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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1answer
47 views

How can I fit an unknown quantum channel?

Suppose that I have one noisy channel $\mathcal{E}$ and I want to fit it with another one $\mathcal{E}_0(p)$ that depends on some fitting parameter $p$. As both of this processes for me are ...
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1answer
67 views

Simulate a quantum channel with a certain fidelity

I am looking for an easy-to-use framework for simulating a quantum channel that can accept the desired average fidelity of the channel as input. For example, if I want a channel with 98% average ...
1
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1answer
48 views

What are the matrices in the POVM for measuring the first $m$ qubits?

Suppose you have a quantum state $|w\rangle$ consisting of $m + n$ qubits, and you set up a measurement that measures the first $m$ qubits in the standard basis. What are the matrices in the ...
3
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1answer
29 views

Qiskit PauliWeightedOperator in the matrix representation?

Suppose we have a PauliWeightedOperator object from Qiskit. Is there any built-in method to convert it to the matrix representation in the computational basis? My search in the docs was not successful....
4
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1answer
67 views

Unitary over bipartite states that can turn a non-product state into a product state

Consider a bipartite quantum state $\rho_{AB}$ over a product of finite-dimensional Hilbert spaces $\mathcal{H}_A \otimes \mathcal{H}_B$. Does there exists a unitary $U$ over $\mathcal{H}_A \otimes \...
2
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2answers
49 views

When is the Choi matrix of a channel pure?

For a quantum channel $\mathcal{E}$, the Choi state is defined by the action of the channel on one half of an unnormalized maximally entangled state as below: $$J(\mathcal{E}) = (\mathcal{E}\otimes I)\...
7
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1answer
83 views

Positive semidefinite relationship after partial trace

Let $\rho_{ABC}$ and $\sigma_{C}$ be arbitrary quantum states and $\lambda\in \mathbb{R}$ be minimal such that $$\rho_{ABC}\leq \lambda \rho_{AB}\otimes\sigma_C$$ We assume there are no issues with ...
5
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1answer
71 views

Unknown quantum circuit symbol

I was reading DiCarlo, L., Reed, M., Sun, L. et al. Preparation and measurement of three-qubit entanglement in a superconducting circuit. Nature 467, 574–578 (2010). https://doi.org/10.1038/...
5
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1answer
63 views

Stinespring dilation: Size of environment

Let $\mathcal{E}_{A\rightarrow B}$ be a quantum channel and consider its $n-$fold tensor product $\mathcal{E}^{\otimes n}_{A^n\rightarrow B^n}$. Any isometry $V_{A\rightarrow BE}$ that satisfies $\...
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1answer
36 views

Prove that different Kraus decompositions are related through a unitary, using the Choi isomorphism

I consider a process $\mathcal{E}$ that is at least CP and hermitian preserving. I know that the Choi matrix then has the form: $$ M = \sum_k |M_k \rangle \rangle \langle \langle M_k | $$ Where $|M_k \...
2
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0answers
38 views

How to define POVM measurement operators for a composite quantum state

I have an evolved quantum composite state $\hat{\rho}^{\otimes{N}}$ that I retrieved from a quantum channel $\mathcal{E}$, Now I do know how to define a POVM for the evolved states $\hat{\rho}$ that ...
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0answers
41 views

Getting Choi-matrix of a subsystem

In Qiskit, for a given QuantumCircuit object, you can computed its Choi-matrix via the corresponding Choi object, for example: <...
2
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2answers
59 views

Does it make sense to sum two Choi operators?

I am very new to the theory of the Choi representation of quantum processes and I am learning it all by myself from research papers (especially this https://arxiv.org/abs/1111.6950) as I didn't find ...
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1answer
48 views

How does the CPTP constraint reflect on the matrix representation of a qubit channel in the Pauli basis?

Let us write the possible states of a qubit in the Bloch representation as $$\newcommand{\bs}[1]{{\boldsymbol{#1}}}\rho_{\bs r}\equiv \frac{I+\bs r\cdot\bs \sigma}{2},$$ where $\bs\sigma=(\sigma_1,\...
2
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2answers
109 views

Transforming $|01 \rangle + |10 \rangle - |11 \rangle \to |01 \rangle - |10 \rangle + |11 \rangle$

How to convert from current state: $$|\psi \rangle =\dfrac{ |01 \rangle + |10 \rangle - |11 \rangle}{\sqrt{3}}$$ into a target state $$|\phi \rangle = \dfrac{|01 \rangle - |10 \rangle + |11 \rangle}{\...
2
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2answers
112 views

How can I find a quantum channel connecting two arbitrary quantum states?

Given two arbitrary density matrices $\rho, \sigma\in \mathcal{H}$ (they have unit trace and are positive), how do I go about finding a possible quantum channel $\mathcal{E}$ such that $\mathcal{E}(\...
4
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1answer
97 views

How to perform this $d$-dimensional unitary operation on IBM Q?

$U_{a,b}=\sum^{d-1}_{x=0}\omega^{bx}|x+a\rangle\langle x|$,$\omega=e^{\frac{2\pi i}{d}}$,$a,b\in\{0,1,2,...,d-1\}$ Can someone please give me the pic of the quantum circuit?
3
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1answer
80 views

Why do the constant operations in a quantum computer need second qubits?

From what I'm reading, building the constant-1 and constant-0 operations in a quantum computer involves building something like this, where there are two qubits being used. Why do we need two? The ...
4
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1answer
53 views

Alternative definition of the coherent information of a quantum channel

Let $T: M_n \to M_n$ be a quantum channel. If I understand Definition 13.5.1 of the book "Quantum information theory" of Wilde, the coherent information $Q(T)=\max_{\phi_{AA'}} I(A \rangle B)...
5
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1answer
167 views

Are perfectly LOCC-indistinguishable states necessarily identical?

Let $\rho,\sigma\in\text{L}(\mathcal{H}_{XAB})$ be given by $$ \rho = \sum_x |x\rangle\langle x|\otimes p_x\rho_x, \quad \sigma = \sum_x |x\rangle\langle x|\otimes q_x\sigma_x, $$ and consider ...
2
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0answers
24 views

What applications does single-shot state conversion have?

Many quantum processes are formulated in a resource theoretic approach like entanglement, athermality, asymmetry, coherence, etc. Some of its topics have obvious applications, like distillation where ...
5
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1answer
83 views

Can local projections increase entanglement?

Consider a generic bipartite pure state $\newcommand{\ket}[1]{\lvert #1\rangle}\ket\Psi\equiv \sum_k \sqrt{p_k}\ket{u_k}\otimes\ket{v_k}\in\mathcal X\otimes\mathcal Y$, where $p_k\ge0$ are the Schmidt ...
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0answers
38 views

Deformation of the Bloch sphere and contraction of its planes under the action of channels

On pg 376-377 of N&C, it gives 3 different diagrams showing how the various axis of the Bloch sphere will be contracted under the action of the channels, limiting the possible states after it's ...
0
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1answer
117 views

How to code a projector operator in qiskit?

I'm new to qiskit and I want to know how do I define a projector operator in qiskit? Specifically, I have prepared a 3 qubit system, and after applying a whole lot of gates and measuring it in a state ...
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1answer
32 views

Is entanglement nonincreasing on average by local operations for all possible ensemble decompositions?

We know for a pure state conversion $|\psi \rangle \rightarrow_\textrm{LOCC} |\phi \rangle$ via local operation and classical communication (LOCC), an entanglement monotone should not increase, that ...
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1answer
41 views

Equivalence of two ways to recover a map from its Choi state

Let $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ be a quantum channel, $\Phi:\mathrm{Lin}(\mathcal X)\to\operatorname{Lin}(\mathcal Y)$. We define its Choi representation as the operator $J(\Phi)$ ...
3
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1answer
103 views

How to find the unitary operation of a depolarizing channel?

Suppose we have a depolarizing channel operation $$E(\rho)=\frac{p}{2}\textbf{1}+(1-p)\rho$$ acting on a Spin$\frac{1}{2}$ density matrix of the form $\rho=\frac{1}{2}(\textbf{1}+\textbf{s}\cdot\...
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0answers
30 views

Is there a physical interpretation for the diagonal terms of a $\chi$ matrix?

I assume $\mathcal{E} \in \mathcal{L}(\mathcal{L}(H))$ is a CPTP map. I call $\{B_i\}$ an orthonormal basis for Hilbert-Schmidt scalar product of $\mathcal{L}(H)$ This quantum map can be decomposed as:...
1
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2answers
103 views

Can I switch $\alpha_0$ and $\alpha_1$ conditionally to $\alpha_0>0$ in a state $\alpha_0|0\rangle+\alpha_1|1\rangle$?

I have a single qubit $a$ in state $$ |s\rangle = \alpha_0|0\rangle + \alpha_1|1\rangle $$ $\alpha_0$ may be 0 whereas $\alpha_1$ is always positive and above $0$. Almost always $$\alpha_0 << \...
1
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1answer
28 views

Are two outputs of a quantum operation (CPTP map) themselves related by a quantum operation?

$\def\ket#1{|#1\rangle} \def\bra#1{\langle#1|} \def\mt#1{\mathrm{#1}} \def\E{\mathcal E} \def\F{\mathcal F}$ Let $\ket\phi$ and $\ket\psi$ be pure states on the same quantum system, so that $\ket\psi=\...
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0answers
21 views

Why the chi-matrix fidelity of the process is the fidelity of the chi-matrix noise map

I am following this paper, and I am stuggling with a derivation. Basically, I consider an orthonormal basis $\{B_i \}$ with respect to Hilbert-Schmidt scalar product, on the density matrix space $\...
2
votes
1answer
53 views

Finding the irreducible representation of Kraus operators of a dephasing channel

I would like to understand an example of finding a noiseless subsystem of a quantum channel from the irreducible representation of its Kraus operators. Assume we have $2$ dephasing channels acting on ...
2
votes
3answers
69 views

Why can every $|X\rangle\in H_1\otimes H_0$ be written as $|X\rangle=(X\otimes I_{H_0})|\Omega \rangle$ for some $X\in\mathcal L(H_0,H_1)$?

In A theoretical framework for quantum networks is proven that a linear map $\mathcal{M} \in \mathcal{L}(\mathcal{H_0},\mathcal{H_1})$ is CP (completely positive) iff its Choi operator $M$ is semi ...
0
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1answer
73 views

Some questions to get intuition on Choi isomorphism [closed]

I have some questions about the Choi-Jamiolkowski isomorphism. I remind how it can be defined. First, we define $|\mathcal{I}_{H_0} \rangle \rangle \in H_0 \otimes H_0$ $$|\mathcal{I}_{H_0} \rangle \...
2
votes
2answers
138 views

How is it not a contradiction that it is possible to build fault tolerant circuits with strictly contractive (e.g.: depolarizing noise) channels?

This paper discusses strictly contractive channels, i.e. channels that strictly decrease the trace distance between any two input quantum states. It is shown that if a quantum circuit is composed of ...
3
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1answer
63 views

What is the certain error rate in a quantum channel?

Quantum error correction is a fundamental aspect of quantum computation. I have read some material about "Quantum Channel" and "Quantum error correction". I have known the formula ...
1
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1answer
79 views

What is the standard noise channel that is applied in simulations?

I know there are various quantum noise channels, which include the depolarizing channel, the dephasing channel and the bit-flip channel; We can apply them in simulators easily. However, is there any ...
1
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1answer
40 views

Is the restriction of a strictly contractive channel (SCC) to a subspace necessarily still SCC? (impossibility of perfect QEC for SCCs)

This paper shows the impossibility of perfect error correction for strictly contractive quantum channels, i.e., for channels such that $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
2
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2answers
59 views

Calculate probability of a state after depolarization

Let's say I have a particle in the quantum state $|+\rangle = \frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)$, represented as a density operator (1st matrix) that went through a depolarizing chanel (2nd ...
3
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1answer
103 views

Prove that $A\preceq B$ implies $A=\Psi(B)$ for some channel $\Psi$

Define $\newcommand{\PP}{\mathbb{P}}\newcommand{\ket}[1]{\lvert #1\rangle}\newcommand{\tr}{\operatorname{tr}}\newcommand{\ketbra}[1]{\lvert #1\rangle\!\langle #1\rvert}\PP_\psi\equiv\ketbra\psi$, and ...
4
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2answers
99 views

Why do quantum operations need to be reversible?

Why do quantum operations need to be reversible? Is it because of the physical implementation of the qubits or operators neesd to be unitary so that the resultant states fit in the Bloch sphere?
3
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1answer
33 views

Given a channel $\Phi(X)=\sum_k c_k(X)\sigma_k$, are there always $F_k\ge0$ such that $\Phi(X)=\sum_k \operatorname{tr}(F_k X)\sigma_k$?

Fix a finite number of states $\sigma_k$, and consider a channel of the form $$\Phi(X)=\sum_k c_{k}(X)\sigma_k.$$ For $\Phi$ to be linear and trace-preserving we must have: $$c_k(X+X') = c_k(X) + c_k(...
2
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1answer
54 views

Can any channel be written as $\Phi(X)=\operatorname{Tr}_{\mathcal Z}[U(X\otimes \sigma)U^\dagger]$ for any state $\sigma$?

We know that every CPTP map $\Phi:\mathcal X\to\mathcal Y$ can be represented via an isometry $U:\mathcal X\otimes\mathcal Z\to\mathcal Y\otimes\mathcal Z$, as $$\Phi(X) = \operatorname{Tr}_{\mathcal ...
1
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1answer
50 views

What is the Kraus representation of quantum-to-classical channels?

As discussed in Watrous' book, quantum-to-classical channels are CPTP maps whose output is always fully depolarised. These can always be written as $$\Phi_\mu(X) = \sum_a \langle X,\mu(a)\rangle E_{a,...
5
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3answers
134 views

What does the adjoint of a channel represent physically?

Given a quantum channel (CPTP map) $\Phi:\mathcal X\to\mathcal Y$, its adjoint is the CPTP map $\Phi^\dagger:\mathcal Y\to\mathcal X$ such that, for all $X\in\mathcal X$ and $Y\in\mathcal Y$, $$\...
3
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0answers
71 views

Representing a von Neumann measurement as $[\mathcal{I} \otimes P_i] U(\rho_s \otimes \rho_a)U^{-1} [\mathcal{I} \otimes P_i]$, how do we choose $U$?

Given the state of a system as $\rho_s$ and that of the ancilla (pointer) as $\rho_a$, the Von-Neumann measurement involves entangling a system with ancilla and then performing a projective ...
2
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4answers
126 views

Can a Kraus representation act as the identity on any operator?

In the textbook “Quantum Computation and Quantum Information” by Nielsen and Chuang, it is stated that there exists a set of unitaries $U_i$ and a probability distribution $p_i$ for any matrix A, $$\...
1
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1answer
49 views

Confused regarding explanation of Schumachers compression in N&C

On page 547 of N&C, for $|\psi_{0}\rangle=|0\rangle$ and $|\psi_{1}\rangle=(|0\rangle+|1\rangle)/\sqrt{2}$ and for $|\tilde{0}\rangle=\cos(\pi/8)|0\rangle+\sin(\pi/8)|1\rangle$ and $|\tilde{1}\...
2
votes
2answers
112 views

Find the Kraus operators of a combined amplitude and phase damping channel

I am going through the paper Surface code with decoherence: An analysis of three superconducting architectures and I have a doubt about how the authors get what they refer to as the combined channel ...
2
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0answers
62 views

Coherent Information and Entanglement Breaking channels

The book by John Watrous, "The Theory of Quantum Information" is an exciting read for anyone wanting to research quantum information theory. The following question presumes some background ...