Finite-Time Entropy Measurements

Entropy is a nonlinear functional of a density matrix, not the outcome of one fixed measurement on one Hawking-radiation state. Estimating a Page curve requires an access model: independent preparations, a radiation algebra, measurements or controlled permutations, accuracy and confidence targets, and finite time and energy budgets. The benchmark below isolates ordinary sampling bias for a known two-level spectrum; it is executable, but deliberately much easier than an unknown gravitational radiation state.

Required background. Quantum Information for Black-Hole Physics supplies density matrices, entropy, and measurements; What the Equivalence Principles Do and Do Not Imply supplies physical clocks, detectors, and disturbance.

Helpful background. Semiclassical Gravity supplies evaporation and backreaction scales; Quantum Fields in Curved Spacetime supplies radiation observables; and Black-Hole Information supplies Page-curve interpretation.

Why entropy is not a single-copy observable

For a state $\rho$, the von Neumann entropy is

$$S(\rho)=-\operatorname{Tr}(\rho\log\rho).$$

If one fixed Hermitian operator $H$ measured entropy on every single copy, its expectation $\operatorname{Tr}(\rho H)$ would be linear in $\rho$. Entropy is not. Two orthogonal pure states each have zero entropy, while their equal mixture has entropy $\log 2$. No such universal single-copy $H$ exists.

Multiple copies change the question. A cyclic permutation $V_n$ obeys

$$\operatorname{Tr}(V_n\rho^{\otimes n}) =\operatorname{Tr}(\rho^n),$$

so ideal controlled-permutation measurements access Rényi moments. The swap $V_2$ is Hermitian with outcomes $\pm1$. For $n\ge3$, the cyclic permutation is generally unitary but not Hermitian; its real and imaginary parts are measured through separate interferometric settings. General entropy estimation can instead use collective or adaptive measurements. Acharya and collaborators give finite-dimensional algorithms in Theorems 1–4, matching lower bounds for integral Rényi orders in Theorem 1, and lower bounds on the empirical estimator in Theorems 7–8 Acharya et al. 2020, § 1.3 and Theorems 1–4, 7–8. These results assume independent copies; sample complexity does not include collection, storage, gate complexity, or gravitational backreaction.

Benchmark access model and estimator

Take a known binary spectrum

$$\rho=0.8\lvert0\rangle\langle0\rvert +0.2\lvert1\rangle\langle1\rvert.$$

Its exact entropy is

$$S_{\mathrm{ref}} =-0.8\log 0.8-0.2\log 0.2 =0.5004024235381879\ \text{nats}.$$

Each independent copy is measured in the known eigenbasis. If $k$ of $N$ outcomes are $0$, the plug-in estimator is

$$\widehat S_N =-\frac{k}{N}\log\frac{k}{N} -\left(1-\frac{k}{N}\right) \log\left(1-\frac{k}{N}\right),$$

with $0\log0=0$. The code enumerates the exact binomial distribution of $k$; there is no Monte Carlo seed. It reports the mean estimator, absolute bias, root-mean-square error, and exact coverage probability of a first-order delta-method 95% interval. Probability-mass residual is the numerical tolerance diagnostic.

This algorithm is a benchmark for finite-sample behavior, not a general tomography algorithm. Knowing the eigenbasis removes the hard task of learning eigenvectors and reduces entropy estimation to classical spectrum sampling.

Reproduce the calculation

Environment. CPython 3.11.9, standard-library-only, using only the Python standard library; no third-party package is imported. Arithmetic is IEEE-754 binary64. The algorithm uses math.lgamma to evaluate binomial probabilities without enormous factorials. The acceptance tolerance is $5\times10^{-12}$ for probability mass, and both absolute bias and RMSE must decrease under every listed refinement.

Save the Python fence exactly as entropy_copies.py. The script must write the reference value, four refinement rows, confidence coverage, and residuals, and must exit nonzero if the benchmark, tolerance, or monotone-refinement check fails.

from math import exp, lgamma, log, sqrt

P = 0.8
REFERENCE = 0.5004024235381879
SIZES = (64, 256, 1024, 4096)
Z95 = 1.959963984540054
TOL = 5e-12


def binary_entropy(x):
    if x <= 0.0 or x >= 1.0:
        return 0.0
    return -x * log(x) - (1.0 - x) * log(1.0 - x)


def binomial_weight(k, n, p):
    return exp(
        lgamma(n + 1.0) - lgamma(k + 1.0) - lgamma(n - k + 1.0)
        + k * log(p) + (n - k) * log(1.0 - p)
    )


def evaluate(n):
    samples = []
    for k in range(n + 1):
        weight = binomial_weight(k, n, P)
        estimate = binary_entropy(k / n)
        samples.append((weight, estimate, k / n))
    mass = sum(weight for weight, _, _ in samples)
    mean = sum(weight * estimate for weight, estimate, _ in samples) / mass
    mse = sum(
        weight * (estimate - REFERENCE) ** 2
        for weight, estimate, _ in samples
    ) / mass
    coverage = 0.0
    for weight, estimate, p_hat in samples:
        if p_hat <= 0.0 or p_hat >= 1.0:
            low = high = estimate
        else:
            slope = abs(log((1.0 - p_hat) / p_hat))
            standard_error = slope * sqrt(p_hat * (1.0 - p_hat) / n)
            low = max(0.0, estimate - Z95 * standard_error)
            high = min(log(2.0), estimate + Z95 * standard_error)
        if low <= REFERENCE <= high:
            coverage += weight
    return (
        n,
        mean,
        abs(mean - REFERENCE),
        sqrt(mse),
        coverage / mass,
        abs(mass - 1.0),
    )


rows = [evaluate(n) for n in SIZES]
print(f"reference_entropy_nats={REFERENCE:.15f}")
print("copies mean_entropy abs_bias rmse coverage95 mass_residual")
for row in rows:
    n, mean, bias, rmse, coverage, residual = row
    print(
        f"{n:6d} {mean:.12f} {bias:.6e} "
        f"{rmse:.6e} {coverage:.6f} {residual:.3e}"
    )

if abs(binary_entropy(P) - REFERENCE) > 5e-15:
    raise SystemExit("reference-value check failed")
if max(row[-1] for row in rows) > TOL:
    raise SystemExit("probability-mass tolerance failed")
if any(
    new[2] >= old[2] or new[3] >= old[3]
    for old, new in zip(rows, rows[1:])
):
    raise SystemExit("refinement check failed")

Run the exact saved program with:

python3 --version
python3 -m py_compile entropy_copies.py
python3 entropy_copies.py

Refinement and confidence behavior

The comparison axes are copy count, mean entropy, absolute bias, RMSE, confidence coverage, and probability-mass residual. They separate statistical accuracy from numerical normalization and from the physical cost of preparing copies.

The retained output is:

Copies $N$	Mean entropy (nats)	Absolute error or bias	RMSE	95% coverage	Mass residual
64	0.492475583694	7.926840e-03	7.069108e-02	0.939714	2.953e-14
256	0.498442518495	1.959905e-03	3.482400e-02	0.949462	4.519e-14
1024	0.499913723466	4.887001e-04	1.734936e-02	0.949215	3.511e-13
4096	0.500280327124	1.220964e-04	8.666921e-03	0.951481	1.204e-12

Quadrupling $N$ reduces the leading plug-in bias by about four and the RMSE by about two, consistent with bias $O(N^{-1})$ and statistical width $O(N^{-1/2})$. The 95% delta-method interval undercovers at $N=64$ and approaches nominal coverage under refinement. This is a behavior check, not a finite-$N$ coverage theorem. All probability-mass residuals pass the stated tolerance.

From one spectrum to a Page curve

A Page curve requires the entropy at many retarded times. If $L$ time points are treated independently and each uses $N$ copies, the simple experiment already consumes $LN$ preparations. Correlated bins from one evaporation are not independent copies of one state. Unknown eigenbases, large radiation dimension, small spectral weights, losses, and collective-measurement control increase the resource count. Sample-optimal tomography bounds make the state- dimension dependence explicit Haah et al. 2017, Theorems 1–2, pp. 5628–5632.

Antonini and collaborators argue that Page- and Rényi-type observables may be defined for radiation in specified gravitating and massless-gravity settings through sufficiently many preparations Antonini et al. 2025, §§ 2–4. That frontier proposal is not a claim that an astrophysical observer can manufacture the required copies. It assumes a carefully defined algebra and background symmetries, perturbative control, and potentially enormous copy number and precision.

Failure modes and physical interpretation

Rare-spectrum bias. Eigenvalues with $Np_i\lesssim1$ are often unseen, driving plug-in entropy downward. The binary benchmark does not probe the many tiny eigenvalues of a large radiation state.

Wrong access model. Measuring in an unknown eigenbasis is unavailable; tomography or collective measurements are then required. Preparation drift or correlations violate the independent-copy binomial model.

Ill-conditioned inference. Entropy is sensitive to small eigenvalues. Finite precision, loss, detector dark counts, and an incorrectly truncated algebra can dominate the stated statistical error.

Vanishing first derivative. At a binary spectrum $p=1/2$, $h'(p)=\log[(1-p)/p]=0$. The first-order delta method then misses the leading quadratic fluctuations: $h(1/2+x)=\log2-2x^2+O(x^4)$. The displayed interval is calibrated for $p=0.8$ and must not be exported to the maximally mixed case without a second-order or finite-sample construction.

Resource overflow. Copy complexity differs from gate complexity. Each preparation needs radiation collection and storage; collective operations need coherent control. The cumulative energy budget and apparatus stress-energy can backreact on the source. Repeating a simulator is not the same as repeating a black hole with the same microscopic state.

The executable table therefore validates one estimator under one generous access model. It neither measures a gravitational Page curve nor supplies a universal lower bound. A credible protocol must add preparation, algebra, collection, gate, timing, and backreaction accounts before interpreting entropy as evidence for microscopic unitarity.

Exercises

Use the refinement table to estimate the copy count needed for absolute plug-in bias below $10^{-4}$ in this binary benchmark. Then state why that estimate cannot be used as a copy bound for a generic Hawking-radiation state.

Solution

The product $N\lvert\text{bias}\rvert$ is approximately $0.5$ in every row, so the leading model gives $N\gtrsim0.5/10^{-4}=5000$ copies. This extrapolation applies only to the known two-level spectrum measured in its eigenbasis. A generic radiation state has a much larger effective dimension, an unknown eigenbasis, many rare eigenvalues, gravitational dressing, loss, and possibly only correlated time bins. Its estimator and sample-complexity lower bound must be derived for that actual access model.

References

• Acharya, Jayadev, Ibrahim Issa, Nirmal V. Shende, and Aaron B. Wagner. 2020. “Estimating Quantum Entropy.” IEEE Journal on Selected Areas in Information Theory 1, 454–468. DOI. arXiv:1711.00814.
• Antonini, Stefano, Chang-Han Chen, Henry Maxfield, and Geoff Penington. 2025. “An Apologia for Islands.” arXiv:2506.04311. arXiv.
• Haah, Jeongwan, Aram W. Harrow, Zhengfeng Ji, Xiaodi Wu, and Nengkun Yu. 2017. “Sample-Optimal Tomography of Quantum States.” IEEE Transactions on Information Theory 63, 5628–5641. DOI. arXiv:1508.01797.