Wednesday, May 04, 2016

Measurements are fundamentally irreversible

I was recently exposed to the brutal misconception that "measurements could or should be fundamentally reversible" promoted by several well-known folks but let me only mention two prominent ones: Leonard Susskind's latest paper and a 2002 talk by Freeman Dyson who argued that quantum mechanics was "incomplete" at Wheeler's 90th birthday party.

These men and others realize that the founders of quantum mechanics have postulated that the measurement is irreversible (e.g. whole chapters of books by von Neumann were dedicated to the irreversibility of the measurement); and they seem to realize that all the folks who talk about decoherence acknowledge and assume that decoherence has to be irreversible. But they're not satisfied and want all processes, including the measurements, to be "reversible". This "dream" partly motivates the vague Everettian movement, too. Some people find the measurement and/or the collapse to be "ugly" and want to eliminate it.

But this is a complete misunderstanding what measurements are or can be – and whether they exist at all (be sure that they do). It's a misunderstanding of the basic difference between the future and the past. We remember the past but can't remember the future, we may be afraid of the future and feel that we can change it but we can't change the past, and so on. Human beings and even animals get these basic things already as "babies". Does it mean that all adult physicists have noticed?

What is reversible or time-reversal-symmetric (or at least CPT-invariant in general enough quantum field theories) is the unitary evolution of the physical systems from one moment of time or another. Let me assume that you know how to complex conjugate the wave functions to achieve the time reversal.

But the unitary evolution in time isn't the only thing we need to do physics. To do any physics (or science) or to simply live and feel the life, we also need to make measurements – to find and prepare the initial conditions; and to verify the final state. And by the very basic logic of quantum mechanics, these measurements just aren't and can't be reversible.

In a similar way, decoherence is obviously irreversible, too. The very word means "the loss of coherence" and the word "loss" refers to the "decrease in time". In particular,
the off-diagonal elements of the density matrix \(\rho_{ij}\) behave like \(\exp(-B\exp(Ct))\) or so.
They're decreasing functions of time. So there obviously is some time-reversal-asymmetry, some irreversibility. We don't have any "spontaneously increasing quantum coherence".

What happens when we observe something, e.g. the spin \(j_z\) of an electron? Before the measurement, at \(t\lt 0\), the electron is in some state. For a while, it interacts with the \(j_z\)-measuring device. After the measurement, for \(t\gt 0\), we know that the state is an eigenstate of \(j_z\), either the "up" or "down".

Note that the electron interacts with the apparatus which measures \(j_z\) and some eigenstate of \(j_z\) is relevant for the description. But the eigenstate of \(j_z\) is only relevant for \(t\gt 0\), not for \(t\lt 0\). The state for \(t\lt 0\) doesn't have to be an eigenstate of \(j_z\), the operator associated with the operator! That's quite an asymmetry between \(t\lt 0\) and \(t\gt 0\).

When the measurement takes place at \(t=0\), the information about \(j_z\) is produced. At the same moment, the apparatus gets entangled with the electron. At \(t\gt 0\), both of them are either in the "up" state or in the "down" state. The probabilities of "up" and "down" are calculable from the Born rule as \(|c_\uparrow|^2\) and \(|c_\downarrow|^2\), respectively.

But what is the right symbol for these probabilities? Imagine that we have the state \(I_i\) before a measurement and \(F_f\) after the measurement. The probability of the transition (collapse) \(P(I_i\to F_f)\) that is calculable by the Born rule is nothing else than the conditional probability\[

P(I_i\to F_f) = P(F_f | I_i) =\dots

\] It's the conditional probability that the future state at \(t\gt 0\) will be the eigenstate \(F_f\) given the assumption that the state at \(t\lt 0\) was the state \(I_i\). Again, this conditional probability is given by the Born rule\[

P(F_f | I_i) = \abs{ \bra{F_f} I_i\rangle }^2.

\] But now, a key point you should realize is that this formula is totally and absolutely time-reversal-asymmetric because the conditional probability cares about the order. As many kids know,\[

P(F_f | I_i) \neq P(I_i | F_f)

\] The conditional probability of \(A\) given \(B\) is not the same thing as the conditional probability of \(B\) given \(A\). Do I really need to repeat some pages from mathematics textbooks for basic schools? OK. An example. The probability that a person is a billionaire given the assumption that he owns an iPhone isn't the same as the probability that a person owns an iPhone given the assumption that he or she is a billionaire. If these two conditional probabilities were the same, people would surely start to buy iPhones in order to become billionaires.

The conditional probability may be written as a ratio\[

P(F_f | I_i) = \frac{ P(F_f \text{ and } I_i ) }{ P(I_i) }

\] The numerator is symmetric with respect to \(F_f\) and \(I_i\) but the denominator is not. This asymmetry of the formula is no small detail, no deviation that may be approximated by zero. The interpretation of the squared probability amplitudes that are calculable from the unitary evolution in quantum mechanics is always in terms of
conditional probabilities of the final state (eigenstate) assuming an initial state
and never the other way around. The arrow of time is irrevocably incorporated to the Born rule. The order is absolutely critical, to confuse the order means to make the result and all claims about science wrong by \(O(100\%)\) or so, and people who just think that the order doesn't matter are incredibly sloppy. The fact that only the conditional probabilities \(P(F_f | I_i)\) are calculable from the laws of physics – and not the opposite ones – reflects the fact that the laws of physics tell you how the future, \(F_f\), evolves from the past, \(I_i\), but not the other way around.

Quantum mechanics can't produce any fixed values of the "reverse" conditional probabilities\[

P(I_i| F_f)

\] at all. They differ from the calculable \(P(F_f | I_i)\) by the factor of \(P(I_i)/P(F_f)\) but this factor simply cannot be determined by the laws of physics, especially because \(P(I_i)\) is a "prior" in the Bayesian sense, and may therefore be chosen more or less arbitrarily by the observer.

If you wish, all these asymmetries may also be seen in the frequentist surveys of the Universe. In the Universe, you may look for all labs ever measuring \(j_z\) of an electron that was prepared in the state \(j_x=+1/2\) before the measurement. You will find that 50% of the lab measurements with the \(j_x=+1/2\) initial state have \(j_z=+1/2\) at the end. But it is not true that 50% of the measurements that have \(j_z=+1/2\) as the final state have \(j_x=+1/2\) as the initial state. All the labs in the Universe could have very well chosen \(j_x=+1/2\) to start with. There can't be any law of physics that would prevent them from doing so. But there is a law of physics that guarantees that the final states are divided to 50-50 according to \(j_z\).

So yes, the physical processes related to a measurement may be described "mechanically" as a unitary evolution by an external observer. But if these processes are described in this way, they are by definition not constituting a measurement. By definition, the measurement has to include the logical or psychological component – an observer is learning something about the physical systems around him, his state of the knowledge is changing – and the Born rule governing this key, logical or psychological, component of the measurement is absolutely and unavoidably time-reversal-asymmetric. Without the logical or psychological component of the measurement (and as the conditional probabilities show, they absolutely distinguish the past and the future), we can't talk about the Born rule, about the probabilities it predicts, or about the collapse of the wave function that it necessitates.

Everything may be imagined to be a "perfect undisturbed Universe" that is always evolving according to the reversible unitary laws of evolution. But if that's so, no one can ever make any statements about the state of the physical objects or the Universe as a whole (e.g. the statement that the Moon exists). To make statements, an observation or observations are simply needed, and they happen according to rules that are absolutely irreversible or time-reversal-asymmetric.

The irreversibility explained above – that ultimately boils down to the asymmetry of the conditional probabilities – holds completely analogously in statistical physics (even classical statistical physics), too. It's this logical or psychological arrow of time that is the true origin from which the second law of thermodynamics can be derived. I've done it in many previous "anti-Carroll" posts. The simplest way to proceed is to realize that the probabilities for ensembles are summed over the final microstates but averaged over the initial microstates. The latter adds a past-future asymmetric factor of \(1/N_i\) and this factor is the reason why the processes with \(N_f\geq N_i\) are favored while the opposite, decreasing-entropy, processes are banned. This asymmetry showing itself as the \(1/N_i\) factor is equivalent to the asymmetry of the conditional probabilities discussed above. It's always the same logical or psychological arrow of time that is behind these asymmetries of the probability calculus.

I am amazed how many people – including famous people – are deeply confused about these absolutely elementary things these days. It couldn't be like that in the past. In his fifth 1964 Messenger Lecture addressed to the Cornell undergraduates, Richard Feynman's first sentence was:
It's obvious to everybody that the phenomena of the world are evidently irreversible.
There were no protests against this first sentence and Feynman continued to talk about the arrow of time for 45 minutes. Well, it seems obvious to me that the irreversibility of the phenomena in the world is not obvious to Leonard Susskind and Freeman Dyson, among others! The very first sentence of an undergraduate physics talk is already too hard or too controversial for some men. I used to think that this complete denial of the fundamental irreversibility of the phenomena of the world must be just some idiosyncratic stupidity of Sean Carroll but this stupidity is evidently much more widespread.

Cups don't unbreak, waves don't arise from spontaneously accumulated foam, and people laugh when movies run backwards. Three examples of the obvious asymmetry.

Incidentally, everything else that Dyson said in that "provocative" 2002 talk was totally wrong, too. It makes no sense to try to make decoherence "reversible". And all the "limitations" of quantum mechanics and "paradoxes" allegedly implied by the incompleteness of quantum mechanics were totally spurious, too.

At the beginning, Dyson said that quantum mechanics may be embraced – just like the U.S. Constitution – either in the literal way, or the broad way. Great. But he conflated the degree of the "literal" interpretation with something different, namely whether quantum mechanics is a complete theory and applicable to all parts of the Universe and the whole Universe as well. These are completely different questions.

Moreover, Dyson has claimed that Bohr has adopted a strict, "literal" attitude (so far so good) which Dyson however identified with the claim that "quantum mechanics only applies to small objects, not the whole Universe". The latter claim is completely wrong. Bohr has never claimed that quantum mechanics was inapplicable to some objects or phenomena in the world. And in fact, when it comes to completeness of quantum mechanics, Bohr has argued with Einstein for many years – the famous Bohr-Einstein debates – and at some moment (when Einstein's attempts to show an inconsistency within QM were hopelessly defeated), the arguments were shifted to the question whether QM was a complete description of Nature. Einstein incorrectly argued "No" (and the famous EPR paper had the phrase "incompleteness of quantum mechanics" in the title) while Bohr has obviously argued "Yes".

These are such well-known facts that it seems extremely puzzling why Freeman Dyson would claim that Bohr has defended a limited applicability of quantum mechanics which Bohr has clearly never done. I think that the explanation of this Dyson's confusion is that Dyson completely fails to understand the subjective, observer-dependent character of the quantum description of Nature. So he thinks that if Bohr recommended to think about the observer and the apparatus in classical terms, it meant that quantum mechanics wasn't capable of describing the apparatuses and human beings.

But that doesn't follow in any way and Bohr or other founders have never defended such limitations. The point is that the "Heisenberg cut" – the boundary between the observer and the observed – isn't some objective boundary expressing the limitations of quantum mechanics. Instead, the location of the "Heisenberg cut" is a choice made by an observer. Another observer may decide to place the first observer on the quantum side of the "Heisenberg cut" – i.e. describe the first human beings in the quantum mechanical way. The fathers of quantum mechanics have always emphasized these things. So all this hopeless confusion by Dyson is due to the misguided attempts to "objectify" all the things that are actually defined with respect to a particular observer only, or subjective, or chosen by himself. They just aren't objective. It's the main new key point brought us by the quantum mechanical revolution that all such claims only make sense relatively to an observer and the set of observations he has made and will do. They're intrinsically subjective in this sense.

The talk gets even weirder later. Dyson basically claims that there is a logical contradiction within quantum mechanics that may be proven in a very simple way – when you study an electron propagating through a series of slits. I was trying to decode the precise reasons why he would say such a thing but the derivation was way too sloppy so I gave up. But whatever the derivation of the contradiction was, it's clear that it was wrong.

Dyson used the laws for the propagation of a free particle in between the slits – something that obeys well-known rules, e.g. the Schrödinger's equation – and the Heisenberg inequality for uncertainties. But all these statements are easily derivable. We know the equations governing the wave function, we can calculate what the wave function will be, what is the magnitude of the uncertainties of \(x\) and \(p\), and other things. So it's easy to decide whether any particular inequality in Dyson's sequence is right or wrong. He could have overlooked the fact that the detectors affect the electron, that the detectors represent different boundary conditions for the wave function, confused the longitudinal and transverse directions, and so on. I really can't isolate his exact mistake but that doesn't prevent me from being absolutely certain that his reasoning was wrong.

If we study the electron propagating through some slits with some geometric parameter satisfying \(L\lt L_0\), whatever \(L\) and \(L_0\) exactly are, it is spectacularly obvious that there cannot be a valid derivation of \(L\gt L_0\) which would mean that there is a mathematical contradiction in the axioms. There can't be any contradiction. It's just a trivial differential equation (Schrödinger's equation) with some boundary conditions at the surface of the macroscopic material. And all the expectation values used by Dyson – if you carefully define them – are easily calculable from the solution to the differential equation. How could there be a logical contradiction?

At 24:10, Nobel prize winner Bob Laughlin (condensed matter physics) asked Dyson what happens when you do the Čerenkov calculation – basically sequential ionization, Laughlin thinks – it's a basic freshman undergraduate problem in QM. Laughlin asks where is the contradiction, making it rather clear that he also thinks that it's totally obvious that there can't be any contradiction or limitation of QM as a theory. And Dyson replies with a crazy yet calm and superficially "intelligent" monologue claiming that QM is inapplicable to most things, including pairs of slits. Holy cow. Does Dyson also have a proof that the wheel cannot work?

Before that exchange, around 24:00, Dyson made fun final remarks. He basically said: We don't need an observer, we only need "someone who does this and that [a definition of an observer followed]". LOL.

Decades ago, I couldn't understand why it took so much time for the people to "believe" some self-evident things such as the observation that the Sun is at the center of the Solar System, the species have evolved, and many other things. How did the mental forces that have managed to prevented many people including top intellectuals from understanding very simple things work? I no longer think that there was ever something particularly strange or dumb about the people in the Middle Ages etc. Elementary things such as the self-evident consistency of quantum mechanics and the irreversibility of the phenomena of the world are simply clashing with some totally irrational but deeply held ideas about "how the world should function" and because of these strongly held beliefs, even famous people are willing to say extremely stupid things all the time.

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