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.
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.
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.
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’.