6.5 Entanglement
The key aspect of quantum computing is entanglement, which allows for quantum states to encode more information than the sum of their individual components. We explain this in detail here and provide examples for two-qubit systems.
6.5.1 Quantum entanglement and why it matters
An n-qubit system can exist in any superposition of the 2n basis states:
with
If such a state can be represented as a tensor product of individual qubit states, then the qubit states are not entangled. For example, it is easy to check that
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=  ⊗ ⊗ , |
(6.5.1) |
so that the quantum state is not entangled (only in superposition). An entangled state cannot be represented as a tensor product of individual qubit states.
For example, the two-qubit state
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does not allow a tensor product decomposition. Namely, for any a,b,c,d ∈ℂ such that |a|2 + |b|2 = |c|2 + |d|2 = 1, we have
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We notice that we need 2n probability amplitudes to describe...
