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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3 votes
2 answers
107 views

Converting $T_1$ and $T_2$ decay rates to noise supported by stim

Stim only supports Pauli noises like DEPOLARIZE1, DEPOLARIZE2, X_ERROR, ...
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2 votes
1 answer
41 views

Question regarding the monotonicity of Mutual Information of a tripartite state under multiple non-local commuting unitaries

Given a system $\rho_{AB}\otimes\rho_{C}$, and a unitary interaction $U_{BC}$, due to the monotonicity of the relative entropy under the actions of the partial trace map, $$I(A:B)=I(A:U_{BC}(B,C))\ge ...
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5 votes
1 answer
64 views

Why is error correction very different for circuits compared to channels?

Background Suppose one wishes to communicate information using a noisy channel $N$ instead of an ideal channel $I$. The general framework to communicate reliably is Take $n$ copies of $N$. For some ...
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3 votes
1 answer
123 views

How to recover the original density matrix from the output of amplitude damping channel?

For amplitude damping, we have the below expression $$\xi(\rho)=E_0\rho{E_0}^\dagger + E_1\rho{E_1}^\dagger.$$ How can I perform a matrix inverse operation on $\xi(\rho)$ at the receiver to get back ...
9 votes
1 answer
233 views

How does the number of copies affect the diamond distance?

Suppose we are given two maps $\Phi$ and $\Psi$ such that $$\|\Phi-\Psi\|_{\diamond}\leqslant\varepsilon.$$ What can we say about $\left\|\Phi^{\otimes t}-\Psi^{\otimes t}\right\|_{\diamond}$? Is it ...
1 vote
1 answer
14 views

Creating a gate which have multiple register as an input

I'm trying to create a new gate in qiskit which the input is multiple register but the program output an error that says: ...
4 votes
1 answer
187 views

What is the adjoint of the depolarizing channel?

Consider the single qubit depolarizing noise channel given by $$\Phi(\rho) = \frac{\lambda}{d} \mathbb{I} + (1- \lambda) \rho.$$ What might be the adjoint $\Phi^{*}(\cdot)$ of this channel? In ...
3 votes
1 answer
39 views

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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3 votes
1 answer
85 views

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 ...
2 votes
0 answers
48 views

Given a POVM, what's the channel that optimally preserves coherence in the post-measurement outcomes?

It is well-known that a POVM $\boldsymbol\mu\equiv (\mu_a)_{a\in\Sigma}$ describes outcome probabilities, but not post-measurement outcomes, which in many scenarios exist and are of interest. To ...
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7 votes
1 answer
85 views

Is LOCC equivalence the same as LU equivalence?

I'm currently learning on LOCC transformations. In the Dur, 2000 paper, there is a statement that (...) two pure states $|\psi\rangle$ and $|\phi\rangle$ can be obtained with certainty from each ...
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0 votes
0 answers
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Question about quantum simulation of the Jaynes-Cummings Hamiltonian

Can you help me with solving exercise 8 (4) in the lecture notes Dynamics and Control of Open Quantum Systems which is about finding the unitary operator corresponding to the Jaynes-Cummings ...
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3 votes
1 answer
33 views

Why is it safe to ignore the phase factor when working with unitary operations? (and potentially elsewhere?)

After not understanding the explanation of the no-cloning theorem proof in my lecture notes I turned to Wikipedia, this explanation made more sense to me however it had an extra phase factor that is ...
1 vote
0 answers
20 views

noise and classical bits (measurements ) in qiskit

I am using qiskit and LocalNoisePass to add custom noise to the gates. But there is a problem when there are classical bits in the circuits. The localNoisePass give this error ...
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1 vote
0 answers
54 views

Why is dual-rail encoding called an error-detecting code for amplitude damping?

Exercise 8.23 : Suppose that a single qubit state is represented by using two qubits, as $|\psi\rangle=a|01\rangle+b|10\rangle$. Show that $\mathcal{E}_{AD}\otimes\mathcal{E}_{AD}$ applied to this ...
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1 vote
2 answers
157 views

Expectation value of Pauli Z for locally rotated Bell state

Suppose we have a Bell state $\frac{\lvert 00 \rangle + \lvert 11 \rangle}{\sqrt{2}}$. The expectation value of the Bell state with respect to $Z \otimes I $ is $\langle Bell|Z_1|Bell\rangle = 0$. Now,...
7 votes
1 answer
58 views

Is there a CPTP map that takes $\rho_{AB}$ to $\rho_A\otimes\rho_B$?

Given some joint state $\rho_{AB}$, one can find either the marginal state $\rho_A$ or the marginal state $\rho_B$ through a CPTP map. The proof being that partial tracing is indeed CPTP. Is a CPTP ...
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1 vote
0 answers
134 views

Prove $\beta=\Lambda\otimes\Lambda$, where $\Lambda=\dfrac{1}{2}\begin{bmatrix}I&X\\X&-I\end{bmatrix}$ for single qubit tomography

In the Section on single qubit quantum process tomography, Box 8.5, Page 393, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, and in Prescription for experimental ...
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2 votes
1 answer
71 views

Nielsen and Chuang: Solving equation of motion for amplitude damping

I would like to know how to obtain a solution to the equation of motion given in Section 8.4.1 Master equations of Nielsen and Chuang, 10th edition. The equation of motion that allows getting the ...
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1 vote
1 answer
52 views

Why is $\chi$ not uniquely determined by $\sum_{mn}\beta_{jk}^{mn}\chi_{mn}=\lambda_{jk}$?

The mathematical construct of the Quantum process tomography is given in Page 391, 392, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, as follows Let a fixed set of ...
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2 votes
0 answers
55 views

Can channels be generalized to linear maps from $\mathbb{C}^{n^k}$ to $\mathbb{C}^{n^k}$?

First let's set some terminology. Recall that a quantum channel is in particular a linear map $\Phi : \text{L} ( \mathcal{X}) \rightarrow \text{L} ( \mathcal{Y})$ where $\mathcal{X}$ and $\mathcal{Y}$ ...
0 votes
0 answers
77 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
0 answers
44 views

Why is combined amplitude and phase damping considered sufficient for noise modeling?

In QECC literature, I often come across the "combined amplitude and phase damping channel" as being representative of a realistic noise model which makes sense (as amplitude damping and de-...
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1 vote
1 answer
32 views

How to create Parametrized Operator

How to transform the U3 gate of parameters (theta, phi, lambda) to an Operator. (in qiskit) The following code don't work ...
1 vote
1 answer
69 views

Number of independent parameters in $\chi$ 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 $d^4−d^2$ ...
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3 votes
2 answers
173 views

Why does the $\chi$ matrix have $d^4-d^2$ independent parameters?

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 $d^4−d^2$ ...
  • 607
1 vote
2 answers
67 views

How to check if a mapping is unitary?

In the case of the No-cloning theorem, it is argued that a unitary $U$ that is capable of performing coping does not exist. Specifically, for any two unknown states $|\psi_1\rangle$ and $|\psi_2\...
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1 vote
1 answer
62 views

Does a noiseless quantum channel violate no-cloning theorem?

A quantum channel is defined as CPTP map from Alice to Bob $\mathcal{N}: \mathcal{H}_A \to \mathcal{H}_B$. In particular, a noiseless quantum channel is such that $\mathcal{N}(\rho_A) = \rho_A$. My ...
2 votes
2 answers
86 views

What is the Bloch sphere representation of $\rho\to\mathcal{E}(\rho) = |+\rangle\langle+|ρ|+\rangle\langle+| + |−\rangle\langle−|ρ|−\rangle\langle−|$?

Suppose a projective measurement is performed on a single qubit in the basis $|+\rangle, |−\rangle$, where $|±\rangle \equiv (|0\rangle\pm |1\rangle)/\sqrt{2}$. In the event that we are ignorant of ...
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1 vote
1 answer
45 views

What is the definition of twirled superoperator?

I am trying find the definition of twirled (super)operator. One such is Definition 2.3.16 on p. 33 of Christoph Dankert, Efficient Simulation of Random Quantum States and Operators. However, the ...
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0 votes
0 answers
103 views

Qiskit: Kraus decompostion of phase amplitude damping error is not appropriate

The phase_amplitude_damping_error(param_amp, param_phase, excited_state_population=0, canonical_kraus=True) in Qiskit allows only the parameter range $p_a + p_p \le ...
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0 votes
0 answers
86 views

Affine map of single qubit quantum operations

In my reference, Page 375, Chapter 8, Quantum Computation and Quantum Information by Nielsen and Chuang, it is given that Lemma: The Pauli matrices, along with the identity matrix $I$, form an ...
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1 vote
1 answer
43 views

How $X\otimes X^{1/2}$ propagates through $CZ$

I am trying to find out how the $X\otimes X^{1/2}$ operator propagates through a $CZ$ one. Below a circuit example.
3 votes
1 answer
53 views

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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0 votes
1 answer
130 views

Kraus operators required for operations on $d$ dimensional Hilbert space

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 ...
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1 vote
0 answers
25 views

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 ...
2 votes
1 answer
162 views

Depolarizing channel for $n$ qubits: why is there a trace term?

The depolarizing channel for an n-qubit quantum circuit is defined as $$ \mathcal{E}(\rho) = \frac{pI}{2^n}\text{Tr}(\rho)+(1-p)\rho,\quad\text{where} \quad\rho \equiv\sum_ip_i|\psi_i\rangle\langle\...
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5 votes
1 answer
61 views

Action of a channel on an "unphysical" state

Suppose we are given a rule $\Phi$ which is completely positive and trace preserving operation takes an input qubit state $\rho$ to an output qubit state $\rho^\prime$ (as an example of such rule see ...
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2 votes
0 answers
32 views

What is the complementary map of a serial concatenation of quantum channels?

I have been studying serial concatenations of quantum channels, i.e. $\mathcal{N}_{A\rightarrow B}=\mathcal{N}_1\circ\mathcal{N}_2=\mathcal{N}_{B'\rightarrow B}\circ\mathcal{N}_{A\rightarrow B'}$. ...
1 vote
1 answer
102 views

How can I represent the completely mixed state as $\frac I2=\frac14(\rho+X\rho X+Y\rho Y+Z\rho Z)$?

Consider the completely mixed state $I/2$. The equation comes from Eq.(8.101) of Nielsen's book: $\frac{I}{2}=\frac{\rho+X\rho X+Y\rho Y+Z\rho Z}{4}$, How comes this equation?
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2 votes
1 answer
145 views

How to obtain the unitary operator to get specific partial trace?

Is there a unitary $U_{AB}$ such that, for any density operator $\rho$, we have $${\rm {Tr}}_A \left[U_{AB} \left(\frac{I_A}{2} \otimes \rho_B\right)U_{AB}^{\dagger}\right]= \frac{\rho_B}{2}+\frac{I_B}...
2 votes
1 answer
126 views

Why can any quantum channel be represented as a matrix?

In this PDF (page 43), it is argued that, given an arbitrary quantum channel with Kraus decomposition: $$ E(\rho) = \sum_{j} K_j \rho K_j^{\dagger} $$ Such map can be represented with a matrix in $\...
2 votes
1 answer
99 views

How to calculate the action of a channel on part of a quantum state?

As the title shows, but I think we can restrict ourselves into a more specific example. Let's consider depolarizing channel $\varepsilon$: $$\varepsilon(\rho)\equiv p\frac{I}{d}+(1-p)\rho\tag{1}$$ ...
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4 votes
2 answers
186 views

Are continuous probability distributions over quantum channels possible?

I am not an expert in the subject and apologize in advance for a strange question and (possible) abuse of the terminology. I have learned that any convex combination of quantum channels (CPTP maps, ...
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2 votes
2 answers
87 views

Find an operator-sum representation for a depolarizing channel acting on 2qubit

In Nielsen and Chuang (page:379), it shows how to represent a 1 qubit depolarizing channel in operator-sum representation. $$ \mathcal{E}_1(\rho)=pI/2+(1-p)\rho =(1-3p/4)\rho+p/4(X\rho X+Y\...
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8 votes
2 answers
408 views

Prove that a channel is close to acting on only one system

Background Suppose I have a quantum channel $\Phi:B(\mathcal{H}_1)\rightarrow B(\mathcal{H}_1)\otimes B(\mathcal{H}_2)$, such that there is some small $\epsilon$ such that for any two input states $\...
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0 votes
0 answers
31 views

Does anybody know what a low-degree Markov field is?

In the paper Fast Estimation of Sparse Quantum Noise I saw the following description: quantum devices approaching the fault-tolerant regime will have very few significant errors (and therefore are ...
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1 vote
1 answer
248 views

Why Kraus operator is not a number?

Quantum operations can be represented in an elegant form known as the operator-sum representationn, namely $\mathcal{E}(\rho)=\sum_kE_k\rho E_k^\dagger$, where $E_k=\langle e_k|U|e_0\rangle$, the ...
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1 vote
0 answers
31 views

What are channels for which entanglement at the encoder improves communication rates?

In discussing the classical capacity of quantum channels, as e.g. mentioned in Wilde's book (see section 20.6), it is possible that using entanglement at the encoder stage can improve transmission ...
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0 votes
0 answers
108 views

What is the Pauli channel characterizing?

I saw from the article :Noise is almost inevitable in all quantum information processing tasks and currently one of the main obstacles to achieving large-scale quantum compu- tation. For any given ...
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