The MatterGravity Entanglement Hypothesis
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Abstract
I outline some of my work and results (some dating back to 1998, some more recent) on my mattergravity entanglement hypothesis, according to which the entropy of a closed quantum gravitational system is equal to the system’s mattergravity entanglement entropy. The main arguments presented are: (1) that this hypothesis is capable of resolving what I call the secondlaw puzzle, i.e. the puzzle as to how the entropy increase of a closed system can be reconciled with the asssumption of unitary timeevolution; (2) that the black hole information loss puzzle may be regarded as a special case of this second law puzzle and that therefore the same resolution applies to it; (3) that the black hole thermal atmosphere puzzle (which I recall) can be resolved by adopting a radically differentfromusual description of quantum black hole equilibrium states, according to which they are total pure states, entangled between matter and gravity in such a way that the partial states of matter and gravity are each approximately thermal equilibrium states (at the Hawking temperature); (4) that the Susskind–Horowitz–Polchinski stringtheoretic understanding of black hole entropy as the logarithm of the degeneracy of a long string (which is the weak string coupling limit of a black hole) cannot be quite correct but should be replaced by a modified understanding according to which it is the entanglement entropy between a long string and its stringy atmosphere, when in a total pure equilibrium state in a suitable box, which (in line with (3)) goes over, at strongcoupling, to a black hole in equilibrium with its thermal atmosphere. The modified understanding in (4) is based on a general result, which I also describe, which concerns the likely state of a quantum system when it is weakly coupled to an energybath and the total state is a random pure state with a given energy. This result generalizes Goldstein et al.’s ‘canonical typicality’ result to systems which are not necessarily small.
Keywords
Mattergravity entanglement Information loss String theory approach to black hole entropy Gravitational decoherence Second law of thermodynamics Canonical typicality1 The Second Law Puzzle
There are longestablished and wellknown arguments—see the discussion of ‘branch systems’ in [1] as also reviewed e.g. in [2])—that other statements of the second law, in terms of what can and cannot happen with heat engines, refrigerators etc. follow from the above statement. As also explained in these references, the above statement leads to an explanation of time asymmetry; i.e. why, for example, it is commonplace to observe wineglasses fall off tables and smash into pieces, but we never see lots of smashed pieces assemble themselves into wineglasses and jump onto tables (Fig. 1).The entropy of the universe begins low and increases monotonically.
But how do we define the entropy of a closed system? And why does it increase?
A standard way of answering this (essentially due to Boltzmann around 1870) might be to consider for example what will happen if one starts with a system of N gas molecules in the left half of a box (see Fig. 2) and removes a partition, allowing the particles to diffuse into the right half of the box.
However, this definition of entropy and this argument for its increase depends, unsatisfactorily, on the need to make judgments about what we can distinguish. For example, if (see again Fig. 3) after previously ignoring such fine distinctions, we were to take the view that we can distinguish a state where, say, 48% of the particles are in the left half of the box and 52% in the right half from a state with roughly equal proportions^{2} then, at times for which the system’s microstate lies in the accordinglydefined new macrostate (obviously a subregion of the previously discussed large macrostate) then Eq. (1) would ascribe a different value to the entropy.
Moreover, this unsatisfactory arbitrariness and vagueness in the definition of entropy is even more of a problem if we want to account for the version of the second law with which we began. For we are not even present to make any distinctions in the early universe!
 This partial trace is characterized by the property that, if O is a (selfadjoint) operator on \(\mathcal {H}_\mathrm {A}\), then$$\begin{aligned} \mathrm{tr}(\rho _\mathrm {A} O)_{\mathcal {H}_\mathrm {A}}= \langle \varPsi (O\otimes I)\varPsi \rangle _{\mathcal {H}_{\mathrm {total}}}. \end{aligned}$$
 Both reduced density operators have equal von Neumann entropies:and this common value is often known as the A–B entanglement entropy of the total statevector \(\varPsi \).$$\begin{aligned} S^{\mathrm {vN}}(\rho _A)=S^{\mathrm {vN}}(\rho _\mathrm {B}) \end{aligned}$$(3)

It only offers a notion of entropy for open systems.

There are lots of ways of decomposing a given \(\mathcal H\) as \(\mathcal {H}_\mathrm {A}\otimes \mathcal {H}_\mathrm {B}\). How we choose to decompose it depends on subjective choices and, again, we are not around in the early universe to make those choices.
In support of this, we note that the decomposition has to be meaningful throughout the entire history of the universe: E.g. we could not identify A with photons and B with nuclei + electrons because these notions are not even meaningful until the photon epoch. We content ourselves, though, with going back to just after the Planck epoch; we assume that a lowenergy quantum gravity theory holds there and throughout the entire subsequent history of the universe and that this is a conventional (unitary) quantum theory with \(\mathcal {H}=\mathcal {H}_{matter}\otimes \mathcal {H}_{gravity}\). We will also assume that the initial degree of mattergravity entanglement is low. (We leave it for a future theory of the prePlanck era to explain that.)
These assumptions then appear to be capable of offering an explanation of the second law in the form stated at the outset since one can argue that an initial state with a low degree of mattergravity entanglement will, because of mattergravity interaction, get more entangled, plausibly monotonically, as time increases. At least the question of whether the second law holds becomes a question which, in principle, can be answered mathematically once we specify the (lowenergy) quantum gravity Hamiltonian (i.e. the generator of the unitary timeevolution) and the initial state. What we have called the second law puzzle would then be resolved because once we define entropy as mattergravity entanglement entropy (rather than as the von Neumann entropy of the total state) there is no conflict between its increase and a unitary timeevolution.
2 The Information Loss Puzzle (Hawking 1976)
As Hawking explained in that work, one expects that such a radiating black hole will lose mass, increasing further its temperature, and eventually evaporate.
During this whole process of collapse to a black hole and subsequent evaporation, one expects the entropy of the total system to increase monotonically.^{3}
The version of the information loss puzzle [8] that I shall adopt here is the puzzle as to how this entropy increase can be reconciled with an assumption of unitary time evolution.
Stated in this way, I think it is clear that the information loss puzzle is nothing but a special case of our Second Law Puzzle; we recall here that this is the puzzle that, if one equates \(S^\mathrm {physical}\) with \(S^{\mathrm {vN}}(\rho _\mathrm {total}\)), then \(S^\mathrm {physical}\) must be constant.
I suggested in [3, 4] that the resolution to the information loss puzzle is simply the special case of the above proposed resolution to the second law puzzle. Namely, \(S^\mathrm {physical}\) is not \(S^{\mathrm {vN}}(\rho _\mathrm {total})\). Rather \(S^\mathrm {physical}\) is the total state’s mattergravity entanglement entropy. As I already said in the more general context in Sect. 1, this is not a unitary invariant and—it is reasonable to assume—would increase, thus offering to resolve the puzzle. That it also offers this resolution to the information loss puzzle lends, is, in my view, further evidence that our mattergravity entanglement hypothesis is on the right track.
3 The Thermal Atmosphere Puzzle

It is the entropy of the gravitational field (so mostly ‘residing’ in the black hole).

It is the entropy of the thermal atmosphere (so apart from the graviton component, consisting mainly of matter).

It is the sum of the above two entropies.
We further suggest, in line with our mattergravity entanglement hypothesis, that \(S^\mathrm {Hawking}\) is really this state’s mattergravity entanglement entropy. This offers to resolve the puzzle in the following way: The first entropy can be regarded, according to the environment paradigm, as the entropy of the open system consisting of the gravitational field due to its matter environment; the second the entropy of the open system consisting of the matter due to its gravity environment. But, by (3), these are actually equal and so, in this environmentparadigm sense, both statements are therefore true, without contradiction. On the other hand, there is no reason why the third statement should be true in any sense and in fact, on our hypothesis it is clearly not true—the total entropy being, by (4) not the sum of the first two, but rather, equal to each of them.
The fact that it seems capable of providing this resolution to the thermal atmosphere puzzle provides further support for the validity of our mattergravity entanglement hypothesis.
4 The Weak StringCoupling Limit of BlackHole Equilibrium States and Black Hole Entropy
Some of the most interesting work towards computing (in certain cases) or, at least, gaining a better understanding of, black hole entropy has been within string theory. Here I shall briefly recall the basic idea due to Susskind [11] and one particular line of development by Horowitz and Polchinski [12, 13] which leads to an explanation of how the entropy of spherically symmetric black holes scales with \(M^2\) (the square of the blackhole mass), albeit the argument is semiqualitative and does not tell us the constant term (so does not explain the factor of 1 / 4 in (6)).
The SHP argument [12, 13] is in two steps^{4}: First (see Fig. 6) one argues that, as one scales the string couplingconstant, g, down and the string length, \(\ell _s\) up, keeping Newton’s constant \(G=g^2\ell _s^2\) fixed, a black hole goes over to a long string. This will have density of states (i.e. number of states per unit energy, where we use \(\epsilon \) to denote energy) \(\sigma _\mathrm {long string}(\epsilon )\) approximately of the form of a constant times \(e^{\ell _s\epsilon }\).
Secondly, one equates the entropy, \(S_\mathrm {black hole}\), with “\(k\log (\sigma _\mathrm {long string}(\epsilon ))\)” \(=k\ell _s\epsilon \) at \(\epsilon =\) constant times M when \(\ell _s=\) constant times GM whereupon \(S_\mathrm {black hole} =\) constant times \(kGM^2\).
Our criticism of this is that it is not correct to equate an entropy with the logarithm of a density of states. (Nor indeed, in other string theory work, with the logarithm of a degeneracy—see [6, 15].) Indeed it only ever makes sense in physics to take the logarithm of a dimensionless quantity but a density of states has of course the dimensions of inverse energy!
I have demonstrated (see Sect. 5 for a discussion of the proof) that:
Theorem 1
For any pair of weakly coupled systems (to be called here ‘system’ and ‘bath’) with densities of states as in (7) a randomly chosen pure equilibrium state with total energy E will, with very high probability, have a systembath entanglement entropy approximately equal to \(k\ell _s E/4\). It will also be such that the reduced states of system and bath separately each have energy E / 2 and are each approximately thermal at temperature \(T=1/k\ell _s\)
Applying this theorem and reading ‘long string’ for ‘system’ and ‘stringy atmosphere‘ for ‘bath’ (or vice versa) and equating the black hole mass, M, with a constant times E and the entanglement entropy of this theorem with the mattergravity entanglement entropy of the black hole equilibrium state at \(\ell _s=\) constant times GM (as in the unmodified argument) the latter entropy will thus be a constant times \(kGM^2\). Thus we achieve a corrected string explanation of this formula for the black hole entropy which is not subject to the criticism we made of the original SHP approach. Moreover making the same substitution, \(\ell _s=\) constant times GM, the temperature formula for the reduced states of the long string and of its stringy atmosphere goes over to the temperature formula T = a constant times 1 / kGM, which agrees with the Hawking temperature formula (5) (up to a constant).^{5}
That ends my discussion of my mattergravity entanglement hypothesis and of how it offers a resolution to the three puzzles: the second law puzzle, the black hole information loss puzzle, and the thermal atmosphere puzzle and, finally, in this section, of how it enables a modification of the SHP string approach to black hole entropy which is free from the criticism^{6} which I made of the original SHP approach.
In the remainder of the talk I would like to supply some of the details about how I proved the above theorem.
5 Explanations of Thermality: Traditional and Modern
Theorem 1 in fact relies on a general theorem—which is stated below as Theorem 2—which I obtained [16] in a general setting where one has a total system (in [16] I abbreviate this with the the term ‘totem’ and I shall follow that terminology here) consisting of a (quantum) system weakly coupled to an energy bath.
5.1 Thermality in the Case the System is Small
The GLTZ explanation is itself a modern replacement for the earlier traditional explanation of the thermality of a small system in contact with a heat bath, so let me recall that first (Fig. 8).
The advantage of the “modern” over the “traditional” point of view is that it bases a theory of how systems get themselves into (approximate) Gibbs states on the same foundational assumption that we usually make for the foundations of quantum mechanics—namely that the total state of a full closed system is a pure (vector) state.
5.2 What Happens When System and Energy Bath are of Comparable Size?
One might think that one could apply the GLTZ result directly to the case our totem is the string equilibrium state illustrated in Fig. 7, identifying, say, the long string with our ‘system’ and the stringy atmosphere with our ‘energy bath’. However, neither of these can be regarded as small with respect to the other. Here we should clarify that ‘small’ in this context would mean having much more widely spaced energy levels, i.e. having a much lower density of states. Instead both densities of states are (ignoring the powerlaw prefactors I mentioned earlier) of the exponentially increasing form (7).
It turns out in general, that when the system and the energy bath are of comparable size, then—on both the traditional assumption of a totem microcanonical ensemble and the modern assumption of a random total pure state with energy in a small band—it is no longer necessarily the case that either system or energy bath will probably be in a thermal equilibrium state. However, I have shown [16] with regard to the modern approach:
Theorem 2
(And similarly with system \(\leftrightarrow \) energy bath).
But it is important to realize that when system and energy bath are of comparable size, \(\rho ^\mathrm {modapprox}_\mathrm {system}\) is not always thermal. (And neither, by the way, is the reduced state of the system thermal when the total state is in a traditional microcanonical ensemble.)
5.3 The Special Nature of Exponential Densities of States
However, it is shown in [16], regarding the modern approach^{7}
Theorem 3

\(\rho ^\mathrm {modapprox}_\mathrm {system}\) and \(\rho ^\mathrm {modapprox}_\mathrm {bath}\) are (close to^{8}) thermal at temperature \(T=1/k\ell _s\). (And each have mean energy E / 2.)

Also, the systemenergy bath entanglement entropy, S, \((=S^\mathrm {vN}(\rho ^\mathrm {modapprox}_\mathrm {system}) =S^\mathrm {vN}(\rho ^\mathrm {modapprox}_\mathrm {bath}))\) is approximately \(k\ell _s E/4\).^{9}
Theorem 1 of Sect. 4 clearly follows immediately from Theorems 2 and 3.
Footnotes
 1.
This article is a written version of a talk given at the 18th UK and European Conference on Foundations of Physics (16–18 July 2016, LSE, London)
 2.
These numbers were not entirely randomly chosen, the talk being given shortly after the June 2016 Brexit referendum.
 3.
Without wishing to imply that they are necessarily exactly additive, we note that while the entropy of the black hole (given by (6)) will decrease because the horizon area will decrease, one expects that this will be more than compensated by the increased entropy of the sphere of emitted Hawking radiation which is growing in size at the speed of light and within which, moreover, the later radiation will be hotter than that emitted earlier.
 4.
 5.
 6.
To provide further perspective on that criticism, let us recall that the attempt to provide a microscopic explanation of thermodynamical behaviour in terms of a classical statistical mechanics has often been criticized because it requires the introduction of an ad hoc quantity with the dimensions of action in order to provide a unit of volume in phase space. It has been said that this shortcoming of classical statistical mechanics is overcome in quantum statistical mechanics where a suitable power of the quantity \(\hbar \) effectively provides the right volume element. One might reexpress the main thesis of this section by saying that, in a similar way, the need to introduce an ad hoc dimensionful quantity as in the SHP approach to black hole entropy and the resolution of that difficulty along the lines explained in the main text indicates that, to have a satisfactory microscopic explanation of thermodynamical behaviour, a quantum statistical mechanics is also insufficient and what is needed, instead, is a quantumgravitational statistical mechanics based on our mattergravity entanglement hypothesis.
 7.
A similar result to Theorem 3 holds for the traditional (microcanonical) approach, except that (now neglecting logarithmic terms) in place of \(k\ell _s E/4\) one finds [16] that the system and the energy bath have entropy \(k\ell _s E/2\). The difference between these two results is interesting since it demonstrates that, in general, the traditional and modern approaches do not give the same results. (It is also interesting since the “right value for the Hawking entropy” mentioned in Footnote 5 depends on the denominator being 4—rather than 2).
 8.
See [16] for the sense in which these states are close to thermal.
 9.
The exact result [16, Endnote 29] is \(k\ell _s E/4 + k\log (c_\mathrm {S}c_\mathrm {B}E^2)/2  k(\log (c_\mathrm {S}/c_\mathrm {B}))^2/4E\).
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