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5 September (2026)
What I find strange, if you ask me, is that in the freedom one has to chose the atoms of the thermal measure, one can always select $C_{1/4}$ as one of them, and counterbalance this critical—most extreme—case, that brings you into non self-averaging and all, with some other moot element of the family. It shows that no matter how good is your champion in the team, the collective becomes moot if paired with average other members.
Life is a juggling act with a cerebral aneurysm and a heart attack.
3 September (2026)
🌑
Well I'd best go over there and do whatever that thing over there I was going to do was.— Dominic in The Banshees of Inisherin.
New month, old collaborations getting kicked up again...
With Camilo back on our crusade to define the criterion for SPS. We had to remember everything—again—and got to revive Camilo's beautiful analytical expression for the $g^{(2)}$ plateau with filtering the $N$-circular cascade, and this conundrum we keep forgetting and rediscovering of how does higher-order correlators factorize (or fail to) in terms of two-photon correlations. They are needed to work out probabilities that more than one photon is detected in the detection time of a physical detector. Hopefully this time we go somewhere. One new line running in parallel, though...
And I also finally fixed the placement of day & time-of-day (both standard and AF times). The code was indeed in the editing gadget so was lost for the rest of the world. Not that the rest of the world missed it very much but at least Daniel & Jacob should have it right now. They'll probably get confused as now it's really LIFO with time flowing up, as I always intended it. Scrolling down goes backward in time. The current day is always visible in its canonical position: in the past. I can call this a night.
From the after-midnight session (in Madrid; it's before Ju-Jitsu session time in Colombia), the HG → same distribution as LG was established by Daniel. There's some correspondence from one to the other, which I don't detail here as it's too voluminous, but it confirms that there is a single distribution that covers for those two basis. The gauging out of the phase removes their geometric distractions and they get pinned to the fundamental object, the same distribution for everybody. It remains to establish whether the quantum state gets transported from one basis to the other, e.g., thermal states of HG gives the same push-forwarded distribution than thermal LG; sounds reasonable, however how about, say, $C_\alpha=1/10$? is the quantum state the same? Daniel is looking into it. Plus generalization to all bases, plus to all photon numbers. And we can call it a day!
2 September (2026)
The transfer of date-pinning to caltoc is challenging me. It's tough to come back to things that were designed two months ago. Today's been a rotten day. Nothing substantial done...
Daniel's been answering the two questions I put to him, though. Reading him at least feels productive. I also asked him to help Matilde face this horrible machinery that is CSIC's administration, and he laid out for her a detailed document that the CSIC should put on their intranet. I've never seen anything so beautifully articulated in relation to their maze of horror.
Anyway, regarding the first question of 05MZV, he confirms that any two LG$_{\pm1}$ quantum states $C_\alpha$ and $C_\beta$ on the two-photon farfalle are rescaled versions of the same distribution: $${g_\alpha^{(2)}-1\over g_\beta^{(2)}-1}={C_\alpha\over C_\beta}\,.$$ So there is only one distribution, which can be stretched more or less. Or not at all, if $C_\alpha=0$ (not correlated), then $g_\alpha^{(2)}=1$. I believe this should be general, i.e., apply to $g^{(n)}$ on its $(n-1)$-dimensional farfalle.
Taking the most correlated case as a reference:
$$g_\alpha^{(2)}(\Delta\theta)=1+2C_\alpha\cos(4\ell\Delta\theta)$$
$C_\alpha={1\over4}$ is the critical value over which the state becomes quantum. He checked with two-mode squeezed vacuum, for which he finds:
$$g_\xi^{(2)}(\Delta\theta)=1+2{\bar n+{1\over2}\over2\bar n+{1\over2}}\cos(4\ell\Delta\theta)$$ with min $g^{(2)}$: $$\min g_\xi^{(2)}={2\bar n \over4\bar n+1}>0\,.$$
That settles nicely that the distribution touching impossibility is a particular case of quantumness: it implies it, but this is not necessary. That makes the contraposite of my theorem a bit moot: reduced to its extreme.
The proof is general an relies on cancellation of anomalous correlators through the phase re-alignment.
It's quite exciting because also the (related) second question gets answered positively. HG structures, once re-aligned, provide the same result: there is only one support and only one distribution on it. It only needs to be generalized (at least to $N>2$). Also on ℤ ⟶
With Daniel's meeting today I realized that browsing ℤ from month to month is inconvenient (also a bit slow, even locally), and that my nice layout of the current day being pinned at the bottom—to keep track of where one is in time—is not in effect for non-logged-in users (which is the rest of the world but me). So I have this highly crafted touch on how to look at time passing and it's only for my benefit. At least poor Daniel could have that, if it's only him using those notes. The problem is that I embedded that with the editor gadget rather than, say, the caltoc. I'll debug it later today.
For now, we need to check what happens with HG geometries on the farfadelles (my hunch is that it could be the same, the geometry will not transpire on the very robust multiphoton structure which we already know is the same, or in its distribution once homogenized). The other thing is whether *all* quantum states have the same distribution modulo some contrast. That would also be quite a thing. That would weaken our theorem that $g^{(n)}=0\implies$ no classical de Finetti representation, because there'd be essentially one case only that realizes the strictly impossible configurations: the maximally correlated one. All the others will spill beyond the classical limit (RPCS) but remain possible everywhere. So my obsession with impossibility would then be wrong: it doesn't need be impossible, just less likely than a critical threshold. Buh!
It's still nice but I wanted things to be impossible, not merely very difficult.
I initially made a statement on the different self-averaging of the variance (I retracted it now, not to confuse myself later, as I was incorrectly stating that variance doesn't self average). I was commenting that self-averaging properties of the variance could be notable and maybe related to spin ice and observables in strongly-correlated condensed matter, where fluctuations are the main concern. This still applies.
I had produced this beautiful figure, for fixed $N$ (hence not telling much about self-averaging):
The figure is correct but its interpretation wasn't: variance does self average.
I don't think I'll look much into the variance as the self-averaging of rays is enough for now. But still, for the record (and later comeback to this), every width self-averages only as $1/N$, while the donut's perimeter manages $1/N^2$ so there is some tension between them (which maybe is captured by self averaging of the mean, though).
1 September (2026)
And now for the big one of today: we introduce the ergodicity-breaking parameter, the relative scatter—across collapses—of the quantity estimated from one of them: \begin{equation}\label{eq:054CV}\mathrm{EB}=\mathrm{Var}(\overline{\delta^2})/\langle\overline{\delta^2} \rangle^2\,.\end{equation} This comes from standard the time-averaged ergodic theory. $\mathrm{EB}\to0$ is ergodic while $\mathrm{EB}\to$ a non-zero constant is broken ergodicity.
Transcribed to multiphotonics, and using the deficit $\delta$ again as well as its variance: \begin{equation} \mathrm{EB}(p_{\mathcal{C}}) =\frac{\mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})}{\langle\delta\rangle^2} =R_N, \qquad \mathrm{EB}(v_{\mathcal{C}}) =\frac{\mathrm{Var}_{\mathcal{C}}(v_{\mathcal{C}})}{E[v_{\mathcal{C}}]^2}, \label{eq:EB} \end{equation} Note that the second looks closer to the original tracking parameter, both being relative scatters of a second moment measured from one realisation.
As was the case for the related self-averaging, we have ergodicity at fixed $k$, as a function of $N$. But criticality breaks it for $k/N$, and the manifestation of this is a beautiful ladder: a factor $10$ increase of $N$ (and thus of $k$) results in a factor 10 decrease of the EB for non-critical cases. The critical case, however, remains essentially exactly constant:
To gloss for ergodicity-breaking parameter:
- Random Time-Scale Invariant Diffusion and Transport Coefficients. Y. He, S. Burov, R. Metzler and E. Barkai in Phys. Rev. Lett. 101:058101 (2008). → EB introduced in this form.
- Ergodic properties of fractional Brownian-Langevin motion. W. Deng and E. Barkai in Phys. Rev. E 79:011112 (2009). → EB for fractional Brownian motion.
- Strange kinetics of single molecules in living cells. E. Barkai, Y. Garini and R. Metzler in Physics Today 65:29 (2012). → popularization.
- Absence of Self-Averaging and Universal Fluctuations in Random Systems near Critical Points. A. Aharony and A. Harris in Phys. Rev. Lett. 77:3700 (1996).
- Finite-Size Scaling and Lack of Self-Averaging in Critical Disordered Systems. S. Wiseman and E. Domany in Phys. Rev. Lett. 81:22 (1998).
An interlude to collect exact results regarding perimeters $\langle p\rangle$ and deficits $\langle\delta\rangle$ of $k$-gons on the unit circle. Jacob was wondering about them. They are not needed to compute the exact deficit since this is obtained as an average of differences from chords to arcs, but, in principle they are needed for the ensemble-average limits. And it's nice to have them.
By definition, $\langle\delta\rangle=2\pi-\langle p\rangle$.
The cases $k=2$ and $k=3$ are accessible for all states $C_\alpha$: $$ \langle p\rangle=\frac{4k}{\pi}\Bigl(1-\frac{2C_\alpha}{15}\Bigr),$$
The donut $C_\alpha=0$ is also available for all $k$ as a series: $$\langle\delta\rangle_{\rm donut}(k) =\sum_{n\ge1}\frac{(-1)^{n+1}\,2\,\pi^{2n+1}} {(k+1)(k+2)\cdots(k+2n)}\,,\ \qquad k\ge2 $$ The leading term is $\langle\delta\rangle\simeq2\pi^3/[(k+1)(k+2)]$.
Off the donut, there are no exact results for $k\ge4$ but asymptotics are interesting: \begin{align} \langle\delta\rangle&\simeq\frac{2\pi^3}{k^2(1-4C_\alpha)^{3/2}}\,, \quad(C_\alpha<1/4), \\ \langle\delta\rangle&\simeq\frac{A}{k},\ \ A=5.698\,, \quad(C_\alpha=1/4). \label{eq:asympt} \end{align}
We see again criticality behaving qualitatively differently.
This is the closed-form solution of an integral which popped up in Daniel's latest computation of the volumes of $N=4$ and 5 farfadelles. He could compute everything exactly but this one:
$$K=\int_0^{\pi/2} x\cot x\,\ln(1+2\sin x)\,dx\,.$$
Claude (Fable), however, could compute it:
\begin{equation}\label{eq:05d6l}\begin{split}K=\frac{35\pi^3}{288}&+\frac{\pi}{4}\ln^2 2+\frac{\pi}{8}\ln^2(2+\sqrt3)+\frac{\ln(2+\sqrt3)}{2}\operatorname{Cl}_2\!\left(\frac{\pi}{3}\right)+\frac{\pi}{2}\operatorname{Li}_2\!\left(-\frac12\right)\\&-2\,\mathrm{Im}\!\left[\operatorname{Li}_3\!\left(\frac{1+i\tan\frac{\pi}{12}}{2}\right)+\operatorname{Li}_3\!\left(\frac{1+i\tan\frac{5\pi}{12}}{2}\right)\right]\end{split}\end{equation} with $\operatorname{Cl}_2$ the Clausen function and Li$_s$ the polylogarithm (so we have dilogarithms and trilogarithms here).
That makes $$K=0.7315710425072642965\ldots$$ which is what I remember from his numerical evaluation of it. It does rely on special functions—which always look like a cheat—but at least that should exclude other nicer functions. Continue on ℤ ⟶
Self-averaging is the last piece I need to assemble Natalia's paper. This is a property of quenched disordered systems, in our case, brought by the collapse. A quantity computed from one such frozen collapse is self-averaging if its collapse-to-collapse relative fluctuation vanishes as the sample grows, $$ R_N=\frac{\mathrm{Var}(X_N)}{E[X_N]^2}\longrightarrow0 \qquad(N\to\infty),$$ and strongly so if $R_N\propto1/N$.
This is the condition for one large-enough sample to represent the ensemble average (ergodicity), and repeating the experiment tells us nothing we didn't already have. When it fails, every sample gives a different picture and only an ensemble means anything. Right our Natalia's ergodicity business!
But now to make it precise and quantify when it holds or not.
For the perimeter itself, we trivially have self-averaging from Eq. (1) at fixed $k$ and large $N$, $$\mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})\simeq\frac{k^2\zeta_1}{N} \quad(\zeta_1>0), \qquad\qquad \mathrm{Var}_{\mathcal{C}}(p_{\mathcal{C}})\simeq \frac{k^2(k-1)^2\zeta_2}{2N^2}\quad(\zeta_1=0),$$ so $p_{\mathcal{C}}$ is strongly self-averaging off the donut and super-self-averaging at it—the standard deviation falling as $N^{-1}$ instead of $N^{-1/2}$. The only notable thing here is that the donut is much more self-averaging than the rest: this comes from its $\zeta_1=0$, i.e., rotation invariance, which says that no single photon carries information about the perimeter.
More than the perimeter, the deficit to that of the circle is more interesting, because instead of $2\pi$, the denominator becomes an increasingly smaller quantity.
The obvious self-averaging at fixed $k$ is not the most interesting: every $C_\alpha$ self-averages because adding photons to a collapse averages a fixed-size observable over ever more subsets. So we find self-averaging, although, again, with a different slope (super self-averaging for donuts):
It is more insightful to consider the self-averaging properties of, say, the biggest subset $\binom Nk$ which is when $k=\lfloor N/2\rfloor$. Or let's even generalize to any ray $k\propto N$. Then we find a breakdown of self-averaging for critical cases ($C_\alpha=1/4$):
