Fine-Grained versus Coarse-Grained Information

Fine-grained entropy characterizes a declared quantum state on its complete algebra; coarse-grained entropy characterizes what remains after subsystems, correlations, or observables have been discarded. Under closed unitary evolution the complete von Neumann entropy cannot change. Radiation entropy can nevertheless rise because the radiation is a reduced system, not because the complete state has ceased to be pure.

Required background. Quantum Information supplies density operators, Schmidt decomposition, channels, and entropy inequalities used in the proof.

Helpful background. Semiclassical Gravity supplies the evaporating-background regime; Quantum Fields in Curved Spacetime supplies the Hawking state whose exterior restriction is considered; Causality and Global Structure supplies the regions on which algebras are defined.

Fine-grained entropy of the complete state

Let $\rho_X$ be a normalized density operator on a finite-dimensional regulator of the complete system $X$. Its von Neumann entropy is

$$S(X)_\rho=-\operatorname{Tr}(\rho_X\ln\rho_X).$$

The regulator makes the trace and spectrum well defined. In continuum quantum field theory, local algebras need not admit density matrices and entanglement entropy is ultraviolet divergent; a cutoff, subtraction, relative entropy, or algebraic replacement must then be stated. Nothing in the following finite-dimensional proof removes that requirement.

Proposition. If the complete system evolves as $\rho_X'=U\rho_XU^\dagger$ with $U$ unitary, then $S(X)_{\rho'}=S(X)_\rho$.

Proof. Unitary conjugation preserves the characteristic polynomial and hence every eigenvalue of $\rho_X$. Equivalently, for every positive integer $n$,

$$\operatorname{Tr}\!\left[(\rho_X')^n\right] =\operatorname{Tr}\!\left[U\rho_X^nU^\dagger\right] =\operatorname{Tr}(\rho_X^n).$$

Because $-x\ln x$ depends only on the eigenvalue $x$, summing it over the common spectrum proves the claim. The same argument preserves all Rényi entropies. This is a statement about the complete state and the complete algebra; it does not say that the entropy of each part is constant.

Entropy moves into correlations

Suppose $X=AB$ and the complete state is pure. A Schmidt decomposition gives

$$\lvert\Psi\rangle_{AB}=\sum_i\sqrt{\lambda_i}\, \lvert i\rangle_A\lvert i\rangle_B, \qquad \lambda_i\geq0, \qquad \sum_i\lambda_i=1.$$

The reduced operators $\rho_A=\operatorname{Tr}_B\lvert\Psi\rangle\langle\Psi\rvert$ and $\rho_B$ have the same nonzero eigenvalues $\{\lambda_i\}$. Therefore

$$S(AB)=0, \qquad S(A)=S(B), \qquad I(A{:}B)=S(A)+S(B)-S(AB)=2S(A).$$

A unitary can increase $S(A)$ while keeping $S(AB)$ zero by creating mutual information. The “missing” local information is encoded in correlations. The entropy identities and their black-hole use are reviewed with explicit subsystem assumptions in Harlow 2016, Appendix C; the distinction between Hawking’s coarse-grained answer and fine-grained radiation entropy is developed in Almheiri et al. 2021, §§ 3–5.

For evaporation, take $A=R(t)$, the radiation collected so far, and $B=B(t)$, the remaining black hole together with any other purifier. Unitary evolution of $RB$ permits $S(R)$ to rise while entanglement is created and later fall as the purifier becomes smaller. A monotonically increasing thermodynamic entropy can coexist with this behavior because it is defined after a different coarse graining.

Declaring a coarse-graining map

Coarse graining is not one unique operation. One possibility is a fixed CPTP map $\mathcal C$ from the microscopic state to the retained system. Examples include a partial trace or dephasing in a specified pointer basis,

$$\mathcal D(\rho)=\sum_j P_j\rho P_j,$$

where the orthogonal projectors satisfy $\sum_jP_j=I$. Once $\mathcal C$ is named, one may compare the microscopic entropy with $S(\mathcal C(\rho))$ only with care: the output may live on a different system.

A different construction assigns the maximum-entropy state compatible with selected expectation values. This is an inference rule and is generally nonlinear, so it must not automatically be called a quantum channel. For example, retaining both marginals of a bipartite state gives the maximum-entropy assignment $\rho_{AB}\mapsto\rho_A\otimes\rho_B$. Applied separately to $\lvert00\rangle$ and $\lvert11\rangle$ it changes nothing, but applied to their equal mixture it gives $I_{AB}/4$. Averaging first and inferring first therefore disagree. Its entropy cannot be smaller than that of the original state because the original state is one of the admissible candidates.

For finite-dimensional dephasing, $\mathcal D$ is unital and makes the eigenvalue vector more mixed, so $S(\mathcal D(\rho))\geq S(\rho)$. An arbitrary CPTP channel does not obey this inequality; amplitude damping can reduce entropy by transferring it to an environment. Thus “coarse graining raises entropy” needs either a specific unital conditional expectation or a maximum-entropy construction. The robust statement is narrower: discarding information can make the retained description more mixed, even when a dilation remains unitary.

Fine-grained entropy is also not the same as an observer’s ability to decode a state. A density matrix may have small entropy while the correlations that purify it are computationally inaccessible. Conversely, a thermal-looking set of low-point observables does not determine the full density matrix. Operational access, coarse observables, and microscopic purity are three distinct axes.

Thermodynamic entropy introduces still another choice: a macrostate and the ensemble of microstates compatible with it. Its growth may follow from repeatedly discarding correlations or from an autonomous kinetic approximation. That useful arrow of time does not contradict invariant fine-grained entropy, because the two quantities are functionals of different descriptions. A valid black-hole argument must not exchange them midway through an inequality.

Bell-pair counterexample

Start with $\lvert00\rangle$ and apply a Hadamard gate to $A$ followed by a controlled-NOT. The final state is $\lvert\Phi^+\rangle=(\lvert00\rangle+\lvert11\rangle)/\sqrt2$. Direct diagonalization gives

$$S(AB)=0, \qquad S(A)=S(B)=\ln2, \qquad I(A{:}B)=2\ln2.$$

The complete unitary has destroyed no information. If instead one dephases $AB$ in the computational basis, the state becomes

$$\rho_{\mathrm{cl}}=\tfrac12\lvert00\rangle\langle00\rvert +\tfrac12\lvert11\rangle\langle11\rvert.$$

Now $S(AB)=\ln2$, both marginals still have entropy $\ln2$, and $I(A{:}B)=\ln2$. Dephasing removed the phase correlation while preserving a classical correlation. A maximally mixed replacement gives $I(A{:}B)=0$ and $S(AB)=2\ln2$. These three states have identical one-qubit marginals but different complete information. They are a counterexample to any attempt to infer fundamental loss from radiation thermality alone.

Limits of the proposition

The proof assumes one normalized complete state, a unitary on that same system, and a regulator for spectral entropies. It does not establish that gravity factorizes exactly into inside and outside factors, that a semiclassical radiation algebra is complete, or that entropy is operationally measurable within the evaporation time. Generic entropy theory belongs to Quantum Information; continuum renormalization belongs to quantum field theory; stable thermodynamic entropy belongs to Semiclassical Gravity. Explicit microscopic string and holographic constructions are outside this comparison.

Exercise

For the three-qubit GHZ state $\lvert\mathrm{GHZ}\rangle=(\lvert000\rangle+\lvert111\rangle)/\sqrt2$, compare $S(ABC)$, $S(A)$, and $S(AB)$. Where is the information that is absent from each one-qubit marginal?

Solution

The complete state is pure, so $S(ABC)=0$. Tracing over $BC$ gives $I_A/2$, hence $S(A)=\ln2$. Tracing over $C$ gives the equal mixture of $\lvert00\rangle$ and $\lvert11\rangle$, hence $S(AB)=\ln2$. The phase distinguishing the GHZ superposition from the corresponding classical mixture is stored in a genuinely three-party correlation. No individual marginal contains it, although the complete state does.

References

• Almheiri, Ahmed, Thomas Hartman, Juan Maldacena, Edgar Shaghoulian, and Amirhossein Tajdini. 2021. “The Entropy of Hawking Radiation.” Reviews of Modern Physics 93: 035002. DOI.
• Harlow, Daniel. 2016. “Jerusalem Lectures on Black Holes and Quantum Information.” Reviews of Modern Physics 88: 015002. Open text.