6 votes

Does a quantum channel always preserve the identity matrix?

No, such maps are referred to as unital maps. A counterexample is the replacement map $$ \mathcal{E}(X) = \mathrm{tr}(X)\sigma $$ defined for some density matrix $\sigma$.
Rammus's user avatar
  • 5,503
5 votes

Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

This solution uses the spectral theorem and some elementary linear algebra computations. If you let $P = \sum_{i = 1}^{d'} \lambda_i \vert{\psi_i}\rangle\langle{\psi_i}\vert$ be the spectral ...
user2533488's user avatar
4 votes

Does a quantum channel always preserve the identity matrix?

No. When this is the case the channel is said to be unital. A necessary and sufficient condition for a map $\mathcal E$ being unital is its adjoint $\mathcal E^\dagger$ being trace-preserving. For ...
glS's user avatar
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4 votes
Accepted

How many free real parameters in a general CPTP map?

An easier way of counting is to consider the Choi state: It is a hermitian $d_Ad_B\times d_Ad_B$ matrix, and thus has $(d_Ad_B)^2$ real parameters. The condition that the reduced state of $A$ is the ...
Norbert Schuch's user avatar
4 votes
Accepted

Prove that if Kraus operators of $\Phi$ form an ONB then $\Phi$ is the replacement map

Distraction The matrix $P$ is a distraction, because the $dd'$ matrices $A_k$ are an orthonormal basis with respect to the inner product $\langle A,B\rangle_P:=\mathrm{tr}(PA^\dagger B)$ if and only ...
Adam Zalcman's user avatar
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4 votes
Accepted

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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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
Accepted

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
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
3 votes
Accepted

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
Accepted

Does a quantum channel always preserve the identity matrix?

No, no reason it should. For instance, the amplitude damping $$ \varepsilon_{AD}(\rho)=E_0\rho E_0^\dagger+E_1\rho E_1^\dagger $$ with $$ E_0=\begin{bmatrix}1 & 0\\ 0 & \sqrt{1-\gamma}\end{...
Pierre-Paul T.'s user avatar
2 votes

Find the Kraus operators for the amplitude damping channel from its isometric representation

You're asking how to relate the unitary/isometric representation of a channel with its representation in terms of Kraus operators. An isometric representation of a channel $\Phi$ is a writing of the ...
glS's user avatar
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2 votes

Does a quantum channel always preserve the identity matrix?

Long comment related to something I really like about the accepted answer: By Kraus' theorem any completely positive map can be written in the form $$ \mathcal{E}(\rho)=\sum_{k} E_k \rho E_k^{\dagger} ...
Ian Gershon Teixeira's user avatar
2 votes

Does a quantum channel always preserve the identity matrix?

No. Just try some of the examples at Canonical examples of quantum channels, such as #5 or #7.
Norbert Schuch's user avatar
2 votes

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

I think you're just confused by the wording. $U N | \psi \rangle $ means apply N, then apply U. $UNU^\dagger U|𝜓 \rangle$ means apply U, then apply $UNU^\dagger$ So $U N | \psi \rangle = UNU^\dagger ...
Peter-Jan's user avatar
  • 1,378
2 votes
Accepted

What is the meaning of complex expected values?

This is a floating point error. The way computers work with floating point numbers, very small errors can happen. You can check it by trying to see if ...
Skilopsaros's user avatar
2 votes

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

Indeed there is some unclarity in the wording "the operator $UNU^\dagger$ acts on states in the same way that $N$ did". Better would be: "the operator $UNU^\dagger$ acts on states $U|\...
Jos Bergervoet's user avatar
1 vote
Accepted

Two-photon N00N state through Mach-Zehnder interferometer

There are some corrections to the calculations, but it is more important to focus on the motivation: why are you sending a NOON state through a Mach-Zehnder interferometer? The idea of the NOON state ...
Quantum Mechanic's user avatar

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