When two initially uncorrelated quantum systems interact through a weak coupling, no entanglement is generated at first order in the coupling constant. This is a useful and very easy to prove fact that I haven’t seen pointed out anywhere, although I assume someone has. I’d love a citation reference if you have one.
Suppose two systems and evolve under where the Hamiltonian coupling them is of the form
with and as usual. We’ll show that when the systems start out uncorrelated, , they remain unentangled (and therefore, since the global state is pure, uncorrelated) to first order in . First, note that local unitaries cannot change the entanglement, so without loss of generality we can consider the modified unitary
which peels off the unimportant local evolution of and . Then the Baker–Campbell–Hausdorff formula gives
where the first few ‘s are given by
The key feature here is that every commutators in each of the ‘s contains at least one copy of , i.e. all the ‘s are at least first order in . That allows us to write
for some new that is independent of . Then we note just that a general Hamiltonian cannot produce entanglement to first order:
This is potentially a very important (negative) result when considering decoherence detection of very weakly coupled particles. If the coupling is so small that terms beyond first order are negligible (e.g. relic neutrinos), then there is no hope of being sensitive to any decoherence.
Of course, non-entangling (unitary) effect may be important. Another way to say this result is: Two weakly coupled systems act only unitarily on each other to first order in the coupling constant.… [continue reading]
This is a follow up on my earlier post on contextuality and non-locality. As far as I can tell, Spekken’s paper is the gold standard for how to think about contextuality in the messy real world. In particular, since the idea of “equivalent” measurements is key, we might never be able to establish that we are making “the same” measurement from one experiment to the next; there could always be small microscopic differences for which we are unable to account. However, Spekken’s idea of forming equivalence classes from measurement protocols that always produce the same results is very natural. It isolates, as much as possible, the inherent ugliness of a contextual model that gives different ontological descriptions for measurements that somehow always seem to give identical results.
I also learned an extremely important thing in my background reading. Apparently John Bell discovered contextuality a few years before Kochen and Specker (KS).This is according to Mermin’s RMP on contextuality and locality. I haven’t gone back and read Bell’s papers to make sure he really did describe something equivalent to the KS theorem.a More importantly, Bell’s theorem on locality grew out of this discovery; the theorem is just a special case of contextuality where “the context” is a space-like separated measurement.
So now I think I can get behind Spekken’s idea that contextuality is more important than non-locality, especially non-locality per se. It seems very plausible to me that the general idea of contextuality is driving at the key thing that’s weird about quantum mechanics (QM) and that — if QM is one day more clearly explained by a successor theory — we will find that the non-local special case of contextuality isn’t particularly different from local versions.… [continue reading]
I’m still trying to decide if I understand this correctly, but it looks like coherent wavepacket spreading is sufficient to produce states of a test-mass that are highly sensitive to weak forces. The Wigner function of a coherent wavepacket is sheared horizontally in phase space (see hand-drawn figure). A force that perturbs it slightly with a small momentum shift will still produce an orthogonal state of the test mass.
The Gaussian wavepacket of a test mass (left) will be sheared horizontally in phase space by the free-particle evolution governed by
. A small vertical (i.e. momentum) shift by a weak force can then produce an orthogonal state of the test mass, while it would not for the unsheared state. However, discriminating between the shifted and unshifted wavepackets requires a momentum-like measurement; position measurements would not suffice.
Of course, we could simply start with a wavepacket with a very wide spatial width and narrow momentum width. Back when this was being discussed by Caves and others in the ’80s, they recognized that these states would have such sensitivity. However, they pointed out, this couldn’t really be exploited because of the difficulty in making true momentum measurements. Rather, we usually measure momentum indirectly by allowing the normal free-particle () evolution carry the state to different points in space, and then measuring position. But this doesn’t work under the condition in which we’re interested: when the time between measurements is limited.The original motivation was for detecting gravitational waves, which transmit zero net momentum when averaged over the time interval on which the wave interacts with the test mass. The only way to notice the wave is to measure it in the act since the momentum transfer can be finite for intermediate times.… [continue reading]
Back in December Gell-Mann and Hartle (G&H) posted their latest paper on consistent histories, “Adaptive Coarse Graining, Environment, Strong Decoherence, and Quasiclassical Realms”. Here are my thoughts.
The discussion of adaptive coarse graining was brief and very much in agreement with previous work.
G&H then give a name and formal description to the idea, long part of the intuitive lore, of a history being defined by the values taken by a particular variable over many time step. (This might be the position of an object, which is being recorded to some accuracy by a environment that decoheres it.) The key idea is that all the Schrödinger-picture projectors at different times defining the history commute:
This they call the narrative condition. From it, one is able to define a smallest set of maximal projectors (which they call a common framework) that obey either or for all . For instance, if the ‘s are onto spatial volumes of position, then the ‘s are just the minimal partition of position space such that the region associated with each is fully contained in the regions corresponding to some of the , and is completely disjoint from the regions corresponding to the others.
For times steps
, the history projectors at a given time project into subspaces which disjointly span Hilbert space. The narrative condition states that all history projectors commute, which means we can think of them as projecting onto disjoint subsets forming a partition of the range of some variable (e.g. position). The common framework is just the set of smaller projectors that also partitions the range of the variable and which obey
… [continue reading]
I wanted to understand Rob Spekkens’ self-described lonely view that the contextual aspect of quantum mechanics is more important than the non-local aspect. Although I like to think I know a thing or two about the foundations of quantum mechanics, I’m embarrassingly unfamiliar with the discussion surrounding contextuality. 90% of my understanding is comes from this famous explanation by David Bacon at his old blog. (Non-experts should definitely take the time to read that nice little post.) What follows are my thoughts before diving into the literature.
I find the map-territory distinction very important for thinking about this. Bell’s theorem isn’t a theorem about quantum mechanics (QM) per se, it’s a theorem about locally realistic theories. It says if the universe satisfies certain very reasonable assumption, then it will behave in a certain manner. We observe that it doesn’t behave in this manner, therefore the universe doesn’t satisfy those assumption. The only reason that QM come into it is that QM correctly predicts the misbehavior, whereas classical mechanics does not (since classical mechanics satisfies the assumptions).
Now, if you’re comfortable writing down a unitarily evolving density matrix of macroscopic systems, then the mechanism by which QM is able to misbehave is actually fairly transparent. Write down an initial state, evolve it, and behold: the wavefunction is a sum of branches of macroscopically distinct outcomes with the appropriate statistics (assuming the Born rule). The importance of Bell’s Theorem is not that it shows that QM is weird, it’s that it shows that the universe is weird. After all, we knew that the QM formalism violated all sorts of our intuitions: entanglement, Heisenberg uncertainty, wave-particle duality, etc.; we didn’t need Bell’s theorem to tell us QM was strange.… [continue reading]
[This is akin to a living review, which may improve from time to time. Last edited 2023-4-10.]
This post will summarize the various consistency conditions that can be found discussed in the consistent histories literature. Most of the conditions have gone by different names under different authors (and sometimes even under the same author), so I’ll try to give all the aliases I know; just hover over the footnote markers.
There is an overarching schism in the choice of terminology in the literature between the terms “consistent” and “decoherent”. ManyI used to say “most authors” here, but some quick Google Scholar searches suggest “consistent” is now more popular than “decoherent”, and that might always have been true.a authors, including Gell-Mann and Hartle, now use the term “decoherent” very loosely and no longer employ “consistent” as an official label for any particular condition (or for the formalism as a whole). Zurek and I believe this is a significant loss in terminology, and we are stubbornly resisting it. In our recent arXiv offering, our rant was thus:
…we emphasize that decoherence is a dynamical physical process predicated on a distinction between system and environment, whereas consistency is a static property of a set of histories, a Hamiltonian, and an initial state. For a given decohering quantum system, there is generally a preferred basis of pointer states [1, 8]. In contrast, the mere requirement of consistency does not distinguish a preferred set of histories which describe classical behavior from any of the many sets with no physical interpretation.
(See also the first footnote on page 3347 of “Classical Equations for Quantum Systems”Gell-Mann and Hartleb which agrees with the importance of this conceptual distinction.)… [continue reading]
andrelaszlo on HackerNews asked how someone could draw a reasonable distinction between “direct” and “indirect” measurements in science. Below is how I answered. This is old hat to many folks and, needless to say, none of this is original to me.
There’s a good philosophy of science argument to be made that there’s no precise and discrete distinction between direct and indirect measurement. In our model of the universe, there are always multiple physical steps that link the phenomena under investigation to our conscious perception. Therefore, any conclusions we draw from a perception are conditional on our confidence in the entire causal chain performing reliably (e.g. a gravitational wave induces a B-mode in the CMB, which propagates as a photon to our detectors, which heats up a transition-edge sensor, which increases the resistivity of the circuit, which flips a bit in the flash memory, which is read out to a monitor, which emits photons to our eye, which change the nerves firing in our brain). “Direct” measurements, then, are just ones that rely on a small number of reliable inferences, while “indirect” measurements rely on a large number of less reliable inferences.
Nonetheless, in practice there is a rather clear distinction which declares “direct” measurements to be those that take place locally (in space) using well-characterized equipment that we can (importantly) manipulate, and which is conditional only on physical laws which are very strongly established. All other measurements are called “indirect”, generally because they are observational (i.e. no manipulation of the experimental parameters), are conditional on tenuous ideas (i.e. naturalness arguments as indirect evidence for supersymmetry), and/or involve intermediary systems that are not well understood (e.g.
… [continue reading]