Is the Decoherent Histories Approach Consistent?

One particular approach of interpreting quantum mechanics is Decoherent Histories. All major non-Copenhagen approaches have serious issues:
- MWI has the issue of the very meaning of probability and without a non-circular derivation (impossible in my opinion) of Born rule does not qualify for anything but a "work in progress" status.
-GRW-type theories make different predictions than quantum mechanics which soon will be confirmed or rejected by ongoing experiments. (My bet is on rejection since later GRW versions tuned their free parameters to avoid collision with known experimental facts instead of making a falsifiable prediction)
-Bohmian approach has issues with "surreal trajectories" which invalidates their only hard ontic claim: the position of the particle.

Now onto Decoherent Histories. I did not closely follow this approach and I cannot state for sure if there are genuine issues here, but I can present the debate. On one hand, Robert Griffiths states:

"What is different is that by employing suitable families of histories one can show that measurement actually measure something that is there, rather than producing a mysterious collapse of the wave function"

On the other hand he states:

"Any description of the properties of an isolated physical system must consists of propositions belonging together to a common consistent logic" - in other words he introduces contextuality.

Critics of decoherent (or consistent) histories use examples which are locally consistent but globally inconsistent to criticize the interpretation.

Here is an example by Goldstein (other examples are known). The example can be found in Bricmont's recent book: Making Sense of Quantum Mechanics on page 231. Consider two particles and two basis for a two-dimensional spin base $$(|e_1\rangle\, |e_2\rangle), (|f_1\rangle, |f_2\rangle))$$ and consider the following state:

$$|\Psi\rangle = a |e_1\rangle|f_2\rangle + a|e_2\rangle|f_1\rangle - b |e_1\rangle|f_1\rangle$$

Then consider four measurements A, B, C, D corresponding to projectors on four vectors, respectively: $$|h\rangle, |g\rangle, |e_2\rangle, |f_2\rangle$$ where:

$$|g\rangle = c|e_1\rangle + d|e_2\rangle$$
$$|h\rangle = c|f_1\rangle + d|f_2\rangle$$

Then we have the following properties:

(1) A and C can be measured simultaneously, and if A=1 then C=1
(2) B and D can be measured simultaneously, and if B=1 then D=1
(3) C and D can be measured simultaneously, but we never get both C and D = 1
(4) A and B can be measured simultaneously, and sometimes we get both A and B = 1

However all 4 statements cannot be true at the same time: when A=B=1 as in (4) then by (1) and (2) C=D=1 and this contradicts (3).

So what is going on here? The mathematical formalism of decoherent histories is correct as they predict nothing different than standard quantum mechanics. The interpretation assigns probabilities to events weather we observe them or not, but does it only after taking into account contextuality. Is this a mortal sin of the approach? Nature is contextual and I don't get the point of the criticism. The interpretation would be incorrect if it does not take into account contextuality. Again, I am not an expert of this approach and I cannot offer a definite conclusion, but to state my bias I like the approach and my gut feeling is that the criticism is without merit.

PS: I'll be going on vacation soon and my next post will be delayed: I will skip a week.

1. The thought experiment is just a tiny variation of the so-called Hardy's paradox – so the "credits" that you, Bricmont, and Goldstein offer is deceitful (Mermin never tries to steal ideas in this way, however). Search e.g. for "Hardy's paradox kills all realistic models of quantum phenomena" to see some comments of mine.

Needless to say, there is no paradox, no inconsistency in Copenhagen, and no inconsistency in consistent/decoherent histories. The latter is complete nonsense because in this thought experiment, one isn't discussing measurements at several moments at all, so Copenhagen and Consistent Histories boil down to exactly equivalent things.

If you can't figure out whether Hardy's paradox is an inconsistency, if you're "not an expert on this thing", then you're obviously not an expert on anything that depends on quantum mechanics in any way. This is really a straightforward homework exercise for undergrads.

1. Which part of "I like the approach and my gut feeling is that the criticism is without merit" is unclear?

Of course there is no inconsistency, but the issue is different: as an interpretation, to what degree does CH claim any sort of realism? I don't know how to answer this since I did not closely follow CH and I am not intimately familiar with all of their claims (to the amount that they do not claim realism I am agreeing with them). All I can state is that to me it looks like there is no issue whatsoever. And in stating this I am disagreeing with Bricmont's position from his book.

On relationship with Hardy's paradox, your charge is nonsensical. There is another example by Hardy about CH.

Hardy is in my opinion one of the best thinkers about QM and he is on par with late Asher Peres, Zurek, and late Viacheslav Belavkin.

2. See: http://fmoldove.blogspot.com/2015/05/interview-with-anti-quantum-zealot-on.html

"Regarding consistent histories, it is a bit inaccurate to lump it in with Copenhagen (at least I'll have to deal with another long email from Bob Griffith if I do so again). I think Omnes looks at it this way, but Griffith wants to view it as a realist interpretation, just with the "single-reality" criterion thrown out. It is more difficult to tell what Gell Mann and Hartle intend, particularly as they keep revising their interpretation by adding exotic probabilities and such like."

3. I have learned the meaning of Consistent Histories from Omnes - it's what makes sense to me. I've talked to Jim Hartle and he's more unclear about what he thinks about "Copenhagen" and other things.

But what can I do. The whole point of having consistent histories and decoherent histories is that one cannot ask about the truth value of the propositions 1-4 in your list simultaneously. For example, the statements 1 and 2 are inconsistent histories with each other (or they're not decoherent from each other) and they're not allowed in the interpretation at the same time.

This is true for all champions of the consistent histories approach.

Omnes probably realizes, just like I do, that this inconsistency of histories plays *exactly* the same role as Bohr's complementarity - it's really the same thing. Others may fail to realize it, I don't know, but it surely has the same effect on the resolution of Hardy's paradox.