Bell pair
The Bell states are a concept in quantum information science and represent the simplest examples of entanglement. They are named after John S. Bell because they are the subject of his famous Bell inequality. An EPR pair is a pair of qubits (or quantum bits) that are in a Bell state together. Because of the entanglement, measurement of one qubit will assign a value to the other qubit immediately in one of four ways, where the value assigned depends on which Bell state the two qubits are in. This behaviour is not subject to relativistic limitations such as the speed of light, but the no-communication theorem prevents this behaviour to be used to transmit information faster than light. The phenomena can be used to agree upon random numbers in less time than it would take to communicate the numbers at the speed of light over the distance between the peers – this was believed to be a result of an error in quantum physics and was named the EPR paradox. Originally, the paradox was resolved by giving up the assumption that the principle of locality is true, but other interpretations have also emerged.
The Bell states are four specific maximally entangled quantum states of two qubits.
The degree to which a state in a quantum system consisting of two "particles" is entangled is measured by the Von Neumann entropy of either of the two reduced density operators of the state. The Von Neumann entropy of a pure state is zero—also for the Bell states, which are specific pure states. But the 2x2 density matrix corresponding to the Bell states can be formed as usual, and the von Neumann entropy of this density operator of the Bell states is positive and maximal, if the matrix does not degenerate to a projector.
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