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Entangled states

Last updated Jun 23, 2022

Entangled states are those quantum states that cannot be factored into Product states. 1 For example, with two qubits, $$\ket{\phi^+} = \frac{1}{\sqrt2}(\ket{00} = \ket{11}),$$ cannot be written as $\ket\psi_1 \ket\psi_0$.

Since an entangled state cannot be factored, a general entangled state of $n$ qubits would have $N = 2^n$ amplitudes, $c_0$ through $c_{N−1}$ : $$\ket\psi = \sum_{j = 0}^{N-1} c_j\ket{j} = c_0\ket0 + c_1\ket1 + \dots + c_{N-1}\ket{N-1} = \begin{pmatrix}c_0 \\ c_1 \\ \vdots \\ c_{N-1}\end{pmatrix}.$$
Entangled states are the cause of the Entanglement property in quantum computing.