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My question is very basic. I am studying quantum gates from various books. I read that by putting a CNOT gate, the qubits are entangled. Is this correct? What exactly entanglement means and what is its benefit and how the two entangled qubit appears on the Bloch sphere?

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    $\begingroup$ Does this answer your question? does CNOT gate cause entanglement? $\endgroup$ Apr 8, 2022 at 17:51
  • $\begingroup$ Thank you @MartinVesely. I still donot understand the benefit of entanglement. $\endgroup$ Apr 9, 2022 at 1:52
  • $\begingroup$ The entanglement is one factor is QC higher performance in comparison with classical computers. However, so far the reason why it is so is not completely understood. $\endgroup$ Apr 9, 2022 at 7:16
  • $\begingroup$ Thank you @MartinVesely $\endgroup$ Apr 9, 2022 at 14:17
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    $\begingroup$ see also physics.stackexchange.com/q/54975/58382 and links therein $\endgroup$
    – glS
    Apr 10, 2022 at 12:54

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There are many different levels of explanation for what entanglement really is: the following answers might help:

I read that by putting a CNOT gate, the qubits are entangled. Is this correct?

Yes, this is 'correct' (see above answers for completeness). And it is already answered and explained elsewhere in the website. For instance start a two-qubit state in $\vert 00 \rangle$ then apply a Hadamard followed by a CNOT operation. What is the state at the end?

What exactly entanglement means

There is entanglement in states and entanglement in measurements as well. Let me just focus on entanglement in states, and you could check answers here for more. Entanglement is a property of different systems: it means in a very rough way, that the two systems can only be described as a whole, and so measurement on one affects directly the state of the other. In a separate state, in contrast, the state of one state is independent of the state of the others. The mathematical description that is general was first given by Werner in 1989, and most of the time, it is defined using a 'negation-like' definition, i.e., we define entanglement as what is not,

Def. A state $\rho$ is said to be separable when is a convex combination of product states. $$\rho = \sum_i p^i \rho_1 \otimes \rho_n.$$

Def. A state $\sigma$ is said to be entangled when it is not separable.

So, entangled states are those for which it is impossible to find a separable states that recover the state as a convex decomposition. States that are separable have this property of being statistically independently related.

What is its benefit?

Entanglement is a necessary condition for quantum computation being universal. For instance we know that arbitrarily little entanglement suffices for universal quantum computation. We also "presence of an unbounded amount of multi-partite entanglement is necessary for exponential speed-ups in circuit-based pure state quantum computation because every protocol that does not exhibit this property can be simulated efficiently on a classical device", result from Ref.. Therefore we know that entanglement is a necessary important resource in quantum computation, but that it is not the resource since we may have entanglement and still be capable of efficient simulation when we restrict computation over Cliffords.

How the two entangled qubit appears on the Bloch sphere?

Entanglement is a property of more than one system and the qubits are the smallest quantum system possible. Hence, one can talk about entanglement between at least two qubits which is not directly represented in a Bloch sphere. Nevertheless there are generalizations of representations of Bloch-like inspiration in higher dimensions (since entangled states live in the composite Hilbert space we have $\dim_{\mathbb{C}}(\mathbb{C}^2 \otimes \mathbb{C}^2) = 4$), this question/answer is interesting in this perspective. But if one can draw a representation over the Bloch sphere itself of an entangled pair of qubits I don't know and maybe other answers might help you (and me) with this part of the question.

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