Unlike classical physics and general relativity, which deal with the deterministic evolution of physical variables such as position and momentum, quantum mechanics deals with an abstract entity called the state vector. In general, the state vector resides in an infinite-dimensional complex Hilbert space. However, in the world of quantum information and quantum computing one deals mostly with state vectors that are finite-dimensional. For example, the spin of a particle or the direction of a superconducting current or energy state of a trapped ion has a state vector that is simply a vector residing in a 2-dimensional complex Hilbert (inner product) space. By the Born rule (or Born postulate), the probability of an outcome during measurement of the physical variable (for example spin) is given by the square of the norm of the state vector. The evolution of the state vector can simply be represented by a 2 X 2 complex unitary matrix. In the absence of measurement, subjecting the system to conservative force fields simply results in the transformation of the state vector by a suitable unitary 2 X 2 matrix. The transformation of a state vector by these unitary matrics is called a quantum logic gate and can be represented graphically. One of the most unique and intriguing aspects of quantum mechanics is the phenomenon of entanglement. It deals with non-local correlations between measurements of complementary observables (such as position and momentum or spin directions) performed on parts of a system that are physically separated by a "large" distance. In the language of state vectors, it simply represents an indecomposable vector in the tensor product of two complex Hilbert spaces. The phenomenon of entanglement was first discussed by Einstein-Podolsky-Rosen in the famous EPR paper with a clearly stated goal of demonstrating the incompleteness of quantum mechanics as a theory of physical reality. EPR demonstrated that quantum mechanics had non-local effects, an anathema for Einstein as it seemed to violate special relativity. Actually, EPR only showed that quantum mechanics implies non-local correlations between measurements, but such a correlation is so counter-intuitive that it seemed to imply that there was more to quantum mechanics than the Copenhagen interpretation of quantum mechanics. Little did Einstein know that John Bell would later show that non-locality was an essential component of quantum mechanics. Bell showed that any local hidden variable theory would have to satisfy an inequality (known as Bell's inequality), which quantum mechanics did not satisfy. Bell wrote a wonderful paper called "Bertellman's Socks and the Nature of Reality", explaining the crux of the EPR paradox and Bell's inequality.
Since then many people have recast the EPR paradox and Bell's discovery into different formats to convey the non-classical, counter-intuitive (Mermin's "Local Reality Machine"), and computationally powerful (CHSH Nonlocal game) nature of entanglement. Today, entanglement forms the foundation of modern quantum information theory and has applications to cybersecurity via schemes like quantum key distribution and quantum cryptography. The following presentation tries to give a flavor of the history and applications of quantum entanglement. I gave this presentation in an evening class on quantum computing that I took at the UW Physics department.
In future posts, I hope to explain each of the slides on quantum entanglement (including Bell's paper on "Bertellman's socks") in simple terms. Stay tuned.
No comments:
Post a Comment