Transforming quantum entanglement into local correlations
Let’s start with an entangled state of two photons, a state commonly used in discussions of quantum mechanics and entanglement:
| \boldsymbol \Psi \left( \nu_1, \nu_2 \right) \rangle = \frac{1}{\sqrt{2}} \left(| \mathbf x_1, \mathbf x_2\rangle + |\mathbf y_1, \mathbf y_2\rangle\right)
Here, |\mathbf x\rangle and |\mathbf y\rangle represent orthogonal polarization states, for instance, horizontal and vertical polarizations.
Now, consider an eavesdropper, Eve, who intercepts each photon and performs a measurement. Eve chooses to measure in a basis \{|\mathbf d \rangle, |\mathbf d_\perp \rangle\} defined by an angle \theta. The states are given by:
|\mathbf d\rangle = \begin{bmatrix} \cos(\theta) \\ \sin(\theta) \end{bmatrix}, \quad |\mathbf d_\perp\rangle = \begin{bmatrix} -\sin(\theta) \\ \cos(\theta) \end{bmatrix}
Eve’s measurement projects each photon into one of these states. The key point is to understand how this measurement affects the entanglement between the photons as observed by Alice and Bob, who are the intended recipients of these photons.
After Eve’s measurement, the overall quantum state can be described as a statistical mixture. The correlations between Alice and Bob’s photons, after Eve’s intervention, can be described using local hidden variables.
This means that the correlations can be expressed in the form:
\mathcal C(\mathbf a,\mathbf b) = \int \mathrm d\lambda \,\rho(\lambda)\,A(\mathbf a,\lambda)\,B(\mathbf b,\lambda)
where \lambda represents the hidden variable (in this case, related to Eve’s measurement outcomes), \rho(\lambda) is a probability distribution, and A(\mathbf a,\lambda) and B(\mathbf b,\lambda) are deterministic outcomes (\pm 1) for Alice and Bob’s measurements given the hidden variable \lambda and their chosen measurement directions \mathbf a and \mathbf b.
Correlations of this form are constrained by Bell’s inequalities. Quantities like S, defined as:
\begin{aligned} | S | = & | A(\lambda, \mathbf a) B(\lambda, \mathbf b) - A(\lambda, \mathbf a) B(\lambda, \mathbf b^\prime) \\ & + A(\lambda, \mathbf a^\prime) B(\lambda, \mathbf b) + A(\lambda, \mathbf a^\prime) B(\lambda, \mathbf b^\prime)| \end{aligned}
will satisfy |S| \le 2. This is in contrast to quantum mechanics, where entangled states can violate Bell’s inequalities, leading to values of |S| > 2.
Eve’s measurement transforms the initial non-local quantum correlations into local correlations. The measurement process collapse the wave function, making the subsequent correlations between Alice and Bob consistent with local hidden variable theories.
For more insights into this topic, you can find the details here.