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When a layman has no chance to comprehend the bit-qubit difference

I am often dreaming about being able to extract 1/2 of my brain and donate parts to others. Why?



Because, as Sheldon Cooper has observed, being stupid isn't a reason to cry. Being sad is a reason to cry. For example, I am sad because other people are so stupid!

The Internet events that maximally influenced this beautiful sunny Sunday morning were comments by the Kansas-based user AP under the June 2016 blog post Leaning of information, not an interaction, is what causes the collapse. As you may recall, and as you can see by thinking about the title, the main point of the blog post was to say that ordinary small quantum objects' evolution – including interactions that make them entangled – doesn't cause any collapse, any irreversible change of the wave function, anything that we associate with the observations.




This crisp and simple point was clearly – and loudly – made in that blog post, in numerous comments beneath the blog post, in hundreds of other TRF blog posts, in textbooks and talks on quantum mechanics that were repeatedly referred to and embedded etc. But it's just totally impossible for certain people to get this point. It seems that they're not even capable of understanding or remembering a rudimentary, unambiguous sentence with several words, such as "One qubit is not an observer".

The discussions with certain people who just aren't getting absolutely anything is much more frustrating than Sheldon's efforts to teach physics to Penny. One is just getting absolutely nowhere. After these people have demonstrably heard the same simple statement (e.g. that \(1+1=2\)) 50 times in various ways, with various proofs, they still keep on repeating things like "surely \(1+1=3\), right?" And after you repeat the trivial information 60 times, they start to infinitesimally notice and tell you things like "Surely aren't you suggesting that \(1+1=2\), are you?" or "Don't tell me it's your idiosyncratic personal belief that \(1+1=2\)".

Holy cow, I haven't been "suggesting" anything and I haven't talked about any "beliefs". I've been making statements about well-known principles of physics in a completely unambiguous way, very loudly, using the language that even dogs should be able to understand. Why would someone be so arrogant to dismiss these elegant, deep, and important insights – and my unbelievable patience and self-sacrifice with which I am trying to explain these things – as "beliefs" or "suggestions"? It's just so disgusting.

It was obvious that I had to ban AP because he quickly became such an incredible sink of time. Talking to people like that is effectively equivalent to talking to a wall, with one difference: Unlike most walls, these people repeatedly make noise that sounds like a human language and fools you into believing that they have a brain inside – and that it makes sense to talk to them. But it's a trap. They don't have any brain.




OK, so AP has repeated about 6 times that observers are surely not needed, physics can't depend on observers, and so on. Sorry but the main conceptual advance of physics that took place in the mid 1920s – and that can never be undone – is the discovery that the role of observers can't be minimized arbitrarily. Observers always matter. It matters whether something is observed or not. An observation always affects the observed system. And only when an actual observation – a process in which some observer learns some information about the physical systems surrounding them – takes place, an irreversible collapse of the wave function takes place.

The previous sentence may sound counter-intuitive to a person trained within classical physics but it's basically a trivial tautology. The wave function is a template for all probability distributions and we know from Bayesian reasoning that probabilities are "collapsing" – or irreversibly and discontinuously changing – when and only when a person has learned the information about some new evidence. The collapse of the wave function plays exactly the same role as the learning of new evidence in Bayesian reasoning. The wave function subjectively collapses because the observer has learned some new information.

One may find it counter-intuitive for a while but soon afterwards, he should get it, realize that this new foundation of physics is absolutely consistent, beautiful, elegant, agrees with everything we know about physics, and it is irrevocable: There's just no way in which the successes of quantum mechanics could be reproduced by a theory in which observers don't matter. The theories in which observers don't matter have simply been falsified over 90 years ago and falsification is irreversible.

Even if I imagine that AP would be capable of understanding these elementary insights of modern physics after 30 more iterations, I just wouldn't be happy about the result because I feel that AP wouldn't be happy about learning important and elegant facts about Nature. Everything indicates that he – and lots of others – is basically equivalent to a retarded 12-year-old kid who still believes that gifts are brought by Santa Claus and who would immensely suffer if he learned that Santa Claus didn't exist. Does it make sense to try to change the "world view" of such people?

Defining features of qubits

So this AP would repeat about 10 times, in pretty much equivalent words, that if you have two qubits, one of them always automatically measures the other just because they get entangled. A qubit is enough as an observer, he repeated about 10 times. There surely isn't need for any "bigger" observer. And two qubits evolve irreversibly because they measure each other.

Holy cow. It's just unbelievable how reliably incorrect statements these people are capable of making. A machine that randomly combines nouns and verbs from the quantum mechanical jargon would produce valid propositions much more frequently than they do – by chance. There is something inside these people that makes them reliably wrong, that guarantees that every sentence they make is wrong – pretty much the negation of an important and true statement.

Even if you read a few sentences in any definition of a qubit, you must be able to understand things better than AP as long as you have mastered some high school physics and your IQ is at least 100.

Nominally, a qubit is the same "amount" of information as a classical bit but the word "qubit" indicates that it behaves according to different laws than a "classical bit". In particular,

  • qubits, when interacting with each other only, always evolve according to unitary i.e. perfectly reversible transformations of the Hilbert space. They cannot measure themselves or their neighbors or cause any irreversible change
  • when the rest of the quantum computer or the observer observes them, it or he can effectively (because of the arbitrary unitary operations before the measurement) observe any observable and most of them are not functions of \(J_{i,z}\) and all of the observables are "equally good"
  • the relative phases of the wave function always matter as much as the absolute values: the absolute values of the amplitudes in a different basis depend on the relative phases in the first basis and vice versa
So again, every step of a quantum computation – with the exception of the final step, the measurement – is a unitary transformation acting on the \(2^N\)-dimensional Hilbert space. A unitary transformation is obviously reversible because its defining property is\[

U^{-1} = U^\dagger.

\] If or when the evolution of these "qubits" failed to be unitary during the evolution, the machine would cease to be a quantum computer, the information would cease to be qubits, and it wouldn't be just a matter of terminology: The gadget would lose its comparative advantage over classical computers. It would lose the ability to effectively compute certain tasks that are easy with a quantum computer but incredibly time-consuming with a classical computer.

So the reversibility of the quantum calculation is not only true but matters – it's really a necessary condition for our meaningfully talking about quantum mechanics, quantum computers, or quantum bits (qubits) at all. If you assume that the information in some gadgets decoheres or collapses in the middle of the calculation, it's simply no longer a quantum computer and the information in it shouldn't be called qubits. It's the point of quantum computers that the full laws of quantum mechanics have to be applicable throughout the calculation – which means that no classical approximation that neglects the reversibility, importance of relative phases, equality of different bases, non-commutativity of observables can possibly be legitimate to discuss the quantum computation.

If you don't understand one of these overlapping points – that the instructions in a quantum computer don't include and don't allow any measurement or "effective measurement"; any irreversibility; any "privileged observables" or "privileged bases" – then you understand exactly zero percent of quantum computation and qubits (and quantum mechanics). You just shouldn't be using these words at all because if you are using them, it shows that you are a pompous fool who simply emits fancy words to sound smarter than he is but he actually understands none of them whatsoever.

But I need to emphasize that these frictions cannot be reduced to some humanities-based differences in philosophy or wording. Whoever misunderstands the things I mentioned must unavoidably misunderstand everything about quantum mechanics and quantum computers at the "purely technical" or mathematical level, too. So for example, AP started by this paragraph:
I've got a question. If we prepare a qubit in \(\ket \uparrow + \ket\downarrow\) and then entangle it with another qubit to get \(\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}\), then it seems we've performed an experiment that is capable of determining "which way" the first qubit went. Yet, we can reverse this entanglement, in which case the possibility of interference is restored.
As I stressed very many times, by an operation on two qubits, one hasn't performed any observation. The preparation of the initial entangled state is just a unitary transformation performed on a Hilbert space – and the unitary transformations are exactly what a measurement is not. So no "which way" (value of the bit, the eigenvalue of \(J_z\)) was measured and it's important that the subsequent behavior is different from the behavior if the "which way" information were measured.

But you may see that AP is clueless in between the lines, too. He is preparing the singlet state\[

\frac{\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}}{\sqrt{2}}

\] by "starting with the first qubit" in the state\[

\frac{\ket\uparrow+\ket\downarrow}{\sqrt{2}}

\] and then entangling it with the second qubit. (I've changed the a relative sign and permutation conventions for the second qubit and added the usual normalization factor of \(1/\sqrt{2}\) – it's not changing anything about the substance.)

But this whole "algorithm how to prepare the entangled state" is just silly. There is no reason why the preparation of the singlet, entangled state should start with \[

\frac{\ket\uparrow+\ket\downarrow}{\sqrt{2}}.

\] After all, the state of the two spins we want to get is the unique (up to the overall complex normalization) state that is rotationally invariant. So all directions are equally good. But the superposition of "up" and "down" above is nothing else than the eigenstate of \(J_x\) with a positive eigenvalue, namely \(\ket\rightarrow\). And \(\ket\rightarrow\) only differs e.g. from \(\ket\uparrow\) by a simple rotation around the \(y\)-axis by 90 degrees.

So AP could have started with the initial state of the single qubit \(\ket\uparrow\), too. It wouldn't make his bringing of the two-qubit system to the maximally entangled state any harder – or any less natural. If you ask what's the difference or distance between the desired singlet state\[

\frac{\ket{\uparrow\downarrow} - \ket{\downarrow\uparrow}}{\sqrt{2}}

\] on one side, and the un-entangled initial states\[

\ket{\uparrow\uparrow}

\] or\[

\frac{ (\ket\uparrow + \ket\downarrow)\otimes (\ket\uparrow +\ket\downarrow) }{2},

\] on the other side, the answer is that the difference or distance is exactly the same. After all, the last displayed expression is nothing else than \(\ket{\rightarrow\rightarrow}\) and only differs from \(\ket{\uparrow\uparrow}\) by a rotation by 90 degrees, e.g. a switched convention for what we call the \(x\)-axis and what we call the \(z\)-axis!

What does it mean when someone apparently thinks that the non-entangled state\[

\frac{ (\ket\uparrow + \ket\downarrow)\otimes (\ket\uparrow +\ket\downarrow) }{2},

\] is closer to the desired entangled singlet state than the simple \(\ket{\uparrow\uparrow}\)? It just shows that he doesn't understand the mathematics of the 2-qubit Hilbert space at all. Effectively, he doesn't understand the difference between the classical bits and qubits at all. He thinks that "what matters" are just the probabilities that the first qubit is "up" or "down", and that the second qubit is "up" or "down".

But that's simply not the case. Even in classical physics with some uncertainty about the final value of the bits, we also have correlations – encoded e.g. in probabilities that the two qubits are the same. And in quantum mechanics, we have many more properties and correlations we may measure because aside from \(J_{z,1}\) and \(J_{z,2}\), we can also measure the \(x\)-components, \(y\)-components, or their arbitrary combinations.

So when AP assumed that his particular initial state – which had 50% probabilities to be up and down for both bits – is "closer" to the desired final, entangled singlet state, he showed that he envisions that only some very specific operations are going to be performed with the two bits. They're just the classical operations – measurements that look at the \(z\)-components of the spins. Moreover, his operations are only those that measure these \(z\)-components separately; they are not allowed to depend on any correlations between the qubits.

However, if you restrict the operations and conditions in this way, you're not using the qubits as qubits at all. The calculation you are thinking about isn't a quantum computation at all and if you only allow these operations, the gadget has no right to be called a quantum computer. Again, it's not just about some terminology and pride about the titles. The machine won't be capable of completing the difficult tasks we expect from quantum computers. So AP has thrown the baby out with the bath water. He pretends to talk about quantum mechanics, quantum computers, and qubits, but in reality, it's absolutely obvious that he is thinking about classical physics, classical computers, and classical bits all the time!

If AP hasn't understood how quantum computers differ from classical ones, then – I believe – he must know it. He must be constantly asking: Why are you using the terms quantum mechanics, quantum computers, and qubits when classical physics, classical computers, and classical bits are the only ones that may exist? It's clear that this is what he should ask if he were honest because he hasn't started to understand the novelties of quantum mechanics, quantum computers, and qubits at all!

And it just drives me up the wall when similarly 100% clueless people who have understood absolutely nothing about modern physics are trying to promote their opinions and claim that they're just equally good opinions and modern physics is just another set of beliefs, opinions, and suggestions. Sorry, it's not the case. What you believe, AP, is just worthless crap that is wrong about everything and that totally misses the point. What physics has learned in the mid 1920s is one of the most valuable gems that the mankind has ever found and it is supported by the most solid body of evidence that we have. If you can't distinguish the status and value of your idiocy from the status and value of the most important principles of modern science, then it is your fault and please don't try to brag about this idiocy of yours – it is extremely stupid and offensive.

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