Entropy of Radiation Subsystems

“Entropy of Hawking radiation” is incomplete until the radiation degrees of freedom and everything traced out have been named. In a bath model, radiation may be a spatial region of a nongravitating system. In asymptotically flat evaporation, it may be an algebra of outgoing modes collected up to a retarded time. The remaining black hole, uncollected fields, reference systems, and gravitational sectors then determine the purifier.

Required background. Quantum Information supplies partial traces, Schmidt decomposition, Rényi entropy, mutual information, and entropy inequalities used in the proof.

Helpful background. Semiclassical Gravity supplies the evaporation regime and entropy renormalization context; Quantum Fields in Curved Spacetime supplies the outgoing-mode algebra and Hawking state; Black-Hole Information supplies the complete-system questions to which the radiation entropy is applied.

Declaring the radiation system

With a finite-dimensional regulator, suppose the complete state lies on

$$\mathcal H_X=\mathcal H_R\otimes\mathcal H_C,$$

where $R$ is the collected radiation and $C$ is its complement. The complement may include the remaining black hole, interior partners, uncollected exterior modes, a bath complement, and a reference. For a normalized global density operator $\rho_{RC}$,

$$\rho_R=\operatorname{Tr}_C\rho_{RC}, \qquad S(R)=-\operatorname{Tr}(\rho_R\ln\rho_R).$$

The trace is not a verbal instruction to “ignore the interior”; it is a map fixed by the tensor product. In a gauge theory or gravity, one may instead specify a radiation subalgebra $\mathcal A_R$ and restrict the state to it. If that algebra is not type I, no intrinsic density matrix need exist. A split regulator, relative entropy, or another algebraic quantity must then replace the naive expression.

Continuum entanglement entropy contains ultraviolet divergences near the boundary of $R$. A meaningful curve must keep one regulator and subtraction scheme fixed as $R$ changes, or use a finite difference in which the local divergences cancel. The choice of a nongravitating bath often makes factorization cleaner, but it is still part of the model.

Three common radiation definitions are not interchangeable. A wave-packet definition selects outgoing modes within frequency and retarded-time windows; an algebraic definition selects all observables supported in a spacetime region; a detector definition retains only records produced by a specified interaction. The first can omit soft or late modes, the second inherits continuum and gravitational-dressing subtleties, and the third includes detector noise and finite efficiency. A curve should state which definition is used and where its complement lives. Otherwise a change caused by enlarging a frequency window can be mistaken for physical entropy production.

When a subtraction is used, it must also be applied to the same geometric boundary and reference state at every comparison point. A finite answer obtained by subtracting two divergent entropies is not automatically the entropy of a density operator and need not satisfy every finite-dimensional identity. The purification proof below therefore applies directly to the regulated factors; any renormalized continuum limit needs its own justification.

Purification theorem

Statement. Under the hypotheses that $\lvert\Psi\rangle_{RC}$ is a normalized pure state on a finite-dimensional tensor product, the reduced density operators $\rho_R$ and $\rho_C$ have identical nonzero spectra. Consequently every spectral entropy agrees:

$$S(R)=S(C), \qquad S_n(R)=S_n(C), \qquad S_n(R)=\frac{1}{1-n}\ln\operatorname{Tr}\rho_R^n.$$

Proof. Regard the coefficients of $\lvert\Psi\rangle$ in product bases as a matrix $M$. Singular-value decomposition gives the Schmidt form

$$\lvert\Psi\rangle_{RC}=\sum_{i=1}^{r}\sqrt{\lambda_i} \lvert i\rangle_R\lvert i\rangle_C, \qquad \lambda_i\gt0, \qquad \sum_i\lambda_i=1.$$

Taking either partial trace yields a diagonal operator with eigenvalues $\{\lambda_i\}$ plus possible zeros. Any entropy depending only on nonzero eigenvalues is therefore equal. The finite-dimensional statement and related inequalities are collected in Harlow 2016, Appendix C; its use for radiation regions and gravitational entropy is explained in Almheiri et al. 2021, § 4.

The equality is not guaranteed if the global state is mixed: for example, $\rho_{RC}=\lvert0\rangle\langle0\rvert_R\otimes I_C/2$ has $S(R)=0$ and $S(C)=\ln2$. It also cannot be invoked for two overlapping algebras or for two reduced states obtained from different global states. Those are discriminating limitations, not small technicalities.

Entropy relations that track correlations

For a bipartite state, mutual information

$$I(R{:}C)=S(R)+S(C)-S(RC)\geq0$$

measures total correlations. For a pure state it equals $2S(R)$. For three named systems, subadditivity and strong subadditivity constrain how radiation can transfer entanglement:

$$S(RC)\leq S(R)+S(C),$$ $$S(RB)+S(BC)\geq S(B)+S(RBC).$$

These inequalities require a common regulated state and the declared factors. A negative conditional entropy, such as $S(R\vert C)=S(RC)-S(C)\lt0$, is a quantum correlation diagnostic; it is not a negative number of radiation modes.

The radiation curve $S(R(t))$ changes both because the state evolves and because the algebra $R(t)$ grows. Comparing times therefore requires an embedding of earlier radiation into the later algebra. If a mode is moved from $C$ to $R$, the operation is a change of subsystem assignment as well as time evolution.

Product, Bell, and GHZ tests

Three states expose the assumptions.

For $\lvert00\rangle_{RC}$, both reductions are pure and $S(R)=S(C)=0$. For the Bell state $(\lvert00\rangle+\lvert11\rangle)/\sqrt2$, both reductions are $I/2$, so $S(R)=S(C)=\ln2$ although $S(RC)=0$. Radiation can thus be thermal-looking while the complete state is pure.

For $\lvert\mathrm{GHZ}\rangle_{RCQ}=(\lvert000\rangle+\lvert111\rangle)/\sqrt2$, the full purifier of $R$ is $CQ$. One finds $S(R)=S(CQ)=\ln2$, but $S(C)=\ln2$ as well only accidentally; the equality $S(R)=S(C)$ is not guaranteed because $RC$ is mixed after tracing $Q$. Tracing an additional purifier changes which equality is valid.

As an algebra test, retain only diagonal observables of a Bell qubit. The restricted state is indistinguishable from a classical fair bit even though the global phase correlation is present. Changing the algebra changes the operational entropy without changing the vector state.

Limits of the subsystem description

The theorem does not choose the correct radiation algebra in gravity, remove ultraviolet divergences, or establish unitary evaporation. It provides the exact bookkeeping once those choices are made. Generic entropy theory belongs to Quantum Information, fixed-background mode construction to Quantum Fields in Curved Spacetime, and stable thermodynamic entropy to Semiclassical Gravity. Explicit string or holographic factorizations are outside this page.

Exercise

Let $\lvert\psi\rangle_{RCQ}=(\lvert000\rangle+\lvert110\rangle+\lvert011\rangle+\lvert101\rangle)/2$. Compute $S(R)$. Can either physical qubit $C$ or $Q$ purify $R$ alone? How large a logical purifier is needed inside $CQ$?

Solution

Group terms by $R$ to obtain

$$\lvert\psi\rangle =\frac{\lvert0\rangle_R\lvert\Phi^+\rangle_{CQ} +\lvert1\rangle_R\lvert\Psi^+\rangle_{CQ}}{\sqrt2},$$

where $\lvert\Phi^+\rangle=(\lvert00\rangle+\lvert11\rangle)/\sqrt2$ and $\lvert\Psi^+\rangle=(\lvert01\rangle+\lvert10\rangle)/\sqrt2$ are orthogonal. Thus $\rho_R=I/2$ and $S(R)=\ln2$. Both $RC$ and $RQ$ are rank-two mixed states, so neither physical qubit alone purifies $R$. The two-dimensional Schmidt support spanned by these Bell states is nevertheless a single logical purifier qubit inside $CQ$. A joint unitary on $CQ$ can extract it into one qubit and leave the other in a fixed pure state. The physical complement and the minimal purification dimension are different notions.

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.