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.
438
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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 ...
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1
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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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How to get the Kraus operator $M_0=\sqrt{1-p}\, I$ for the depolarizing channel, from its isometric representation?
I am confused as to how we get $M_{0} = \sqrt{1-p} I$ and the following $M_{1}, M_{2}, M_{3}$.
The above notes say that we should partially trace over the environment in the $|{0}\rangle, |{1}\rangle, ...
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0
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Kraus decomposition in the infinite-dimensional case
The action of a quantum channel $\Lambda$ on a generic state $\rho_S$ coupled to a pure environment state $|0\rangle_E$ can be written in the Stinespring representation as
$\Lambda(\rho_S) = \text{tr}...
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1
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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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What is the Choi matrix of the $H$ gate?
I see the definition of Choi matrix is:
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\...
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Physical description of trace of ancilla state yields a depolarising channel
Let's start with
$Tr_{\Omega}[|0,\Omega_{0}\rangle\langle0,\Omega_{0}|U^{\dagger}] = \sum_{\alpha}E_{\alpha}|0\rangle\langle0|E_{\alpha}^{\dagger}$
where $U$ be a unitary operator. The trace operator ...
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How to unfold the $E= (|I⟩⟩⟨⟨I|+|X⟩⟩⟨⟨X|)^{⊗2}$ and aquire the following equation?
The standard notation of $|X⟩⟩$ for
the $4^n$-dimensional vector representing $X$ in the space acted on by superoperators. $E= (|I⟩⟩⟨⟨I|+|X⟩⟩⟨⟨X|)^{⊗2}$.
From this definition, How comes
$E|I⟩⟩=(Tr(I) ...
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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, ...
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1
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indeterministic knowledge on unknown state using a Kraus operator
Suppose Alice transmits a qubit in either of two states $|\psi\rangle_{1}, |\psi \rangle_{2}$.
Bob has 3 Kraus operators: $\hat{E1}, \hat{E2}, \hat{E3}$ such that the average measurement value of $\...
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How to build the quantum circuit ansatz to implement a diagonal unitary operator with just 1 and -1 elements?
Let's consider a set of $N = 2^n$ binary values $S_i \in \left\{-1, 1\right\}$ and define the diagonal matrix $W$ as a quantum unitary operator acting on a system of $n$ qubits:
$$
W =
\begin{pmatrix}
...
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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 ...
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Validity of quantum channel given pairs of inputs and outputs
Given finitely many pairs of pure states $|x_1\rangle,|y_1\rangle,\ldots,|x_k\rangle,|y_k\rangle\in\mathcal{H}_n$, we can decide if there exists a unitary operator $U$ such that $U|x_i\rangle=|y_i\...
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Is there a known deterministic counterexample for non-additivity of minimal output entropy?
Hastings has proved that the minimal output entropy is not additive: it may happen that $S_{\mathrm{min}}(\Phi_1 \otimes \Phi_2) < S_{\mathrm{min}}(\Phi_1)+S_{\mathrm{min}}(\Phi_2) $ for quantum ...
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160
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Unital channel which is not mixed unitary
How to prove that for a multi-qubit system a unital channel is not necessarily mixed unitary? This is Problem 8.3 in Nielsen and Chuang. Here's a snippet of the text:
Shall I need to take two ...
2
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0
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Is there a Qutip equivalent of "expand_operator" for superoperators?
I want to calculate a superoperator for a small noisy subsystem consisting of $k$ qubits, and expand it to $n > k$ qubits, where the remaining $n-k$ qubits are not subject to any noise. ...
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What are "completely positive" and "CPTP" quantum maps?
I am studying quantum computing a little bit by myself, and I have simple questions.
I didn't find a clear definition of what is a completely positive and trace-preserving (CPTP) map. The best I've ...
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0
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Error in repeated applications of a quantum channel?
Suppose I have two quantum channels. Assume they they consist of $r\in \mathbb{Z}$ applications of unitaries, $U$ and $V$ respectively. Let the error between the channels acting on some state $\rho$ ...
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Coherent information is a lower bound of channel capacity. What about coherent information based on Renyi entropies?
It is known that coherent information defined in terms of von Neumann entropies is a lower bound of quantum channel capacity. If we define coherent information in terms of $\alpha$-Renyi entropies, ...
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142
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Stinespring Representation
I need to conduct a numerical analysis of a quantum channel using SDP and therefore, I need the Stinespring representation of this quantum channel:
$$\Phi=\mathrm{Tr}_{\mathrm{DE}}\Big((\mathrm{Id}^{\...
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1
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59
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Implementing swap test on quantum register
I'm interested in implementing a controlled-swap operation on quantum registers. However, it appears that Qiskit's documentation primarily focuses on single qubits rather than registers. Could you ...
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How does the dual rail qubit conceptually detect erasures for the amplitude damping channel?
I think I am missing something in my understanding of the amplitude damping and erasure channels when it comes to the dual-rail encoding.
I have the following dual-rail photonic qubit labeled by ...
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How is a quantum error correcting code $C(E)=\langle\psi|E^\dagger E|\psi\rangle$ diagonalized?
UIn page 6 of the following paper: https://arxiv.org/pdf/0904.2557.pdf, in the proof of theorem 3:"Suppose equation (30) holds. We can diagonalize $C_{ab}$. This involves choosing new basis $\{...
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2
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Proof that two sets of quantum maps are equivalent only when they are related by a unitary transformation
I am trying to show that the two different quantum maps $\rho'=\sum_{\alpha} K_{\alpha} \rho K_{\alpha}^{\dagger}$ and $\rho''=\sum_{\beta} L_{\beta} \rho L_{\beta}^{\dagger}$ are equivalent i.e. $\...
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Improving Quantum State Distinguishability through Embedding
Consider the following map, $\mathcal{E}:\mathcal{L}(H_A) \rightarrow \mathcal{L}(H_{AB})$,
$$
\mathcal{E}(\rho_A|U_{AB}, U_{AC}) = {\rm Tr_C} \left[ U_{AC}U_{AB}(\rho_A\otimes |0_B\rangle\langle 0_B|\...
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How to prove the "damage lemma" for gentle measurements?
Let $\rho$ be a mixed state. For all $i \in [m]$, let $S_i$ be quantum operation, which is a two-outcome POVM measurement, and "accept" a state $\sigma$ with probability $tr(S_i(\sigma))$ ...
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An Inequality related to the Trace Norm
Let $\rho, \sigma$ be two states of a qubit, and let $U$ be the $CX_{12}$-gate (control on 1st qubit, target on 2nd qubit).
Prove that, for an arbitrary CPTP Map $\mathcal{E}$,
$$
||\mathcal{E}(\rho -...
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1
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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 ...
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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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0
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130
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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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Existence of Hamiltonians such that the time evolution unitary becomes identity
Can we always find a set of coefficients ${k_i}$ (where not every $k_i = 0$) for a given Hamiltonian $H = \sum k_i H_i$, such that the unitary operator becomes the identity operation: $e^{-iH} = e^{i\...
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1
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Is there a notion of approximate entanglement breaking (EB) channels?
Is there a notion of approximate entanglement breaking (EB) channels? Say, e.g. the output is always close to a separable state. If so, do the nice properties of the EB channels, such as additive ...
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1
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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 ...
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1
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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{...
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40
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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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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 $x$.
...
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2
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Action of a CPTP map on Identity
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^+ =\...
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0
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47
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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\!\...
0
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1
answer
60
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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 &...
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1
answer
99
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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
1
answer
71
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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 ...
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2
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1
answer
209
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Diamond norm distances between some channel and the identity
I'm currently working with the continuity result by Kretschmann-Schlingemann-Werner (arXiv version) for Stinespring isometries (more precisely, the following corollary to their result, cf. Appendix C ...
4
votes
1
answer
65
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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, ...
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