Harold Ollivier has put out a nice paper generalizing my best result:
Ollivier does a good job of summarizing why there is an urgent need to find a way to identify objectively classical variables in a many-body system without leaning on a preferred system-environment tensor decomposition. He also concisely describes the main results of my paper in somewhat different language, so some of you may find his version nicer to read.a
From exact to approximate case
My paper concentrated on the (simple) case of projective measurements, exact records, and pure states, which in this blog post I will call “the exact case”. Ollivier considers the more general case of POVM measurements, approximate records, and mixed states, which I will call “the approximate case”.
Ollivier argues convincingly for the following generalization of the redundantly recorded condition to the approximate case: Given a state , a POVM on a spatial region is said to -approximately record another POVM on a disjoint region when
for all values of the index (which labels the operators in the POVM). His Lemma 1 shows that this equivalent to redundant records in the exact case.
I am somewhat less confident in the naturalness of the specific definition he uses to generalize the notion of compatibility — that objective classical observables should commute when acting the wavefunction — to the approximate case. In part, this is because I haven’t thought about it enough. After considering and discarding the alternative notions of commutativity, non-disturbance, and coexistence, Ollivier settles on the criterion of -approximate joint-measurability as a way to generalize the notion of compatibility. A set of POVMs is -approximately jointly measurable if there exists a single POVM from which they can all be recovered (up to errors of size ). That is, given a state , then there exists a POVM such that for all we have
where is, for each fixed and , some probability distribution over . In other words, one can sample from each by first sampling from and then, conditional on the outcome , sampling from .
One concern I have: Ollivier is explicitly motivated by questions of what observers can measure and consequently eschews more restrictive conditions relating to the (approximate) commutativity of observables. In contrast, I’m in search of a formulation of quantum mechanics that treats observers as just another physical system. Thus, the dynamical question of approximate commutativity of observables might interest me more. But I don’t have a strong opinion.
Theorems 1 & 2: Compatibility of candidate variables
Ollivier first provesb in his Theorem 1 that two sets of POVMs that are -approximately redundantly recorded are -approximately jointly measureable if they are not pair-covering. He also cites Heinosaari and Wolf to show that joint-measurability is equivalent to compatibility in the case of projective measurements and . Thus, he generalizes my main result from the exact to mixed case for the situation when there are only two candidate classical variables.
However, unlike my main result in the exact case, his Theorem 1 cannot be directly extended to an arbitrary number of candidate classical variables. The reason is that pairwise joint measurability does not imply global joint measurability, as previously shown by Heunen, Fritz, & Reyes.
So instead, Ollivier considers the condition of “non tuple-covering”c : this mean that given an arbitrary number of sets of POVMs, and given a primary choice of one POVM (corresponding to one spatial region) from each set, it most be possible to make a secondary choice of one POVM from each set that does not spatially overlap with any of the primary or secondary choices. In his Theorem 2, Ollivier shows that an arbitrary number of sets of POVMs, where each are -approximately redundantly recorded, is jointly measurable if the regions associated with the sets of POVMs are non tuple-covering. In this sense he generalizes the main result from my paper (for any number of candidate classical variables), except that he must assume the condition non tuple-covering instead of non pair-covering.
Theorem 3: Re-framing the Darwinian channel theorems
Next Ollivier says
The non pair-covering condition has an appealing property of being rather simple and allowing the recovery of objectivity for usual many-body physics experiments: pair-covering is too delicate to maintain for macroscopic systems containing possibly millions or billions of microscopic sites so that they would necessarily be exhibiting only usual classical properties.
On the contrary, the non tuple-covering seems a more complex, if not harder, condition to achieve. This, in turn, weakens considerably the above argument and, as a consequence, the reach of quantum Darwinism for quantum many-body systems. Yet, we prove below that this is not the case, and that quantum Darwinism is a ubiquitous mechanism to explain the emergence of a single set of approximately jointly-measurable POVMs
However, the theorem he goes on the prove does not, as far as I can tell, achieve this.
The framing of Theorem 1 from Qi & Ranard [2001.01507] (a strengthening of Brandao, Piani, & Horodecki [1310.8640]) in terms of maps (as opposed to states, as in his Theorem 2) is, in my opinion, best motivated by the intuition that certain microscopic degrees of freedom get amplified destructively to macroscopic levels such that they may leave records all over the place without the system that hosted the original information surviving, e.g., the electron in a Stern-Gerlach experiment. (In other words, non-demolition measurements are rare.) The use of CP maps in particular is justified by the assumptions that (1) the system that is amplified begins unentangled with the rest of the universe and (2) the system that we can feasibly analyze is smaller than the entire universe.
Qi & Ranard’s Theorem 1 says, essentially: “Outside of a Markov blanket, subsets of are in states determined by the results of classical information about that could be inferred from measurements in some basis/POVM”. Ollivier’s Theorem 3 says, essentially: “Outside of a Markov blanket, the results of measurements on subsets of are inferable by the results of measurements on the Markov blanket”. I would have thought Ollivier’s Theorem 3 could be derived starting from Qi & Ranard’s Theorem 1, but Ollivier looks to have proved it, using their techniques, starting from Proposition 1 (which I guess gives a tighter bound than if he has worked from the statement of the Theorem 1). I haven’t read the proof carefully.
What I don’t understand is what this means for the existence of a preferred set of classical variables. Ollivier claims that Theorem 3 is a way for making up for the fact that his Theorem 2 relies on a stronger assumption of “non tuple-covering”, which he argues is forced by the move to POVMs and mixed states, but which is not as well justified as my “non pair-covering” assumption. But I really don’t follow his reasoning at all on why his Theorem 3 helps.
In the concluding section, Ollivier writes:
The last section shows that generic evolutions of quantum many-body systems do systematically generate Markov blankets that capture all correlations between fragments of . As a consequence, measurement results obtained by observers measuring fragments of outside Markov blankets can be explained using classical correlations only. This implies that the non tuple-covering condition is generically satisfied for all partitions of that contain the Markov blanket.
Although the non tuple-covering condition implies that measurements of fragments can be explained using only classical correlations, the converse isn’t true. (There exists partitions of that are tuple-covering yet which, for some states, features only classical correlations.)
I suspect by “generically”, Ollivier must be arguing that fragments accessed by realistic observers usually do not overlap with the Markov blanket. Again, this wouldn’t guarantee non tuple-covering, but it would at least guarantee classical correlations in the fragments.
However, I don’t really see how we could argue in this way to conclude that there exists a preferred (and jointly measurable) set of objective properties. Any given many-body system will have many possible subsystems so that, for a given evolution from an initial time to a later time, there will be many CPTP maps that correspond to the reduced dynamics from each subsystem at the initial time to part/all of the many-body system at the later time. Is Ollivier’s claim that none of these are likely to have Markov blankets that overlap with a given observer’s fragment? If so, why is this so and can it be made quantitative? If not, how are we to rule out most of these other CPTP maps without essentially smuggling in a preferred-subsystem assumption whose avoidance is (I think) a goal of this work?
Here is a slightly different way to state my concern: How could we use Theorem 3 to identify a preferred, jointly measurable set of all objective properties without appeal to a preferred subsystem? In the case of a pure state and perfect records, could we identify preferred wavefunction branches?
(↵ returns to text)
- A minor quibble: Although this is of course a matter of taste, I disagree that the Shor code example was the “core of the main result” of my paper. In my opinion, the key idea was that there was a sensible way of defining redundancy at all in a way that allowed for proving statements about compatibility without recourse to a preferred non-microscopic tensor structure. The Shor-code example is more important for showing the limits of what redundancy can tell you (which is saturated in a weak sense).↵
- For reasons I don’t understand, Ollivier’s Theorem 1 (but not Theorem 2) is stated in terms of a set of density matrices, rather than a single state, but the theorem does not seem to make non-trivial use of it: Theorem 1 could be re-written in terms of a single state and, if the hypothesis applies to each , then so would the conclusions, of course. The importance of preferred sets of states only arises later in the paper.↵
- The “non tuple-covering” might be misleadingly named because, I think, the non tuple-covering condition does not imply the non pair-covering condition, even in the case of two redundantly recorded observables and . To see this, suppose there are qubits and the regions for the observables and are, respectively, and . pair-covers with the pair of records on and , but and are are non tuple-covering.↵