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The set of separable states is closed. Thus, around any entangled state - not necessarily pure - there is an $\epsilon$-ball which lies entirely within the entangled states. Or, in the language of your question: All entangled states are "robust". (As illustrated by DaftWullie's answer, the size of this ball can depend on the state: There are pure ...

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For a fixed $\epsilon_0$, why not simply consider $$|\psi\rangle=\cos\theta|00\rangle+\sin\theta|11\rangle?$$ Since it's a two-qubit state, entanglement can be determined using the PPT criterion. Hence, $$\rho=(1-\epsilon)|\psi\rangle\langle\psi|+\epsilon I/4$$ is entangled if $\epsilon<2\sin(2\theta)/(1+2\sin(2\theta))$. Any $\epsilon$ you give me, ...

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You are analysing the case where you know a unitary has definitely been applied on the first qubit. In that case, it should not be surprising that there's no change in entanglement. You can take a couple of perspectives: single qubit unitaries do not change entanglement. To change entanglement with a unitary requires a two-qubit unitary. If you know what ...

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Given a separable bipartite state as $|\psi\rangle\otimes|\phi\rangle$, you "get" the states of the single systems but taking only the corresponding state, e.g. here $|\psi\rangle$ or $|\phi\rangle$. More generally, you might not know the structure of the state, and you might have entanglement between the different subsystems, in which case the ...

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I would like to add to keisuke.akira answer. The Noise Model in which only a Single Qubit Flips is correct. However we can assume a more general Noise Model which may be more realistic and still see the use of Bit Flip Code. Since Quantum Circuits are analog, hence it is rare that a qubit flips completely. It is more likely that there is a small coherent ...

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We're trying to build a code to protect against single bit flips. That is, we are assuming the noise model. By assumption, it has the form $\sigma_x \otimes \mathbb{I} \otimes \mathbb{I}$, therefore, it only flips one of them. Of course, in general, the noise does whatever it wants, and therefore, we need to build codes that can protect against more general ...

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While designing your circuit are you sure 6 Qubits were really required for X gate . As i am observing in your above plot there are two possible probabilities hopefully that might be right , I would suggest you to start designing the above circuit with 2 Qubits and observe the outcome.

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An example of a state which yeilds an entangled state when you trace one qubit out, is the 3-qubit "W" state: $$\lvert W_3 \rangle = \tfrac{1}{\sqrt3} \Bigl( \lvert 100 \rangle + \lvert 010 \rangle + \lvert 001 \rangle \Bigr)$$ Taking the outer product with itself, we obtain $$\lvert W_3 \rangle\!\langle W_3 \rvert = \tfrac{1}{3} \Bigl( \!\... 6 Symmetric Werner states in any dimension n\geq 2 provide examples. Let's take n=2 as an example for simplicity. Define \rho\in\mathrm{D}(\mathbb{C}^2\otimes\mathbb{C}^2) as$$ \rho = \frac{1}{6}\, \begin{pmatrix} 2 & 0 & 0 & 0\\ 0 & 1 & 1 & 0\\ 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 2 \end{pmatrix}, $$which is ... 0 To see intuitively why this protocol increases the entanglement after each iteration, we can work out an example where our initial state is say \lvert 00\rangle+\lvert 11\rangle. Upon passing through a 50:50 Beam Splitter, we get:$$|00\rangle+|11\rangle = |00\rangle+ a_{0}^{+}a_{1}^{+}|00\rangle \hspace{3mm}transforms\hspace{3mm}|00\rangle + \frac{1}{2}(...

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I don't know the details of this paper, although there are much better things that you can say about the GHZ case (in general, properties of GHZ states are much easier to analyse than W states). I'll summarise the key result in this context below, but further details are available in my paper, here. There are some very simple entanglement criteria that one ...

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Apparently, the specific question posed here has been answered in the affirmative--at least (first, we point out) through numerical means--by Arsen Khvedelidze and Ilya Rogojin in Table 2 of their 2018 paper, "On the Generation of Random Ensembles of Qubits and Qutrits: Computing Separability Probabilities for Fixed Rank States" ArsenIlya They ...

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