Questions tagged [quantum-channel]
In quantum information theory, a quantum channel is a communication channel which can transmit quantum information, as well as classical information. An example of quantum information is the state of a qubit. An example of classical information is a text document transmitted over the Internet. More formally, quantum channels are completely positive (CP) trace-preserving maps between spaces of operators. If appropriate, also use the [quantum-operation] tag.
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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
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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 ...
2
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46 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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56 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
82 views
Prove that 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
$$C^{(1)} \geq S(\frac{\varepsilon(|{\psi}\rangle\langle{\psi}|) +\varepsilon(|{\varphi}\rangle\langle{\varphi}|)}{2} - \frac{1}{2}\varepsilon(|{\psi}\rangle\langle{\psi}|)-...
3
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1answer
41 views
Mutual information of Choi state=0, what would that imply about the quantum channel?
Classically, if the mutual information between the input and output of some channel or circuit $= 0$, it means the output is independent of the input, and the circuit is in a way 'useless'.
For the ...
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1answer
29 views
Quantum operation to get rid of small but nonzero eigenvalues
Updated and edited question:
Let $N_{\delta}:P(\mathcal{H}_A)\rightarrow P(\mathcal{H}_B)$ be a completely positive trace nonincreasing map from the set of positive semidefinite operators in $\...
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How to build a quantum computer in your house?
As still quarantine is going on for some of us. I was wondering how to make a Quantum Computer in your garage. What may be the total cost for building one?
Was inspired by this youtube video.
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2answers
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What happens when you send a Bell state through depolarizing channel?
For noise parameter $Q$ and a density matrix $\rho$, we know that the depolarization channel $\mathcal{E}$ would act like:
$$
\mathcal{E}(\rho) = (1 - Q)\rho +Q\frac{I}{2},
$$
where $I$ is the ...
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6answers
139 views
Show that $I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$ for all states $\rho$
I am trying to show that for any qubit state p, the following holds:
$$I = \frac{\rho + \sigma_x\rho\sigma_x +\sigma_y\rho\sigma_y + \sigma_z\rho\sigma_z}{2}$$
I have tried different manipulations,...
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1answer
65 views
How to compute the tensor product of the depolarizing channel with the identity?
Consider two quantum systems A and B, B goes through a depolarizing noise channel, while A is not changed, i.e., they go through the channel $\mathbb{I}_A \otimes \mathcal{E_{\text{depol}}} $. If the ...
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2answers
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$M(\rho)=\operatorname{Tr}_2\left(\ U\ \rho\otimes\rho_2\ U^{\dagger}\right)$is unitary $\iff\ U=U_1\otimes U_2$, a product of $2$ unitary operators?
Let $\rho : V_1 \to V_1 $ and $\rho_2 : V_2 \to V_2 $, where $V_1$ and $V_2$ are Hilbert spaces.
Suppose that $U:V_1\otimes V_2 \to V_1\otimes V_2$ is a unitary operator.
Define a map $M : L(V_1, ...
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1answer
93 views
What does it mean to take the Choi-Jamiolkowski of a quantum channel?
The Choi-Jamiolkowski of a channel $\newcommand{\on}[1]{\operatorname{#1}}\Lambda : \on{End}(\mathcal{H_A}) \xrightarrow{} \on{End}(\mathcal{H_B})$ is obtained through an isomorphism of the form:
$$
...
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22 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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1answer
70 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
$$I_{\rm{coh}}(...
2
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1answer
78 views
How is quantum error applied to the qubits?
I am trying to check the way qiskit has the noise implemented. I have read how is this theoretically done using quantum channels (see Nielsen and Chuang chapter 8) and I want to verify if qiskit ...
2
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1answer
125 views
Partial trace condition in Choi state
Consider Hilbert spaces $\mathcal{X}, \mathcal{Y}$. For any quantum channel $\mathcal{E}_{\mathcal{X}\rightarrow \mathcal{Y}}$, the bipartite Choi state $J(\mathcal{E}) \in L(\mathcal{Y}\otimes\...
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1answer
104 views
Optimizing over quantum channels
I am given fixed quantum states $\rho_X$ and $\sigma_Y$ and some function of the form $\text{Tr}(N_{X\rightarrow Y}(\rho_X)\sigma_Y)$. I would like to maximize this function over all completely ...
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1answer
40 views
Kraus decomposition for non trace preserving operation: shouldn't we have $0 \leq \sum_k E_k^{\dagger} E_k \leq I$
In N&Chuang, on page 368 is written the following theorem:
The map $\mathcal{E}$ satisfies axioms A1,A2,A3 if and only if
$$\mathcal{E}(\rho)=\sum_k E_k \rho E_k^{\dagger}$$
Where $\sum_k ...
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101 views
Does higher channel fidelity imply higher entanglement fidelity?
Consider two noisy quantum channels (CPTP maps), $\Phi_1^A$ and $\Phi_2^A$, acting on a system $A$. Suppose that for any pure state $\left|\psi\right>\in \mathcal H_A$,
$$
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 ...
2
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2answers
45 views
Can Eve perform this operation?
I am a beginner in quantum computing. Please consider the following scenario:
Suppose Alice wants to send
$\frac{1}{\sqrt{N}}\sum_{j=0,1,2,..N-1} |j\rangle$ to Bob.
Eve has intercepted the state ...
4
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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?
...
3
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2answers
189 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 ...
2
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1answer
69 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 ...
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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
votes
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
91 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 ...
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1answer
169 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
$$\Phi(X)=...
4
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2answers
311 views
Do the Kraus operators of a CPTP channel need to be orthogonal?
Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map.
Any such channel admits a Kraus decomposition of the form
$$\Phi(X)=\sum_a A_a X A_a^\dagger,$$
for a set of operators $A_a\in\mathrm{Lin}(\...
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1answer
70 views
Holevo quantity and mutual information
On this page, it is stated that the Holevo quantity is an upper bound to the accessible information of a quantum state. In the scenario where Alice encodes classical information into a quantum state ...
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1answer
271 views
Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?
If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that
$$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$
in the ...
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1answer
110 views
Derive phase damping quantum operation
I am reading about the phase damping quantum operation on page 384 of Nielsen & Chuang's Quantum Computation and Quantum Information (10th Anniversary Edition).
Nielsen & Chuang derived the ...
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61 views
Operation Elements for Amplitude Damping Channel
To find operation elements for the Amplitude Damping channel, Nielsen and Chuang (in Section 8.3.5 of my copy) use the action of a beamsplitter on an initial state $ \alpha |0\rangle + \beta |1\rangle$...
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1answer
302 views
Does the Choi-Jamiolkowski isomorphism really establish a connection between kinematics and dynamics?
I understand the mathematical construction of the Choi-Jamiolkowski isomorphism aka channel-state duality. It all makes sense formally, yet I still struggle to grasp its physical (or quantum-...
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1answer
427 views
Quantum fidelity simplified formula while both of the density matrices are single qubit states
I have a question while reading the quantum fidelity definition in Wikipedia Fidelity of quantum states, at the end of the Definition section of quantum fidelity formula, it says Explicit expression ...
7
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2answers
120 views
Quantum channel representation of projective measurement
Let $P$ be a projector and $Q = I-P$ be its complement. How to find probability $p$ and unitaries $U_1, U_2$ such that for any $\rho$, $P\rho P + Q\rho Q = p U_1\rho U_1^\dagger + (1-p)U_2\rho U_2^\...
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1answer
498 views
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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0answers
39 views
How does a map being “only” positive reflect on its Choi representation?
We know that a map $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ being completely positive is equivalent to its Choi representation being positive: $J(\Phi)\in\operatorname{Pos}(\mathcal Y\otimes\mathcal ...
2
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1answer
100 views
Direct derivation of the Kraus representation from the natural representation, using SVD
$\newcommand{\Y}{\mathcal{Y}}\newcommand{\X}{\mathcal{X}}\newcommand{\rmL}{\mathrm{L}}$As explained for example in Watrous' book (chapter 2, p. 79), given an arbitrary linear map $\Phi\in\rmL(\rmL(
\X)...
3
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1answer
63 views
State of a system after the second qubit of a Bell state sent through a bit flip error channel
The second qubit of a two-qubit system in the Bell state
$$|\beta_{01}\rangle= \frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$$ is sent through an error channel which introduces a bit flip error with ...
4
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1answer
116 views
Degradable channels and their quantum capacity
Note: I'm reposting this question as it was deleted by the original author, so that we do not lose out on the existing answer there, by Prof. Watrous. Further answers are obviously welcome.
I have ...
3
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1answer
81 views
Quantum channel cannot increase Holevo information of an ensemble
I need to prove the fact that a quantum channel (a superoperator) cannot increase the Holevo information of an ensemble $\epsilon = \{\rho_x, p_x\}$. Mathematically expressed I need to prove
$$\begin{...
3
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1answer
52 views
Counting channel uses of the lossy bosonic channel or definition of channel uses
The PLOB-bound ("Fundamental Limits of Repeaterless Quantum Communications") gives an asymptotic upper bound on the secret-key rate per used lossy bosonic channel.
However, I'm not sure how to count ...
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0answers
65 views
Implementing a depolarizing channel for 2 qubits on IBM Q
I am trying to use IBM Q to perform the following depolarizing channel on a state of 2 qubits $\rho=|\psi \rangle \langle \psi |$:
$$\rho \to (1-\lambda)\rho + \frac{\lambda}{4}I$$
This is within ...
4
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1answer
1k views
What's the difference between Kraus operators and measurement operators?
It is said in a lecture note[1] by John Preskill that,
Equivalently, we may imagine measuring system $B$ in the basis $\{|a\rangle\}$, but failing to record the measurement outcome, so we are ...
4
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1answer
162 views
Understanding classical vs. quantum channel capacities
The classical channel capacity ($C_{ea}$) and the quantum channel capacity ($Q$) as defined here (eqs. 1 and 2) are given by
\begin{equation}
C_{ea} = \text{sup}_{\rho} \Big[S(\rho) + S(\Phi_t \rho) -...
