Questions tagged [entanglement-witness]
For questions about entanglement witnesses: observables whose negative expectation value certifies the entanglement of measured states
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Is purity an entanglement witness?
Suppose an n-qubit mixed state $\rho$
with subsystems $a_1, a_2,..., a_n$
If purity of an arbitrary qubit $a_i$ is 1, then can i conclude its separatable from the entire system?
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What are examples of weakly optimal witnesses?
While discussing witnesses, in https://arxiv.org/abs/0811.2803 the authors mention (page 16 of the arxiv version, below Eq. (32)) that a necessary condition for a witness $W$ to be optimal is that it ...
glS♦
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Prove that an entanglement witness is optimal iff it's zero on a spanning set of product states
I am reading about entanglement witnesses from here.
In section 2.5.2, it is written that
Furthermore, a witness $\mathcal{W}$ is called optimal, if there is no other witness, which is finer than $\...
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Prove that an entanglement witness satisfies $\operatorname{tr}(W)>0$ and $\operatorname{tr}(W)^2\ge \operatorname{tr}(W^2)$
If $W$ is an entanglement witness ($W \neq 0$), prove that
(a) $ tr(W) >0$
(b) $ tr(W)^2 > tr(W^2)$
For (a), by definition, since $ |ab\rangle$ is separable, thus $tr(\rho W)=\langle ab | W| ab \...
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Can we characterise the general structure of two-qubit witness operators?
Consider a two-qubit space, and a Hermitian operator $R\in\mathrm{Herm}(\mathbb{C}^2\otimes\mathbb{C}^2)$ in this space.
The operator is positive semidefinite iff $\langle u,Ru\rangle\ge0$ for all $u\...
glS♦
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What it means to say that SWAP gate has maximum operator entanglement entropy?
In this (freely available) article, just below equation 14, it says
The SWAP operator has the maximum operator entanglement entropy...
The operator entanglement entropy, denoted by $E(U)$ for an ...
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Understanding the $M$ upper bound in the paper: "Multipartite entanglement and high-precision metrology"
This paper is a paper in 2012 and cited by a lot of papers. And there does not exist comment in arxiv or error statement in PRA. But when I reading this paper, I think the right part of the eq(23) ...
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Entanglement Witnesses close to GHZ states
Consider page 2 of Toth's paper 'Entanglement detection in the stabilizer formalism (2005)'. To detect entanglement close to GHZ states, they construct entanglement witnesses of the form $$\mathcal{W} ...
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Finding a class $C$ of bipartite PPT states such that entanglement of $\rho \in C$ implies entanglement of $\rho + \rho^{\Gamma}$
Consider an entangled bipartite quantum state $\rho \in \mathcal{M}_d(\mathbb{C}) \otimes \mathcal{M}_{d'}(\mathbb{C})$ which is positive under partial transposition, i.e., $\rho^\Gamma \geq 0$. As ...
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How to prove the following bosonic entanglement expression?
Based on the article given by J. L. Ball, I. Fuentes-Schuller, and F. P. Schuller, Phys. Lett. A 359, 550 (2006)
had used the following expression of von-Neumann entropy
\begin{equation}
S = - \...
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How does a map being "only" positive reflect on its Choi representation?
We know that a map $\Phi\in\mathrm T(\mathcal X,\mathcal Y)$ being completely positive is equivalent to its Choi representation being positive: $J(\Phi)\in\operatorname{Pos}(\mathcal Y\otimes\mathcal ...
glS♦
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How are witness operators physically implemented?
Let's take an example of an entanglement witness of the form $W = | \phi \rangle \langle \phi | ^{T_2}$ where $ | \phi \rangle $ is some pure entangled state.
If I wanted to test some state $\rho$, I ...
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Explicit 16⨯16 matrix representations of two-qudit entanglement witnesses
I have a set of $16 \times 16$ two-qudit density matrices. I would like to study the bound-entanglement for this set, making use of entanglement witnesses for which explicit matrix representations are ...
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Are entanglement witnesses of this form optimal?
One can make an entanglement witness by taking the partial transpose of any pure entangled state.
Consider $|\phi \rangle $ as any pure entangled state.
Then $W = | \phi \rangle \langle \phi |^{T_2} ...
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Is there a two-qudit Choi entanglement witness $W^{(+)}$?
Example 2 in arXiv:1811.09896 states that the "Choi EW (entanglement witness) $W^{(+)}$ obtained from the Choi map in $d=3$ $\ldots$ is given by
\begin{equation}
W^{(+)} = \frac{1}{6} \left( \...
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How is the expression for the optimal entanglement witness derived?
In the Bertlmann 2009 paper in the Annals of Physics (here), an optimal witness operator for an entangled state $\rho$, given that the closest separable state to it is $\rho_0$ is given by:
$$A_{\...
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