From Abstruse Goose “Moment of Clarity (?)–part 2″

A related but more tractable question is: why SU(2) when we can already represent rotations by SO(3)? Daniel Larson has an answer. In short, representations of SU(2) are projective representations of SO(3) (which is possible because in quantum mechanics we are in projective Hilbert spaces i.e. we don’t care about the phase). As a bonus, Larson goes on to discuss whether the choice of SO(3) or SU(2) makes any difference.

As for my opinion of it, I think one of the Amazon reviews is spot on. “A history” and “a flood”, from the book’s subtitle, are a bit boring for me, while “a theory” is excellent. For that reason, I recommend a preview of some of those boring parts How We Know, by Freeman Dyson. For me, the book starts to pick up from chapter 6 “New Wires, New Logic” when Shannon, Hartley, and Nyquist enter the scene. Following are a fraction of what everyone can learn from this book

• why we have the mantra “meaning is irrelevant” and the notion of coding and an abstract bit
• the relationship between Shannon and Turing and the parallel relationship between Shannon’s information theory and the Turing machine, which ultimately resulted in the Kolmogorov-Chaitin algorithmic entropy
• the role of chance and information in fundamental laws of physics

For further technical readings of specific topics, I recommend taking a look at

• Information theory, Evolution, and the Origin of Life by Hubert Yockey
• Maxwell’s Demon 2 by Harvey Leff and Andrew Rex
• Quantum Processes Systems, and Information by Benjamin Schumacher and Michael Westmoreland, especially the notion of an informationally isolated system that is tied to the ultimate physical limit of privacy, error correction, and duplication of information (copying)
• Quantum Mechanics as Quantum Information (and only a little more) by Christopher Fuchs
• From Classical to Quantum Shannon Theory by Mark Wilde for an official account of quantum Shannon theory
• for this one, the name alone might blow your mind–Quantum Darwinism, a research program by Wojciech Zurek et al.

Other related and interesting topics are probabilitistic paradoxes and Bayes’ theorem which are featured more in Hans Christian von Baeyer book’s Information, but that’s for another time.

My first thought when I saw the table of content was “Hey, this book could have initiated my whole scientific career!” I began to have serious interest in science in high school, where, unfortunately, the maths and physics were to me mere fun and games without any purpose. Biology was more like storytelling. I became interest in consciousness, evolution, and genetics and picked biology major in college, and was thoroughly disappointed by lack of clear thinking from most professors and fellow students. (The last class that I enjoyed was organic chemistry. Everything went downhill after that when all classes became purely “biology.”) That was when I discovered Schrödinger’s What is Life? and started to discuss with physics professors and friends about quantum mechanics (and not surprisingly, Schrödinger’s cat). Once I bit the apple, there was no turning back.

I really want to buy many copies of this book and send it to my cousins and friends, if only they are fluent in English, at least only to show the unity of science. It is such a sad state that most Thai students I have met never imagine that mathematics can be used in biology, that one can bring physics and biology together not in an ad hoc way, or that abstractly biology is about information. (It’s still a mystery to me why in high school the naive scientific methods–make a hypothesis, do experiment, analyze data, conclude–is exclusively belong to biology courses. I think it encourages the attitude “why think when we can do more experiments?”)

I recommend Weinberg volume 1 for a very logical presentation of the material (be warned it is harder than most qft texts precisely b/c of this fact). If that still does not satisfy you, well seek out Streeter-Wightman, but do keep in mind that material is sort of a siren call and merely a first pedestrian step into a tunnel that is quite dark.

…even though they are rigorous formulations of QFTs, maybe there is a better mathematical language that expresses QFT rigorously and concisely, rather than rigorously with nine billion pages of measure theory as is currently the case.

From the thread “Interacting theory lives in a different Hilbert space,”

PCT, Spin and Statistics and all that by Streater and Wightman, it’s basically a good summary of the general properties of QFTs and also describes what exactly fields are mathematically.

The books that I have my eye on right now are the following:

• Srednicki, Mark A. Quantum Field Theory.
• Maggiore, Michele. A Modern Introduction to Quantum Field Theory.
• Weinberg, Steven. The Quantum Theory of Fields, Volume 1: Foundations.
• — “What is Quantum Field Theory, and What Did We Think It Is?” arXiv:9702027
• Streater, Raymond F. and Arthur S. Wightman. PCT, Spin and Statistics, and All That.

Given the hype, it is not so surprising that it caught the eye of someone that I know outside of the physics community as well. The opportunity to share my enthusiasm about quantum theory with them was the main motivation for this blog post, but soon I realize that this paper is not so interesting if you do not know about the Bohr-Einstein debate, the EPR paper, and the Bell’s theorem before. So I suggest a truly interested one to read those up as well. Personally, my main resources are Quantiki, Scholarpedia, Stanford Encyclopedia of Philosophy, and Asher Peres’ Quantum theory: concepts and methods.

What do they mean by statistical interpretation?

One reason that makes quantum theory difficult to grasp intuitively is its probabilistic aspect. Formal prediction of experimental result (by means of so-called “observables”) are probabilistic (although the probability can be zero or one). Other than the quantity we choose to observe, the probability is also determined by a quantum state (or “wave function” if you prefer) that the system is in via Born’s rule. Max Born was given a Nobel prize in physics precisely because of “his statistical interpretation of the wavefunction.” These are all accepted by every working physicist, and this is not what PBR refuted. Some might think of Ballentine’s ensemble interpretation, but that is not the target of PBR too. The target of PBR is the view that there is some underlying reality and quantum states only reflects our own knowledge about a given physical system; the point of view that was prominently advocated by Einstein as quoted in the paper itself but quoted here again for your convenience

“… I incline to the opinion that the wave function does not (completely) describe what is real, but only a (to us) empirically accessible maximal knowledge regarding that which really exists [...] This is what I mean when I advance the view that quantum mechanics gives an incomplete description of the real state of affairs.”  Albert Einstein

Why state of knowledge?

One motivation is that if you are already a classical Bayesian and think that probability is only in the mind (this is not problematic, for example, in classical statistical mechanics), then there is a chance that quantum probability is no different. But this depends on the interpretation of probabilities, which is another story.

A stronger motivation comes from the analogy between the “wave function collapse” and information updating in probability theory. In orthodox (“Copenhagen”) QM, collapse supposes to happen instantaneously and you should start to smell a rat. If two observers Alice and Bob far apart each make a measurement on an EPR pair, in one frame of reference, Alice will collapse a state before Bob does, and in another frame of reference, Bob will collapse a state before Alice does. If collapse is a physical process (implied by the reality of the quantum state), then causality is apparently violated. This problem is dissolved if the quantum state, even a pure state, merely represents an observer’s state of knowledge. That is, we do not have to presume any local cause in this scenario.

Nevertheless, it seems like what we do when we take the epistemic interpretation seriously is just pushing the mystery further. True; Gleason’s theorem gives Born’s rule, but why should we be in this complex Hilbert space in the first place? This is similar to a trivial and natural proof of a mathematical theorem making use of complicated and seemingly ad hoc definitions. However, the epistemic approach has been a motivation for proving important results that shed some light on this issue like the quantum de Finetti theorem by Caves et al, which requires complex numbers in QM.

The ultimate question still remains untouched: “states of knowledge about what?” This leads us to what is traditionally known as hidden variables.

A classification of epistemic views of quantum state

The definitive reference here is Harrigan and Spekkens (HS). In fact, everyone who wants to understand what the PBR result is all about should read it because it provides the context of the definitions that PBR use.

The idea is to take Bell’s approach to hidden variable theories that he used to prove his celebrated inequality. Suppose that there is a set of variables collectively denoted that we can imagine live in what we might call the ontic state space which does not have to be the same as Hilbert space. determines the probability of every measurement outcome that one can make via a set of positive functions , where labels a measurement and labels an outcome. (To get a contextual theory, we demand that depends on other observables that are simultaneously measured as well.)¹ If measurement outcomes are discrete, then for fixed and

for all , since probabilities must add up to one. This simple requirement is the ingredient used in the final step of the PBR proof.

Let a projector associated with outcome of an observable . The probability of a getting outcome upon measuring in state is by quantum theory. But now with the hidden variables

where is the distribution of given a state . A precise classification of hidden variable theories then comes from looking at the map between the ontic state space and quantum state space and measurements in that theory.

Akin to the value-laden word “reality,” whether something is ontic or epistemic is, up to some degree, a matter of convention. So we define an ontic theory and an epistemic theory precisely in order to prove theorems about them. There is and always be a debate going on whether these definitions are justified, but this does not diminish the value of the PBR result. As with everything in life, we frame our concepts to reason about them. We do not fabricate a rule of reason first and then arbitrarily changing the meaning of its elements. And I think that the HS definition is useful enough to be an effective definition.

• In a -complete theory, we want to be able to say that quantum states determine everything that is to be said. Mathematically, this demands the ontic state space to be isomorphic to the standard Hilbert space of quantum theory, and every ontic state is realizable by means of some preparation method.

• If a theory is not -complete, it is said to be -incomplete. That theory may happen to be a -augmented theory like the de Broglie-Bohm theory.

• An ontic theory is -ontic if is reflected in quantum states unambiguously i.e. when we compare any two ‘s, their set of are disjoint.

• If a theory is not -ontic, it is said to be -epistemic.

You can see that these definitions are natural within this framework by trying to come up with definition of -complete, -incomplete, -ontic, and -epistemic theory on your own without looking at the definitions by HS.

By their definitions, a -complete model is ontic. By logic then a -epistemic model is incomplete, so a set of -incomplete model is larger than a set of -epistemic model, for example.

The PBR result

According to PBR, “If the quantum state is statistical in nature (the second view), then a full specification of need not determine the quantum state uniquely.”

What PBR calls a statistical interpretation of quantum state is precisely our -epistemic model, and PBR shows that such a model is impossible. To prove that, it is sufficient to reach a contradiction by assuming that for any pair of , their sets of overlap. PBR proves this by generalizing from a special case where are and of a two-level system that I shall call a qubit.

Suppose that we have two qubits labeled 1 and 2. Assume that the distributions and overlap. If we specify that lies in the support of both distributions of qubit 1, then the qubit is either in the state or . Do the same with and qubit 2. The joint state of both qubits is then one of the four possible states:

Now, we want to show that such does not exist. PBR gives a four-outcome joint measurement such that for each and every outcome, the post-measurement state is orthogonal to one of these four possible pre-measurement state.

It is easy to see that forms an orthonormal basis and that for all since is also an orthonormal basis.

But is specified by alone and not the quantum state, so . (A crucial assumption is that the distribution of is “well-behaved under a tensor product” e.g. the distribution of the product state is simply , the product of the distributions.) Then it is immediate that , which is a contradiction.

Does this come as a surprise? Bell’s theorem and Bell-Kochen-Specker theorem was motivated by the question of whether the peculiar features of the de Broglie-Bohm theory namely nonlocality are necessary or not, and both theorem apply to the class of ontic models we have talked about. So we already know since 1966 that -epistemic ontic models within the Bell framework have to be nonlocal and contextual. But PBR go further and rule out all such -epistemic ontic models. So we may think that this actually shakes the foundations of quantum physics. It turns out, however, that this approach is not seriously taken by the foundations community anymore according to Matt Leifer. Nevertheless, an explicit proof is always a nice thing to have.

Implications of the PBR result

“…what is proved by impossibility proofs is lack of imagination.” John Stewart Bell.

Einstein was known to emphasize the incompleteness of QM by this dilemma:

“[T]he paradox forces us to relinquish one of the following two assertions:

(1) the description by means of the -function is complete

(2) the real states of spatially separate objects are independent of each other

Even if we are convinced that the first option is right, he did not tell us how to complete it. The ontic models discussed above is such an attempt. So the dilemma posed by QM is usually phrased as a conflict between realism and locality since if quantum states are real, the locality condition makes no sense because quantum states are not even separable in space.

For those who wants to hold on to locality (and believe that a measurement has an outcome), the old solution still works: thinking of quantum states as epistemic and being agnostic of anything beyond that. A description (quantum state) may be nonlocal but information is localized. It is just that now if we are excused of our lack of imagination and confine ourselves only to the Bell’s framework, then the “underlying reality” is not just nonlocal, it does not even exist.

1.  Spekkens, R. W. (2005). “Contextuality for preparations, transformations, and unsharp measurements.” Phys. Rev. A 71, 052108.  arXiv:0406166

A proof of a very important theorem in quantum theory that two self-adjoint operators can be simultaneously diagonalized if and only if they commute is presented, with the hope that it is less obscured than the standard proof in physics textbooks that separates the nondegenerate and the degenerate cases. This is a continuation from the last post.

Discussion

First of all, let limit ourselves to discussions about finite-dimensional vector spaces.

We talked about adjoint operators last time. Now, it is easy to define a self-adjoint operator. It is an operator such that

They play a very important role in quantum theory, partly because they have real eigenvalues so that they are “observable.” However, I want to point out another very important property of self-adjoint operators; they admit the spectral decomposition. This simply means that they are “diagonalizable” (as a matrix) and that the eigenvectors form a complete set of orthogonal basis (which is usually assumed in a physics textbook). The matrix

has real eigenvalues. Nevertheless, it is not diagonalizable, since if it were diagonalizable, then its diagonal form would be the identity matrix because its eigenvalue is .

Before thinking about the spectral theorem for self-adjoint operators, let me quickly “remind” you (meaning that if you did’t know this before you shouldn’t expect to understand it) how one usually arrives at the spectral theorem in general. Recall that associated with every linear map is the characteristic polynomial . The Cayley-Hamilton theorem states that a matrix of the linear map itself is also a root of . (Of course, just pluggin in in place of to get does not make any sense! We need to think of as a polynomial of the ring of matrices instead.) Then is in the kernel of the evaluation map . By a standard result in ring theory, the kernel is an ideal of and every ideal of a polynomial ring is generated by only one element. We call this element the minimal polynomial of . The spectral theorem then guarantees that

  is a product of distinct linear factors is diagonalizable.

Usually this is the most direct way to argue that some operator is diagonalizable (instead of working with matrices). This theorem is one of the high points of any first abstract linear algebra course where you get to play around with structures of -invariant subspaces.

Theorem. A self-adjoint operator in a finite-dimensional vector space over with the standard inner product is diagonalizable in an orthonormal basis.

Proof. By the fundamental theorem of algebra, has at least one eigenvector. (The fact that splits into linear factors does not imply that is diagonalizable!) Choose a vector such that , and (by normalization).  Let . One can prove that by, say, the rank theorem. Since the standard inner product is positive definite, does not contain itself. Thus

Now, for the induction step, use the important fact that is -invariant. Indeed, if , then . Therefore, is the standard inner product, and is self-adjoint.

By induction, is diagonalizable in an orthonormal basis . Take to be the basis for . It is orthonormal and is diagonal in this basis by construction.

Here comes a series of interesting exercises.

Let be a finite-dimensional vector space.

1. Given , show that and have the same eigenvalues. (Hint. consider the case and separately. The case does not work in an infinite-dimensional space.)

2. Let and suppose that . Prove that and have a common eigenvector.

Solution. Assume . Then . Does this help? Yes. It means that is -invariant. Then because we are in has at least one eigenvector in , which is, of course, an eigenvector of too.

3. Let has an additional standard inner product structure. If and are self-adjoint, prove that there exists a basis in which and are simultaneously diagonal.

In the context of quantum mechanics, we say that  and are compatible observables; they can be measured jointly. Note that a physics textbook like Sakurai proves the cases of nondegenerate and degenerate eigenvalues separately, which is unnecessary here by virtue of the result of the previous exercise.

4. Show that any real matrix can be written as the sum a symmetric matrix and an anti-symmetry matrix.

Analogously, Show that any complex matrix can be written as where are self-adjoint.

5. A normal operator is an operator that commutes with its own adjoint; i.e., . Use the results from previous exercises to show that

is normal is diagonalizable.

Takeaway

Do the exercises.

The existence of the standard basis in a vector space is so natural that when we learn linear algebra for the first time we are unaware that we have chosen a basis whenever we write down a column vector. Likewise, physics students often learn the definition of self-adjoint matrices without the emphasis that whether a matrix is self-adjoint or not depends on the inner product we are using. The purpose of this post is to clarify what an adjoint operator is, and how we can think of it in a higher level of abstraction.

Discussion

Self-adjoint (“Hermitian”) operators play a very important role in quantum theory, partly because they have real eigenvalues so that they are observables. However, I want to point out the most important property of self-adjoint operators; they admit the spectral decomposition. This simply means that they are diagonalizable and that the eigenvectors form a complete set of orthogonal basis (which is usually assumed in a physics textbook). The matrix

has real eigenvalues. Nevertheless, it is not diagonalizable, since if it were diagonalizable, then its diagonal form would be the identity matrix because its eigenvalue is . (It is already in the Jordan normal form.)

To learn what self-adjoint operators are, we need to learn a bit about dual spaces first.

For a warm up, let be a set and the collection of all maps (functionals) from to . For , define and . Then becomes a vector space. Suppose there is another set with the vector space of functionals on and a map . Then naturally gives rise to a map so that the diagram below commutes. Note that there does not seem to be a natural mapping “in the same direction” as . If you want to use to create a linear functional on from , you need to compose with which may or may not exist.

Now, let the dual space   (resp. be a space of all linear functionals on the vector space (resp. ). Then

and we define a map by the formula

We end up constructing a map between dual spaces with what we have at hand.

In other words, given and both the categories of (say, real) vector spaces. There is a functor betweens them. For , let be . Then we need to construct a rule for functorial morphism , which in this case, makes contravariant¹ since for a diagram in category , . This reverse of order is the familiar property of the adjoint of a linear map.

You might recall that and its dual have the same dimension. Thus they are isomorphic as abstract vector spaces. However, there is no natural (basis-independent) way to identify one with another. Normally, one has to pick a basis in to construct the canonical dual basis in , but this is completely arbitrary.

Anyway, an inner product , a positive-definite bilinear form, defines a linear map from by where the dot denotes the slot for a vector in . The Riesz representation theorem (there are many verstions of them), which is easy to prove in finite dimensions, states that this maps is an isomorphism. That is, every linear functional on can be represented as an inner product of a fixed vector with another vector. Although in infinite dimensions, the map is merely injective; can only represent a continuous linear functional.²

Equipped with inner products with the matrix of bilinear form on and with the matrix of bilinear form on , we can define a map  which makes the diagram commutes.

Both sides are linear functionals on .

by the definition of .

Somehow, we have represented a map in the dual spaces as what is called an adjoint map  in the same spaces as the original (but mapping in the opposite direction). People usually denote this map by since, after all, with the inner products, the vector spaces and their duals are the same.

For (the standard inner product), what is the matrix of this with respect to the matrix of  ?

Suppose that we are on and the vector space is finite dimensional. On the one hand, by the commutative diagram above,

( are column vectors.) On the other hand,

Therefore, the matrix of  is the complex conjugate of the transpose matrix of .

as we all remember. (If , then .)

In Dirac notation, the action of an operator to the left on the bra is defined via the action to the right on the ket. (The Wikipedia entry is good on the bra-ket manipulation.)

It actually means that is a new functional . All of the above constructions are implicitly assumed when we write

Takeaway

1. Dual spaces exist naturally. An inner product identifies each continuous linear functional with each vector of the vector space.

2. A linear map between vector spaces induces a linear map between their duals in the opposite direction. Only with the specification of an inner product then one can define an adjoint of the linear map.

3. Diagrams help organizing seemingly arbitrary formulas and contructions into something more meaningful.

1. Covariant and contravariant tensors are physicists’ terms which have nothing to do with category theory.

2. For a proof, see theorem 11.4 on page 43 of Reed and Simon volume 1.