Weinberg’s «deconstruction of the measurement» isn’t wrong but it’s pointless

Luboš Motl, March 22, 2016

Steven Weinberg has published a new paper about the foundations of quantum mechanics (QM) in PRA:

What Happens in a Measurement?

The paper implicitly suggests that some conceptually important problem may be solved by comments involving the Lindblad equation. I think that in this case, he never says things that are sharply wrong (at most distortions of the history). But in between the lines, he seems to write that certain steps are useful to clarify the situation even though they aren’t.

I think that even though Weinberg avoids clearly wrong statements, relatively to the clear and groundbreaking insights and formulations of the Copenhagen school, his paper is a step backwards away from the proper and problem-free understanding of quantum mechanics. The main problem is that he tries to «reduce» the measurement. But it only makes sense to «reduce» things that are composite.

The measurement is ultimately an irreducible, fundamental process required to apply any QM theory, so it just makes no sense to try to «decompose» it, to try to write whole papers about «what is happening inside».

The first problems arise when Weinberg offers his ideas what the Copenhagen Interpretation means. He claims that the Copenhagen Interpretation says that the initial density matrix undergoes the collapse

\[ \rho_{\rm initial} \mapsto \rho_{\rm final} = \sum_\alpha p_\alpha \Lambda_\alpha \] into a combination of Hermitian projection operators where probabilities are the weights. Now, you can say that the claim about physics is true to some extent or in some sense or context – except that one should better say that it’s the «von Neumann pre-measurement», not the collapse itself (the collapse occurs only when the thing is observed and, equivalently, all the numbers \(p_\alpha\) become either zero or one).

But what is misleading is the suggestion that this claim is a part, let alone a fundamental part, of the Copenhagen Interpretation.

Weinberg’s reference #1 is a 1928 Nature paper by Bohr which Weinberg agrees to consider the «original Copenhagen Interpretation». Note that the paper based on a 1927 lecture by Bohr has its own web page at Nature. You can find the full text at Google Books.

Just show me some non-unitary transformation of the density matrix in Bohr’s paper that would be claimed to «be» the measurement or the collapse. I think that you won’t even find any reference to the density matrix at all. After all, the density matrix was introduced by von Neumann in 1927 and it could have been after Bohr gave his talk. It was a new stuff in 1927 – and not essential stuff.

The hard truth is that none of these things is really needed or fundamental to formulate the rules of QM. The density matrix may be viewed as a supplementary mathematical tool to deal with the extra uncertainty or ignorance that has existed in classical physics as well. You don’t really need density matrices to formulate the fundamental QM laws at all – you may assume that QM is fundamentally about pure states. This assumption is completely analogous to the assumption that in classical physics, some values of \(x(t),p(t)\) objectively exist even if people are ignorant about these values.

The knowledge of \(\ket\psi\) in QM or \(x(t),p(t)\) in classical physics represent the maximally known i.e. least uncertain states that the apparatuses of the theories allow. We may also deal with the less certain descriptions of the physical objects in terms of the phase space distribution function or density matrix \(\rho\) but we don’t have to. Both in classical physics and QM, the probabilistic laws for \(\rho\) can be straightforwardly derived by applying the universal probability calculus to the dynamical laws governing the maximum-knowledge states, either \(x(t),p(t)\) classically or \(\ket\psi\) in QM. The uncertainty implied by the uncertainty principle is already present in predictions based on pure states \(\ket\psi\) and is unavoidable in QM; on the other hand, the extra uncertainty or ignorance that the density matrix formalism adds is avoidable and therefore non-fundamental.

A similar comment holds even more strongly for the Lindblad equation. (Incidentally, Weinberg has been playing with this equation for some time – but he had to withdraw the previous paper on the equation four months ago because of a mistake.) This equation is even less fundamental than the concept of the density matrix. We just don’t need it at all to do physics. The Lindblad equation is the most general linear equation that governs the evolution of the density matrix \(\rho\) in a way that preserves the trace and the positivity of the eigenvalues. For almost all choices of the parameters, the evolution of \(\rho\) is nonunitary i.e. can’t be written as \(\rho\to U \rho U^\dagger\) for some \(U\). The unitary evolution of \(\rho\) may be derived from an action on the «simpler» Hilbert space of column (ket) vectors which may be imagined to be \(N\)-dimensional; the general non-unitary evolution resulting from the Lindblad equation can transform the \(N^2\) components of the density matrix more generally i.e. access the matrix entries «individually», largely ignoring their arrangement in a matrix.

But at the fundamental level, when dealing with the exact description of the physical systems, we only need to consider closed systems; and we only need to consider their unitary evolution. Any non-unitary evolution is not only ugly but it causes additional problems. For example, in a 1984 paper published in NPB (300 cits), Banks, Peskin, and Susskind showed that any non-unitary equation for \(\rho\) violates either locality or the energy-momentum conservation. It’s simply bad as a fundamental equation for Nature. That’s also why nothing such as the Lindblad equation has ever been used by the QM founding fathers to define the basic rules of the game. It would be silly to do so.

When it comes to its meaning in physics, the Lindblad equation is at most some effective approximate equation that may be deduced from the fundamental equations; and that may be useful to deal with open systems. But it just can’t be fundamental and it can’t be needed to formulate the fundamental postulates of QM.

Imagine that you measure the spin \(s_z\) of an electron. The electron may be in a pure state

\[ \ket\psi = 0.6 \ket{\uparrow} + 0.8 i \ket{\downarrow}. \] The density matrix is pure, \(\rho_{\rm initial} = \ket\psi \bra\psi\). The «collapse» postulated by Weinberg replaces this pure density matrix by the mixed one

\[ \rho_{\rm final} = 0.36 \ket{\uparrow}\bra{\uparrow} + 0.64 \ket{\downarrow}\bra{\downarrow}. \] This density matrix has eigenvalues \(0.36,0.64\) instead of the \(0,1\) eigenvalues of the original pure density matrix. It’s because the mixed terms \(\ket{\uparrow}\bra{\downarrow}\) and its complex conjugates have been erased. They were erased by the interaction with the apparatus that is only sensitive to \(s_z\). Because of this erasure, the probabilities for all quantities refusing to commute with \(s_z\), e.g. \(s_x\), have been modified by the interaction with the apparatus. That’s how we can see that the measurement has changed the state of the electron – which is unavoidable.

On the other hand, this is only true if we take «Weinberg’s» mixed density matrix to be the real deal. In principle, we may describe the electron and the apparatus exactly. If we do so, the density matrix of the whole system remains pure at all times. Fine correlations inside the apparatus remember the relative phase of «up» and «down». We could in principle reverse all the interactions with the apparatus and measure \(s_z\) just like if no interaction between the electron and the apparatus has ever taken place.

This contradicts the outcome of the Lindblad equation which says that the «pure to mixed» transition is irreversible. But the right appraisal of the contradiction is that in principle, the Lindblad equation just isn’t exact. The «pure to mixed» transformation and the accompanied loss of information about the relative phases is an artifact of approximations. That’s why no paper assuming the Lindblad equation is studying the fundamental laws. The Lindblad equation may be useful in practice – especially if the «ignored» system is a hopelessly messy thermal bath. But sometimes some correlations inside this system may be traced and reversed in which case the Lindblad equation will produce wrong predictions for the system of interest.

But my main complaint isn’t that the founding fathers had nothing to do with the Lindblad equation which is simply not fundamental and it is not needed. Much more generally, the efforts of «decompose» the measurement are pointless. It seems that Weinberg actually realizes that when he writes

Of course, this view of the Copenhagen interpretation just pushes the hard problems of interpreting quantum mechanics to a larger scale.

OK, if he knows that he’s just pushing (what he considers) the problems around, why is he doing these things? The important fact he can’t circumvent is that in QM, an observation by an observer is ultimately needed, anyway. Instead of measuring the voltage directly by his skin, the observer may look at the pointer of the voltmeter. The technical protocol for the measurement may change if we insert the apparatus in between the observer’s senses and the measured object but nothing changes qualitatively about the fact that an observer is needed at the end and those who (irrationally) consider the observation «mysterious» will always do so. The extra inserted layers can in no way reduce the «amount of mystery» hiding in the process of the observation.

Weinberg’s approximate Lindblad-equation-based description of the interactions between the measured system and the measured apparatus may explain why the information about the measured system is transferred to the apparatus as well. But this transfer is just pushing things around. The possible final states of the apparatus will still be predicted just probabilistically – just like the state of the measured electron – so there still has to be an observer who just looks.

Now, if Weinberg or someone else analyzes why and how a particular measurement apparatus works, it would be useful. We need to know what’s happening inside a telescope or LIGO or a voltmeter to understand these machines and to be able to improve them, fix them, or construct better ones. But if one only studies all the measurement apparatuses in general, the physical content of this research never goes beyond the nearly tautological fact that «the relevant information about the measured object is imprinted to the apparatus in some way». Great. It does. It’s true pretty much by definition of the measurement apparatus. The voltmeter is defined as a physical system that can translate the voltage between two points of a wire to the location of a pointer. As I said, it may make sense to show that a voltmeter exists, and how it works. This special exercise is also a proof that at least one measurement apparatus exists. But it’s vacuous to try to discuss «the existence of measurement apparatuses» in general.

At the end, the basic reason why futile exercises such as Weinberg’s are repeated all the time is that many folks, apparently including Weinberg himself, are just not willing to reconcile themselves with the fact that QM is fundamentally different than classical physics and it predicts observations for observers. QM just cannot work without observers. When we talk about observers, it means that they realize that they’re observers and they’ve observed something, and they realize what they just observed, and this realization must be assumed to exist – as an elementary building block for physics to be possible. It’s this perception that QM (probabilistically) predicts. The perceptions are the ultimate client who uses the laws of QM. If this basic setup looks too idealist to you, it’s your problem. It’s surely how Nature works.

Even if and when Weinberg avoids downright wrong claims about physics (he doesn’t avoid wrong claims about the history), his paper is still a part of the misguided efforts to find some «classical wheels and gears» inside QM. But there are no classical wheels and gears inside QM. QM describes how Nature works at the fundamental level. The claim that «the squared absolute value of a complex probability amplitude calculates the probability of one outcome of an observation or another», it’s a clear, unambiguous, true, and complete law of physics. There is nothing «deeper» to be found here. A teacher may try to explain this statement very slowly to a student. But it’s silly to write whole research papers attempting to claim that there are tens of pages of explanations that have to be «added» for the sentence to become great science. It’s simply not true. Nothing needs to be added here.

The distortions of the views of the QM founding fathers seems to be omnipresent in Weinberg’s paper – and similar papers. For example, Weinberg writes:

In the original formulation [1] of the Copenhagen interpretation it was simply accepted that the change in a system during measurement in principle departs from quantum mechanics.

Oh, really? Just open the full 1928 paper by Bohr and show me where he says that there exist changes during the measurement that depart from quantum mechanics.

If you read the actual paper, Bohr, on the contrary, explains very clearly that Planck’s quantum of action, \(\hbar\), quantifies the unavoidable departure of the laws of Nature from the framework of classical physics, namely from the assumption that properties of physical objects exist independently of observers and observations. They just don’t exist independently of those. To prove that a quantity has a meaning in science, one has to formulate an operational method to talk about it, at least in principle. But because in QM, it’s guaranteed that any observations alter the physical system, it’s clear that no values of any dynamical variables can objectively exist without observers in the same sense as they did in classical physics.

Bohr says that QM requires the «quantum postulate» which equips any individual system with the discontinuity, or rather individuality, that is completely unknown to classical physics. Bohr is arguably not very clear about what the «quantum postulate» exactly says but he is extremely clear about what it does not say.

QM and Bohr’s text (which is not such a bad or incomprehensible summary of the character of QM, I think) is all about the inadequacy of classical physics, not the breakdown of quantum mechanics. I can’t believe that Weinberg can’t read. Someone who actually reads Bohr’s paper must see that Bohr wrote pretty much the opposite than what Weinberg tries to ascribe to him.

The only «similar» thing that Bohr says is that the process of measurement/observation yields the information of the same kind that we knew in classical physics. But that’s not a violation of QM. It’s just a statement that classical physics and QM mean exactly the same thing by the result of a measurement, e.g. the measurement of \(s_z\) I mentioned above. Classical physics has no monopoly over the «outcomes of observations» and their definition. On the contrary, QM happily uses the same concept – and the concept is more important in QM than it was in classical physics.

But classical physics and QM have completely different methods to predict the outcomes of the measurement – and, consequently, different values of the probabilities that they actually predict (sometimes very different, sometimes slightly different). Classical physics assumes that independently of any observer, there exists an objective state of Nature – a point on the phase space – while QM postulates that such a thing doesn’t exist, all information must be carefully reduced to observations, and those depend on what we mean by the observer which cannot be defined in any objective way because the observation is ultimately a subjective process (an observer knows that he has seen something or learned something, but others can’t strictly speaking prove that the observer felt anything at all). QM calculates the probabilities of outcomes of observations «directly» using a formula involving the complex matrix elements of linear operators on the Hilbert space – those are associated with the measurable information. That’s how Nature works. If classical physics were correct, one could calculate the probabilities from probabilistic laws that may be classically derived from a set of classical deterministic (or deterministic plus random noise) equations. But QM states that it isn’t possible. There don’t exist any «classical laws» beneath the quantum ones.

When people like Weinberg – and especially people who are much more confused than he is – incorrectly say that the measurement violates unitarity or the laws of QM (it doesn’t do either), they mean that

if the dynamical equation governing the evolution of the wave function or the density matrix were a classical evolution equation, one would have to add modifications of this equation during the process of the observation/measurement/collapse.

But what they actually say isn’t true because the condition «if» in the quote just isn’t satisfied in Nature. Neither the wave function nor the density matrix is a collection of classical degrees of freedom. Instead, they are packages of probability amplitudes whose only physical meaning is that probabilities of outcomes may be calculated from them (as bilinear functions of the wave function; or linear functions of the matrix elements of the density matrix). Consequently, Schrödinger’s equation for the wave function (or the related Liouville-von Neumann equation for the density matrix) isn’t a classical dynamical equation for observables, either.

When we observe a particular outcome of a measurement, e.g. that «the spin is up», we are just confirming a postulate of QM that observations exist (and for every linear Hermitian operator, there exists a possible «type of a measurement»). We are not «debunking» QM. Instead, we are confirming its postulate. What Schrödinger’s or equivalent equations in QM say is how the probabilities may be calculated. And these predictions for the probabilities may be verified e.g. by repetitions of the same experiment. But the very fact that individual experiments have sharp outcomes isn’t a violation of QM. It’s a confirmation of QM and the outcomes are sharp according to QM because only sharp outcomes are said to occur with nonzero probabilities by the rules of QM.

I think that Weinberg is a smart man but it seems obvious to me that when it comes to «conceptually really new» insights that had to be made when QM was discovered, relatively to Bohr, Heisenberg, Born, and others, Weinberg is an intellectual dwarf. He couldn’t have possibly made the transition from classical physics to quantum physics because he has some problems with it even now, 90 years after the discovery.

Weinberg’s paper also discusses one evergreen from similar paper – the suggestion that he may derive the Born rule. But all these games in all similar papers are self-evidently examples of muddy circular reasoning. The point is that a basic postulate of QM is that the probabilities – something that may be «measured» by the frequentist formula if we repeat experiments many times – is a linear function of the density matrix,

\[ {\rm Prob}_A = {\rm Tr} (\rho P_A) \] where \(P_A\) is the projection operator corresponding to the «Yes» statement. This statement of QM is a necessary link to connect the mathematical symbols with the actual observations by the experimenters. QM says that only probabilities of outcomes may be predicted; and it just gives you the right formulae that quantify (predict) these probabilities.

If you omit a law that connects the probability (something that experimenters know from their very practical life) with the wave function or density matrix (something that the workers with the mathematical apparatus of QM know, especially from their work on paper), the mathematical apparatus of QM ceases to have any relevance for natural science.

So to make the equations of QM physical, you simply need to say that probabilities are connected with the mathematical formulae in the known way. You need to add the Born rule. Once you add it, you can rederive it in many other ways, but as a «derivation» of the Born rule, any such exercise is clearly just a case of circular reasoning. There can’t be anything more fundamental about the «derivation» of the probabilistic character of QM than the statement that the probabilities (known to experimenters) may be calculated as bilinear functions of the wave function or linear functions of the density matrix.

You need this assumption – the Born rule or call it whatever you want – to do physics and this assumption can’t be «simplified» or «reduced» further. And once you assume that probabilities (that experimenters know) are linear functions of the density matrix, then it follows that it’s true at all times because all the transformations that the density matrix undergoes are linear, too. There’s nothing nontrivial and new to be found here. It’s a pure waste of time.

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