For questions about entanglement witnesses: observables whose negative expectation value certifies the entanglement of measured states
Questions tagged [entanglement-witness]
19 questions
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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) should be $2M+(N-M)(N-M+2)$ instead of…
narip
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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: $J(\Phi)\in\operatorname{Pos}(\mathcal Y\otimes\mathcal X)$, as shown for example in Watrous' book,…
glS
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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 separability of $\rho$ is equivalent to…
mathwizard
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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 would have to perform $\mathrm{Tr}(W \rho)$. I…
Mahathi Vempati
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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 available or can be generated.
So to be fully…
Paul B. Slater
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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} $ is an entanglement witness.
However, is there…
Mahathi Vempati
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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( \sum_{i=0}^{2} [ 2| ii \rangle \langle ii | + | i,i-1…
Paul B. Slater
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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:
$$A_{\text{opt}} = \frac{\rho_0 - \rho - \langle \rho_0,…
Mahathi Vempati
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Can we characterise the general structure of two-qubit witness operators?
Consider a two-qubit space, and a Hermitian operator $R\in\mathrm{Herm}(\mathbb{C}^2\otimes\mathbb{C}^2)$ in this space.
The operator is positive semidefinite iff $\langle u,Ru\rangle\ge0$ for all $u\in\mathbb{C}^2\otimes\mathbb{C}^2$. We can…
glS
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Entanglement Witnesses close to GHZ states
Consider page 2 of Toth's paper 'Entanglement detection in the stabilizer formalism (2005)'. To detect entanglement close to GHZ states, they construct entanglement witnesses of the form $$\mathcal{W} := c_0 I - \tilde{S}_{k}^{(GHZ_N)} -…
John Doe
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Inconsistent values of $S$ (CHSH inequality) in my Qiskit implementation of the E91 protocol
I'm currently experimenting with Qiskit and I wanted to implement the Ekert's E91 protocol. I had no problem writing the code, but then I got to the point of calculating the $S$ value to detect the presence of an eavesdropper. In all the papers…
Lorenzo
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Entanglement Negativity and Reduced Density Matrices
The negativity of a density matrix is defined as
$$
\mathcal{N}(\rho_{AB}) = \frac{1}{2} \bigl(\|\rho_{AB}^{T_A}\|_1 - 1\bigr),
$$
where $\rho_{AB}$ is a density matrix on the bipartite system $A \otimes B$,
$T_A$ denotes the partial transpose with…
zeroknowledgeprover
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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 touches the set of separable states, meaning…
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 $\mathcal{W}$. This implies that for any positive…
Anindita Sarkar
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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 = - \operatorname { Tr } \left( \varrho \log _ { 2 }…
EnthusiastiC
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