The mathematical foundations of quantum mechanics

Von Neumann’s 1932 book “The mathematical foundations of quantum mechanics” was a cornerstone for the development of quantum theory, yet his insights have been mostly ignored by physicists. Many of us know that the book exists and that it formalizes the mathematics of operators on a Hilbert space as the basic language of quantum theory, but few have bothered to read it; feeling that as long as the foundations are there, we don’t need to examine them. One reason the book has never been popular as a textbook is that von Neumann does not Dirac’s notation. He admits that Dirac’s representation of quantum mechanics is “scarcely to be surpassed in brevity and elegance” but then goes on to criticize its mathematical rigor, leaving us with a mathematically rigorous treatment but a notation that is difficult to follow and a book that nobody ever reads.

Today, von Neumann’s book is brought up often in connection with foundations research, which is perhaps not surprising given the author’s original intent as stated in the preface:

the principle emphasis shell be placed on the general and fundamental questions which have arisen in connection with this theory. In particular , the difficult problems of interpretation, many of which are even now not fully resolved, will be investigated in detail.

Two particular results are often cited

  1. Von Neumann’s (presumably incorrect) proof of impossibility for a hidden variable model of quantum theory.
  2. The von Neumann measurement scheme, which is the standard formalism for describing a quantum measurement.

The description of the measurement scheme is (or at least was to me) a little surprising. The usual textbook description of von Neumann’s work is to start with the Hamiltonian that couples the measurement device to the system under measurement, and to show that it produces the right dynamics (up to the necessity for state collapse). It turns out that von Neumann’s motivation was in some sense the opposite.

He begins by considering a classical measurement and showing that the observer must be external to the measurement, i.e there are three distinct objects, the system under observation, the measurement device, and the observer. Keeping this as a guide, von Neumann provides a dynamical process where the observer is not quantum mechanical while the measurement device is. The important point is that the precise cut between the (classical) observer and the (quantum) measurement device, is not relevant to the physics of the system, just as in classical physics.

Von Neumann does not suggest that he solves the measurement problem, but he does make it clear that the problem can be pushed as far back as we want, making it irrelevant for most practical purposes, and in some ways just as problematic as it would be in classical physics. Many of us know the mathematics, and could re-derive the result, but few appreciate von Neumann’s motivation: understanding the role of the observer.

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Bell tests

The loophole free Bell experiments are among the top achievements in quantum information science over the last few years. However, as with other recent experimental validations of an a well accepted theory, the results did not change our view of reality. The few skeptics remained unconvinced, while the majority received further confirmation of a theory we already accepted. It turns out that this was not the case with the first Bell tests in the 1970s and 1980s (Clauser, Aspect etc. )

Jaynes, a prominent 20th century physicist who did some important work on light matter interaction did not believe that the electromagnetic field needs to be quantized (until Clauser’s experiment) and did extensive work on explaining optical phenomena without photons. As part of our recent work on modeling a quantum optics experiment using a modified version of classical electrodynamics (and no `photons’) we had a look at Jaynes’s last review of his neo-classical theory (1973). This work was incredibly impressive and fairly successful, but it was clear (to him at least) that it could not survive a violation of Bell’s inequalities. Jaynes’s review was written at the same time as the first Bell test experiments were reported by Clauser. In a show of extraordinary scientific honesty he wrote:

If it [Clauser’s experiment] survives that scrutiny, and if the experimental result is confirmed by others, then this will surely go down as one of the most incredible intellectual achievements in the history of science, and my own work will lie in ruins.

Misconceptions about weak measurements: 1. Weak ‘measurements’.

It seems that I am somehow drawn to controversial subjects. Maybe it’s my nature as an Israeli. Much of my Phd research has been around the controversial topic of ‘quantum discord‘.  Now I find myself working hard on the subject of my Master’s thesis, ‘weak measurements’, yet another topic which is both controversial and misunderstood.  Unlike quantum discord which became controversial mainly due to it’s popularity (aka the discord bubble),  weak measurements were controversial from day one.   This controversy  is, at leas in part, due to both misunderstandings, different interpretations, and choice of words; in particular the word `measurement’.

I first realized that the use of the word measurement may  cause of confusion after watching a recorded lecture by Anthony Leggett.  But the problem really sank in after many discussion with Marco Piani who helped me clarify my thoughts about the subject.  At one point Marco’s reaction reminded me of the phrase “You keep using that word. I do not think it means what you think it means.“.

Let me explain.

In a discussion of measurements, Asher Peres, one of the main critics of weak measurements (and my academic `grandfather’) wrote `The “detector clicks” are the only real thing we have to consider. Their observed relative frequencies are objective data.’ This is the usual sense we think of measurements in quantum information, a measurement  is a channel that takes a quantum state as input and gives probabilities (the relative frequencies) as an output.  Let us call this the quantum information approach. In more technical language a measurement is completely specified by the POVM elements. Roughly speaking any set of POVM elements that sum up to the identity can describe a measurement i.e probabilities for the various detectors clicking. However, a weak measurement cannot be described in this way, the POVM elements are at best, only part of the picture.

 

click

The quantum information approach. Quantum in – classical out

 

The first time I encountered the term POVM was during the time between my B.Sc and M.Sc when I started learning the basics of quantum information. As  an undergad, I was taught about a different type of quantum measurement. The quantum world, I was told, is made up of quantities that are observable; these correspond to Hermitian operators. The results of the measurements are eigenvalues and after the measurement the measured system will change its state to the relevant eigenvector, the so-called collapse of the wave function. I will call this the textbook approach. The big difference between this and the `quantum information approach’ (above) is that the channel has a classical input and a classical (eigenvalue) and quantum (eigenvector) output. This is closer in spirit to the `measurement’ in a weak measurement. However this framework does not have any variable strength.

TB

The textbook approach. Quantum in- classical [eigenvalue] and quantum [eigenvector] out.

 

The `textbook approach’ is unsatisfactory in two ways. First it allows a limited class of measurements that do not necessarily correspond to realistic situations. Second it does not include a dynamical picture: measurements simply happen. Although a full dynamical picture is still an (if not the) open problem, von Neumann gave the a reasonable dynamical picture for the measurement which is know as the von Neumann scheme. The measurement is described as a coherent interaction between the measured system and a (quantum) meter initially in a state |0>.  The interaction Hamiltonian is set up so that: if the system is in an eigenstate a of the desired observable, the meter will shift accordingly i.e it will end up in the state |a>.  Generally the system-meter state will be entangled after the measurement. With the right choice of interaction Hamiltonian the local picture will be  a mixed state that gives the right statistics for the textbook measurement. A slightly more elaborate picture can be used to describe more general measurements.

vN

The von Neumann scheme. Quantum [product system-meter] in- quantum [entangled system-meter] out.

 

A weak measurement is a measurement in this sense, i.e it is a channel that has a system-meter (quantum) input and a system-meter (quantum) output. The measurement can be followed by a readout stage where a single detector `clicks’, but this part simply tells us something about the meter and only indirectly about the system whose state has changed.To complete the transition from the von Neumann scheme to a weak measurement we simply need to make the interaction Hamiltonian weak. It should be so weak that, after the measurement, the shifts corresponding to different eigenvalues will strongly overlap. The first  advantage of this  method is that the system state is virtually unchanged by the measurement process. Other, surprising  advantages follow, especially when one considers the fact that this measurement process is symmetric with respect to time.

I hope I convinced you that the term `measurement’ means different things to different people; While quantum information theorists say measurement and mean `a quantum to classical channel’ the weak measurement community think of a `quantum to quantum channel’. I believe this is major source of confusion that leads to controversy around weak measurements. My advice to people in the quantum information community is: either stop thinking about weak measurements as measurements, or read the literature and try to convince yourself that this channel represents the closest thing we have to a measurement in quantum theory.  Either way stop trying to understand weak measurements simply in terms of POVM elements.

 

In upcoming posts I will try to clarify some other misconceptions including the difference between `noisy measurements’ and `weak measurements’, and an explanation of what is anomalous about ‘anomalous weak values’.