Linked Questions

6 votes
1 answer
511 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}^...
FDGod's user avatar
  • 2,401
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's user avatar
  • 25.5k
5 votes
1 answer
282 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's user avatar
  • 25.5k
5 votes
1 answer
708 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\...
user1936752's user avatar
  • 3,033
2 votes
2 answers
219 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\...
Takimoto.R's user avatar
5 votes
2 answers
251 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, ...
GouldBach's user avatar
  • 151
3 votes
2 answers
276 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's user avatar
  • 25.5k
1 vote
1 answer
227 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|...
user1936752's user avatar
  • 3,033
3 votes
1 answer
368 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's user avatar
  • 25.5k
5 votes
1 answer
236 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 ...
JSdJ's user avatar
  • 5,539
4 votes
1 answer
147 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 ...
Frederik vom Ende's user avatar
2 votes
1 answer
153 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 $n$ ...
user1936752's user avatar
  • 3,033
1 vote
1 answer
309 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 ...
Sudhir Kumar's user avatar
3 votes
1 answer
138 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 $t\geq 0$ ...
Frederik vom Ende's user avatar
5 votes
1 answer
63 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} - \...
usermm's user avatar
  • 91

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