5.5 Separable or entangled?

Most vectors in \mathcal{H}_a\otimes \mathcal{H}_b are entangled and cannot be written as product states |a\rangle\otimes|b\rangle with |a\rangle\in\mathcal{H}_a and |b\rangle\in\mathcal{H}_b.

In order to see this, let us write any joint state |\psi\rangle of \mathcal{A} and \mathcal{B} in a product basis as \begin{aligned} |\psi\rangle &= \sum_{ij} c_{ij}|a_i\rangle\otimes|b_j\rangle \\&= \sum_i|a_i\rangle\otimes\left(\sum_j c_{ij}|b_j\rangle\right) \\&= \sum_i|a_i\rangle\otimes|\phi_i\rangle \end{aligned} \tag{5.5.1} where the |\phi_i\rangle=\sum_j c_{ij}|b_j\rangle are vectors in \mathcal{H}_{\mathcal{B}} that need not be normalised. For any product state, these vectors have a special form. Indeed, if |\psi\rangle= |a\rangle\otimes|b\rangle then, after expanding the first state in the |a_i\rangle basis, we obtain |\psi\rangle = \sum_{i}|a_i\rangle\otimes\left(\sum_i\alpha_i|b\rangle\right). This expression has the same form as Equation (5.5.1) with |\phi_i\rangle=\alpha_i|b\rangle, i.e. each of the |\phi_i\rangle vectors in this expansion is a multiple of the same vector |b\rangle. Conversely, if |\phi_i\rangle = \alpha_i|b\rangle for all i in Equation (5.5.1), then |\psi\rangle must be a product state.74 So if we want to identify which joint states are product states and which are not, we simply write the joint state according to Equation (5.5.1) and check if all the vectors |\phi_i\rangle are multiples of a single vector. Needless to say, if we choose the states |\phi\rangle randomly, it is very unlikely that this condition is satisfied, and we almost certainly pick an entangled state.

Quantum entanglement is one of the most fascinating aspects of quantum theory. We will now explore some of its implications.

  1. Even though an entangled state cannot be written as a tensor product, it can always be written as a linear combination of vectors from the tensor product basis. In fact, any state of n qubits |\psi\rangle can be expressed in the standard product basis. In general, given n qubits, we need 2(2^n-1) real parameters to describe their state vector, but only 2n to describe separable states.↩︎