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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19 views

What has been done to conserve quantum resources?

In resource intensive or scarce scenarios, we obviously want to maximize efficiency with limited resources. So in quantum computing, and quantum information processing in general (especially one-way ...
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63 views

How to perform this d-dimension 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?
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Why do the constant operations in a quantum computer need second qbits?

From what I'm reading, building the constant-1 and constant-0 operations in a quantum computer involves building something like this, where there's two qbits being used. Why do we need two? The ...
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49 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)...
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149 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 ...
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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 ...
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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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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 ...
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57 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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31 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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36 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 $J(\Phi)$ ...
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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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General interpretation of 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:...
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101 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 << \...
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27 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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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 $\...
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1answer
43 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 ...
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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 ...
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1answer
64 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 \...
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131 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 ...
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1answer
58 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 ...
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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 ...
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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 $||\mathcal{E}(\rho)-\mathcal{E}(\sigma) ||\leq k ||\rho-\sigma||$...
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56 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 ...
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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 ...
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91 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?
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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(...
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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 ...
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42 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,...
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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 $Y\in\mathcal Y$, $$\...
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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 ...
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4answers
117 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, $$\...
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48 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}\...
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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 ...
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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 ...
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58 views

How to compute the capacity of a quantum channel from its Kraus operators?

Is there a working rule to compute the capacity of a quantum channel described by a set of Kraus operators $\{K_i\}$?
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Prove that the depolarizing channel is completely positive

In two dimensions, for a density operator $\rho$ and probability $\lambda$, a depolarizing channel can be written as: $$\mathcal{E}(\rho) = (1-\lambda) \frac{\mathbb{I}}{2} + \lambda\rho$$ In ...
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1answer
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Confusion over HSW theorem depicted in Nielsen and Chuang

On page 560, it states that $$C^{(1)} \geq S(\frac{\varepsilon(|{\psi}\rangle\langle{\psi}|) +\varepsilon(|{\varphi}\rangle\langle{\varphi}|)}{2} - \frac{1}{2}\varepsilon(|{\psi}\rangle\langle{\psi}|)-...
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Mutual information of Choi state=0, what would that imply about the quantum channel?

Classically, if the mutual information between the input and output of some channel or circuit $= 0$, it means the output is independent of the input, and the circuit is in a way 'useless'. For the ...
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31 views

Quantum operation to get rid of small but nonzero eigenvalues

Updated and edited question: Let $N_{\delta}:P(\mathcal{H}_A)\rightarrow P(\mathcal{H}_B)$ be a completely positive trace nonincreasing map from the set of positive semidefinite operators in $\...
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What happens when you send a Bell state through depolarizing channel?

For noise parameter $Q$ and a density matrix $\rho$, we know that the depolarization channel $\mathcal{E}$ would act like: $$ \mathcal{E}(\rho) = (1 - Q)\rho +Q\frac{I}{2}, $$ where $I$ is the ...
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Physical Interpretation of Non-Trace Preserving Quantum Operations

In Chapter 8 of Nielsen and Chuang's Quantum Computation and Quantum Information, a mathematical framework is developed to describe the dynamics of open quantum systems. Suppose the initial state of ...
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How to compute the tensor product of the depolarizing channel with the identity?

Consider two quantum systems A and B, B goes through a depolarizing noise channel, while A is not changed, i.e., they go through the channel $\mathbb{I}_A \otimes \mathcal{E_{\text{depol}}} $. If the ...
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$M(\rho)=\operatorname{Tr}_2\left(\ U\ \rho\otimes\rho_2\ U^{\dagger}\right)$is unitary $\iff\ U=U_1\otimes U_2$, a product of $2$ unitary operators?

Let $\rho : V_1 \to V_1 $ and $\rho_2 : V_2 \to V_2 $, where $V_1$ and $V_2$ are Hilbert spaces. Suppose that $U:V_1\otimes V_2 \to V_1\otimes V_2$ is a unitary operator. Define a map $M : L(V_1, ...
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When can a non-completely-positive evolution of a state be physical?

Definitions: a map $\Phi$ is called positive if $\Phi(\rho)$ is positive semidefinite for any positive semidefinite $\rho$, and completely positive (CP) if $\Phi \otimes \mathrm{Id}$ is a positive map ...
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What does it mean to take the Choi-Jamiolkowski of a quantum channel?

The Choi-Jamiolkowski of a channel $\newcommand{\on}[1]{\operatorname{#1}}\Lambda : \on{End}(\mathcal{H_A}) \xrightarrow{} \on{End}(\mathcal{H_B})$ is obtained through an isomorphism of the form: $$ ...
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57 views

Understanding Quantum Channel and Choi Jamiolkowski Notation [closed]

I am given the following $\newcommand{\on}[1]{{\operatorname{#1}}}$ Let $|i\rangle$, $1 \leq i \leq \on{dim}\,\mathcal{H_A}, |s\rangle$, $1 \leq s \leq \on{dim}\,\mathcal{H_B}$ be unitary bases. ...
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26 views

How can I obtain the lineally independent Kraus operators for the composition of two quantum channels?

Imagine we have two quantum channels, the bit flip and the depolarizing ones (for the corresponding noise models). These two quantum channels have 2 and 4 Kraus operators, respectively. We could ...
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1answer
70 views

Confusing notation in Wikipedia's quantum channel article

In the Wikipedia's Quantum channel article, it is said that a purely quantum channel $\phi$ (it's not exactly the same phi calligraphy but it's close), in the Schrodinger picture, is a linear map ...
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24 views

Is quantum deletion via a partial randomization procedure possible?

The paper, Quantum deletion is possible via a partial randomization procedure claims that it is possible to bypass the no-deleting theorem by a procedure called R-deletion. But this seems to go ...