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 ...
Karry's user avatar
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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 ...
DJD's user avatar
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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 ...
nervxxx's user avatar
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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 ...
Manuel E's user avatar
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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, ...
QC123_367's user avatar
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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}...
Quantastic's user avatar
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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 $\Phi^{\ast}$ ...
trurl's user avatar
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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 ...
Michael.Andy's user avatar
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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 $p$)
codeit's user avatar
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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 $\chi$ ...
Karry's user avatar
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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\...
Karry's user avatar
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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 ...
Physkid's user avatar
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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) ...
Karry's user avatar
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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, ...
trurl's user avatar
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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 $\...
Physkid's user avatar
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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} ...
SimoneGasperini's user avatar
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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 ...
TheoryQuest1's user avatar
5 votes
2 answers
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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\...
Wei Zhan's user avatar
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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 ...
Blazej's user avatar
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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 ...
Sudhir Kumar's user avatar
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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. ...
JoJo P's user avatar
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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 ...
X0-user-0X's user avatar
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37 views

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 $\rho$ ...
Hans Schmuber's user avatar
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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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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}^{\...
milenteor's user avatar
1 vote
1 answer
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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 ...
Yuval Idan's user avatar
1 vote
0 answers
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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 ...
TTa's user avatar
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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 $\{...
Star21's user avatar
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2 votes
2 answers
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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. $\...
Anindita Sarkar's user avatar
1 vote
0 answers
27 views

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|\...
Sowmitra Das's user avatar
1 vote
0 answers
61 views

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 $tr(S_i(\sigma))$ ...
Zehong Fan's user avatar
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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 -...
Sowmitra Das's user avatar
3 votes
1 answer
69 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 ...
Peeveey's user avatar
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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)=...
Joseph Van Name's user avatar
1 vote
0 answers
130 views

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 ...
Zehong Fan's user avatar
5 votes
1 answer
62 views

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\...
Hailey Han's user avatar
1 vote
1 answer
53 views

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 ...
Shadumu's user avatar
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1 answer
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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 ...
Anindita Sarkar's user avatar
2 votes
1 answer
59 views

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{...
Anindita Sarkar's user avatar
0 votes
1 answer
40 views

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....
AP110's user avatar
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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 $x$. ...
AP110's user avatar
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1 vote
2 answers
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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^+ =\...
Cain's user avatar
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0 answers
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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\!\...
Saurabh Shringarpure's user avatar
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1 answer
60 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 &...
joy Jaganath's user avatar
1 vote
1 answer
99 views

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 $X_A \equiv \sum_k |A_k\rangle\!\rangle\langle\!\langle A_k|$, ...
Saurabh Shringarpure's user avatar
1 vote
1 answer
71 views

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 ...
Abir's user avatar
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2 votes
1 answer
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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 ...
Frederik vom Ende's user avatar
4 votes
1 answer
65 views

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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