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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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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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
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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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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
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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
68 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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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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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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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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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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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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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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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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1 answer
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Check that a channel implements a particular unitary

Consider a channel $C$ with Kraus operators $\{K_k\}$ and a unitary U. How can I check that $C$ implements $U$ ? One can write that their Choi matrices are equal i.e: \begin{equation} \sum_{i,j}|i\...
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Is there an inverse for Stinespring dilation?

Given a set of Kraus operators we can find a unitary that does the equivalent map on an extended space including the environment using Stinespring dilation. My question is how do we go about doing the ...
1 vote
1 answer
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Can one turn a non-Trace Preserving map into one that is Trace Preserving?

A trace non-preserving quantum channel $\mathcal{A}$ takes a state $\rho$ to $\rho^\prime$, i.e., $\sum_{i=1}^{n} A_i \rho A_i^\dagger = \rho^\prime$, with $\sum_{i}^{n} A_i^\dagger A_i \ne \mathbf{I}$...
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What is the most general way to describe post-measurement states?

Background Generally speaking, the description of post-measurement states associated with a POVM seems to always pass through, in some form or another, the formalism of Kraus operators. For example: ...
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4 votes
1 answer
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Given a state $\rho$ and operator $0\le \Lambda\le I$, what does $\sqrt\Lambda \rho \sqrt\Lambda$ represent?

An expression that is found in a good number of results is $\sqrt\Lambda\rho\sqrt\Lambda$, for some pair of positive semidefinite operators $\rho,\Lambda\ge0$. For example, in the gentle operator ...
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what is Pauli twirling approximation?

In this video, Artur Ekert shows that for a single qubit, 4 Kraus operators can be chosen such that the action on state $\rho$ is given as $\rho \rightarrow \sum_m p_m A_m \rho A_m^\dagger$. We can ...
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Phase damping and cnot gate

Show that a single controlled-NOT gate can be used as a model for phase damping, if we let the initial state of the environment be a mixed state, where the amount of damping is determined by the ...
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Encode Matrix Elements on register

I am trying to construct a circuit which makes the following encoding $$O_H \left| i \right> \left| j \right> \left| z \right> = \left| i \right> \left| j \right> \left| z \oplus H_{ij} ...
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QuTiP lecture: Single-Atom-Lasing

I have tried to run one of the examples in QuTip lectures: single atom lasing. Although I have used the prepared code represented in reference, I have received error and I have not been able to ...
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Commutative operators

I have got a 2-qubit circuit with the following instructions: ...
1 vote
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How to convert tableau representation of random clifford gate into its matrix representation using stim?

I am currently trying to benchmark my code with a Haar circuit and I require to sample clifford gates in matrix form. I know a function "stim.Tableau.random(n)" which does that and gives me ...
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Quantum operation and non-trace-preserving

In Chapter 8 of Nielsen and Chuang's Quantum Computation and Quantum Information, a mathematical framework is developed to provide a realization of the quantum operation $\mathcal{E}$ with operation ...
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1 answer
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Question about the probability of failure of the bit flip code

For the bit flip superoperator is $$\mathcal{E}_{BF}(\rho) = (1-p)\rho + p X \rho X$$ where the first term refers to no bit flip and the second term refers to the bit flipping. A single qubit pure ...
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1 answer
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Cirq: Getting The Instances (samples) of a Quantum Circuit with Probabilistic Unitaries or Mixtures

The wavefunction simulator in Cirq uses a Monte Carlo approach to simulate a certain subset of quantum noise channels, namely through probabilistic/stochastic application of unitary gates. These are ...
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Validity of a quantum operation for generalised state erasure

In this set of notes, pg. 177, the quantum erasure channel (which does nothing to a transmitted state with probability $1-\varepsilon$ and 'erases' it with probability $\varepsilon$) is defined as the ...
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1 answer
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Is there a name for a gate that 'moves' one qubit to a new position via multiple SWAP gates?

Let's say there is a qubit at position $i$, and I want to move it to position $i'$. Without loss of generality, let's say $i < i'$. By 'move it' I mean, perform multiple $SWAP$ operations so that ...
2 votes
2 answers
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what does it mean to "measure a qubit" if measurement is defined on operators

As far as I know "measurement" depends on the two variables : state and operator. So what exactly does measuring a qubit mean? is it implied that the operator is the $Z$ operator on that ...
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1 vote
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Interesting properties of maps whose natural representation is unitary?

Let $\rho \in L(\mathcal{X})$ be a state in the space of linear operators acting on some complex Hilbert space $\mathcal{X}$. I'm interested in linear maps $\Phi: L(\mathcal{X}) \rightarrow L(\mathcal{...
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2 votes
2 answers
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Does the unitary freedom in choice of Kraus operators come from the freedom in the choice of purifications?

Does the unitary freedom in the choice of Kraus operators for a given quantum channel just come from the unitary freedom in choice of purification of a quantum state? Here's what I'm thinking. If I ...
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4 votes
2 answers
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What are the possible channels preserving purity of all input states?

Consider channels $\Phi$ such that $\Phi(|\psi\rangle\!\langle\psi|)$ is pure for all $|\psi\rangle$. Is there a simple way to characterise channels with this property? Let's suppose $\Phi$ acts ...
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1 vote
1 answer
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What is the adjoint of the complementary channel?

Given a channel $\phi$ with the set of kraus operators; $(K_1, K_2,...,K_n)$, I know the complementary channel is; $\phi^c(A)=\sum_{i,j}tr(K^*_jK_iA)E_{ij}$ what will be the adjoint of this ...
2 votes
2 answers
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Why is $\Phi\otimes \operatorname{Id}_n$ being positive on maximally entangled states sufficient to know that $\Phi$ is CP?

(Notation) Let $\Phi$ be a generic quantum map sending states in $\mathbb{C}^n$ into states in $\mathbb{C}^m$. We say that $\Phi$ is positive when $\Phi(X)\ge0$ for any positive linear operator $X\in\...
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6 votes
1 answer
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Why does code switching not allow for universal fault-tolerant quantum computation?

In this paper, the authors briefly mention that one proposed method to bypass the Eastin-Knill theorem is to perform code-switching. That is, given codes $C_1$ and $C_2$ which permit a complementary ...
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Qiskit library.gaussian() does not accept parametric expression

I'm trying to build a gaussian pulse in qiskit where I keep the amplitude as a parameter but for the followig code ...
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Filtering operation is trace decreasing?

Let $\rho$ is a bipartite state. W is a local filtering operation that acts on a subsystem of the state $\rho$. After the local filtering operation $\rho$ emerges into a $\tilde{\rho}$ i.e $\tilde{\...
3 votes
2 answers
112 views

Can any rank-$n$ POVM be realized as a rank-one POVM?

Let, $\mathcal{M}$ is a POVM measurement whose elements are $M_i=\sum_{k=1}^np_{ki}|\phi_{ki}\rangle\langle\phi_{ki}|$ with $p_{ki}\geq 0$ and $\sum_{i=1}^sM_i=I$ where $|\phi_{ki}\rangle$ is a ...
7 votes
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Rotation resolutions in operations for qubits in commercial implementations

I have found information about Honeywell provider supporting operations with high-resolution rotations (i.e. around $\pi/500$) here. What are typical maximal rotation resolution values supported by ...
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How to find the Choi state of a bipartite quantum channel?

The Choi state of a quantum channel $\mathcal{N}_A$ acting on a system $\rho_A \in \mathcal{H}^A$ is given by: $Choi(\mathcal{N}_A) =( \mathcal{I}_{A'} \otimes \mathcal{N}_A)|\Phi^+\rangle \langle\...
1 vote
1 answer
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Characterise, via Naimark's theorem, the POVM corresponding to a PVM in a dilated space

Let $F\equiv\{F^a\}_a$ be a POVM in some finite-dimensional Hilbert space $\mathcal X$. It is well-known that one can always understand $F$ as a projective measurement (PVM) in an isometrically ...
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4 votes
0 answers
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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 ...
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8 votes
3 answers
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What are the possible Kraus operators of the identity channel?

Consider a Kraus representation $\{A_a\}_a$ of the identity channel $\mathcal{I}$ that maps any state to itself. Of course, $\{A_a\}_a$ are not the simplest Kraus operators, which would just be $\{I\}$...
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Quantum channels that commute with any unitary channel

Consider a quantum channel $\Phi$ that maps from density operators $\mathcal{S}(\mathcal{H}_A)$ to itself, that commutes with any unitary channel $\mathcal{U}$ on $\mathcal{S}(\mathcal{H}_A)$, i.e. $\...
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1 answer
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Can one define a Choi state for a a classical channel?

Suppose one has a classical channel $W(y|x)$ that is a conditional probability distribution. Can one define a Choi state for this channel? My guess is that one should think of it as a special case of ...
6 votes
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Verification of local unitary equivalence between two pure states

This might be a non-trivial and hard problem. I've been thinking about this for days but couldn't find a good answer, so I hope any of you could give me a good answer/intuition for me to move forward. ...
3 votes
2 answers
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Existence of a perturbed channel that achieves a perturbed output state

Consider a $d$-dimensional maximally entangled state $\vert\phi\rangle = \frac{1}{d}\sum_{i=1}^d\vert i\rangle_A\vert i\rangle_B$. Let $N_{A\rightarrow A'}$ be a quantum channel and consider $\rho_{A'...
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2 votes
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Local Hermitian operators can be written as sums over local error operators?

In this paper, near the bottom of the left half of page 3, the authors claim that any local Hermitian operator (one which acts only on a single subsystem of a larger composite system) can be expressed ...
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Do quantum states with a single parameter give any theoretical or experimental advantage compared to multi-parameter ones?

If a quantum state is a single parameter two-qubit mixed entangled state then is there any theoretical or experimental advantage compared to a multi-parameter state? suppose, we take a single ...

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