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.

Filter by
Sorted by
Tagged with
-1 votes
0 answers
19 views

Question on State Transfers [closed]

I am beginner to quantum physics and I came across this problem that I cannot figure out[![enter image description here][1]][1]
user avatar
  • 1
4 votes
2 answers
165 views

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 ...
user avatar
  • 18.1k
1 vote
1 answer
79 views

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 ...
user avatar
2 votes
2 answers
71 views

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\...
user avatar
  • 18.1k
6 votes
1 answer
80 views

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 ...
user avatar
  • 287
0 votes
1 answer
15 views

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 ...
user avatar
  • 1
0 votes
1 answer
19 views

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{\...
user avatar
3 votes
2 answers
75 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 ...
user avatar
6 votes
0 answers
39 views

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 ...
user avatar
  • 302
2 votes
0 answers
47 views

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\...
user avatar
1 vote
1 answer
90 views

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 ...
user avatar
  • 18.1k
4 votes
0 answers
87 views

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 ...
user avatar
  • 287
7 votes
3 answers
98 views

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 $\{I\}$...
user avatar
  • 215
4 votes
1 answer
124 views

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. $\...
user avatar
  • 215
1 vote
1 answer
51 views

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 ...
user avatar
6 votes
0 answers
65 views

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. ...
user avatar
3 votes
2 answers
85 views

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'...
user avatar
  • 2,008
2 votes
0 answers
76 views

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 ...
user avatar
  • 173
0 votes
1 answer
73 views

Calculating the expectation value of an observable in Qiskit

Consider the following quantum circuit that consists of a three qubit quantum register and an ancilla qubit: Let $W$ and $V$ be unitary operators. $U(t)$ is an implementation of unitary time ...
user avatar
0 votes
0 answers
68 views

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 ...
user avatar
5 votes
1 answer
93 views

What are examples of the correspondence between channels and their Stinespring dilations?

In this post I read that "quantum measurements are special cases of quantum channels (CPTP maps). Stinespring's dilation states that any quantum channel is realized by partial tracing a unitary ...
user avatar
  • 165
5 votes
2 answers
123 views

maximization of trace between two operators with respect to different norm constraints

I want to maximize $\text{Tr}(XY)$ over $X$ for fixed $Y$, where $X$ and $Y$ are both hermitian (but doesn't necessarily positive) operators, and $X$ is constrained by its p-norm bounded by $1$, i.e. $...
user avatar
0 votes
1 answer
54 views

General Ehrenfest Theorem applied to N-qubit system operator

Please advise if the following short calculation of the derivative of the expectation value of an all spin Pauli-$\hat{Y}$ operator (acting on a $N$-qubit system) is consistent: The general Ehrenfest ...
user avatar
  • 709
1 vote
1 answer
54 views

Data Processing equality variation

Let $\rho_{AB}$ be a state and $T: B \rightarrow C$ be a CPTP map with $\sigma_{AC}= T(\rho_{AB})$. It is well known that $H_{\infty}(A \vert B)_{\rho} \geq H_{\infty}(A \vert C)_{\sigma}$ (aka data ...
user avatar
  • 439
5 votes
1 answer
79 views

What is the Stinespring dilation of $T\otimes I$ for some CPTP map $T$?

Let $T: \mathcal{H}_A \rightarrow \mathcal{H}_B$ be a CPTP map with Stinespring extension $U: \mathcal{H}_{A} \rightarrow \mathcal{H}_{B} \otimes \mathcal{H}_E$. That is $U$ is an isometry such that ...
user avatar
  • 439
3 votes
1 answer
96 views

What is the technique for calculating $\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$?

I am stuck on calculating $\mathcal{E}(\rho)=\text{Tr}_b[{U(\rho\otimes\rho_b)U^{\dagger}}]$. For example, in the case when $U$ is the CNOT matrix $$U=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 &...
user avatar
  • 177
2 votes
1 answer
105 views

How to find the Kraus operators from the process matrix?

I am trying to find the Kraus operator from process matrix. For instance, suppose that for single qubit identity gate, I have the following process matrix: ...
user avatar
  • 406
5 votes
2 answers
388 views

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 ...
user avatar
  • 800
4 votes
1 answer
61 views

Quantum channel between two states with inaccessible reference - when can it be done?

Suppose I have a pair of bipartite states $\rho_{AR}$ and $\sigma_{BR}$. $R$ is a reference system that we do not have access to. It is clear that we cannot always have a channel $N_{A\rightarrow B}$ ...
user avatar
  • 2,008
2 votes
1 answer
63 views

How to comparing a quantum channel with a unitary?

I have a target unitary that I want to implement and I have a known dynamics that results in a quantum channel. I already have some fidelity measures to characterize the overlap between them. (like: ...
user avatar
2 votes
0 answers
47 views

How to write the joint action of a CP map that acts on a single qubit of a bipartite state?

The question Say I have a completely-positive (CP) map $\mathcal{A}_{ij}$ defined in terms of two projectors $\Pi_i = |i\rangle \langle i |$ and $\Pi_j = |j\rangle \langle j |$ that acts on a density ...
user avatar
  • 21
1 vote
1 answer
113 views

Qiskit Encoding of a Hamiltonian

I am not sure if this is a trivial question and I am just stuck or this is worth posting. I have a Hamiltonian, which is written in terms of Pauli matrices, such as H = 4.5$I$ - 16 $X_1$ - 16 $X_2$ - ...
user avatar
  • 35
0 votes
1 answer
54 views

Are quantum gates superoperators? How to write a quantum circuit as superoperator?

I have some questions related to superoperators: What is the differences between quantum operators and superoperators? For instance quantum gates are also unitary operators but can we say quantum ...
user avatar
  • 406
4 votes
0 answers
203 views

What's a "natural" way to show that, for unital channels, $\Phi(X)=X$ iff $[X,A_a]=0$ for all Kraus operators $A_a$?

This is a statement proved in Watrous, Theorem 4.25, page 229 of the online version. Let $\Phi\in\mathrm C(\mathcal X)$ be a unital channel with Kraus representation $\Phi(X)=\sum_a A_a X A_a^\dagger$....
user avatar
  • 18.1k
3 votes
4 answers
329 views

Is there an upper-bound on the operator norm (max-singular value) of the matrix representation of a quantum channel?

Suppose $\Phi$ is a CPTP map with Kraus operators $\phi_n$, so that $\hat{\Phi} := Σ_n (\phi_n ⊗ \phi_n^*)$ is the matrix representation (here $*$ being entry-wise complex conjugate). Is there an ...
user avatar
  • 53
2 votes
2 answers
108 views

How to compute the unitary from the $\chi$ matrix obtained from QPT

I am trying to do quantum process tomography for one qubit and obtain the unitary for the gates that are applied on the qubit. I have studied the theory on process tomography from mike and ike and the ...
user avatar
4 votes
1 answer
130 views

Is the composition of two extremal channels also extremal?

In this question, I follow the terminology and notation of the book of Watrous, most notably chapter two. Extremal channels An extremal channel $\Phi(X) \in C(\mathcal{X},\mathcal{Y})$ is a channel ...
user avatar
  • 4,630
1 vote
1 answer
52 views

What are the measurement operators $F_k$ corresponding to a homodyne measurement?

By definition, a measurement is characterized by a set of positive-semidefinite matrices $\{F_k\}$ satisfying the completeness relation $\sum_k F_k = \textbf{I}$. I am interested in knowing how does ...
user avatar
  • 476
3 votes
1 answer
108 views

Implication of SWAP being not positive in terms of quantum channel

I am going over chapter 3 of Preskill's lecture notes regarding complete positivity. Specifically, on page 19, it is mentioned that since SWAP has eigenstates with eigenvalue -1, it is not positive, ...
user avatar
  • 73
4 votes
1 answer
234 views

What is the root of the non-trace-preserving bit-flip map

I have a quantum channel defined by the Kraus operators: $$ U_1 = \begin{bmatrix} p & 0 \\ 0 & p \end{bmatrix},\quad U_2 = \begin{bmatrix} 0 & p \\ p & 0 \end{bmatrix} $$ i.e. $$ U_1\...
user avatar
  • 147
4 votes
1 answer
110 views

What do commuting quantum channels look like?

Consider two channels, $\Phi,\Psi\in\mathrm C(\mathcal X)$ acting on some space $\mathcal X$, and suppose they commute, that is, $$\Phi(\Psi(\rho))=\Psi(\Phi(\rho))$$ for all states $\rho$. Can ...
user avatar
  • 18.1k
3 votes
1 answer
71 views

Can LOCC operations take product states to non-product states?

Given a product state $\rho^{(1)} \otimes \rho^{(2)}$, can this state become non-product state under LOCC? Can LOCC create correlations between two systems?
user avatar
  • 476
2 votes
1 answer
158 views

How can one check if a given quantum channel is unitary?

A unitary channel is a channel $\mathcal{U}$ of the following form: $\mathcal{U}(\rho) = U\rho U^{\dagger}$. A mixed unitary channel is a channel $\mathcal{U}_m$ of the form: $\mathcal{U}_m(\rho) = \...
user avatar
0 votes
0 answers
30 views

Measuring with a certain probability in Qiskit

I have the following quantum operation on two qubits: $$ \mathcal{E}(\rho) = p \mathcal{T} \circ \mathcal{U}(\rho) + (1-p) \mathcal{U}(\rho) ​$$ where $p$ is some probability, $\mathcal{U}$ is some ...
user avatar
3 votes
1 answer
60 views

Special properties of a channel whose Kraus decomposition contains Identity

I would like to know if there are any special properties of channels that permit a Kraus representation that includes an identity? That is, if I am given a Kraus representation of a CPTP map $\Phi$ ...
user avatar
  • 4,507
2 votes
1 answer
39 views

Can every unitary on $\mathcal{H}\otimes \mathcal{K}$ be modelled by quantum operations on $\mathcal{H}$?

In section 8.2.3 of Nielsen and Chuang, they discuss how unitary dynamics of a system and environment arise from quantum operations (i.e. Kraus operators $E_k$ such that $\sum_k E_k^*E_k=I$). ...
user avatar
  • 1,437
4 votes
1 answer
65 views

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 ...
user avatar
2 votes
0 answers
26 views

What are examples of "LOCC linked" quantum instruments?

Define a quantum instrument $\mathfrak J$ as a collection of completely positive (CP) maps $(\mathcal E_j:j\in\Theta)\subset\mathrm{CP}(\mathcal H)$, such that $\sum_j \mathcal E_j$ is also trace-...
user avatar
  • 18.1k
2 votes
1 answer
52 views

Does ${\rm tr}(\Pi_z\rho\Pi_z)\le p$ imply $\cal E(\rho)$ and $\cal E(\Pi_{-z}\rho\Pi_{-z})$ are close in trace distance?

Suppose I have a quantum operation $\mathcal{E}$ and a state $\rho$ such that: $$ \operatorname{tr}(\Pi_z \rho \Pi_z) \le p $$ for some probability $p$ and some projection $\Pi_z$ onto some subspace ...
user avatar
1 vote
0 answers
28 views

How to detect and correct swap errors in a quantum circuit?

Let's assume that I have a density matrix $\rho$ that consists of $N$ qubits. If this density matrix undergoes an error-channel that swaps any two qubits with an equal probability, i.e. $$ \mathcal{E}(...
user avatar

1
2 3 4 5 6