# About the logical foundations of the second law of thermodynamics

“The law that entropy always increases holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell’s equations – then so much the worse for Maxwell’s equations. If it is found to be contradicted by observation—well, these experimentalists do bungle things sometimes. But if your theory is found to be against the Second Law of Thermodynamics I can give you no hope; there is nothing for it to collapse in deepest humiliation.”

― Arthur Eddington, The Nature of the Physical World, Chap. 4

But why is that so? Why is the Second Law so “special” among the other laws of physics?

Simply because—as we argue in a paper recently published on Physical Review E and freely available on the arXiv—the Second Law is not so much about physics, as it is about logic and consistent reasoning. More precisely, we argue that the Second Law can be seen as the shadow of a deeper asymmetry that exists in statistical inference between prediction and retrodiction, and ultimately imposed by the consistency of the Bayes–Laplace Rule.

A little bit of background. In the past two decades, thermodynamics has undergone unprecedented progresses. These can be traced back to the developments of stochastic thermodynamics, on the one hand, and the theory of nonequilibrium fluctuations, on the other. The latter, in particular, has shown that the Second Law emerges from a more fundamental “balance relation” between a physical process and its reverse. According to such a balance relation, for example, scrambled eggs are not forbidden to unscramble spontaneously—instead, the probability of such a process is just extremely tiny, compared with that of its more familiar reverse. In turn, entropy—i.e. the thing that “no one knows what it really is”, according to the apocryphal exchange between Shannon and von Neumann—precisely is a measure of such a disparity.

In this paper we go one step further and show that the existence of a disparity is not due to some kind of “physical propensity” that irreversible processes have for unfolding in one direction more likely than in the opposite direction—an explanation that would lead to a circular argument—, but to the intrinsic asymmetry that exists between prediction and retrodiction in inferential logic. We thus conclude that the foundations of the Second Law are not to be found within physics, but one step below, at the level of logic.

A nice little piece written by CQT/NUS outreach is also available here.

# Colloquium on Bayesian retrodiction in statistical physics

One month ago I gave a colloquium at the 13th Annual Symposium of the Centre for Quantum Technologies (CQT) in Singapore. I decided to speak about my recent work on the role of Bayesian retrodiction in statistical mechanics — more precisely, in the conceptual foundations underlying fluctuation relations and the second law of thermodynamics.

The preprint is available on the arXiv: https://arxiv.org/abs/2009.02849

# Invited talk at AQIS 2020

Given at the Asian Quantum Information Science Conference 2020 (link).

# “Quantumness” always happens in time—and needs to be programmable

Incompatibility of quantum measurements lies at the core of nearly all quantum phenomena, from Heisenberg’s Uncertainty Principle, to the violation of Bell inequalities, and all the way up to quantum computational speed-ups.  Historically, quantum incompatibility has been considered only in a qualitative sense.  However, recently various resource-theoretic approaches have been proposed that aim to capture incompatibility in an operational and quantitative manner.  Previous results in this direction have focused on particular subsets of quantum measurements, leaving large parts of the total picture uncharted.

A work, which I wrote together with Eric Chitambar and Wenbin Zhou and was published yesterday on Physical Review Letters, proposes the first complete solution to this problem by formulating a resource theory of measurement incompatibility that allows free convertibility among all compatible measurements.  As a result, we are now able to explain quantum incompatibility in terms of quantum programmability; namely, the ability to switch on the fly between incompatible measurements is seen as a resource.  From this perspective, quantum measurement incompatibility is intrinsically a dynamical phenomenon that reveals itself in time as we try to control the system.

Read about this on Physical Review Letters or, for free, on the arXiv.

# Birkhoff-von Neumann Prize

I was delighted to learn that I was awarded with the “Birkhoff-von Neumann Prize” by the International Quantum Structures Association. I feel very honored and humbled — at once! — to join a list including such superb colleagues. Thank you very much!

Francesco Buscemi is Associate Professor at the Department of Mathematical Informatics of Nagoya University, Japan. His results solved some long-standing open problems in the foundations of quantum physics, using ideas from mathematical statistics and information theory. He established, in a series of single-authored papers, the theory of quantum statistical morphisms and quantum statistical comparison, generalizing to the noncommutative setting some fundamental results in mathematical statistics dating back to works of David Blackwell and Lucien Le Cam. In particular, Prof. Buscemi successfully applied his theory to construct the framework of “semiquantum nonlocal games,” which extend Bell tests and are now widely used in theory and experiments to certify, in a measurement device-independent way, the presence of non-classical correlations in space and time.

In such an occasion, it is impossible not to remember Professor Paul Busch, gentleman scientist, President of IQSA until his sudden death, of which I learned almost simultaneously with my award.

# The “No-Hypersignaling Principle”

An important consequence of special relativity, in particular, of the constant and finite speed of light, is that space-like separated regions in spacetime cannot communicate. This fact is often referred to as the “no-signaling principle” in physics.

However, even when signaling is in fact possible, there still are obvious constraints on how signaling can occur: for example, by sending one physical bit, no more than one bit of information can be communicated; by sending two physical bits, no more than two bits of information can be communicated; and so on. Such extra constraints, that by analogy we call “no-hypersignaling,” are not dictated by special relativity, but by the physical theory describing the system being transmitted. If the physical bit is described by classical theory, then the no-hypersignaling principle is true by definition. It is not so in quantum theory, where the validity of the no-hypersignaling principle becomes a non-trivial mathematical theorem relying on a recent result by Péter E. Frenkel and Mihály Weiner (whose proof, using the “supply-demand theorem” for bipartite graphs, is very interesting in itself).

As one may suspect, the no-hypersignaling principle does not hold in general: it is possible to construct artificial worlds in which the no-hypersignaling principle is violated. Such worlds are close relatives of the “box world,” a toy-model theory used to describe conceptual devices called Popescu-Rohrlich boxes. Exploring such alternative box worlds, one further discovers that the no-hypersignaling principle is logically independent of both the conventional no-signaling principle and the information causality principle, however related these two may seem to be with no-hypersignaling.

This means that the no-hypersignaling principle needs to be either assumed from the start, or derived from presently unknown physical principles analogous to the finite and constant speed of light behind Einstein’s no-signaling principle.

The paper was published on Physical Review Letters, but is also available free of charge on the arXiv.

# The Many Facets of the Information-Disturbance Tradeoff in Quantum Theory

Next Wednesday, I will be giving an invited lecture at the National Cheng Kung University in Tainan, Taiwan, about all that I’ve learnt concerning the information-disturbance tradeoff in quantum theory. Keeping a unified viewpoint, I will cover many aspects of the problem: from the difference between physical and stochastic reversibility, to qualitative “no information without disturbance” statements and quantitative balance equations, up to the two-observable approach à la Heisenberg.

Click the drawing above for the PDF.

# Quantum uncertainties defeat Laplace’s demon

I recently gave a colloquium at the Department of Applied Mathematics of Hanyang University in Ansan, Korea, in which I tried to introduce the idea of incompatibility of quantum measurements to students that were not all perfectly fluent in quantum theory.

Incompatibility, in the form of uncertainty relations, is available in many flavours: statistical and dynamical, variance-based and entropy-based, state-dependent and state-independent… As I was asked to share the slides, I’m now making them publicly available (click on the cover below):

The mechanical hybris is defeated!

# Quantum Lorenz curves and Hilbert α-divergences

In 1905, the American economist Max Lorenz introduced a way to graphically represent the concentration of wealth distribution in a country, what is now known as the country’s Lorenz curve. Since then, Lorenz curves have found uncountable applications in a wide range of quantitative sciences, ranging from mathematical statistics to biology and finance. Whenever discrete distributions (including not only probability distributions, but also assets portfolios or biodiversity indicators) appear in the modeling of a problem, Lorenz curves and related ideas such as majorization are likely to play a crucial role.

If wealth were quantum, how would you measure wealth concentration?

Quantum theory deals with objects, quantum states, that from many viewpoints resemble discrete distributions, but with the crucial difference of being non-commutative. In a paper published few days ago on Physical Review A, Gilad Gour and I generalize the definition of Lorenz curves to arbitrary pairs of quantum states, reconstructing the classical theory of majorization in the case commuting states, and discussing applications of this new tool in the emerging fields of quantum thermodynamics and quantum resource theories.

A tool that we introduce (and that may potentially be of general interest) is the family of divergences $H_\alpha(\rho\|\sigma)$ (for $\alpha$ varying between 1 and $\infty$) that we name “Hilbert α-divergences” due to their close kinship with Hilbert’s metric, and which interpolate between the trace-distance $\frac{1}{2} \|\rho-\sigma\|_1=\ln2\lim_{\alpha\to 1}H_\alpha(\rho\|\sigma)$ and the max-relative entropy $D_{\max}(\rho\|\sigma)=\lim_{\alpha\to \infty}H_\alpha(\rho\|\sigma)$.

F. Buscemi and G. Gour, Quantum Relative Lorenz Curves. Physical Review A, vol. 95, 012110 (2017). The paper is available in its journal version (paywall) and on the arXiv (free).