Non-Anthropocentric Natural Units for Area and Information in Holographic Entropy Bounds

In the paper's words

Information-theoretic bounds in gravitational physics are conventionally expressed in bits, a unit that presupposes two human choices: the practice of counting states by integers and the selection of logarithm base 2. This paper asks whether the holographic entropy bound can instead be stated entirely in observer-independent quantities. Working in Planck natural units (c = ℏ = G = k_B = 1), in which area is measured in Planck areas ℓ_P² and is therefore dimensionless, we take the Bekenstein–Hawking value $S = A/4$ (in nats) as the definition of the entropy of a region's horizon.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. We show that the bound $D \leq e^{A/4}$ on the Hilbert-space dimension $D$ of a finite region contains no base-dependent constant; that the bit is recoverable only as the derived, base-dependent quantity $S/\ln 2$ with conversion factor $1/\ln 2 = 1.442695$; and that every base-$b$ representation of entropy is a fixed constant multiple of the same invariant, so that no observable at fixed theory content distinguishes the invariant formulation from the conventional one.

   from Abstract

2. The base $e$ of the exponential is not chosen but forced by the requirement that entropy be additive under composition of independent systems: if two disjoint regions with areas $A_1$ and $A_2$ have independent Hilbert spaces of dimensions $D_1$ and $D_2$, the composite dimension is $D_{12} = D_1 D_2$, and the entropy $S = \ln D$ defined by the Bekenstein–Hawking relation $S = A/4$ is fixed in nats by geometry.

   from 1. Introduction

3. The conjecture this paper develops, following a prior companion paper [12], is that the entire content of the holographic bound can be restated using only the dimensionless entropy $S = A/4$, with the bit recovered as a derived, base-dependent quantity $b = S/\ln 2$.

   from 1. Introduction

4. Its relevance is that Shannon entropy there is handled as an abstract quantity obeying inequalities, independent of any particular logarithm base; homogeneous inequalities are invariant under rescaling of the log base, exactly the invariance structure we formalize in Section 4.3. [4] considers classical communication over a finite-dimensional quantum channel with memory using a separable-state input ensemble and local output measurements, and proposes algorithms for estimating and bounding the information rate of such setups, some based on so-called auxiliary channels, extending prior algorithms.

   from 2. Background and Related Work

5. The finite-dimensional quantum channel is the same algebraic setting in which the Hilbert-space dimension $D$ entering our definition of physical entropy lives; the supplied summary gives no further detail on the forms of the bounds. [5] describes the Gamma Ray Large Area Space Telescope (GLAST) Large Area Telescope (LAT), a pair-production high-energy ($\gt 20$ MeV) gamma-ray telescope built by an international partnership for a satellite launch in 2006, and the collaboration's construction of a balloon flight engineering model.

   from 2. Background and Related Work

6. This is the direct predecessor of the present paper; our contribution is to make the base-independence claim precise, to exhibit the bit as a derived quantity with explicit conversion factor, to prove the empirical-equivalence result of Section 4.3, and to analyze the integer-rounding problem, which [12]'s summary does not address.

   from 2. Background and Related Work

How it is done

1. This paper asks whether the holographic entropy bound can instead be stated entirely in observer-independent quantities.

   from Abstract

2. The holographic entropy bound offers an unusual opportunity.

   from 1. Introduction

3. The supplied summary is truncated and gives no further technical detail. [2] argues that the holographic bound in physics constrains the complexity of life, that the finite storage capability of information in the observable universe requires "protein linguistics" in the evolution of life, and reports finding that the evolution of the genetic code determines the variance of amino acid frequencies and genomic GC content among species, with the linguistic mechanism confirmed by experimental observations (the summary is cut off mid-sentence).

   from 2. Background and Related Work

4. The corresponding bound on the Hilbert-space dimension $D$ of the region is obtained by inverting the thermodynamic relation $S = \ln D$ (valid for a maximally mixed state on a $D$-dimensional Hilbert space): $$D \leq e^{S} = e^{A/4}.$$ This is the central object of the paper.

   from 3.2 The base-free entropy bound

What would falsify it

1. We show that the bound $D \leq e^{A/4}$ on the Hilbert-space dimension $D$ of a finite region contains no base-dependent constant; that the bit is recoverable only as the derived, base-dependent quantity $S/\ln 2$ with conversion factor $1/\ln 2 = 1.442695$; and that every base-$b$ representation of entropy is a fixed constant multiple of the same invariant, so that no observable at fixed theory content distinguishes the invariant formulation from the conventional one.

   from Abstract

2. We identify the rounding of $D$ to an integer as the one genuinely open conventional element and discuss limitations, failure modes, and falsification conditions.

   from Abstract

3. Section 6 discusses limitations, failure modes, and falsification conditions.

   from 1. Introduction

Where the claim stops

1. Second, the illustrative areas inherit their status as assumptions; all Results that depend on them do too.

   from 6. Discussion

2. Third, the derivation assumes the Bekenstein–Hawking relation $S = A/4$ as a definition; a reader who regards it as a derived result requiring quantum-gravity input will find our "base-free" claim conditional on that relation.

   from 6. Discussion

3. Fourth, the additivity argument for the nat (Section 3.2) shows the nat is distinguished among logarithm bases, but it does not show that entropy must be logarithmic at all; that assumption is inherited from the Bekenstein–Hawking framework.

   from 6. Discussion

Where to read

1. Abstract229 words · 1 min
2. 1. Introduction529 words · 2 min
3. 2. Background and Related Work956 words · 4 min
4. 3. Methods379 words · 2 min
5. 4. Analysis987 words · 4 min
6. 5. Results316 words · 1 min
7. 6. Discussion864 words · 4 min
8. 7. Conclusion180 words · 1 min