In winters of 2011, I took a self study project under supervision of Dr. Rajdeep Chaterjee with the aim of acquiring knowledge essential for quantum computing. I first studied linear algebra from the notes of a mathematics major student. Then I surveyed Feynman Lectures Vol III and “Quantum Computation and Quantum Information” by Neilsen & Chaung to get a clear idea abut quantum mechanics and its use in quantum computation. The notes I compiled can be found here. Finally I surveyed EPR paper with the help from my guide.
This turned out to be an important and amazing period filled with lots of surprises and fun. First the knowledge of linear algebra provided a better way to visualize fourier transforms, special functions and matrix algebra. I realised that most of the mathematics that I had been studying is essentially linear algebra. Then the existence of inner product structure resulting in Cauchy-Schwarz inequality showed another way why entities like usual 3 dimensional vectors and complex numbers follow triangle inequality. I realized the importance of inner product structure in defining the angle between the linear vectors, which I had been using since intermediate (3d vectors and complex numbers). But the best part was to see the angles being associated with functions and matrices.
Then the notion of representing a state of a physical system with linear vectors and associating probability with the norm appeared strange and took some time to digest. Till then I was witnessing the superposition of physical waves like string waves or pressure waves. But it was the first time I saw physical states (associated to physical systems like electrons) being superimposed. I saw that this superimposition is responsible for all those “unusual” phenomenon and results like appearance of fringes in electron YDSE, unusual probability in alpha-alpha particle scattering (with notion of identical particles) and discreet marks in Stern-Gerlach experiment.
I started studying quantum computation to get better hold of these foreign concepts. I believed that the exposure to quantum computation would give me confidence with quantum mechanics just like the exposure to daily activities gave me the confidence with Newtonian mechanics. Furthermore, the involvement of algorithms attracted me towards this field. I had studied on wikipedia that quantum algorithms provide speed up for certain cases. So I started reviewing concepts of quantum computation with the hope of doing a research project next summer.
The review of EPR paper almost entirely change my perception of Physics. I realised that nature is independent of physical theory i.e it behaves as it should without any restriction. We humans try to find out the way nature behaves by formulating physical theories which are mere models to explain how nature might be working. And to verify the theory we have to establish one to one correspondence with the reality. The problem (of completeness) noted by EPR shows how careful one needs to be, while formulating a fundamental theory, by taking into account logical reasoning and mathematical consistency. The only thing missing in the paper is the experimental verification of the assumptions made. John S. Bell elegantly devised a formalism in which experiments are allowed to decide the fate of quantum mechanics. Experiments favor quantum mechanics showing that at-least one of the two assumptions, realism and locality, made by EPR is not valid.
Although the problem noted by EPR is resolved (with the help from experiments), the paper still remains as one fine example which teaches to do a through examination of the subject and raise appropriate questions.