but nonzero eigenvalues

Updated and edited question: Let $N_{\delta}:P(\mathcal{H}_A)\rightarrow P(\mathcal{H}_B)$ be a completely positive trace nonincreasing map from the set of positive semidefinite operators in $\...
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2answers
81 views

What happens when you send a Bell state through depolarizing channel?

For noise parameter $Q$ and a density matrix $\rho$, we know that the depolarization channel $\mathcal{E}$ would act like: $$ \mathcal{E}(\rho) = (1 - Q)\rho +Q\frac{I}{2}, $$ where $I$ is the ...
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1answer
73 views

How to compute the tensor product of the depolarizing channel with the identity?

Consider two quantum systems A and B, B goes through a depolarizing noise channel, while A is not changed, i.e., they go through the channel $\mathbb{I}_A \otimes \mathcal{E_{\text{depol}}} $. If the ...
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2answers
57 views

$M(\rho)=\operatorname{Tr}_2\left(\ U\ \rho\otimes\rho_2\ U^{\dagger}\right)$is unitary $\iff\ U=U_1\otimes U_2$, a product of $2$ unitary operators?

Let $\rho : V_1 \to V_1 $ and $\rho_2 : V_2 \to V_2 $, where $V_1$ and $V_2$ are Hilbert spaces. Suppose that $U:V_1\otimes V_2 \to V_1\otimes V_2$ is a unitary operator. Define a map $M : L(V_1, ...
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When can a non-completely-positive evolution of a state be physical?

Definitions: a map $\Phi$ is called positive if $\Phi(\rho)$ is positive semidefinite for any positive semidefinite $\rho$, and completely positive (CP) if $\Phi \otimes \mathrm{Id}$ is a positive map ...
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1answer
103 views

What does it mean to take the Choi-Jamiolkowski of a quantum channel?

The Choi-Jamiolkowski of a channel $\newcommand{\on}[1]{\operatorname{#1}}\Lambda : \on{End}(\mathcal{H_A}) \xrightarrow{} \on{End}(\mathcal{H_B})$ is obtained through an isomorphism of the form: $$ ...
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0answers
48 views

Understanding Quantum Channel and Choi Jamiolkowski Notation [closed]

I am given the following $\newcommand{\on}[1]{{\operatorname{#1}}}$ Let $|i\rangle$, $1 \leq i \leq \on{dim}\,\mathcal{H_A}, |s\rangle$, $1 \leq s \leq \on{dim}\,\mathcal{H_B}$ be unitary bases. ...
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23 views

How can I obtain the lineally independent Kraus operators for the composition of two quantum channels?

Imagine we have two quantum channels, the bit flip and the depolarizing ones (for the corresponding noise models). These two quantum channels have 2 and 4 Kraus operators, respectively. We could ...
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1answer
67 views

Confusing notation in Wikipedia's quantum channel article

In the Wikipedia's Quantum channel article, it is said that a purely quantum channel $\phi$ (it's not exactly the same phi calligraphy but it's close), in the Schrodinger picture, is a linear map ...
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Is quantum deletion via a partial randomization procedure possible?

The paper, Quantum deletion is possible via a partial randomization procedure claims that it is possible to bypass the no-deleting theorem by a procedure called R-deletion. But this seems to go ...
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1answer
51 views

CPTP, Kraus representation and classical registers

What is the best mathematical representation of a quantum system that has some classical registers and some quantum registers? I'm asking because I'm considering any "physical" process $\pi()$ that ...
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1answer
83 views

How does qiskit finally implement a noise model?

I have been reading qiskit documentation for hours and I still don't get how does it implements noise in the circuit. I have understood that it works with a objects of the class QuantumError which ...
2
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0answers
88 views

Forbidden/allowed outputs of a quantum channel

The coherent information of a channel $\mathcal{E}_{A'\rightarrow B}$ is defined as the maximum value obtained by the following function where the maximization is over all input states $$I_{\rm{coh}}(...
2
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1answer
141 views

Partial trace condition in Choi state

Consider Hilbert spaces $\mathcal{X}, \mathcal{Y}$. For any quantum channel $\mathcal{E}_{\mathcal{X}\rightarrow \mathcal{Y}}$, the bipartite Choi state $J(\mathcal{E}) \in L(\mathcal{Y}\otimes\...
3
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1answer
159 views

What does it mean “less than identity” in the operator sum representation?

In Quantum Computation and Quantum Information by Isaac Chuang and Michael Nielsen, section 8.2.3, $\mathcal{E}=\sum_{k}E_k\rho E_k^{\dagger}$ gives the operator-sum representation. In general, it ...
3
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1answer
105 views

Optimizing over quantum channels

I am given fixed quantum states $\rho_X$ and $\sigma_Y$ and some function of the form $\text{Tr}(N_{X\rightarrow Y}(\rho_X)\sigma_Y)$. I would like to maximize this function over all completely ...
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1answer
41 views

Kraus decomposition for non trace preserving operation: shouldn't we have $0 \leq \sum_k E_k^{\dagger} E_k \leq I$

In N&Chuang, on page 368 is written the following theorem: The map $\mathcal{E}$ satisfies axioms A1,A2,A3 if and only if $$\mathcal{E}(\rho)=\sum_k E_k \rho E_k^{\dagger}$$ Where $\sum_k ...
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0answers
113 views

Does higher channel fidelity imply higher entanglement fidelity?

Consider two noisy quantum channels (CPTP maps), $\Phi_1^A$ and $\Phi_2^A$, acting on a system $A$. Suppose that for any pure state $\left|\psi\right>\in \mathcal H_A$, $$ F\big(\psi, \Phi_1^A(\psi)...
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0answers
62 views

Proof that quantum entanglement does not increase the asymptotic capacity of classical channel

Consider a classical channel $N_{X\rightarrow Y}$ which takes every input alphabet $x\in X$ to output alphabet $y\in Y$ with probability $P(y|x)_{Y|X}$. It is stated in many papers that even if the ...
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2answers
56 views

Simulating Classical Probabilistic Transitions with superoperators

I'm working on the following exercise: "Show how a classical probabilistic transition on an M -state system can be simulated by a quantum algorithm by adding an additional M -state ‘ancilla’ ...
3
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1answer
81 views

How to obtain the tensor-product of two quantum operations (superoperators) explicitly?

I have an amplitude damping channel, denoted as a superoperator $\mathcal{E}$ with operator elements \begin{matrix} E_1=\begin{pmatrix} 1 & 0 \\ 0 & \sqrt{1-r} \end{pmatrix},\quad ...
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0answers
30 views

What is the difference between intercept-resend attack and measure-resend attack?

I am going through different types of attacks that eve can perform on the quantum channel. I came across the intercept-resend attack and measure-resend attack. What is the difference between the two? ...
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2answers
202 views

How to define a quantum channel for the partial trace?

I understand that the partial trace is a linear map, a completely positive map and a trace-preserving map. However, I have no idea how to define a quantum channel with the partial trace because ...
4
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1answer
60 views

What goes wrong if I try to simulate a system with a larger Hilbert space with a smaller Hilbert space?

System 1: This has a Hilbert space of dimension $N$. System 2: This has a Hilbert space of dimension $N'$, with the condition that $N' \ll N$. We want to simulate system 1 using the system 2, and so ...
3
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2answers
172 views

If a state is only “close to” an eigenstate of an operator, how many applications of the operator does it take to scramble the state?

Suppose we have an operator $U$, and a register $|\lambda\rangle$ in an eigenstate of $U$ with eigenvalue $\lambda=1$. Repeatedly applying $U$ to $|\lambda\rangle$ does not affect $|\lambda\rangle$ - ...
2
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1answer
70 views

The solution when we transmit a qubit through a Pauli channel?

A Pauli channel is defined as a convex combination of Pauli operators, i.e. $\epsilon_{\text{Pauli}} (\rho)=\sum_{j} q_j\sigma_j\rho \sigma_j$, where $0 \leq q_j \leq 1$ and $\sum_j q_j=1$. Now, I ...
6
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3answers
209 views

Is there a quantum operation whose output is always orthogonal to the input?

I'm trying to show/convince myself the following statement is correct (I haven't been able to find any similar posts): "There is no reversible quantum operation that transforms any input state to a ...
6
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0answers
41 views

Entanglement-assisted hashing bound for asymmetric depolarizing channels

I reading the paper EXIT-Chart Aided Quantum Code Design Improves the Normalised Throughput of Realistic Quantum Devices, which proposes the use of QTCs in order to do quantum error correction for ...
3
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2answers
150 views

Why does entanglement not increase the classical capacity of a channel?

In a recent paper, the authors quote an older work of Bennett, Shor and others and make the following statement While entanglement assistance can increase achievable rates for classical point-to-...
4
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1answer
108 views

Why can any LOCC operation be written as $\sum_k (A_k\otimes B_k)\rho(A_k^\dagger\otimes B_k^\dagger)$?

This statement can be found in Vedral et al. 1997, eq. (1). More precisely, given a bipartite state $\rho_{AB}$, they claim that any operation on $\rho_{AB}$ involving local operations plus classical ...
5
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1answer
186 views

Do the eigenvalues of the Choi matrix have any direct physical interpretation?

Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map, and let $J(\Phi)$ be its Choi representation. As is well known, any such map can be written in a Kraus representation of the form $$\Phi(X)=...
4
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2answers
326 views

Do the Kraus operators of a CPTP channel need to be orthogonal?

Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map. Any such channel admits a Kraus decomposition of the form $$\Phi(X)=\sum_a A_a X A_a^\dagger,$$ for a set of operators $A_a\in\mathrm{Lin}(\...