27 July 2007

Quantum Mechanics in Pictures, or How I spent my summer vacation

After a two-week camping trip, I visited Canada/USA Mathcamp for one week. While there, I studiously avoided any Harry Potter spoilers (upon returning home, I dutifully took a one-day break from the rest of life to read the book), offered an evening activity focused on mind-body-movement-partnering exercises called "Learn to Fly", and hung out with some of the best people on the planet. Oh, and I taught a three-day course on topics in quantum mechanics, with a fun structure: I set myself the challenge of preparing three one-day talks, which, while all part of the same large story, could be understood and appreciated entirely separately. I more-or-less succeeded, although certainly I would edit the talks before giving them again. Partly to preserve them for posterity, and partly so that I could share them with you all, I thought it might be nice to post my lecture notes here; the first is in this post, and the later two will be in posts of their own.

Day One was on "Quantum Mechanics in Pictures", and is based largely on B. Coecke's paper Kindergarten Quantum Mechanics.

We begin by setting up a graphical notation with which to describe science. We have systems = particles, which we draw by vertical lines with upward-pointing arrows, and which we label by the kind of system=particle that the line represents. States, costates, operations, and numbers are, respectively, downward-pointing triangles with single lines coming out the top, upwards-pointing triangles with single lines coming out the bottom, squares with a line coming out each end, and diamonds with no lines. By "line", I always mean labeled directed line i.e. system.

"Numbers" live in some system --- for CM, numbers are 0 or 1; for SM, probabilities; for QM, numbers are complex ---, and we can multiply them by putting them next to each other. We can arrange our system in some state; a "costate" is really a measurement, asking "how much is this system having this property?". Operations are functions: we plug in a state of system A, and get out a state of system B.

Given two systems, we can combine them ("tensor product") by drawing their lines next to each other, and interpreting it as a single line --- like putting two pipes together into one big tube. This corresponds to having a joint system of A and B. In general, we can have "operations" from any number of systems to any other. We can also hook these objects together to form composite objects. Lines are allowed to cross each other --- we don't care about whether on top or on bottom --- and the usual laws hold. In general, we can tell what kind of object we have (A-state, function from A to B, etc.) just by inspecting the incoming and outgoing lines. E.g. a costate is a function from A to "nothing", i.e. to numbers. If I were speaking in abstract language, I'd say that we are working in a symmetric braided monoidal category.

This will be Axiom 0: that we have these types of objects and can combine them in these natural ways. Axiom 0 is required of any type of "physics": any scientific theory should accommodate systems, states, joint systems, manipulations, measurements, etc.

Quantum Mechanics admits two kinds of "turning upside down". The first kind gives us Axiom 1: Given any X: A -> B, we can form its adjoint X† : B -> A. In general, I will be sloppy and drop the dagger when I'm taking the adjoint a state (to get a costate). The adjoint of a number is called its "complex conjugate". Given a state S, its adjoint is the measurement "how S-like is it? We can define the dot product by drawing two systems meeting at a dot, meaning "turn the second one into its adjoint and compose". Taking the adjoint is some kind of "time reversal".

Axiom 2: for any physical evolution --- i.e. time running forward --- i.e. anything we can actually do _in a lab_ --- X:A->B, we have X^{-1} = X†. This is called "unitarity". Pictures: this says that time evolves deterministically, in such a way that relative similarity does not change. Note: This is not true in the physics of _math_. We could make a theory of "functions"; these can be many-to-one, so definitely not unitary.

There's another kind of turning upside down. Adjoints reversed operators, etc., but not spces. Axiom 3 pt. 1: to every system A, a "dual" system A*, which we still label A but draw the arrow pointing down, s.t. A** = A. Axiom 3 pt. 2: Existence of Bell (co)states; pictures; satifying rule.

Then we have a "teleportation protocol": I have a state. I want to transmit all info in state to you. But there's no reason to believe that I can _measure_ all info (c.f. Heisenberg). One protocol: just carry particle to you. Bad: maybe you're in China, and we're just talking by phone. Better: before you leave, we make entangled pair (BFFs sharing dollare bill). Then I _measure_ "in what way is my state like LH part of Bell state" and transmit that classical info. Advertisement: we will explaion this in more detail tomorrow.

_But_: no cloning theorem. (Star Trek). A "cloning protocol" would be a physicial thing we could do X(s,e) = s,s for a fixed e and any s. But then by unitarity and gluing we get a contradiction, unless only numbers are 0 and 1. E.g. CM. If more numebrs are available, no.

Axiom 4: For any two states a and b, can get from one to another "continuously", in the sense that it's cont's when you dot with some (co)state c. This is true in SM. There exist "mixed states" where you don't have complete knowledge. SM allows the following cloning: measure state exactly, then clone it. But this _does not_ clone the "mixed state" unless I perversely then _forget_ the new info. In QM, axiom is amended: Axiom 4': Can cont'sly transmorm from any _pure_ state to any other _pure_ state going through only _pure_ staets.

"Pure" state can be well-defined; intuition is "state of maximum knowledge". We will give example of this tomorrow. It is a theorem that SM (= CM + mixed states = Axioms 0 through 3 plus technicalities) _plus_ Axiom 4' is equivalent to QM (in particular, forces complex numbers), proved in L. Hardy Quantum Theory From Five Reasonable Axioms.

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