# Role of “hierarchic structure” in solving the Measurement Problem

Background

Decoherence solves some aspects of the Measurement Problem by ensuring that at the end of a measurement process off diagonal terms in the density matrix are supressed. However decoherence does not solve the Problem of Outcomes: how to go from a diagonal density matrix to a single definite outcome for an individual measurement run.

The solution in the ABN Paper

The solution to the Problem of Outcomes in the 2013 paper by Nieuwenhuizen et al ( Scientist propose solution to quantum measurement problem, explain further) adopts a Statistical/Ensemble interpretation of QM. Consider a system S (with eigenvalues $s_i$ of some observable) interacting with a measuring aparatus A (with pointer states $A_i$ which can correlate to the $s_i$). We start with a density matrix before measurement of
$$\hat{D}(t_0) = \hat{r} (0) \otimes \hat{R}(0)$$

After the physical measurement process (including decoherence to remove off-diagonal terms) we end up with

$$\hat{D}(t_f) = \sum_i p_i \lvert s_i \rangle \langle s_i \rvert \otimes \hat{R}_i$$

They call this truncation and registration.

The density matrix above applies to a complete ensemble $\mathcal{E}$ of the S+A system (representing multiple runs of the same experiment with a distribution of outcomes given by the $p_i$).

They then argue that after the measurement (where we now have an objective outcome $A_i$ say shown on the measuring device) we can now retrospectively restrict to a subensemble $\mathcal{E}_i$ : for this restriction the density matrix reduces to just

$$\hat{D}_i(t_f) = \lvert s_i \rangle \langle s_i \rvert \otimes \hat{R}_i$$

Hence each experiment has a unique (albeit unpredictable) outcome: Measurement Problem solved!

My question

To justify the final reduction step they claim they need to first demonstrate something they call hierarchic structure (page 15):

We will rely on a property of arbitrary subsets of runs of the measurement, their hierarchic structure. Namely any subset must be described at the final time by a density operator of the form $\sum_i q_i \hat{D}_i$ with arbitrary weights $q_i$.

It seems from this that the full solution to the Measurement Problem hangs on a rather subtle technicality: can anyone either expand on the relevance of this hierarchic structure property or refer to other papers where this property has been used. I am trying to find out if this approach has been generally accepted or if it has been refuted before studying the paper in more detail.

• While I cannot provide an answer concerning the paper you mention, I may say that the "measurement problem" as you describe it was solved almost 30 years before the aforementioned paper in Ozawa 1984. The paper does not mention this work, but another by the same author that is, in my opinion, less poignant. – yuggib Sep 5 '16 at 5:43
• In the paper I just linked, Ozawa proves (in a mathematically rigorous fashion) that for every observable of (non-relativistic) QM, there exists a measurement process (à la von Neumann, consisting of the interaction with a measuring scaled apparatus) that behaves as expected by the postulates of quantum mechanics. In addition, measurements are (weakly) repeatable only if the observable has purely discrete spectrum. – yuggib Sep 5 '16 at 5:49
• @yuggib Thanks, this is the type of feedback I wanted to understand if ABN have done something new. I have downloaded Ozawa 1998 to study. I assume this has similar ideas. The immediate observation is that ABN stress the need to do a full realistic model of the measuring device arguing it is insufficient to describe the apparatus initial state as a pure state. – Bruce Greetham Sep 5 '16 at 7:14
• Well, the later one of 98 seems to be somewhat more complicated than the one I referred to (that unluckily is quite hard to get without paying). Concerning the initial state of the measuring device, if I recall correctly it is not assumed to be in a pure state by Ozawa, but in a general initial quantum state. – yuggib Sep 5 '16 at 7:38
• @yuggib A follow up comment after first read of Ozawa 98: I agree on one level he is trying to achieve a similar aim as ABN of using some Bayes principle to justify reduction. On another level ABN is 200 pages, Ozawa is 6 pages: it seems you get what you pay for - is it fair to deduce from this that quantum measurement theory has developed somewhat in 30 years? All that leaves me unsure about my initial question - I think I'll plough on with ABN - I'm finding it a good read! Final point on Ozawa: I couldn't get how he has done this without decoherence. – Bruce Greetham Sep 5 '16 at 15:49