7 votes
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Do the operators $\Phi(\sigma_k)$ form a basis if $\sigma_k$ do?

No, not necessarily. For example, the channel $\Phi(\rho) = \operatorname{Tr}(\rho) \vert 0 \rangle \langle 0 \vert$ makes $\{\Phi(\sigma_0), \ldots, \Phi(\sigma_3)\}$ linearly dependent. (In fact, ...
John Watrous's user avatar
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7 votes
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What are "completely positive" and "CPTP" quantum maps?

[A] States lie in Hilbert space $\mathcal{H_S}$. $|\psi\rangle \in \mathcal{H_S}\,.$ Operators, density operators lie in the bounded operator space of $\mathcal{H}_S$. $\rho \in \mathcal{B}(\...
FDGod's user avatar
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6 votes
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Equal partial traces

No, assume $\rho_{AB}$ is pure, so that $\rho_{AB} = |u\rangle\langle u|_{{AB}}$. Since it's pure $\rho_{ABC}$ must have the form $\rho_{ABC} = \rho_{AB} \otimes \rho_{C}$. It follows that $\rho_{AC} =...
Danylo Y's user avatar
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5 votes
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Can any channel be represented as $A\rho A^\dagger$ for some $A$?

No, this is not possible in general. To see it, consider for example what happens taking the trace of that expression. You'd get: $$\operatorname{tr}(A^\dagger A\rho)=\operatorname{tr}\left(\sum_j K_j^...
glS's user avatar
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5 votes
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Existence of Hamiltonians such that the time evolution unitary becomes identity

I don't believe that this is always possible. For instance, what if my set of $\{H_i\}$s comprise a single term that I can construct to be arbitrarily awkward? The key feature will be gaps between ...
DaftWullie's user avatar
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5 votes
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Qiskit reverse_bits is not equivalent to swapping qubits

The issue is that the qubit values are being mixed together throughout the circuit, so swapping at the end is not enough. However, if you also swap at the beginning of the circuit, the qubit values ...
Nick Mertes's user avatar
4 votes
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What is the meaning of $\langle e_k|U|e_0\rangle$ when $U$ acts on a larger Hilbert space than that in which $|e_0\rangle$ and $|e_k\rangle$ live?

TL;DR: We can understand the object $E_k=\langle e_k|U|e_0\rangle$ rigorously in two steps. First, think of $\langle e_k|$ and $|e_0\rangle$ as linear functions. Next, treat the implicit operation in $...
Adam Zalcman's user avatar
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4 votes
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Action of a CPTP map on Identity

Let $d$ be the dimension of the system. Since $\sum_i K_iK_i^\dagger = \Phi(\mathbb 1) = d \Phi(\mathbb 1/d)$, it can be $d$ times any quantum state (by, e.g., setting $\Phi$ as a replacement channel)....
Senrui Chen's user avatar
4 votes

Are $n$-qubit Pauli channels $\mathcal E(\rho)=\sum_j p_j P_j \rho P_j$ invertible?

TL;DR: A Pauli channel has a mathematical inverse if and only if it doesn't vanish on any Pauli operator. The inverse is physical if and only if the channel is unitary. The former follows from ...
Adam Zalcman's user avatar
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4 votes

What are "completely positive" and "CPTP" quantum maps?

Definitions Let $\mathcal{H}$ be a complex Hilbert space. It turns out that the set $L(\mathcal{H})$ of all linear operators on $\mathcal{H}$ is also a Hilbert space. Let $I_\mathcal{H}$ denote the ...
Adam Zalcman's user avatar
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Is the "unitary twirling operation" physically realizable?

Question 1: I guess it depends what your understanding of "physical" is. In my understanding, everything you can do in the lab is physical. Thus, twirling is perfectly physical. Note that ...
Markus Heinrich's user avatar
4 votes

Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

Maybe it's easier to see when it's presented like this: $UN\left|\psi\right> = UNU^\dagger U\left|\psi\right>$ means that $UNU^\dagger$ maps $U\cdot \left|\psi\right>$ to $U \cdot N\left|\psi\...
Vladimir Lysikov's user avatar
4 votes
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Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

I think it probably helps to understand what Gottesman is trying to do with the operator $N$ (later in the paper). He wants to start with some state $|\psi\rangle$, but instead of directly describing ...
DaftWullie's user avatar
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3 votes
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What is the explicit form of $T_1$ decay channel?

Relaxation time $T_1$ describes the strength of amplitude damping$^1$ by specifying the mean lifetime$^2$ of the $|1\rangle$ state. More precisely, $\mathcal{N}$ has a Kraus representation $$ E_0=\...
Adam Zalcman's user avatar
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3 votes

Are permutations of the Pauli strings unitary operations?

No. This fails because the operation $U_{g}$ is not necessarily trace-preserving. Suppose $N = 1$ and $g(1) = 0$, i.e. the Permutation that maps $X$ to $\mathbb{I}$. We thus have $\mathbb{I} = \tau_{0}...
JSdJ's user avatar
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3 votes
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Do quantum channels satisfy $\Phi(0)=0$?

Any linear function between any two vector spaces preserves the zero, so yes, any quantum map (and thus also any CPTP map) sends 0 to 0. Note that here "0" means the operator $\mathbb{C}^n\...
glS's user avatar
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3 votes
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How to extract probabilities from Kraus representation?

We can indeed rewrite $\mathcal{E}(\rho)=\sum_iK_i\rho K_i^\dagger$ as $\mathcal{E}(\rho)=\sum_ip(i)\rho_i$ by setting $p(i):=\mathrm{tr}(K_i\rho K_i^\dagger)$ and $\rho_i:=\frac{K_i\rho K_i^\dagger}{...
Adam Zalcman's user avatar
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3 votes

Why is there always a $k$-outcome experiment associated to operators such that $\sum_{i=1}^k M_i^* M_i=I$?

Imagine you want to make a measurement on a state $|\psi\rangle$ (and we will make the solution work for all possible $|\psi\rangle$). We introduce an ancilla system (Hilbert space dimension at least ...
DaftWullie's user avatar
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3 votes

Proof that $\frac{1}{d} | \text{Tr}(\mathcal{P}_i^\dagger \mathcal{E}(\mathcal{P}_j))| \leq 1$ for superoperators

Here's a one-line proof using Hölder's inequality: $$ |\mathrm{tr}\left( \mathcal{P}_i^\dagger \mathcal{E}(\mathcal{P}_i) \right)| \leq \| \mathcal{P}_i^\dagger \|_\infty \|\mathcal{E}\|_{1\...
Markus Heinrich's user avatar
3 votes

How comes the definition of Pauli transfer matrix?

Pauli transfer matrix is usually used in tomography. Quantum channels are often assumed to be a completely positive, trace-preserving (CPTP) map. In the PTM representation of a quantum channel, it ...
cyrie wang's user avatar
3 votes
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Are quantum channels bounded linear maps?

A linear map defined on density operators is uniquely extendible on the set of all linear operators (in finite-dimensional case). Simply because any linear operator is a linear combination of density ...
Danylo Y's user avatar
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3 votes

Validity of quantum channel given pairs of inputs and outputs

The problem can clearly be solved with an SDP. Let $F :=\mathfrak{C}(\Phi)$ be $\Phi$'s Choi matrix. Then the channel exists iff \begin{gather*} \operatorname{tr}_\text{in}[ (\rho_i^T \otimes I)F] = \...
Mateus Araújo's user avatar
3 votes
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What is the Choi matrix of the $H$ gate?

Let $\rho$ be your state. Let $\mathcal{E}$ be the Hadamard map. $$\therefore \mathcal{E}(\rho) = H \rho H^{\dagger} = H \rho H\;.$$ Let $\Upsilon_{\mathcal{E}}$ be the Choi matrix. For this case, $...
FDGod's user avatar
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3 votes

How to characterize the extreme points of the set of CPTP maps?

Cardinality The set $C(\mathcal{X},\mathcal{Y})$ of all channels$^{1,2}$ $\Phi:L(\mathcal{X})\to L(\mathcal{Y})$ has uncountably infinitely many extreme points. To see this, first note that every pure ...
Adam Zalcman's user avatar
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3 votes
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Is a process matrix of rank $1$ unique?

TL;DR: The elements of the process matrix with respect to an operator basis $E_i$ are just the coefficients in the expansion of the channel, viewed as an operator on $\mathcal{H}\otimes\mathcal{H}$, ...
Adam Zalcman's user avatar
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3 votes
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How do you work out the matrix for controlled-U operations?

$\left|0\right>$ and $\left|1\right>$ don't mean $\begin{pmatrix}0 \\ 0\end{pmatrix}$ and $\begin{pmatrix}1 \\ 1\end{pmatrix}$. In bra-ket notation we usually fix some basis and denote by $\left|...
Vladimir Lysikov's user avatar
3 votes

Equal partial traces

As noted in the comment, the state part of the question is precisely the question of symmetric extensions; see for instance the thesis https://arxiv.org/abs/1103.0766 and references therein. Apart ...
helloworld's user avatar
3 votes
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Affine transformation of the Bloch sphere to Kraus representation of qubit channels

TL;DR: The given form of the channel is essentially an obscured way of writing down the channel's Pauli transfer matrix. One set of Kraus operators coincides (up to vectorization) with the ...
Adam Zalcman's user avatar
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3 votes

Evolution of a state vector: Why is the action of $N$ equivalent to the action of $UNU^{†}$?

Indeed mathematically $N$ and $UNU\dagger$ may be interpreted as 2 matrix representations of exactly the same physical operator $n$ (mind uppercase/lowercase), but in 2 different orthonormal basis. ...
Pierre-Paul T.'s user avatar
2 votes
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What are the possible channels preserving purity of all pure input states?

TL;DR: The unitary and reset channels are the only ones that return pure output for every pure input. That's because under Stinespring dilation the requirement that $\Phi$ return pure output for every ...
Adam Zalcman's user avatar
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