## Thursday, December 15, 2016 ... /////

### I approve the message of Aaronson's QM comic

Scott Aaronson is a nutcase when it comes to politically loaded issues. He wanted (and begged a psychiatrist) to chemically castrate himself because he was persuaded that males without castration are weapons of mass destruction (Lily Rebecca Aaronson, you've been very lucky).

After Trump won, he joined a jihadist resistance movement – and his political state of the mind resembles that of Keith Olberman who is giving speeches like Hitler after a meltdown from the frequently recycled and parodied film.

But I think that among the real-world people who are being marketed as experts on foundations of quantum mechanics, he belongs to the 20% sanest folks. (The figure 20% and other positive statements about Aaronson here are meant to be neither excessive compliments nor "damning with faint praise", as TRF+Aaronson reader Zach Cox suggests, but as accurate appraisals as I can find.) This opinion of mine was just strengthened by

Scott Aaronson's and Zach Weinersmith's comic "The Talk" (huge PNG)
at SMBC-comics.com, announced at Aaronson's blog. In the comic, a mother – a smart lady who is no longer a MILF, sadly – is terrified when she discovers that her son is reading some pop-science nonsense about quantum mechanics.

She explains to her son – who no longer wants to be considered a child but he clearly is one – that the usual statements about quantum mechanics, qubits, the clever trick of quantum computers, and the availability of quantum computers that are often described in the pop-science journals aren't really right.

There is one unrealistic aspect of the comic. In the real world, young boys and men are almost never taught quantum mechanics and important facts about it by their mothers. But if we ignore this piece of the feminist fantasy – feminism was guaranteed to affect Aaronson's cartoon in one way or another – I must say that I agree with everything that the comic claims.

First, the son is told that it isn't true that quantum bits are 0 and 1 "at the same time". Instead, one of the options may be measured but the way how they're combined is a "new type of ontology", a complex generalization of the probability calculus. I think that I've used almost the same words many times in the past. The wave functions are closer to probabilities but they're not quite the usual probabilities. Instead, they're probability amplitudes which are complex and also have the ability to constructively or destructively interfere. When one is observing anything, the amplitudes are converted to the usual probabilities only. But when no one is looking, the probability amplitudes evolve as a new entity according to new rules that have no counterparts in classical physics.

Another myth – pumped into her son – that the mother debunks is the idea that the quantum computer is fast because it's nothing else than a classical computer trying all the possible answers in parallel. As I explained e.g. through the mouth of a fake PM Trudeau, this ain't the case. There's no "splitting of the worlds" during a quantum computation. On the contrary, the splitting of the worlds may only make sense after a measurement which can only occur after decoherence – but the quantum computation depends on the absence of any decoherence (I will make the same observation again later). So a key necessary condition for the quantum computer to work – and do some things that are practically impossible on classical computers – is that there's no decoherence and no splitting of the world during the calculation.

The son is also brainwashed by the idea that his classmates are testing quantum interference with frogs (and probably also oil droplets). Well, they aren't, the mother points out. The quantum probability amplitudes are something totally different than any feature of these macroscopic classical objects and experiments we've been familiar with for centuries. When the mother tries to summarize the wisdom, she says:
Relax. It's all just different consequences of one fact: classical events have probabilities, and quantum events have amplitudes. Remember that, and you'll do just fine.
This looks utterly reasonable to me, too. To understand quantum mechanics, you really need to understand that it's some generalization of the probabilistic thinking that's been around for a long time – and it's a generalization in which you have to work not just with the probabilities but also with some semi-baked product (in English, the approximate equivalent is an "intermediate good" but it doesn't quite capture the more-food-industry-focused "polotovar" in Czech), the probability amplitudes, which are complex are have to be manipulated in ways that are analogous (and give rise) to the usual probabilistic calculus but don't quite coincide with any pre-1925 calculation of probabilities.

The last frames of the comic also mock the popular articles (and there exist very similar ones in the real world) that iPhone 8 will be a quantum computer, someone will have a gigaqubit quantum computer soon, and that quantum computing is equivalent to consciousness because "both are weird". I would really endorse every frame of this comic – including the spirit and accents. Weinersmith is a good cartoonist but I guess that "most of the work" according to my counting was done by Aaronson – after all, the mother and her son are just talking to each other all the time.

I find it irresistible to react to some comments by Aaronson's readers:
Hwold quotes: "It’s not the size that matters. It’s the rotation through complex vector space."

Other ideas in this comic are familiar to me, but not this one. Any reference explaining this?
Well, that's unfortunate that readers of heavily quantum mechanical blogs have never heard about the rotations through a complex vector space. All transformations – including the evolution in time – are represented by unitary (linear) transformations of the Hilbert space. That's also why Paul Dirac basically liked to use the term "transformation theory" for the bulk of the mathematical apparatus of quantum mechanics.

Imagine how many times some wrong claims about quantum mechanics are repeated in the popular media. But many people who often read this stuff can never hear about complex rotations of the Hilbert space.

OK, I said it. Every transformation of a physical object (rotation, time evolution etc.) in classical physics may be represented by some rearrangement (permutation) of the points in the classical phase space. In quantum mechanics, the same kind of an operation – a transformation of the physical system (or the whole Universe) – is always represented by a unitary operator/matrix $U$ mapping the Hilbert space onto itself and obeying $U^\dagger = U^{-1}$. The complex coordinates of the vectors in the Hilbert space are the probability amplitudes and they're just being rotated by unitary (=complexified orthogonal, with the dagger) transformations.

So all the possible laws of evolution in time are quantum mechanically represented by different unitary "rotations" of the same infinite-dimensional space and by different ways to choose coordinate systems – labeled by various things that can be measured – on that infinite-dimensional space.
Jacob Aron: As a journalist infecting young minds with filthy quantum articles in magazines, this got a laugh out of me. But the fact that the comic is so long means it can’t quite sustain the joke, which handily illustrates the point of coming up with these incorrect approximations – sometimes you just don’t have room to be right!
Too bad that the comic got a laugh of a filthy journalist like you, Jacob. A more rigorous reaction would be crying through your missing teeth because for your bastardization of the youth, you deserve a proper thrashing.

The cartoon may be long – because they wanted to include lots of useful but essential information. But the full length isn't necessary. Individual frames of the comic could be used as valuable sources of wisdom separately from others. An average frame in this comic is more valuable than an average filthy article that the likes of Jacob Aron are printing in pop-science journals these days.
Jon K.: Very funny and elightening cartoon. I hope it makes its rounds in the popular presses.

But I wonder what would have happened if the kid started asking other questions like, “But I heard complex numbers could just be represented by two ordinary numbers?” or “What do you mean by ‘isolated’?” …Or something else that might have given the mom a little more pause(I’m not sure if those questions would actually do that, as this mom seems pretty smart and quick to respond with enlightening answers).

Scott, what are the hard questions that this kid could have asked his Mom where she would not have been able to give him an answer that he’d be satisfied with?
Scott says that the mother could answer all these questions – the cartoon would get even longer, however. In particular, by an "isolated system", they (and we) mean that the description of that system is in terms of a factor in the tensor product of Hilbert spaces.
Job: [Quantum computers are hard because the calculation is different than the classical computation. ...] In the quantum world it’s difficult to break things down into steps. [...]
It's simply not true. A quantum computation – e.g. one following Shor's algorithm or any similar algorithm people normally talk about – is composed of individual steps just like a classical calculation. It's just the character of the individual steps that is different for classical and quantum computers. For quantum computers, the single step is a unitary transformation discussed previously. But the collection of possible operations envisioned by realistic quantum computers is equally finite – and similarly large – as the collection of operations that a classical computer may do in one step. In fact, close cousins of the classical operations are usually operations done by quantum computers, too. Quantum computers have some additional operations that mix the qubits in quantum logical ways that have no classical counterparts.

Another difference is that we must be careful not to make any measurement during the calculation. Only at the very end, we do a measurement using a quantum computer. This is needed to allow the probability amplitudes to evolve in their new, characteristically quantum way – which is needed for the relative superiority of the quantum computer.

Scott later says the same and correctly points out that the only "new" thing about the steps in a quantum computer is that they need a harder mathematics to be understood – but any mathematics may be hard.

There exist all conceivable confusions about these things. Job says that quantum mechanics doesn't allow "steps". It does allow them. Computation – both classical and quantum computation – envisions computers, objects whose evolution may be pretty much divided to steps (i.e. for which the time is effectively discrete). More general objects in Nature – whether they are described by classical or quantum mechanics – don't allow such steps i.e. quantization of time.

Despite deluded cranks' suggestions that the time has to be discrete in quantum mechanics (or quantum gravity), no implication like that holds. Time is normally continuous, is effectively discrete when we build computers, but the discreteness of time is completely independent from the quantumness of the laws of physics.
Jahan: Scott: Is tunneling, Heisenberg uncertainty, and action at a distance really consequences of complex amplitudes? I would think to predict tunneling you’d need to know Schrodinger’s equation, to understand the uncertainty principle you need to know $[x,p]$, and to get action at a distance you need entanglement. Can’t all those things exist independently of complex amplitudes?
The laymen are mostly hating the foundations of quantum mechanics – and they seem to hate a part of mathematics, complex numbers, equally fanatically. They hope that they're not needed. Someone will ban them etc. But complex numbers are fundamental in physics, especially in quantum mechanics.

Yes, complex numbers are absolutely needed for Schrödinger's equation, the uncertainty principle, and a meaningful $[x,p]$, too. There's no "action at a distance" so let me not discuss Jahan's confusion about the entanglement – I've discussed it many times in the past.

Schrödinger's equation says$i\hbar \frac{\partial}{\partial t} \ket{\psi(t)} = H \ket{\psi(t)}.$ The time derivative of the state vector or wave function $\psi(t)$ is obtained by the action of the Hermitian Hamiltonian operator $H$ on the same state vector – but divided by the purely imaginary constant $i\hbar$. Because this constant is imaginary – i.e. complex, not real – an initial wave function, even if it is real at one moment, is pretty much guaranteed to be complex at the following moment: the time derivative contains some purely imaginary pieces.

Does the coefficient in the Schrödinger equation have to be imaginary? You bet. For an initial state that is an energy eigenstate i.e. obeys$H \ket\psi = E \ket\psi,$ the solution to any Schrödinger-like equation with any coefficient $C$ on the left hand side unavoidably has the form$\ket{\psi(t)} = \exp(Et / C) \ket\psi$ You simply need $C$ to be pure imaginary for the total probability $|\psi|^2$ not to exponentially grow or decrease in time. Exactly when $C$ is pure imaginary, the transformations you get for any time evolution will be unitary, and therefore preserving the total probability.

So complex numbers are absolutely needed for Schrödinger's equation to work. By using the usual proof of the equivalence of the Schrödinger and Heisenberg picture, you may also prove that the same $i$ is absolutely needed in the Heisenberg equations of motion. And the proofs of the equivalence of Heisenberg-or-Schrödinger's pictures with the Feynman integrals similarly tell you that you need an $i$ in the integrand $\exp(iS/\hbar)$ of the path integral, too.

The commutator $[x,p]=xp-px$ that is nonzero and mathematically underlies the uncertainty principle absolutely requires complex numbers, too. The reason is simple. The operators $x$ and $p$ have to be Hermitian for their (measurable) eigenvalues to be real. So we have $x=x^\dagger$ and $p=p^\dagger$. But that's enough to see that$(xp-px)^\dagger = p^\dagger x^\dagger - x^\dagger p^\dagger = px-xp = -(xp-px).$ The Hermitian conjugate of $[x,p]$ is minus itself. We got the minus sign because the two terms in the difference got permuted and this permutation has arisen because $(AB)^\dagger = B^\dagger A^\dagger$ i.e. because (just like the transposition of matrices) the Hermitian conjugation forces you to read the factors from the right to left.

In other words, $[x,p]$ is unavoidably anti-Hermitian i.e. $i$ times a Hermitian operator. The commutator $[x,p]$ is $i\hbar$ i.e. $i$ times a Hermitian operator proportional to the identity matrix. So if you write $x$ as a real matrix, then you are guaranteed that the operator $p$ must contain some complex (often pure imaginary) entries and vice versa. If both $x,p$ were real matrices, their commutator would be a real matrix as well but $i$ times a nonzero real $c$-number can't be a real matrix because of the damn $i$.

One can list several other simple arguments like that which prove that virtually nothing could work in quantum mechanics if you demanded that all coefficients in the equations are real. Complex numbers are absolutely paramount in quantum mechanics. You may childishly imagine that a complex number is a pair of two real numbers – which is a wrong way to think about complex numbers because a single complex number is really "more elementary or fundamental" than a real one (good calculus and representation theory experts in mathematics would surely agree with this physicists' view) – but in that case, the proofs above show that the pairing is absolutely unavoidable whenever you deal with the probability amplitudes. The beef of the complexity of amplitudes us unavoidable.

We may also ask whether the complexity of the amplitudes implies the principles of quantum mechanics such as the uncertainty principle (the opposite implication than discussed above, basically). Well, strictly speaking, No. But if you morally want "something new and useful or interesting to be done with the complex numbers", then yes, quantum mechanics with all the principles is the only new interesting set of ideas that uses makes the complexity helpful for anything. Probabilities can't be complex so they should better be calculated as the squared absolute values of the complex amplitudes. If that's so, the complex numbers – if they are physically present at all – should better be placed as off-diagonal elements of some matrices, starting from the density matrix (the diagonal elements are the real probabilities themselves) and the whole representation of observables as matrices or operators morally follows.
ppnl: I liked the last frame in the cartoon where it takes a swipe at the connection between quantum mechanics and consciousness.

Lubos Motl seems – as best as I can figure out – to be saying that a conscious observer is needed for quantum wave collapse. I tried to point out that there can be no difference between a mindless robot observing a quantum particle and a conscious scientist observing it. There can be no experiment that differentiates between the two. [...]
If this is the best way how you're capable of reading the last frame of the comic, then you're sadly a mental cripple, Mr ppnl. The frame says nothing of the sort. Instead, the frame follows frames saying that "if you don't talk to your child about QM, someone else will" and mocks a pop-science journal that says:
Quantum computing and consciousness are both weird and therefore equivalent.
This statement has absolutely nothing to do with my obviously correct statements about consciousness that are too hard for Mr ppnl. The mocked journal title is a variation of what Roger Penrose likes to say (a collapse in the brain is what allows the brain to calculate, and the brain is therefore working as a quantum computer, and that's why consciousness and quantum computation are equivalent).

So the text in the mocked journal is pretty much equivalent to the kind of silliness that Roger Penrose has been saying for decades – a point observed by another reader of Aaronson's blog, Fazal Majid. (Penrose wasn't the only person who made vaguely similar, silly identifications.) Moreover, it's framed in a way that I know from Lee Smolin. Lee Smolin has once invited himself to Harvard (by painting himself as a victim of a sort, abusing myself, and forcing me to ask our secretary to invite him) and he taught us:
I can't believe that M-theory is hard. Three-dimensional Chern-Simons theory is also simple and therefore three-dimensional Chern-Simons theory and M-theory must be equivalent.
I laughed for quite some time after I heard it – and we did in the Society of Fellows and elsewhere. By that moment, I had known for some 5 years that Smolin was a crackpot but only at that time, I began to appreciate how amazingly stupid he was.

When two things share an adjective, it's very far from a convincing argument (let alone a proof) that they're equivalent. To make things worse, the attribution of the adjective was controversial for one object and almost certainly wrong for the other. So Smolin has applied a logical fallacy in a way that was mostly wrong by itself.

Concerning this particular equivalence, there is really no consciousness during a quantum computation. I have already explained the reason in this very essay and many others. An observer is only conscious about an outcome of an observation after he has made an observation but that requires some decoherence to take place in his mind before that. But a quantum computation depends on the absence of any decoherence, so there's simply no consciousness "taking place" during the quantum calculation.

The mocked statement isn't just proving something that can't be properly proven. It's claiming something that may be disproven. It's the opposite of the truth.

My claims about "robots and humans" that ppnl tries to disagree with are correct, of course. Quantum mechanics doesn't tell you to treat humans and robots differently – after all, the definition of a "human" and a "robot" is a very complex and mostly ill-defined task. So whatever holds for biological humans may hold for machines and vice versa. So far so good, ppnl would agree.

But quantum mechanics does treat and has to treat observers differently than the observed objects. So whether someone or something is a human or a machine, it's a physical system that evolves to complex superpositions of states up to the very moment when it's observed by an external agent, an observer. On the other hand, from the viewpoint of an observer (whether he's biological or a machine or whatever), the observation – the act of changing his knowledge about the world (or objects in it) – is always accompanied by (or inseparable from) the collapse of the wave function as well.

In general, the description of objects (including humans and robots) by quantum mechanics depends on who is describing them, from whose observational viewpoint the description takes place. That agent, the observer, plays a special role in the description. In rather generic situations (often caricatured as the Wigner's friend thought experiment), two observers may use very different wave functions in the same situation. A key point is that the collapse of the wave function is always a subjective event. Heisenberg considered this novelty of quantum mechanics – the dependence of the description on the choice of the observer and, in this sense, "subjectivity" – "sort of obvious", a generalization of the positivist lessons about the relativity (inertial frame-dependence) of some quantities introduced by Einstein's special theory of relativity.

The need to describe the evolution relatively to an observer who (subjectively) knows – independently from any theory – what is an observation and what isn't is absolutely essential for quantum mechanics. Quantum mechanics simply can't be applied without observers (or something equivalent, something that pre-knows what are the relevant questions that are being asked, observables that are being measured). Everyone who talks about quantum mechanics without observers is clueless: it's an oxymoron.

If ppnl were posting this kind of untrue crap on my blog, he would be banned rather soon. Well, I am not just bragging about these credentials.
ppnl: To be fair I’m not really sure what his position is since he banned me for disagreeing without discussing it.
Right, ppnl, you're a piece of lying šit (which is more relevant for your status than some disagreement) and I am afraid that I would immediately make the same conclusion after several seconds even if you tried to obscure your identity by wearing 1,000 condoms on your f*cking stupid head.