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Questions tagged [quantum-operation]

For questions about quantum channels or more generally quantum maps and the related formalism. For questions about unitary operations, please use quantum-gate instead.

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5 votes
1 answer
76 views

What is the meaning of $\sum_i K_iK_i^\dagger$ for a quantum channel with Kraus operators $K_i$?

Let a channel $N$ be given in terms of its Kraus operators $K_i$ as $$N(\rho) = \sum_i K_i\rho K_i^\dagger.$$ Is the sum $\sum_i K_iK_i^\dagger$ a meaningful quantity? I know that $\sum_i K_i K_i^\...
5 votes
2 answers
215 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, ...
0 votes
0 answers
12 views

relationship between helstrom operators acting on different pairs of quantum states

Let $\rho_1, \rho_2, \rho_3, \rho_4$ be arbitrary single-qubit density matrices. Let $A$ be an Hermitian operator and its spectral decomposition as $A = \sum_i \lambda_i \lvert i \rangle \langle i \...
2 votes
1 answer
308 views

How can the depolarizing channel be a quantum operation?

In Quantum Computing: From Linear Algebra to Physical Realizations it states that A quantum operation maps a density matrix to another density matrix linearly But let $\rho\in M_2$ be a density ...
3 votes
1 answer
1k views

How does the Kraus decomposition imply the Stinespring representation?

To show that the Kraus decomposition $\Phi(\rho)=\sum_{k=1}^D M_k\rho_S M_k^\dagger$ implies the Stinespring form $$\Phi(\rho)=\text{tr}_E[U_{SE}(\rho_S\otimes|0\rangle\langle 0|_E)U_{SE}^\dagger]$$ ...
0 votes
1 answer
80 views

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\!\...
8 votes
1 answer
538 views

Is there a comprehensive list of counterexamples in quantum information?

As was already asked about in this phys.SE question many years ago---which, sadly, got closed and never received an answer---is there a collection of counterexamples in quantum information theory, &...
2 votes
3 answers
431 views

Why are entanglement breaking channels, defined as $\Phi(\rho)=\sum_a \operatorname{Tr}(\mu(a)\rho)\sigma_a$, entanglement breaking?

Define an entanglement breaking channel $\Phi$ as a channel (CPTP map) of the form $$\Phi(\rho) = \sum_a \operatorname{Tr}(\mu(a)\rho) \sigma_a\tag A$$ for some POVM $\{\mu(a)\}_a$ and states $\...
1 vote
1 answer
425 views

Matrix Representation of Quantum Channels

I am working on a project and I expect to have expressions of a bunch of quantum channels of interest. The quantum channels will be in matrix form. For example for a 2 qubit system, the quantum ...
5 votes
2 answers
126 views

Are quantum channels bounded linear maps?

I've been reading about quantum channels from a couple of sources and have some doubts regarding some mathematical perspectives and properties of quantum channels. I've listed them below: It is known ...
0 votes
2 answers
128 views

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, ...
2 votes
1 answer
132 views

Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$?

Given the state of a system as $\rho_s$ and that of the ancilla (pointer) as $\rho_a$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective ...
1 vote
1 answer
96 views

If $\text{tr}_B \rho \in A$, then $\rho \in A \otimes B$?

Let our Hilbert space be $H = (A \otimes B) \oplus (A \otimes B)^{\perp}$. If $\rho \in A \otimes B$, then we have $\text{tr}_B \rho \in A$. Is the converse true: if $\text{tr}_B \rho \in A$, then $\...
7 votes
2 answers
85 views

What is the definition of physical gate error rate?

The fidelity of two quantum states $\rho$ and $\sigma$ is a well-defined (up to discussions about a square): $$ F(\rho, \sigma) = \text{Tr}\left( \sqrt{ \sqrt{\rho} \sigma \sqrt{\rho}}\right)^2. $$ ...
4 votes
3 answers
572 views

Can Kraus operators change a mixed state into a pure state?

It seems that Kraus operators cannot change a pure state into a mixed one (wrong). For any pure state can be written as $|\psi\rangle\langle\psi|$, so after the Kraus operators. It becomes $$\sum_l\...
8 votes
1 answer
802 views

Can the Kraus decomposition always be chosen to be a statistical mixture of unitary evolutions?

If $\mathcal{E}$ is a CPTP map between hermitian operators on two Hilbert spaces, then we can find a set of operators $\{K_j\}_j$ such that $$\mathcal{E}(\rho)=\sum_j K_j\rho K_j^\dagger $$ in the ...
5 votes
2 answers
844 views

Determining whether there exists an equivalent set of unitary Kraus operators

I have a CPTP quantum channel $\mathcal{E}$ that I've characterized by an operator sum representation $\{E_i\}$ for $i=1, \dots, m$ which acts on an input state like $$ \mathcal{E}(\rho) = \sum_{i=1}^...
4 votes
2 answers
144 views

What can we say about the eigendecomposition of quantum channels?

It is known that quantum channels, being CPTP maps, map density operators to density operators. And thus, they can be seen as superoperators. Similar to operators, where eigenstates and eigenvalues ...
3 votes
1 answer
53 views

Interesting properties of maps whose natural representation is unitary?

Let $\rho \in L(\mathcal{X})$ be a state in the space of linear operators acting on some complex Hilbert space $\mathcal{X}$. I'm interested in linear maps $\Phi: L(\mathcal{X}) \rightarrow L(\mathcal{...
6 votes
1 answer
686 views

What are examples of the correspondence between channels and their Stinespring dilations?

In this post I read that "quantum measurements are special cases of quantum channels (CPTP maps). Stinespring's dilation states that any quantum channel is realized by partial tracing a unitary ...
6 votes
0 answers
158 views

Is it possible to obtain a closed-form expression of the diamond distance between two single-qubit channels?

I would like to compute the diamond norm of the difference of two single-qubit channels $\Phi_1$ and $\Phi_2$. This difference is equal to, for any $2\times2$ complex matrix $\rho$: $$\...
4 votes
1 answer
73 views

How to get the Kraus decomposition of the amplitude damping channel from its Choi?

I found going from the Choi-matrix of a quantum channel to the Choi-Kraus decomposition a bit difficult. I know that it follows from the eigen-decomposition of the Choi-matrix. But I struggle with ...
2 votes
2 answers
100 views

Resources for understanding non-unitary channels and operators

I need some resources to understand non-unitary channels and operators in depth in order to simulate non-unitary channels instead of unitary ones in some problems. I would appreciate any guidance or ...
1 vote
0 answers
29 views

What is the rank of a superoperator of the form $\Xi (\cdot) = \sum_i^n U_i^\dagger {\cdot}\, U_i$?

Given a superoperator $\Xi$ as $\Xi (\cdot) = \sum_i^n U_i^\dagger \cdot U_i $ where $U_i$ are unitary. What can I say about the image of this map or about the rank of $\Xi$? Also, do you have some ...
4 votes
2 answers
94 views

Show that all extensions of $\rho$ can be obtained as a channel applied to its purification

I am struggling with this exercise here: Let $H:A, H_E$ and $H_{E′}$ denote complex Euclidean spaces. Consider a purification $|ψ_{AE}⟩⟨ψ_{AE}| ∈ D(H_A ⊗ H_E)$ of a quantum state $ρ_A ∈ D(H_A)$ and a ...
1 vote
1 answer
108 views

How to get the quantum process back from its Choi matrix?

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 $\{|k\rangle\}_{k=1}^d$ ...
3 votes
1 answer
152 views

Why is the coefficient-squared the probability, and not just the coefficient itself?

Context: I have decided not to accept the postulates of quantum mechanics blindly as gospel. There must be a way someone arrived at those postulates, and I want to know the basic reasoning behind them,...
1 vote
0 answers
19 views

qml.StronglyEntanglingLayers custom CNOT placement

The qml.StronglyEntanglingLayers function works great for what I need. However, I'd like to modify so that for each layer, only the first qubit is the control and the rest are targets of the control ...
1 vote
1 answer
53 views

What is the smallest environment size that allows to represent every quantum channel in fixed dimensions?

From the Stinespring dilation, we have that the dual or complementary channel can be observed in/expressed with the environment. Can we reconstruct any channel for environments with $\text{dim}>1$ ...
1 vote
0 answers
29 views

Trouble in Depolarizing Error Simulation with Qiskit

I'm currently attempting to simulate depolarizing errors using Qiskit, but I'm encountering an issue where it appears that no errors are being introduced into my simulation. After running the ...
0 votes
1 answer
72 views

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 &...
6 votes
3 answers
399 views

Can a CPTP map increase the purity of a state?

I am wondering if there exist CPTP maps $T$ such that the purity of a quantum state $\rho$ can increase, i.e. $$ \text{tr} ( T ( \rho )^2 ) \geq \text{tr} ( \rho ^2). $$ If so, what are the conditions ...
2 votes
2 answers
176 views

What can we say about $\sum_i K_i K_i^\dagger$ for non-unital CPTP maps?

Suppose we have a CPTP map $\Phi(\rho)=\sum_i K_i \rho K_i^+$, such that, $\sum_i K_i^+K_i=\mathbb{I}$. In case the map preserves Identity, is unital, then we immediately have $\sum_i K_i K_i^+ =\...
3 votes
2 answers
241 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 ...
8 votes
3 answers
603 views

What channels preserve the purity of all pure inputs?

Consider channels $\Phi$ such that $\Phi(|\psi\rangle\!\langle\psi|)$ is pure for all $|\psi\rangle$. Is there a simple way to characterise channels with this property? Let's suppose $\Phi$ acts ...
11 votes
1 answer
972 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 ...
2 votes
1 answer
282 views

Deriving the choi matrix definition of the quantum depolarizing channel

I was reading up on the quantum depolarizing channel (Preskill's Notes) (stack exchaange explanation), and I'm failing to see how the form \begin{align} \sigma &= (\mathcal E \otimes \mathbb I)(|\...
1 vote
1 answer
197 views

Physical Realizability of Outer product

I have a quantum state $|\Psi\rangle$ of $n$ qubits whose exact coefficients/probabilities I do not know. I want to operate another unknown quantum state $|\Phi\rangle$ with $U = a|\Psi\rangle \langle ...
0 votes
1 answer
63 views

What does any 2 Qbit Universal Gate in any Quantum Circuit with an N Qbit Input "operate on" mean mathematically

I am very new to Quantum Computing thus please excuse the layman question. I am aware that just like classical gates Quantum Computation also has a set of universal gates. Moreover, a universal set of ...
0 votes
0 answers
14 views

Multimode unitary channel in terms of action on characteristic function

Consider a set of $M$ signal modes described by the creation operators $\mathbf a^\dagger = (a_1^\dagger,...,a_M^\dagger)$, and let $\Phi_U$ be the channel defined by the conjugation $\Phi_U(\cdot)=U(\...
0 votes
0 answers
18 views

Liouville superoperators of multi-qubit quantum channels : Need for Numerics

As greatly described here, a density matrix can be vectorized in column-major order such that a (unitary) channel can be written $$ \rho \rightarrow |\rho\rangle = vec(\rho) \\ \rho' = U \rho U^{\...
2 votes
1 answer
126 views

How to prove there's no quantum channel that clones all classical states?

Considering a qubit $\scr H =\Bbb C^2$ I have seen a proof of the no-cloning theorem for pure states. I wonder how do you prove it for a classical state? 1)That is, how do I prove that there is no ...
0 votes
1 answer
40 views

How do I prove the following maps are completely positive?

I am trying to prove that the following superoperators are quantum channels, that is completely positive and trace-perserving linear maps 1 $\Psi[M]=WMW^\dagger$ where $W$ is an isometry 2 $\Psi[M_A]=...
1 vote
2 answers
100 views

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

I am currently learning about quantum channels and am sadly stuck at a rudimentary problem, where I don't understand how to find the Kraus matrices of a quantum channel. The amplitude damping channel ...
0 votes
0 answers
70 views

Show that the Choi of a tensor product is the tensor product of the Chois

I have the following problem. Let $N:L(H_A)\rightarrow L(H_A)$ be a quantum superoperator. The quantum state corresponding to this operator (via Choi-Jamiolkowski Isomorphism) is $\Gamma_A^{N}=id\...
9 votes
2 answers
1k views

Is the Kraus representation of a quantum channel equivalent to a unitary evolution in an enlarged space?

I understand that there are two ways to think about 'general quantum operators'. Way 1 We can think of them as trace-preserving completely positive operators. These can be written in the form $$\...
1 vote
1 answer
328 views

Prove that at most $d^2$ Kraus operators are sufficient to represent any quantum operation

All quantum operations $\mathcal{E}$ on a system of Hilbert space dimension $d$ can be generated by an operator-sum representation containing at most $d^2$ elements, $$ \mathcal{E}(\rho)=\sum_{k=1}^M ...
1 vote
1 answer
183 views

SX operator and superposition

I am running some tests using the probabilities we get from statevector to assert values in qiskit. For instance, with two qubits and a hadamard gate on the first one we have: ...
5 votes
1 answer
604 views

What is an example of a separable measurement that is not LOCC?

Could you give me an example of a measurement which is separable but not LOCC (Local Operations Classical Communication)? Given an ensable of states $\rho^{N}$, a separable measurement on it is a POVM ...
4 votes
2 answers
243 views

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

This problem is from a "passing remark" in this lecture notes. With the help of some colleagues I managed to find a way for this supposedly elementary fact, but I would like to see if there ...

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