Is Bell inequality always violated by a quantum system? Can it ever be violated by a classical system?
There is no "Bell's inequality" (there is one in the original paper, but when people talk about Bell's inequality, that is rarely the one they mean), there are many such inequalities. The basic principle for a Bell inequality is that it is never violated by classical systems but that it is violated by certain quantum states.
The modern understanding of Bell inequalities is the following:
- Any physical theory such as classical mechanics and quantum mechanics defines a set of states that the system can be in. Usually this set is a convex set
- Classical states form a convex subset of the convex set of states
- A Bell inequality is just a hyperplane in the state space such that all classical states are in exactly one half of the state space and (usually) at least some quantum state is not on this half of the state space.
- Violating a Bell inequality just means that the state is not on the side of the hyperplane that all classical states are on.
In order to make the connection with inequalities, note that a hyperplane fulfils a linear equation and an inequality would then be valid for any point lying on one of the two halves of the hyperplane.
This picture means in particular that no classical state can violate a Bell inequality and there should always be quantum states that violate a Bell inequality.
How does this connect to measurements?
- For any state, you can measure something (for example position or momentum). A measurement is therefore a function, which gives some value if you apply it to a state, i.e. its a functional.
- Measurements are linear, hence hyperplanes are defined by measurement configurations
Bell's inequality is not always violated. For particular orientations of your detectors you will find that your system follows the inequality (classical or quantum).
For classical systems the derivation will always lead to follow the Bell inequalities. Instead for a quantum systems there will be orientations of your detector that are going to have a large correlation that violate the inequalities.
What is interesting is that you can predict the correlation of your system for any orientation and see if your quantum system goes along, and generally your correlation would take values that are lower or higher, meaning it doesn't even follow the same pattern of the classical prediction.