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.
534 questions
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What does "${\cal M}_{A,\alpha}$ is a measurement operation" mean?
My question regards this paper: https://arxiv.org/abs/1909.07534
If you look at the sentence below equation 8, it says that
Ma,x is measurement operation post-selected with a measurement outcome $x$.
...
CommunityBot
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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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Is there a normal form for completely positive superoperators with rotationally symmetric spectra?
Let $d$ be a natural number. Given $A_1,\dots,A_r\in M_d(\mathbb{C})$, define a linear operator
$\Phi(A_1,\dots,A_r):M_d(\mathbb{C})\rightarrow M_d(\mathbb{C})$ by letting
$\Phi(A_1,\dots,A_r)(X)=...
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Is the Eastin-Knill Theorem incorrect?
I am reading through this paper (the Eastin-Knill Theorem) and there is a step in the proof of the main theorem that I do not understand.
Let $Q$ be a composite quantum system supporting a quantum ...
CommunityBot
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Confusion regarding the interpretation of no-pancake theorem
I am reading about phase damping channel from Preskill's notes on quantum information. It is shown that the channel causes decay of the $x$ and $y$ components of the spin polarization of the density ...
glS♦
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What's the reasoning behind writing the isometric representation of a channel?
I am reading about phase damping channel from Preskill's notes. He writes off the unitary representation of the channel as
Unitary representation. An isometric representation of the channel is
\begin{...
glS♦
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Implementation of two qubit gate decomposition in local operations
My questions is regards to this paper: https://arxiv.org/abs/1909.07534
The above is the decomposition of a two qubit gate into local operations. Please note they are using a Super Operator formalism....
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How does the spectral decomposition of the Choi operator relate to Kraus operators?
In Nielsen and Chuang's QCQI, there is a proof states that
Theorem 8.1: The map $\mathcal{E}$ satisfies axioms A1, A2 and A3 if and only if
$$
\mathcal{E}(\rho)=\sum_{i} E_{i} \rho E_{i}^{\dagger}
$$...
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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 $X_A \equiv \sum_k |A_k\rangle\!\rangle\langle\!\langle A_k|$, ...
glS♦
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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 ...
glS♦
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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, ...
glS♦
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Closeness of unitary dilations of CPTP maps
Let $\Phi_1,\Phi_2 \colon S(\mathcal{H}) \to S(\mathcal{H})$ be CPTP maps on the same Hilbert space $\mathcal{H}$ which are $\varepsilon$-close in diamond norm, and let $U_1,U_2$ be respective unitary ...
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Finding the Clifford circuit that implement a particular mapping of Paulis strings
Denote $P_N=\{\tau \}$ the set of Pauli strings,
composed out of tensor products of Pauli matrices $\sigma_i^\alpha$
acting on $N$ qubits, e.g.
$\tau=\sigma^x_1 \otimes \mathbb{1}_2 \otimes \sigma^y_3 ...
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Non trace-preserving map in axiomatic approach to quantum operations
In Nielsen and Chuang's Quantum Computation and Quantum information there is an axiomatic definition of the quantum operation (as one of the 3 approaches to quantum operations).
A quantum operation is ...
glS♦
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Representing 1 qubit pauli-channels as an averaging effect of random rotations in the bloch sphere. Basic literature?
I am looking into 1 qubit pauli channels, e.g. the dephasing channel
$$\mathcal{E}(\rho)=(1-p)\rho+p\sigma_z\rho\sigma_z.$$ I found out it could be represented as
$$\mathcal{E}(\rho)=\int p(\lambda)\...
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Are $n$-qubit Pauli channels $\mathcal E(\rho)=\sum_j p_j P_j \rho P_j$ invertible?
An $n$-qubit Pauli channel $\mathcal E$ acting on a quantum state $\rho$ is of the form
\begin{equation}\label{PauliChannel}
\mathcal E(\rho)=\sum_jp_jP_j\rho P_j
\end{equation}
where $p_j\in[0,1]$ ...
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Unambiguous discrimination using POVM with highest discriminate probability
I was studying Nielsen&Chuang's textbook (about page 92), and come up with a question that I cannot solve it.
Given one of the two state $|\psi_1\rangle=|0\rangle$ and $|\psi_2\rangle=\frac{1}{\...
glS♦
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Are quantum controlled non-unitary circuit operations possible?
Are quantum controlled non-unitary operations possible?
For instance can I define a controlled-reset where I reset a target qubit to $|0 \rangle$ if a controlled qubit is in the $|1\rangle$ state? How ...
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Optimality of discriminating POVM among all quantum channels
It is proven in the very helpful answer here that using the optimal POVM for unambiguous discrimination of the equally-likely non-orthogonal states $\lvert 0\rangle$ and $\lvert +\rangle$, we can ...
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Why is there always a $k$-outcome experiment associated to operators such that $\sum_{i=1}^k M_i^* M_i=I$?
When we want to observe a system, we make quantum measurements, which are always described by a class of operators $ \{ M_i \}_i $
The probability that we observe the outcome $i$, given that the ...
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Proof that $\frac{1}{d} | \text{Tr}(\mathcal{P}_i^\dagger \mathcal{E}(\mathcal{P}_j))| \leq 1$ for superoperators
I have two Pauli operators $\frac{1}{\sqrt{d}} \mathcal{P}_i$, $\frac{1}{\sqrt{d}} \mathcal{P}_j$, and an arbitrary quantum channel $\mathcal{E}$ (in the superoperator/Liouville representation) all ...
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Does any quantum channel satisfy ${\rm Tr}(\mathcal E^\dagger \mathcal E) \in[0, d^2]$?
I am reading the paper "Direct Fidelity Estimation from Few Pauli Measurements".
According to the paper, the entanglement fidelity between the a unitary channel $\mathcal U$ and a quantum ...
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In general, what is feasible quantum computation?
I don't really understand what is feasible quantum computation, in my book (Lipton and Regan's Quantum Algorithms via Linear Algebra) they said that:
A quantum computation $C$ on s qubits is feasible ...
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What is the explicit form of $T_1$ decay channel?
I see $T_1$ error mentioned in many experimental papers (for example, page $10$ of this paper mentions such a decay.)
How do I model $T_1$ decay theoretically? More concretely, here's my question. Say ...
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How can $\chi(\hat{A},\hat{B}:C) \le \chi(\hat{A},B:C)$ be true?
The holevo information of $\rho_{ABC}$ w.r.t to measurements on A and B (for the sake of this we'll assume local measurements suffice), is given by $$\chi(\hat{A},\hat{B}:C)$$ where $\hat{A}$ and $\...
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Is the map $1^{(P',P)}$ sending $P$ to $P'$ a quantum channel?
Let $\mathscr{P}_n$ denote the $n$-qubit unsigned Pauli operators, and let $P, P' \in \mathscr{P}_n$. On page 10, appendix A of this arXiv paper, $1^{(P^{'}, P)}$ is defined as a linear function on $n$...
glS♦
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How to code a projector operator in qiskit?
I'm new to qiskit and I want to know how do I define a projector operator in qiskit? Specifically, I have prepared a 3 qubit system, and after applying a whole lot of gates and measuring it in a state ...
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What do quantum noise models have in common?
Let us see the one-qubit case of different noise channel, the depolarizing channel is
$\Lambda_1^{depol}(\rho_1)=(1-\frac{4}{3}\epsilon_1)\rho_1+\frac{1}{3}\epsilon_1\sum_{i=0}^{3}\sigma_i\rho_1\...
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CZ-gate in neutral atoms computer with Rydberg pulses
I'm trying to understand the protocol to obtain a CZ-Gate with two qubits and Rydberg pulses : https://queracomputing.github.io/Bloqade.jl/dev/3-level/#pulse-CZ-gate
Apply Rydberg π-pulse on control ...
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How to derive the number of independent parameters in the $\chi$ matrix from the Choi matrix?
In the section on Quantum process tomography, Page 391, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang. it is given that
In general, $\chi$ will contain $d^4−d^2$ ...
glS♦
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How does trace norm of a state change after measurement? Why is the 1-LOCC norm less?
I recently asked this question about the meaning of a quantum classical channel in a paper I read.
The answer I accepted provided an explanation for the 1-LOCC norm (which I asked about) which is ...
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Neumark dilation for qubit tetrahedron SIC-POVM
I would like to know if an analytic solution is known for the Neumark dilation of the qubit tetrahedron SIC-POVM defined by
$$ M_0= \frac{1}{4\sqrt{3}} \Big( \sqrt{3}I + X +Y +Z \Big), \qquad M_1= \...
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Do quantum channels satisfy $\Phi(0)=0$?
Assume $\Phi$ is a quantum channel, which is a CPTP map. Does the following equality holds and why?
\begin{align}
\Phi(0) = 0.
\end{align}
glS♦
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Understanding of the transverse-field Ising model
I want to make sure whether I do understand the transverse Ising model correctly or not. The classical Ising model describes the interaction between spins in a grid and the state of spins can be ...
CommunityBot
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Prove the invariance upon change of variables in the definition of twirled channel
The twirled operation of a quantum channel $\varepsilon$ is defined as
\begin{align}
\varepsilon_T(\rho)
&= \int dU U^\dagger \varepsilon(U \rho U^\dagger)U,
\end{align}
where the integral is over ...
glS♦
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Prove that the twirling operation on a channel gives a decomposition $\int dU\, U^\dagger\mathcal E(U\rho U^\dagger)U=\alpha P+\beta Q$
The twirled operation of a quantum channel $\mathcal E$ is defined as
\begin{align}
\mathcal E_T(\rho)
&= \int dU U^\dagger \mathcal E(U \rho U^\dagger)U,
\end{align}
where the integral is over ...
glS♦
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Why does the twirl of a quantum channel give a depolarizing channel?
I would like to understand in detail why the twirl of a quantum channel gives depolarizing channel, which is the starting point of randomized benchmarking. To be self-contained, let me set up the ...
glS♦
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Why is the quantum capacity quantified by the coherent information?
Most types channel capacities associated to a given quantum channel are quantified using mutual informations (sometimes classical, sometimes quantum, sometimes regularised), which is not surprising ...
glS♦
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Coherent Information and Entanglement Breaking channels
The book by John Watrous, "The Theory of Quantum Information" is an exciting read for anyone wanting to research quantum information theory. The following question presumes some background ...
glS♦
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Complementary channel of binary sum channel
This isn't strictly a quantum question but the idea of complementary channels is the following: Take any channel $N_{A\rightarrow B}$. Take it's Stinespring dilation (which is an isometry) $V_{A\...
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Zero noise extrapolation for error mitigation: Meaning of rescaled density matrix, specifically when there is no local hamiltonian evolution
I have a few questions regarding dynamics rescaling for zero noise extrapolation. In the paper Error mitigation for short-depth quantum circuits, in equation (30), they write
We redefine $T \...
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What is the meaning of $\langle e_k|U|e_0\rangle$ when $U$ acts on a larger Hilbert space than that in which $|e_0\rangle$ and $|e_k\rangle$ live?
In Nielsen and Chuang, 10th Anniversary Edition, there is a definition of the operator sum representation of a quantum operation: $\mathcal{E}(\rho)=\sum_{k}\langle e_k|U[\rho\otimes|e_0\rangle\langle ...
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Can two measurements be represented as a single measurement when they are acted upon sequentially?
Let two different POVM measurements represent as $\mathcal{M}_1=\{\Pi_i\}_{i=1}^k$ where $\Pi_i$ is element of the $\mathcal{M}_1$ measurement and $\mathcal{M}_2=\{E_j\}_{j=1}^n$ where $E_j$ is the ...
glS♦
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Why Qiskit's circuit denotes the top bit as the least significant bit?
The descriptions can be found here:Qiskit Circuit. Where it says:
It really confuses me, because it's very inconvenient to run quantum circuits designed in a classical way in the physics textbook and ...
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Understanding adversarial Channels
The paper here (Definition 1) defines adversarial channels as $N(\rho)= \sum_i A_i \rho A_i$ with the mention that the $A_i$ is chosen only after a communication strategy is decide. This gives the ...
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How to splice Hamiltonians corresponding to channels $\Phi_1$ and $\Phi_2$ so as to obtain a Hamiltonian corresponding to $\Phi_2\circ\Phi_1$?
Suppose I have two quantum channels $\Phi_1:B(\mathcal{H}_1)\rightarrow B(\mathcal{H}_2)$ and $\Phi_2:B(\mathcal{H}_2)\rightarrow B(\mathcal{H}_3)$, and let $\Phi=\Phi_2\circ \Phi_1$.
Stinespring ...
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Is tomography of the Choi state sufficient for channel tomography?
Given that there is an isomorphism between quantum states and quantum channels (the Choi-Jamiolkowski isomorphism) and given that state tomography is well-researched, why is quantum process or quantum ...
glS♦
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How to find the operator sum representation of the depolarizing channel?
In Nielsen and Chuang (page:379), it is shown that the operator sum representation of a depolarizing channel $\mathcal{E}(\rho) = \frac{pI}{2} + (1-p)\rho$ is easily seen by substituting the identity ...
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Does $\Phi(A^\dagger) = \Phi^\dagger(A)$?
Define linear map $\Phi \in T(X)$, linear operator $A \in L(X)$.
Is it true that
\begin{equation}
\Phi(A^\dagger) = \Phi^\dagger(A)?
\tag{1}
\end{equation}
What are the conditions that will let ...
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An inequality involving quantum channels
Consider two quantum circuits $\mathsf{C}$ and $\mathsf{D}$ applied to $|0^n\rangle$. Then, measure in the standard basis and, for $x \in \{0, 1\}^n$, consider two probabilities:
\begin{equation}
p_{x,...
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