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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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
58 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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51 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 ...
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84 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
$$I_{\rm{coh}}(...
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1answer
61 views
How is quantum error applied to the qubits?
I am trying to check the way qiskit has the noise implemented. I have read how is this theoretically done using quantum channels (see Nielsen and Chuang chapter 8) and I want to verify if qiskit ...
2
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1answer
75 views
Partial trace condition in Choi state
Consider Hilbert spaces $\mathcal{X}, \mathcal{Y}$. For any quantum channel $\mathcal{E}_{\mathcal{X}\rightarrow \mathcal{Y}}$, the bipartite Choi state $J(\mathcal{E}) \in L(\mathcal{Y}\otimes\...
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1answer
88 views
Optimizing over quantum channels
I am given fixed quantum states $\rho_X$ and $\sigma_Y$ and some function of the form $\text{Tr}(N_{X\rightarrow Y}(\rho_X)\sigma_Y)$. I would like to maximize this function over all completely ...
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1answer
37 views
Kraus decomposition for non trace preserving operation: shouldn't we have $0 \leq \sum_k E_k^{\dagger} E_k \leq I$
In N&Chuang, on page 368 is written the following theorem:
The map $\mathcal{E}$ satisfies axioms A1,A2,A3 if and only if
$$\mathcal{E}(\rho)=\sum_k E_k \rho E_k^{\dagger}$$
Where $\sum_k ...
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78 views
Does higher channel fidelity imply higher entanglement fidelity?
Consider two noisy quantum channels (CPTP maps), $\Phi_1^A$ and $\Phi_2^A$, acting on a system $A$. Suppose that for any pure state $\left|\psi\right>\in \mathcal H_A$,
$$
F\big(\psi, \Phi_1^A(\psi)...
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60 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
43 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 ...
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29 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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150 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
66 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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38 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
144 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-...
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1answer
81 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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145 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
$$\Phi(X)=...
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2answers
258 views
Do the Kraus operators of a CPTP channel need to be orthogonal?
Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map.
Any such channel admits a Kraus decomposition of the form
$$\Phi(X)=\sum_a A_a X A_a^\dagger,$$
for a set of operators $A_a\in\mathrm{Lin}(\...
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1answer
50 views
Holevo quantity and mutual information
On this page, it is stated that the Holevo quantity is an upper bound to the accessible information of a quantum state. In the scenario where Alice encodes classical information into a quantum state ...
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1answer
229 views
Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?
If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that
$$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$
in the ...
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1answer
98 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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53 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
201 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
335 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 ...
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2answers
113 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
301 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
38 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
91 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)...
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1answer
58 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)$$ is sent through an error channel which introduces a bit flip error with ...
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1answer
107 views
Degradable channels and their quantum capacity
Note: I'm reposting this question as it was deleted by the original author, so that we do not lose out on the existing answer there, by Prof. Watrous. Further answers are obviously welcome.
I have ...
3
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1answer
79 views
Quantum channel cannot increase Holevo information of an ensemble
I need to prove the fact that a quantum channel (a superoperator) cannot increase the Holevo information of an ensemble $\epsilon = \{\rho_x, p_x\}$. Mathematically expressed I need to prove
$$\begin{...
3
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1answer
46 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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55 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 ...
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1answer
898 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 ...
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1answer
158 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) -...
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64 views
Quantum Fisher information
I am reading paper Channel Identification and its Impact on Quantum LDPC Code Performance where the authors discuss the scenario where the decoder of a Quantum LDPC code uses an estimation of the ...
3
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1answer
326 views
How does the vectorization map relate to the Choi and Kraus representations of a channel?
I know that the Choi operator is a useful tool to construct the Kraus representation of a given map, and that the vectorization map plays an important role in such construction.
How exactly does the ...
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1answer
442 views
What is a complementary map?
I have a quantum map described by the following Kraus operators
$$A_0 = c_0 \begin{pmatrix}
1 & 0\\
0 & 1 \end{pmatrix},
\qquad A_1 = c_1 \begin{pmatrix}
...
3
votes
1answer
73 views
Clarifying the relationship of the probabilities of depolarizing channel and relaxation/dephasing times
I am reading the paper Duality of Quantum and Classical Error Correction
Codes: Design Principles & Examples and at the beginning of it, the authors describe how the general Pauli channel is ...
3
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1answer
92 views
Estimating the depolarizing probability of depolarizing channels
When considering quantum error correction over depolarizing channels, the depolarizing probability $p$ such that an error of the kind $X,Y,Z$ will happen is used as a priori information in order to ...
3
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0answers
86 views
Real-life examples of classical-quantum channels
In quantum information theory, classical-quantum channels are considered to be channels whose input is the realizations $x\in\mathcal{X}$ of a classical random variable to a quantum state $\rho_x^B$, ...
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1answer
825 views
Depolarizing channel operator sum representation
In Nielsen and Chuang, it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity matrix with
...
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2answers
230 views
How many Kraus operators are required to characterise a channel with different start and end dimensions?
If we have a quantum channel mapping from a $d$-dimensional state to a $d$-dimensional state, it can be described by at most $d^2$ Kraus operators. Suppose our channel maps instead from a $d_1$-...
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243 views
Dephasing channels
I'm taking a quantum information course and one of my exercises says to find $p,p'$, for which there is a channel $\tilde\Lambda(\Lambda(\rho))=\Lambda'(\rho)$, where $\Lambda$ and $\Lambda'$ are ...
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0answers
57 views
Isometric Extension of an Erasure Channel
Show that an isometric extension of the erasure channel is $$U^N_{A\to BE} =\sqrt{1ā\epsilon}\left(|0\rangle_B \langle 0|_A +|1\rangle_B \langle 1|_A \right)\otimes|e\rangle_E+ \sqrt{\epsilon}|e\...
3
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1answer
234 views
Confusion on the definition of the phase-damping channel
I am reading about the phase damping channel, and I have seen that some of the different references talking about such channel give different definitions of the Kraus operators that define the action ...
3
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0answers
71 views
Quantum channel Holevo information additivity: proof approach
I have an interesting idea for a proof approach that someone might find useful. Here it is.
Suppose we are given a quantum qubit channel $N$ (for example the amplitude damping channel) whose Holevo ...
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0answers
70 views
Three sender quantum simultaneous decoder conjecture
Recently I have started to read about network quantum information theory, where network problems are studied under the classical-quantum channel. For example, capacities of the cq-MAC, cq-broadcast or ...
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1answer
784 views
Deduce the Kraus operators of the dephasing channel using the Choi
I'm trying to deduce the Kraus representation of the dephasing channel using the Choi operator (I know the Kraus operators can be guessed in this case, I want to understand the general case).
The ...
