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Sunday, 8 September 2013

Quantum Entanglement

Although Bell states are systems of two qubits, neither of them can be expressed as the tensor product of any two states in \(\mathcal{H}^2\). That is, for each Bell state \(\left| \beta_{ij}\right>
\neq\left|\psi\right>\otimes \left|\phi\right>\) for any states \(\left|\psi\right>, \left|\phi\right>\in\mathcal{H}^2\). This has the implication of certain correlations that exist when measurements of the individual qubits are made on a Bell state. For instance, consider \[\left|\beta_{00}\right>=\frac{1}{\sqrt2}(\left|00\right>+\left|11\right>),\] and suppose the first qubit in the register is measured. This measurement will yield either \(\left|0\right> \) or \(\left|1\right>\) each with probability \(\frac{1}{2}\). A subsequent measurement on the second qubit in the register would then give a definite state \(\left|0\right>\) or \(\left|1\right>\) with probability \(1\) depending on the state of the first qubit being either \(\left|0\right> \) or \(\left|1\right>\), respectively. However, if it was the second qubit that was observed first, then either \(\left|0\right> \) or \(\left|1\right>\) would be observed with equal probability and the state of the first qubit would be correlated with the second in the same way.

The kind of correlations present in Bell states, where the measurement outcomes of subsystems of a larger composite system are inherently correlated is a property of entangled systems. A state \(\left|\psi\right>\in \mathcal{H}_A\otimes \mathcal{H}_B\) of some composite system is entangled if \(\left|\psi\right>\neq\left|\phi_A\right>\otimes\left|\phi_B\right>\) for any states \(\left|\phi_A\right>\in \mathcal{H}_A\) and \(\left|\phi_B\right>\in\mathcal{H}_B\). Otherwise, if it is possible to express a state \(\left|\psi\right>\in \mathcal{H}_A\otimes \mathcal{H}_B\) as a tensor product of two states \(\left|\psi\right>=\left|\phi_A\right>\otimes\left|\phi_B\right>\), then the state \(\left|\psi\right>\) is called a separable state. The existence of entangled states is a unique feature of quantum mechanics, and such states are exploited in various quantum algorithms and protocols.

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