Un état à deux qubits est un état intriqué si et seulement s'il n'existe deux états d' un qubit | un ⟩ = alpha | 0 ⟩ + ß | 1 ⟩ ∈ C 2 et | b ⟩ = gamma | 0 ⟩ + X | 1 ⟩ ∈ C 2 de telle sorte que | un ⟩ ⊗ | b ⟩ = | ψ , Où ⊗ désigne le produit tensoriel , et .
Donc, pour montrer que l'état de Bell est un état intriqué, nous devons simplement montrer qu'il n'y existe pas deux étatsun qubit| un⟩et| b⟩telle que| Φ+⟩=| un⟩⊗| b⟩.
Nous pouvons maintenant simplement appliquer la propriété distributive pour obtenir
Cela doit être égal à , that is, we must find coefficients , , and , such that
Observe that, in the expression , we want to keep both and . Hence, and , which are the coefficients of , cannot be zero; in other words, we must have and . Similarly, and , which are the complex numbers multiplying cannot be zero, i.e. and . So, all complex numbers , , and must be different from zero.
But, to obtain the Bell state , we want to get rid of and . So, one of the numbers (or both) multiplying (and ) in the expression , i.e. and (and, respectively, and ), must be equal to zero. But we have just seen that , , and must all be different from zero. So, we cannot find a combination of complex numbers , , and such that
In other words, we are not able to express as a tensor product of two one-qubit states. Therefore, is a entangled state.
We can perform a similar proof for other Bell states or, in general, if we want to prove that a state is entangled.
A two qudit pure state is separable if and only if it can be written in the form
To determine if the pure state is entangled, one could try a brute force method of attempting to find satisfying states and , as in this answer. This is inelegant, and hard work in the general case. A more straightforward way to prove whether this pure state is entangled is the calculate the reduced density matrix for one of the qudits, i.e. by tracing out the other. The state is separable if and only if has rank 1. Otherwise it is entangled. Mathematically, you can test the rank condition simply by evaluating . The original state is separable if and only if this value is 1. Otherwise the state is entangled.
For example, imagine one has a pure separable state . The reduced density matrix on is
Meanwhile, if we take , then
If you wish to know about detecting entanglement in mixed states (not pure states), this is less straightforward, but for two qubits there is a necessary and sufficient condition for separability: positivity under the partial transpose operation.