How is entanglement achieved between two particles in quantum computing?

Many texts (especially meant for public consumption) discussing quantum mechanics tend to skim over exactly how entanglement is achieved. Even the Wikipedia article on quantum entanglement describes the phenomenon as follows:

"Quantum entanglement is a physical phenomenon which occurs when pairs or groups of particles are generated, interact, or share spatial proximity in ways such that the quantum state of each particle cannot be described independently of the state of the other(s)..."

This doesn't explain how the process actually comes into being. How are these particles "generated," "interact," or "share spatial-proximity" such that we can claim that two particles are entangled? What is the process?

Josu has given an example of how using quantum gates you can get an entangled state. However, quantum gates are sort of "black-boxes".

The physical methods for creating entangled states and testing for entangled states are covered well on the relevant Wikipedia page:

Entanglement is usually created by direct interactions between subatomic particles. These interactions can take numerous forms. One of the most commonly used methods is spontaneous parametric down-conversion to generate a pair of photons entangled in polarisation. Other methods include the use of a fiber coupler to confine and mix photons, photons emitted from decay cascade of the bi-exciton in a quantum dot, the use of the Hong–Ou–Mandel effect, etc., In the earliest tests of Bell's theorem, the entangled particles were generated using atomic cascades.

It is also possible to create entanglement between quantum systems that never directly interacted, through the use of entanglement swapping. Two independently-prepared, identical particles may also be entangled if their wave functions merely spatially overlap, at least partially.

Testing a system for entanglement:

Systems which contain no entanglement are said to be separable. For $2$-Qubit and Qubit-Qutrit systems ($2 × 2$ and $2 × 3$ respectively) the simple Peres–Horodecki criterion provides both a necessary and a sufficient criterion for separability, and thus for detecting entanglement. However, for the general case, the criterion is merely a sufficient one for separability, as the problem becomes NP-hard. A numerical approach to the problem is suggested by Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement". Leinaas et al. offer a numerical approach, iteratively refining an estimated separable state towards the target state to be tested, and checking if the target state can indeed be reached. An implementation of the algorithm (including a built-in Peres-Horodecki criterion testing) is brought in the "StateSeparator" web-app.

In 2016 China launched the world’s first quantum communications satellite. The $100m Quantum Experiments at Space Scale (QUESS) mission was launched on Aug 16, 2016, from the Jiuquan Satellite Launch Center in northern China at 01:40 local time. For the next two years, the craft – nicknamed "Micius" after the ancient Chinese philosopher – will demonstrate the feasibility of quantum communication between Earth and space, and test quantum entanglement over unprecedented distances. In the June 16, 2017, issue of Science, Yin et al. report setting a new quantum entanglement distance record of$1203$km, demonstrating the survival of a$2$-photon pair and a violation of a Bell inequality, reaching a CHSH valuation of$2.37 ± 0.09\$, under strict Einstein locality conditions, from the Micius satellite to bases in Lijian, Yunnan and Delingha, Quinhai, increasing the efficiency of transmission over prior fiberoptic experiments by an order of magnitude.

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How do I show that a two-qubit state is an entangled state?

How to show that an n-level system is entangled?

You can easily create an EPR pair by means of a circuit consisting of a Hadamard gate on the first qubit, followed by a CNOT gate, having two $$|0\rangle$$ states as an input. The calculations would be:

• $$(H\otimes I)(|0\rangle\otimes|0\rangle)=H|0\rangle\otimes|0\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes|0\rangle$$

• $$\operatorname{CNOT}\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)\otimes|0\rangle=\frac{1}{\sqrt{2}}(|0\rangle\otimes|0\rangle+|1\rangle\otimes|1\rangle)=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)=|\Psi^+\rangle$$

The ending state consequently is the EPR pair called $$|\Psi^+\rangle$$, which is a classical example of an entangled pair.