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4 votes

How to mathematically describe the action of CNOT on the control qubit alone?

The action $\mathcal{E}$ of CNOT on just the control qubit is the composition of two functions: the action \begin{align} \mathcal{C}(\rho)=\text{CNOT}\,\rho\,\text{CNOT}\tag1 \end{align} of CNOT on ...
Adam Zalcman's user avatar
  • 22.9k
4 votes
Accepted

What are kraus operators of a qubit interacting a thermal environment?

One way to model interaction with a thermal environment (of some temperature $T$) is through so-called thermal operations. Given some system's Hamiltonian $H_S$ they're all channels $\Phi$ of the form ...
Frederik vom Ende's user avatar
4 votes

Why do we need/have the operator sum representation (Kraus representation)?

To add to Daniele's answer, $E_k$ is an operator—and not a scalar—because the notation in Nielsen & Chuang is sloppy. What is meant is that $E_k=\iota_k^\dagger U\iota_0$ where $\iota_k:\mathbb C^...
Frederik vom Ende's user avatar
4 votes
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Is the trace of a positive map always positive?

Consider the qubit map $\Phi(\rho):=\sigma_Y\rho^T\sigma_Y$, that is, $$ \Phi\begin{pmatrix}\rho_{11}&\rho_{12}\\\rho_{21}&\rho_{22}\end{pmatrix}=\begin{pmatrix}\rho_{22}&-\rho_{12}\\-\...
Frederik vom Ende's user avatar
4 votes

Definition of a quantum gate

Sometimes people mean unitary-only. Sometimes they mean generically anything you might do to the qubits (like a measurement gate or a reset gate or a dynamical decoupling gate or a leakage removal ...
Craig Gidney's user avatar
  • 37.9k
3 votes
Accepted

Do all Hermiticity-preserving maps generate completely positive maps?

First a basic observation: if all Hermitian preserving $\mathcal L$ gave rise to completely positive dynamics $e^{t\mathcal L}$ for all $t\geq 0$, then so would $-\mathcal L$ (still Hermitian ...
Frederik vom Ende's user avatar
3 votes

Definition of a quantum gate

I wouldn't say it's coupled with the measurement gate. I think they are inherently different: A quantum gate is equivalent to a Unitary matrix. Quantum gates create together a quantum circuit (which, ...
Nati Erez's user avatar
2 votes

Prove $\|{\cal E}(\rho-\sigma)\|_1\leq\|({\cal E}\otimes{\rm id})(U\{(\rho - \sigma)\otimes |0⟩⟨ 0|\}U^{\dagger})\|_1$ with $U$ a CNOT

tl;dr: The inequality in its current form is wrong, although a weaker version of it (where the right-hand side has an additional factor of $2$) holds true. For a counterexample to the inequality as ...
Frederik vom Ende's user avatar
2 votes
Accepted

Validity of a quantum operation for generalised state erasure

Late to the question but tl;dr: your idea almost works, you just have to change the $\oplus$ into a $\otimes$—cf. Equation (1) below. First, the map you proposed is not linear because $\mathcal E(0)=\...
Frederik vom Ende's user avatar
2 votes
Accepted

If states are close together does there always exist a channel close to the identity mapping one to the other?

We shall see that for general mixed states no such upper bound can exist in the following precise sense: For no continuous function $c:[0,2]\to[0,2]$ with $c(0)=0$ does it hold that for all $\|\rho-\...
Frederik vom Ende's user avatar
2 votes

Kraus decomposition of merging in lattice surgery

Lattice merge isn't faithfully captured by a CPTP map description (and therefore a Kraus representation). Tracking $M$, the measurement outcome, conditions what logical correction operation you will ...
rmehta's user avatar
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2 votes
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Is every quantum channel covariant with respect to some non-trivial Hamiltonian?

This quesiton passes all of Norbert Schuch's canonical channel examples except for the last one: consider the entanglement breaking channel $$ \Phi(\rho):={\rm tr}(F\rho)\sigma_1+{\rm tr}(({\bf1}-F)\...
Frederik vom Ende's user avatar
2 votes

Why do we need/have the operator sum representation (Kraus representation)?

The formalism is useful to describe a noisy transformation $\mathcal{E}$ only in the domain of a system of interest $\mathcal{H}$. According to the book, you can select an orthogonal bases $\{e_i\}_i$ ...
Daniele Cuomo's user avatar
2 votes

Resource for geometric representation of quantum channels

Chapters 5.2 and 10.7 in the book "Geometry of Quantum States" by Bengtsson & Zyczkowski (alt link) may be of interest to you. Alternatively, you could have a look at Chapters 2.1.3 and ...
2 votes
Accepted

Is the adjoint of a strictly positive channel again strictly positive?

The answer to this is two-fold: the adjoint of a strictly positive channel (in the Schrödinger picture, i.e. CPTP) is indeed always strictly positive, simply because the adjoint of any channel is ...
Frederik vom Ende's user avatar
2 votes

Why do we need/have the operator sum representation (Kraus representation)?

To add to Daniele's and Frederik's answers: the operator sum representation is even more useful than the Nielson and Chuang derivation might suggest - though, the derivation is very helpful from a ...
qubitzer's user avatar
  • 317
2 votes
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IBM quantum computer backend cycle time and real gate duration

The integers in the InstructionDurations() are the multipliers of the real-system timestep dt. So for example, consider the ...
Shravan Patel's user avatar
1 vote

Are Stinespring unitaries that give rise to the same channel locally unitarily equivalent?

As it turns out the statement in question is wrong. For a counterexample consider the full-rank environment state $$ \omega=\begin{pmatrix} \frac12&0&0\\0&\frac14&0\\0&0&\...
Frederik vom Ende's user avatar
1 vote

How to mathematically describe the action of CNOT on the control qubit alone?

In the ZX calculus, the CNOT gate factors into a Z type node for the control linked to an X type node for the target. The Z type node (the "control part of the operation") has three ports: $...
Craig Gidney's user avatar
  • 37.9k
1 vote
Accepted

Can the spectral radius of a completely positive map exceed the spectral radius of its transition matrix?

While the inequality in quesiton does hold for all channels and all self-adjoint positive maps (as shown in the above question) perhaps surprisingly it fails for general completely positive maps. For ...
Frederik vom Ende's user avatar
1 vote
Accepted

Kraus decomposition of merging in lattice surgery

CP maps for lattice surgery merge with post-selection Post-selecting on $M_{ZZ}=0$, lattice surgery merge is described by the completely positive linear map with Kraus representation \begin{align} \...
Adam Zalcman's user avatar
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1 vote
Accepted

To what extent is the normal form of the Pauli transfer matrix unique?

It turns out that uniqueness can only be guaranteed if $\lambda_1>\lambda_2>|\lambda_3|$; otherwise one can construct counterexamples (as we will do below). In order to understand why ...
Frederik vom Ende's user avatar
1 vote

Solving optimization problems on real quantum hardware

Current quantum hardware based on a quantum circuit model has a limited number of qubits, which are prone to errors and decoherence. So, comparing solutions from classical solvers with those produced ...
MonteNero's user avatar
  • 2,666
1 vote

relationship between helstrom operators acting on different pairs of quantum states

tl;dr: In the single-qubit case the explicit formula $$P(|0\rangle\langle0|-\rho)=\begin{cases} 0&\rho=|0\rangle\langle 0|\\ \frac{|0\rangle\langle0|-\rho}{\sqrt{|\langle 0|\rho|1\rangle|^2+(\...
Frederik vom Ende's user avatar
1 vote

Analyzing the composition of a channel with its adjoint in relation with an identical composition obtained for the channel's complement

The short answer is: yes the inclusion in question is true for all maps $\Phi$ which are completely positive and completely co-positive. In fact, it turns out to hold for the slightly more general ...
Frederik vom Ende's user avatar

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