Quantum Entanglement and why it bothered Einstein?

There is no stronger proof than quantum entanglement that validates quantum mechanics and its fundamental divergence from classical mode of thinking.

Quantum Entanglement

Quantum entanglement is a phenomenon in which two or more particles become interconnected in such a way that their states become intrinsically linked such that, measuring the state of one particle instantaneously affects the state of the other, regardless of the spatial separation between them, their states cannot be described independently. 

But what do we really  mean by measuring ?

Classically measuring should be thought as a way of removal of ignorance about information that already exists, for example if I measure the radius of a football, I now know its radius but before my measurement that ball was still of exact same radius, my measurement does not affect its property. 

But quantum mechanically, measuring a quantum state affects its state, we call it the collapse of wavefunction, before my measurement the state is indeterministic and has various possible measurement outcomes, mathematically speaking,  the measured state is one of eigenstate of the operator linked to that measurement. So it's not that my quantum football was always of same radius before measurement, but it was in superposition of footballs with varied radius, it's only the act of my measurement which narrowed it down to one single value of radius or 'state'.

Spin

Electrons have this unique intrinsic property called as spin (they are not actually spinning but they possess angular momentum, so we attribute it to spin) , an electron can exist in spin up (+1/2 )or spin down( -1/2 ) or a superposition of these two states. This property can be measured in any direction , for convention up arrow indicates spin +1/2 in z direction. A general spin quantum state can be mathematically described as given below: 

                               \[\psi=\alpha|\uparrow\rangle+\beta|\downarrow\rangle\]

Entanglement in context of Spin

Suppose you have two spin half particles, the two particles are said to be entangled if measurement of spin of one particle affects the spin of other particle, these two particles are not independent, the combined quantum state of these particles is called an 'entangled pair'.

\[\psi=\frac{1}{\sqrt{2}}\left(|\uparrow\rangle|\downarrow\rangle- |\downarrow\rangle|\uparrow\rangle\right)\]

Here, the greek letter- psi denotes the quantum state of two particle system, on right hand side the first term denotes a configuration in which first particle is spin up and second particle is spin down, while the second term means the configuration where the first particle is spin down and second particle is spin up .Now, what the subtraction sign means is that the two particle state is in superposition of the states described by two terms. Superposition means that upon measuring the spin of two particles we can get any of the two possibilities /configurations with probability = 1/2

So, if we prepare an entangled state  and then measure the spin of 2nd particle, lets say, we find it spin up, this means that our quantum state has collapsed onto the second term which means we instantly know the spin of 1st particle that is spin down. If I seperate the two entangled particles over a large distance, and then measure spin of 2nd particle, then I would impact the spin of first particle instantaneously. 

Why this bothered Einstein ?

The astute will think that entanglement might violate special relativity as the 'instantaneous affect' no matter the distance might mean faster than light communication which is not possible, but it turns out we are not that much in  trouble since there is no way of using entanglement to transmit information faster than light that is ALICE(holding particle 1) and BOB (holding particle 2 ) can't use entanglement to send desired signals. Because, if Bob observes a spin-down state, he cannot determine whether it occurred due to his own wavefunction collapse or if Alice had already measured and obtained a spin-up result. To understand Einstein's concern, we need to look at another concept called as locality.

Locality  

Locality means that the behavior of a particle at one location should not be influenced by measurements or actions performed on another particle located far away, unless information is transmitted through a classical communication channel at a speed no faster than the speed of light.

However, quantum entanglement is a non local phenomenon.

Einstein was uneasy with this fact and in 1935 research paper with Podolsky and Rosen, they coined the term element of reality. For a physical property to be an element of reality it should be possible to predict with certainty the value that property will have, immediately before measurement. Now we can already see that quantum entanglement theory doesn't include this element of reality since it is impossible to predict with certainty what the 'spin' value will be unless we measure it, all quantum mechanics offers is probabilities .

Lets look at an example, suppose I have a black pen and a white pen, I distribute one pen each to Alice and Bob but only after I have sealed them in boxes, I send Alice and Bob to very far away planets where they open the boxes and see which pen they had been carrying all the way, if Alice finds a black pen then she instantly knows that Bob has a white pen , but there is nothing non-local about this, whatever uncertainty  Alice and Bob had was due to their ignorance and my ability to hide it from them. in brief, the pen in boxes were always of their respective color and never in any kind of undetermined state regarding their color.

Einstein believed in an explanation of quantum entanglement which is analogous to above example, he believed that quantum theory in incomplete since quantum mechanical equations  don't help us predict with certainty what the outcome will be.  EPR(Einstein, Podolsky and Rosen) introduced the idea of hidden variable, a hypothetical piece of information that was there from the beginning and says what the result of measurement will give with complete certainty.

In this picture, probabilities of quantum mechanics simply stem out of our ignorance of these hidden variables and all uncertainties in quantum mechanics are no different from uncertainty in knowing which pen is in the box, the cause of uncertainty is attributed entirely to lack of knowledge of these variables. By this EPR called for  a classical, local realism view which is consistent with logical thinking.Keep in mind that the lack of knowledge of these variables is different from their non existence and the two predict different results while analyzing physical phenomena.

 So, who was wrong here ? Was Einstein right in saying that quantum mechanics is an incomplete theory and does his idea of hidden variable stand correct? It turns out that after 30 years, an experiment proved that the picture of the world painted by EPR was wrong, Nature experimentally agreed with quantum mechanics and its postulates. The main key to experiment was Bell's inequality, we derive Bell's inequality about a system, first using notions of common sense (EPR) and then using quantum mechanical analysis which is not consistent with common sense analysis . We then perform the real world experiment to determine which analysis was correct and it turns out that quantum  mechanics came out victorious.

Nature doesn't care whether it makes sense to us or not and it has no obligations to follow whatever grandiose, beautiful theories predict. In the end what matters is whether our theory is experimentally validated or not, however we should always keep in mind to not ignore the theories that were proved wrong later, physical theories have always been hit and trial throughout the history and the failed attempts often prove successful in providing a new insight in our our understanding of nature.