# 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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### Trouble understanding operator sum representation

I am having lot of trouble trying to understand the operator sum representation in Nielsen and Chuang: I get the very first equation in above but how does that translate to 2 and how does the 3rd ...
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### Tests for unitarity of a quantum channel that don't require the Choi state

I am aware that the rank of the Choi state is a test for whether or not a channel is unitary. Are there any other equivalent tests that we can do to test if a channel is unitary without involving a ...
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### Are peripheral eigenvalues of a completely positive map always semisimple?

It is known that all peripheral eigenvalues (i.e. all eigenvalues $\lambda\in\mathbb C$ such that $|\lambda|$ equals the spectral radius) of positive trace-preserving or positive unital maps are ...
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### What is the domain of the dual map of a quantum channel?

Possibly a naive question...if the dual map of a quantum channel gives the evolution of the system in the Heisenberg picture by acting on observables, and observables are self-adjoint operators on the ...
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### Finding a succinct representation for the CPTP map ${\cal N}^{\otimes n}$ such that ${\cal N}(I)=I+pZ$ and ${\cal N}(Z)=(1-p)Z$

Consider a single qubit CPTP map $\mathcal{N}$ such that $$\mathcal{N}(I) = I + pZ,~~~~~~\mathcal{N}(Z) = (1-p)Z,$$ where $I$ and $Z$ are Pauli operators. For an $n$ qubit Pauli operator $P$, made ...
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It is well known that for if any two linear maps $V_1,V_2:\mathbb C^n\to\mathbb C^k\otimes\mathbb C^m$ (isometry or not) satisfy $${\rm tr}_{\mathbb C^m}(V_1(\cdot)V_1^\dagger)={\rm tr}_{\mathbb C^m}(... • 2,588 3 votes 1 answer 151 views ### Do all Hermiticity-preserving maps generate completely positive maps? I am confused about what kinds of maps are valid infinitesimal generators of completely positive maps. I know that any Markovian completely positive map can be written in the form e^{t \mathcal{L}}, ... • 73 0 votes 1 answer 30 views ### Given \Psi completely positive when do there exist K_1,K_2 such that K_2\Psi(K_1^\dagger(\cdot)K_1)K_2^\dagger is also trace preserving? In quantum information it occasionally happens that one ends up with a completely positive but not yet trace-preserving map \Psi which one wants to make trace preserving somehow; this often comes up ... • 2,588 1 vote 2 answers 161 views ### Definition of a quantum gate A quantum gate is usually defined as a unitary transformation, like the definition found in "Mathematics of Quantum Mechanics" by Scherer. According to this definition, can we consider a ... • 427 4 votes 3 answers 230 views ### Why do we need/have the operator sum representation (Kraus representation)? I am reading through Nielsen & Chuang, and I am on the section about operator sum representation. They performed this derivation. Why is it important and useful for us to bundle together the ... 0 votes 1 answer 45 views ### Resource for geometric representation of quantum channels I was wondering if anyone knows about any good resources on representing unital/quantum channels by using rotations/pauli matrices. It is mentioned in Nielsen&Chuang on p774, but i feel it is ... 2 votes 1 answer 52 views ### Is the adjoint of a strictly positive channel again strictly positive? Building on the concept of positive definite operators{}^1—denoted A>0—a linear map \Phi:\mathbb C^{n\times n}\to\mathbb C^{k\times k} is called strictly positive if \Phi(A)>0 for all A&... • 2,588 1 vote 0 answers 16 views ### Solving KnapSack problem on D Wave hybrid CQM I am solving 01 KnapSack problem for 500k items with the help of hybrid CQM solver of D Wave. And for comparison I solved same problem with CPLEX. Solution quality of CPLEX solver is better than D-... 3 votes 3 answers 262 views ### How to mathematically describe the action of CNOT on the control qubit alone? Basically the title. From what I know, starting from the |{+0}\rangle state where the reduced density matrix of the first qubit |+\rangle is \frac{1}{2}\begin{bmatrix} 1 & 1 \\ 1 & 1 \end{... • 594 0 votes 0 answers 45 views ### How to interpret result for an OR problem solved in Qiskit I solved "01 KnapSack problem" in Qiskit optimization module. Though the iterations doesn't gets converged but I got an answer (may be infeasible). But my question is that how to interpret ... 3 votes 1 answer 105 views ### What are kraus operators of a qubit interacting a thermal environment? Suppose a qubit that interacting a thermal environment. The thermal environment can be a thermal field for example. What is the kraus operators for this case? • 771 1 vote 1 answer 78 views ### Can the spectral radius of a completely positive map exceed the spectral radius of its transition matrix? Recalling the spectral radius r(T):=\max_{\lambda\in\sigma(T)}|\lambda| of a linear map T (where \sigma(T) refers to the spectrum of T), it is known that every quantum channel \Phi:\mathbb C^{... • 2,588 4 votes 0 answers 43 views ### What are examples of channels whose Holevo capacity can be computed explicitly? Given a channel \Phi:\operatorname{Lin}(\mathbb{C}^n)\to\operatorname{Lin}(\mathbb{C}^m), we define its Holevo capacity as$$\chi(\Phi) = \sup_\eta \chi(\Phi(\eta)),$$with the sup taken with ... • 26.1k 2 votes 2 answers 146 views ### Kraus decomposition of merging in lattice surgery I am reading about lattice surgery from this paper. I am interested in the merge operation which takes 2 qubits to 1 qubit. I want to understand the logical-level Kraus operation that the merge does. ... • 23 0 votes 0 answers 25 views ### Transforming spin operators into fermionic operators and finding their anticommutation relations The Jordan-Wigner transformation (JWT) is a method used in quantum mechanics to map spin operators, which are typically associated with spin-1/2 particles, to fermionic operators, which describe ... 5 votes 1 answer 250 views ### For how many different times do I have to know that e^{tL} is a quantum channel to conclude that L is of Lindblad form? As first shown by Gorini, Kossakowski, Sudarshan and Lindblad given some linear map \mathcal L:\mathbb C^{n\times n}\to\mathbb C^{n\times n}, e^{t\mathcal L} is a quantum channel for all t\geq 0 ... • 2,588 1 vote 1 answer 39 views ### Is every quantum channel covariant with respect to some non-trivial Hamiltonian? When asking whether every channel is covariant with respect to some non-trivial unitary channel I mean the following: Does there for every CPTP map \Phi:\mathbb C^{n\times n}\to\mathbb C^{n\times n}... • 2,588 1 vote 1 answer 70 views ### To what extent is the normal form of the Pauli transfer matrix unique? In order to properly state the question let me be precise about the object at the core of this question's title. First, given any orthonormal basis of G:=\{G_j\}_{j=1}^{n^2} of \mathbb C^{n\times n}... • 2,588 0 votes 1 answer 45 views ### Solving optimization problems on real quantum hardware I want to know how to solve a 01 Knapsack problem (or any optimization problem) on real Quantum Hardware only. I don't want to use an Application class or any classical simulation technique. If anyone ... 1 vote 1 answer 79 views ### IBM quantum computer backend cycle time and real gate duration I am new to dynamical decoupling and is trying to study this from qiskit: https://docs.quantum.ibm.com/api/qiskit/qiskit.transpiler.passes.PadDynamicalDecoupling . There, they are specifying the ... 5 votes 1 answer 206 views ### Is the trace of a positive map always positive? Obviously, positive semi-definite operators always admit a positive trace as {\rm tr}(A)=\|A\|_1\geq 0 whenever A\geq 0. This motivates the following "lifted" question: Given any ... • 2,588 2 votes 1 answer 50 views ### If states are close together does there always exist a channel close to the identity mapping one to the other? Question: Given states \rho,\omega\in\mathbb C^{n\times n} and \varepsilon>0 such that \rho and \omega are \varepsilon-close in trace norm does there exist a channel \Phi with \Phi(\... • 2,588 2 votes 0 answers 59 views ### How to systematically find the kernel of a channel from its Kraus operators? A quantum channel is a completely positive trace-preserving map. Given a quantum state \rho and channel N, let the output be N(\rho). Given the Kraus operators of the channel, how can one find ... • 3,117 0 votes 0 answers 18 views ### Quantum Channel with least disturbance for any input and output dimensions Let n and m be two arbitrary dimensions of the input Hilbert space and output Hilbert space respectively. What is the quantum channel that preserves information as much as possible (i.e. with the ... • 321 1 vote 1 answer 134 views ### How to view operator norms on open-system representation of quantum channels I know how the operator norms \| X\|_{1},\| X\|_{2}, and \| X\|_{\infty} are defined for any operator X\in B(\mathcal{H}). My question is about how to view\| T(X)\|_{1},\| T(X)\|_{2}, and ... 0 votes 1 answer 30 views ### Can any separable \rho=\sum_i p_i\sigma_i\otimes\tau_i be written as \rho=(I\otimes T)(\sum_ip_i\sigma_i\otimes|i⟩\!⟨i|) for some channel T? I am struggling with the following exercise, and was wondering if anybody had any good tips on how to attack the problem/where to begin: Given a separable quantum state$$\rho_{AB'}=\sum_{i=1}^{k}p_{i}...
It is well known that $\|\mathcal{E} \circ \mathcal{F} - \mathcal{E}\|_\lozenge \leq \|\mathcal{F} - \mathcal{I}\|_\lozenge$. What if I have \$\|\mathcal{A} \circ \mathcal{E} \circ \mathcal{F} - \...