### Quantum Entaglement & Cryptography

Today, I had an interesting meeting with my physics advisor, Professor Jean Bellissard. We were discussing the relation between Steinsberg and Kraus theorems, and how coherency with redundancy of copiable number of equivalent shared states cannot be achieved in quantum computing. So, I take it upon myself to prove it to him that it is possible -- and come up with an ultra-complex long-winded proof on how in the case of Bose-Einstein Condensates (BECs), this redundancy can be achieved since the quantum states are similar, and all the Hilbert spaces can be extended to a super-space (well, extending Steinsberg's theorem). He looks at me, gives me a smile. And then goes on to the board. And writes a simple equation. And tells me -- "what you proved using that complex means is nothing but theoretical proof for the existence of quantum entanglement". And I open my mouth wide open, and he extends my theory and proves that entanglement is not just a sufficient consequence of QC, but a necessary one too, for BECs. This isn't the interesting part. The interesting part was, I told him that if that were true, then by the extension of Krauss theorem for limited finite vector spaces, I can estimate the randomness created by any RNG that uses the common seed, especially since I could extend the Hilbert space within all the possible finite transformations without changing the vector, and using only Unitary and phase transformations. At which point, he smiles again and says that Professor Nicholas Gisin has already done pioneering research in this area. Hmm, so is there a way around this dilemma? Dr. Bellissard suggested one idea -- the use of a Brownian motion source coupled with the matrix transformations for formulation, thereby adding a truly random source -- more or less. I have a few ideas, too, but the problem is in mathematically proving them to be random. Ahh, the pleasures of working with mathematically oriented physicists. Who am I kidding? My family is full of those.

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