The Question

What differs the quantum theory from classical ones? The uncertainty principle, the Hilbert space formalism, the probabilistic description, to name a few.

However, they are all built upon the Hilbert space formalism of quantum states, which is merely mathematical definitions. From our examples, the uncertainty principle comes from the Cauchy-Schwartz inequality regarding the Hilbert space inner product, which is identified with the probability amplitude of one quantum state with respect to another.

So what does such mathematical formalism tell us about the nature of quantum mechanics, or more distinctively, the physical system described by quantum mechanics? How do we associate the theory (which is based on mathematics) with the actual physics? This is a very hard question to answer directly, we will gradually see what bizarre non-classical features come up from those basic foundamental principles.

Our Approach

Here we briefly lay out our plan for this blog post series, which shall be constituted by two parts, the first one focusing on quantum foundations while the second shifted towards the application of it in quantum information theory.

In the first part of this blog series, we shall discuss three famous non-classical features of quantum theory: the Bell non-locality, the Kochen-Specker contextuality, and the Pusey-Barrett-Rudolph theorem.

Bell nonlocality, briefly described, states that the reality of a quantum state is not local in the classical sense, i.e., there could be long-range correlations, resolved the question proposed by Albert Einstein, Boris Podolsky, and Nathan Rosen in their 1935 paper.

Kochen-Specher contexuality, naively put, states that there can be correlations among quantum states in a similar but not the same manner like that appears in Bell nonlocality, rised from the contexts of the given physical system. Or put another, the contexts in which the systems are operating, have an impact on the correlations among the states.

The PBR theorem, as I currently understand it, states that quantum states are more than mere information of the actual physical system.

A disclaimer, quantum fundation is an active research field to this day, where scientists have been making great progress, but with the level of mathematics that I am not be able to comprehend. So my understanding of it should be bounded by the mathematics that I know, there are some concise reformulations of the same problem but with more advanced mathematical frameworks. If I’m aware of them I shall point that out in the reference section, otherwise please understand my ignorance.

In the second part of this blog series, I shall shift a bit towards the broader sense of quantum mechanics and indulge my curiosity, for example, PT-symmetric quantum mechanics (aka non-Hermitian QM) and dissipative quantum mechanics (i.e., with an open system).

More on Motivations

Why do we care? Of course we care, as much as we care about this world. For it is the very same first principles that constitute our current world.

Additionally, the foundational approach reformulated in an information-theoretical perspective seems like a more feasible way to understand blackholes and to quantize gravity.