Linked Questions
22 questions linked to/from Counterexamples in quantum information theory
7
votes
1
answer
749
views
If a quantum error correcting code can correct every single-qubit $X$ and $Z$ error, can it also correct every single-qubit $Y$ error?
Let $\mathcal{C}$ be a given arbitrary $n$ qubit quantum error correcting code which can correct any single qubit $X$ error and any single qubit $Z$ error, i.e., $\{X_i\}_{i=1}^n$ & $\{Z_i\}_{i=1}^...
- 2,593
12
votes
1
answer
1k
views
How does the invertibility of a quantum map reflect on its Kraus operators?
Consider a quantum map $\Phi\in\mathrm T(\mathcal X)$, that is, a linear operator $\Phi:\mathrm{Lin}(\mathcal X)\to\mathrm{Lin}(\mathcal X)$ for some finite-dimensional complex vector spaces $\mathcal ...
glS♦
- 26.9k
5
votes
1
answer
298
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
$$\Phi(X) = \operatorname{Tr}_{\mathcal ...
glS♦
- 26.9k
5
votes
1
answer
873
views
Trace distance between mixed state and pure state vs trace distance between their purifications
Let $\rho$ be a mixed state and $\vert\psi\rangle\langle\psi\vert$ be a pure state on some Hilbert space $H_A$ such that
$$\|\rho - \vert\psi\rangle\langle\psi\vert \|_1 \leq \varepsilon,$$
where $\|A\...
- 3,169
2
votes
2
answers
229
views
Does proximity of two bipartite states in a norm force high overlap between the elements of the Schmidt bases?
I want to know that there is a relation between the distance of two vectors and the corresponding elements of the Schmidt bases.
We assume that two bipartite vectors $|\phi\rangle^{AB}$ and $|\psi\...
- 63
5
votes
2
answers
288
views
$M(\rho)=\operatorname{Tr}_2[U(\rho\otimes\rho_2)U^{\dagger}]$ is unitary $\iff U=U_1\otimes U_2$, a product of $2$ unitary operators?
Let $\rho : V_1 \to V_1 $ and $\rho_2 : V_2 \to V_2 $, where $V_1$ and $V_2$ are Hilbert spaces.
Suppose that $U:V_1\otimes V_2 \to V_1\otimes V_2$ is a unitary operator.
Define a map $M : L(V_1, ...
- 151
3
votes
2
answers
348
views
What are examples of quantum maps with complex eigenvalues?
Chapter 6 of Michael Wolf's notes (MichaelWolf/QChannelLecture.pdf) discuss the structure of the spectrum of quantum maps and channels. However, it seems like the only explicit example given in the ...
glS♦
- 26.9k
1
vote
1
answer
240
views
Can the fidelity $F(\rho,\sigma)$ be computed knowing only $\rho - \sigma$?
The motivation for this question comes from trace distance. For any two states $\rho, \sigma$, the trace distance $T(\rho, \sigma)$ is given by
$$T(\rho, \sigma) = |\rho - \sigma|_1,$$
where $|\cdot|...
- 3,169
3
votes
1
answer
454
views
How does the CPTP constraint reflect on the matrix representation of a qubit channel in the Pauli basis?
Let us write the possible states of a qubit in the Bloch representation as
$$\newcommand{\bs}[1]{{\boldsymbol{#1}}}\rho_{\bs r}\equiv \frac{I+\bs r\cdot\bs \sigma}{2},$$
where $\bs\sigma=(\sigma_1,\...
glS♦
- 26.9k
5
votes
1
answer
259
views
Is the composition of two extremal channels also extremal?
In this question, I follow the terminology and notation of the book of Watrous, most notably chapter two.
Extremal channels
An extremal channel $\Phi(X) \in C(\mathcal{X},\mathcal{Y})$ is a channel ...
- 5,739
5
votes
1
answer
275
views
Is the trace of a positive map always positive?
Obviously, positive semi-definite operators always admit a positive trace as ${\rm tr}(A)=\|A\|_1\geq 0$ whenever $A\geq 0$. This motivates the following "lifted" question:
Given any ...
- 3,474
2
votes
1
answer
162
views
Permutation covariant channels and their Stinespring dilations
I am interested in a quantum channel from $A^{\otimes n}$ to $B^{\otimes n}$ denoted as $N_{A^{\otimes n} \rightarrow B^{\otimes n}}(\cdot)$. Let $\pi(\cdot)$ be a permutation operation among the $n$ ...
- 3,169
1
vote
1
answer
454
views
Can a unital channel not be mixed unitary?
How to prove that for a multi-qubit system a unital channel is not necessarily mixed unitary? This is Problem 8.3 in Nielsen and Chuang. Here's a snippet of the text:
Shall I need to take two ...
- 45
5
votes
1
answer
291
views
For how many different times do I have to know that $e^{tL}$ is a quantum channel to conclude that $L$ is of Lindblad form?
As first shown by Gorini, Kossakowski, Sudarshan and Lindblad given some linear map $\mathcal L:\mathbb C^{n\times n}\to\mathbb C^{n\times n}$, $e^{t\mathcal L}$ is a quantum channel for all $t\geq 0$ ...
- 3,474
5
votes
1
answer
74
views
Does monotonicity of diamond distance hold for intermediate channels?
It is well known that $\|\mathcal{E} \circ \mathcal{F} - \mathcal{E}\|_\lozenge \leq \|\mathcal{F} - \mathcal{I}\|_\lozenge$.
What if I have $\|\mathcal{A} \circ \mathcal{E} \circ \mathcal{F} - \...
- 101