# Questions tagged [quantum-operation]

For questions about completely positive (CP) linear maps between quantum states. Can also be used for trace-preserving CP maps (quantum channels). For questions about unitary operations, please use quantum-gate instead.

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### Is a process matrix of rank $1$ unique?

It is said that when an unknown process is unitary, its $\chi$ matrix is rank-$1$ and possesses only one positive eigenvalue. See eg https://arxiv.org/abs/2306.07867. So when the process matrix has ...
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### How to characterize the extreme points of the set of CPTP maps?

The set of CPTP maps is convex, therefore, it is enough to perform the needed optimizations over the set of extreme points. Is there any way of characterizing the said extreme points that would lend ...
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### Question about Nielson & Chuang Problem 9.2

I am working on the following problem from the book "Quantum Computation and Quantum Information" by Nielsen and Chuang. Problem 9.2: Let $\mathcal{E}$ be a trace-preserving quantum ...
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### What is known about the size of the spectral gap of unital quantum channels?

I am interested in the spectrum of unital quantum channels $\Phi$ (which act on finite dimensional spaces). The literature is extremely vast on such objects so I hope some experts can point me along ...
1 vote
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### Derivation of Choi-Jamiolkowski isomorphism

I'm following the course Mathematical Methods of Quantum Information Theory by Reinhard Werner. In lecture 6, he gave a derivation of Choi-Jamiolkowski isomorphism, and I'm struggling to understand ...
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1 vote
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### What are the singular values of a quantum channel?

I have tried to find the explicit definition of them but was not able to. My guess is that they are eigenvalues of the superoperator $\Phi^{\ast}(\Phi)$, where $\Phi$ is the channel and $\Phi^{\ast}$ ...
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### Infidelity as distance measure

Let $\mathcal{X} \in {\rm CP}(\mathcal{H}, \mathcal{K})$ and unital (compositive positive and unital maps). Let $\mathcal{Y} \in {\rm CPT}(\mathcal{H}, \mathcal{K})$(complete positive and trace ...
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### How can I get the process $\mathcal{E}$ from Choi matrix and Choi-Jamiolkowski isomorphism?

The unnormalized maximally entangled bipartite state between a quantum system $S$ and an ancilla system $A$ is $|\psi\rangle=\sum_{k=1}^d|k\rangle_A|k\rangle_S$ , where $\{|k\rangle\}_{k=1}^d$ ...
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### Get matrix for an X gate for a given fidelity p

Wanted to check on how to mathematically obtain the matrix of an X gate which has fidelity/probability $p$? (i.e. it acts as an $X$ gate with probability $p$)
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### When should I use the Choi matrix and when should I use the $\chi$ matrix?

A quantum map on a $d$-dimensional space has the general representation: $$\mathcal{S}(\rho)=\sum_{\alpha,\beta}^{d^2}\chi_{\alpha\beta}\Gamma_{\alpha}\rho \Gamma_{\beta}^{\dagger},$$ where $\chi$ ...
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### Yet another condition for a map to be completely-positive and trace-preserving

Surely, these conditions are all well-defined and well-known (via the Choi, Kraus, and Stinespring presentations). Is the following 'definition' valid? Does it make sense? "The map is CPTP if, ...
1 vote
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### Commute partial trace operator and measurement operator

Suppose I have a general measurement $M$ applying on n-qubit registers. So we are able to use the POVM notation, where $\sum_m M_m = I$ and $M_m = E_m^\dagger E_m$. And I want to know the exact ...
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### Interconversion between different representations of quantum channels

I was reading TQI-notes by Watrous where they introduce different representations for quantum channels and wondering how to go from one to the other. I have: \begin{align} &|\Phi(\rho)\rangle\!\...
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### How to perform below operation in Qiskit?

I want to implement the below equation in Qiskit. $(A \otimes B).\rho.(B^\dagger \otimes A^\dagger)$ where $\rho$ is a density matrix and $A$ and $B$ are CNOT gates.  A=\begin{bmatrix} 1 & 0 &...
1 vote
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### What is the relation between the Choi matrix and the Liouville space (superoperator) representations of a channel?

A.S. Fletcher, P. W. Shor, and M. Z. Win Phys. Rev. A 75, 012338 (2007) says the Choi matrix for the operation $\mathcal{A}$ is given by $X_A \equiv \sum_k |A_k\rangle\!\rangle\langle\!\langle A_k|$, ...
1 vote
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### How to justify the conclusion $|E_{sq}(\rho)-E_{sq}(\sigma)|\le f(\epsilon)$, when proving the continuity of the squashed entanglement?

I am following the paper by Christandl and Winter introducing squashed entanglement. My question is particularly on the continuity proof of squshed entanglement mentioned after conjecture 14 and ...
### Can any channel be represented as $A\rho A^\dagger$ for some $A$?
Consider an arbitrary quantum operation defined by a series of Kraus operators $\sum_j K_j\rho K_j^\dagger$ over the density matrix of the system $\rho$. The operation might or might not be unitary, ...