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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questions
2
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1answer
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About a quantum channel using 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
votes
1answer
65 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
votes
0answers
30 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
votes
2answers
117 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
votes
1answer
57 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
votes
1answer
87 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
$$\Phi(X)=...
3
votes
2answers
151 views
Do the Kraus operators of a CPTP channel need to be orthogonal?
Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a CPTP map.
Any such channel admits a Kraus decomposition of the form
$$\Phi(X)=\sum_a A_a X A_a^\dagger,$$
for a set of operators $A_a\in\mathrm{Lin}(\...
2
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0answers
33 views
Holevo quantity and mutual information
On this page, it is stated that the Holevo quantity is an upper bound to the accessible information of a quantum state. In the scenario where Alice encodes classical information into a quantum state ...
4
votes
1answer
161 views
Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?
If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that
$$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$
in the ...
1
vote
0answers
44 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$...
7
votes
1answer
105 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-...
4
votes
1answer
173 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
votes
2answers
100 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^\...
5
votes
1answer
152 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 ...
2
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0answers
35 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
votes
1answer
66 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
votes
1answer
55 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)$$ is sent through an error channel which introduces a bit flip error with ...
3
votes
1answer
94 views
Degradable channels and their quantum capacity
Note: I'm reposting this question as it was deleted by the original author, so that we do not lose out on the existing answer there, by Prof. Watrous. Further answers are obviously welcome.
I have ...
3
votes
1answer
78 views
Quantum channel cannot increase Holevo information of an ensemble
I need to prove the fact that a quantum channel (a superoperator) cannot increase the Holevo information of an ensemble $\epsilon = \{\rho_x, p_x\}$. Mathematically expressed I need to prove
$$\begin{...
3
votes
0answers
15 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 ...
3
votes
0answers
48 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
votes
1answer
457 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
votes
1answer
136 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) -...
3
votes
0answers
56 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
votes
1answer
203 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 ...
2
votes
1answer
333 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
69 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
votes
1answer
76 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
votes
0answers
67 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
534 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
...
5
votes
2answers
129 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$-...
2
votes
0answers
192 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 ...
3
votes
0answers
50 views
Isometric Extension of an Erasure Channel
Show that an isometric extension of the erasure channel is $$U^N_{A\to BE} =\sqrt{1−\epsilon}\left(|0\rangle_B \langle 0|_A +|1\rangle_B \langle 1|_A \right)\otimes|e\rangle_E+ \sqrt{\epsilon}|e\...
3
votes
1answer
159 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
votes
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 ...
2
votes
0answers
66 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 ...
5
votes
1answer
623 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 ...
7
votes
1answer
246 views
Depolarizing channel implementation on IBM Q
Given a single qubit in the computational basis, $|\psi\rangle =\alpha |0\rangle + \beta|1\rangle$, the density matrix is $\rho=|\psi\rangle\langle\psi|=\begin{pmatrix} \alpha^2 & \alpha \beta^*\\
...
7
votes
1answer
352 views
Advances in Quantum Channel Capacity
I have been reading about the Quantum Channel Capacity and it seems to be an open problem to find such capacity in general. Quantum capacity is the highest rate at which quantum information can be ...
3
votes
1answer
219 views
Tensor product properties used to obtain Kraus operator decomposition of a channel
I work on a Quantum Information Science II: Quantum states, noise and error correction MOOC by Prof. Aram Harrow, and I do not understand which property of tensor products is used in one of the ...
4
votes
1answer
63 views
How to find the fidelity between two state when one is an operator?
I am going through Nielsen and Chuang and am finding the chapter on error-correction particularly confusing. At the moment I am stuck on exercise 10.12 which states
Show that the fidelity between the ...
8
votes
1answer
313 views
Quantum Channel Models
The so called depolarizing channel is the channel model that is mostly used when constructing quantum error correction codes. The action of such channel over a quantum state $\rho$ is
$\rho\...
11
votes
3answers
374 views
Is acting with a positive map on a state not part of a larger system allowed?
In the comments to a question I asked recently, there is a discussion between user1271772 and myself on positive operators.
I know that for a positive trace-preserving operator $\Lambda$ (e.g. the ...
6
votes
2answers
182 views
Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?
I understand that there are two ways to think about 'general quantum operators'.
Way 1
We can think of them as trace-preserving completely positive operators. These can be written in the form
$$\...
16
votes
2answers
2k views
Significance of The Church of the Higher Hilbert space
The term "Church of the Higher Hilbert Space" is used in quantum information frequently when analysing quantum channels and quantum states.
What does this term mean (or, alternately, what does the ...
4
votes
1answer
689 views
What is the difference between signaling and non-signaling quantum correlations, and what is a signaling channel?
This article talks about correlation and causality in quantum mechanics. In the abstract, under Framework for local quantum mechanics, it says (emphasis mine):
The most studied, almost epitomical, ...
