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When a notion of energy coincides with some information

There are at least two interesting hep-th preprints today that have links to quantum information. Juan Maldacena wrote a nice gift to Andy Strominger's 60th birthday (happy birthday, Andy!)

A model with cosmological Bell inequalities
where he presents an example of an inflationary model where the violation of Bell's inequalities by quantum mechanics is observable on a natural enough quantity. If you want to know in advance, Juan talks about a singlet state constructed from isospin-doublet particles whose mass is variable and axion-dependent.

One of the virtues of cosmic inflation is that quantum fluctuations are stretched to astronomical or cosmological length scales and later become observable and "classical": The nonuniformities that were needed for galaxies to be born owe their origin to quantum fluctuations in the early Universe.

But all of quantum mechanics may be important – and very important, not just a small correction – in the early stages of the evolution of our Universe. You can see that the agreement with this elementary point is one of the things that distinguishes top physicists like Maldacena – in whose blood systems quantum mechanics circulates all the time – from bottom physicists like the "quantum foundations community" who surely prefer to look for wrong excuses for their incorrect claims that quantum mechanics can't be relevant in the Universe etc. (The last third of this blog post is another example, after all.)

The second paper I want to mention is
Canonical Energy is Quantum Fisher Information
where Lashkari and Van Raamsdonk argue in favor of the equivalence of one type of "energy" and one type of "information".

They look for a ball in a boundary conformal field theory in a holographic correspondence and construct the corresponding Rindler wedge in the bulk i.e. in the anti de Sitter space. The "information" is easily seen inside the boundary; the equivalent "energy" is defined in the bulk. Their relationship is another specific result in the line of research that visualizes the spacetime geometry (in this case the canonical energy which determines something about the curvature etc.) as some features of the quantum information (in this case the Quantum Fisher Information).

The information they discuss is called the Quantum Fisher Information and it may be defined using an intermediate concept, the relative entropy:\[

S(\rho||\sigma) = {\rm tr}(\rho\log\rho) - {\rm tr} (\rho\log \sigma).

\] This quantity is a form of "information" that measures how well the physical system – described by the density matrix \(\rho\) – is from a benchmark density matrix called \(\sigma\). Note that all expectation values and entropies are written as \({\rm tr}(\rho \dots)\) where the "dots" stand for something – either a linear operator or \(-\log \rho\), if we talk about the von Neumann entropy.

In this "relative entropy" case, the "dots" are defined as \(\log\rho-\log\sigma\). An interesting feature of this \(S(\rho||\sigma)\) is that it is positively definite (and zero only for \(\rho=\sigma\)). Try to prove it.

Also, we may Taylor expand the relative entropy in the vicinity of \(\rho\approx \sigma\). The leading absolute term is zero. The linear term vanishes. And the first nontrivial, quadratic term defines a quadratic form on the space of variations of the density matrix \(\delta \rho\) around \(\rho=\sigma\). The corresponding quadratic function of \(\delta\rho\) is the Quantum Fisher Information.

They present the evidence that this quantity should be identified with the canonical energy in the bulk.

You may protest because the energy and the information have different units; the energy comes in "joules" (or \(\GeV\)) while the information is dimensionless (or in "nats" which are really "one"). Well, there is no "unit problem" here because the canonical energy in the Rindler wedge is generated not by translations but by "boosts", the Minkowskian counterparts of rotations, and the generator is therefore dimensionless, too.

You may replace the "boost" by a "translation" if you are far enough from the fixed point of the boost. Then the conversion factor between the normal, joule-based energy and their canonical energy is nothing else than the temperature \(T\). So their claim, schematically \(E=S\), is really a version of \(\Delta E=TS\), a relationship you know from thermodynamics, with some "careful" definition of \(\Delta E\) and a careful special type of \(S\), too.

I want to mention one more paper that was posted 2 weeks ago,
Comment on "Universal decoherence due to gravitational time dilation"
by Bonder, Okon, and Sudarsky. They join my blog post published a month before them explaining why a recently hyped paper is wrong. That paper had claimed that to sit in the gravitational field is enough for decoherence which is enough for the quantum-to-classical transition.

First of all, I agree with their basic claim that the paper (recently published in Nature Physics) is wrong. Many details of their arguments – with the focus on the equivalence principle – slightly differ from my main criticisms but I do endorse over 90 percent of the Bonder et al. criticism, too. Also, I fully endorse their point that decoherence doesn't "localize" objects. It creates a statistical mixture of different values of a quantity, weighted by probabilities.

Those probabilities may be interpreted "classically" because the information about the relative phases disappears when decoherence occurs (it doesn't occur in this gravitational field context, however). But they're still probabilities and decoherence in no way turns the ambiguous results to unequivocal, deterministic ones. This basic point was contradicted by Pikovski et al. as well and it's good that someone else dares to criticize this widespread conceptual mistake.

Bonder, Okon, and Sudarsky are among the more reasonable people who write similar articles even though I would disagree at least e.g. with some "sentiments" behind their papers criticizing the consistent histories approach to quantum mechanics. I agree with them that the "consistent histories" don't make the departure of quantum mechanics from the objectivist, classical physics any more "explained" than before. But I disagree with the suggestions that there's really any outstanding problem left by either of the approaches, so the "failure" of the consistent histories to achieve "progress" is not a problem.

Their explicit support for objective collapse models is unfortunate, of course, but no surprise here – this whole "foundations" community is deluded about fundamental questions.

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