To understand the concept of Quantum Entanglement one has to focus his telescope in taking a basic purview of what is a classical entanglement. In other words let’s first assume our particles are classical.
For that end we take the example of coins. Good old coins. Yea the ones that you toss in air and let them spin and wobble so you can see the fun. It can even be useful, eg in deciding a tie or something. (to untie it)
When I toss a coin into air, what happens to all the other coins in the vicinity? Nothing. They are physically independent of each other.
The physical state or condition of one coin, eg whether its spinning or resting on the side against a wedge or lying horizontally are totally independent of the physical condition or state of another coin, if we decide to take two such coins into consideration.
So a physical state is a more formal and physics-wise description, quantitatively, of a physical condition or configuration. So as to say that a physical state is a more aptly and specifically defined condition of the physical object, here the coin.
Eg one can describe the physical condition of the coin by its various channels of gaining or losing energy. Eg spinning coin has an added degree of energy gain or loss. And a coin present in a gravitational field or accelerating or rotating in particular ways or references are other possible energy channels of the coin. By taking into account all possible energy states or possibilities we determine the state of the coin.
In Classical Mechanics this, the state of a physical system, is merely a quantitative description of its total energy; which are always expressed in two different but useful ways, the Lagrangian or the Hamiltonian. One is the difference of Potential and Kinetic energy and the other is the summation of the two energies.
To be more approximate one often gives the physical state of a system in terms of various customary variables, which are equivalent to stating the total energy. Eg Pressure and Volume in studying gaseous phase of matter along with certain other variables suffices to state the state of the system thoroughly.
In Physical Kinematics eg in the motion of elementary particles one describes a system via what’s called Phase-Space which is nothing but a detailed quantification of the state of the system of particles usually through the momentum and energy of each possible element of the system which are then concisely listed into the 4-vectors of the elements.
Eg the 4-vector of an electron will merely state its total energy and total momentum in all possible directions. This then gives the state of the electron, but once we take all electrons in the system and specify their individual states we have what’s called a Phase Space of the system under consideration.
So a coin has a particular classical state described completely by the physical variables (such as spin and energy) which are totally independent of the other coins, if they are not connected by means of a physical bondage. This is where the concept of the Quantum Entanglement comes. As long as, and its quite possible, we do not force the coins through a rigid or otherwise physical connection, their physical states are totally independent or uncorrelated. We do something to the coin eg toss it or throw it, its trajectory is totally independent of the state or physical condition of another chosen coin. They don’t influence each other if they are not physically connected.
But take an example of a quantum mechanical particle eg an electron or say two electrons, no matter how we propose the electrons to be physically-not-connected, as long as they are in each other’s vicinity they form an inherently correlated or physically connected system, that is their physical state or phase space will have components that are not independent of each other. That is so intuitively, because they are very subtle waves whose prongs can be affecting or intertwined with each other in more ways than we can even theoretically envisage.
So as to say, the states given by complex numbered functions of each other’s physical variables are dependent functions of each other at some level. If I specify the quantum mechanical state of one electron by its required wave-functions, that is already a mixation of the wave functions of other electrons in the vicinity. That is because its nonsensical to say “one electron” in quantum mechanics.
Given we say an electron is a wave represented by its wave function giving its various probabilities of physical variables (eg location, energy, etc) to demarcate ONE electron is to violate that fundamental principle of statistical uncertainty. Therefore two electrons are merely a physical wave-function of an uncertain number of electrons, where we can-not arbitrarily minimize; such an uncertainty. We only make sure under a particular situation what is the best likelihood, of number of electrons is and so on.
This fact is inherently, nothing but the concept of Quantum Entanglement. Unlike classical particles which are completely independent of each other the quantum sized objects are not. For physical intuition they are at a very small and subtle level intertwined with each other as if threads are kind of tied onto each other. We can-not toss an electron because the whole concept of one electron completely independent of all other makes sense only when all other electrons are not interacting because they are arbitrarily far away.