Entropy, Degeneracy, and Observable Distinguishability

Orthogonal black-hole microstates are perfectly distinguishable by some global measurement, but that statement says nothing about the precision, duration, copy count, or operator complexity available to a realistic observer. In an entropy-sized sector, adjacent energy levels can be separated by order $e^{-S}$ of a macroscopic bandwidth, while simple-observable differences often scale as $e^{-S/2}$ under ETH-like assumptions. Resolving the former by frequency spectroscopy requires exponential observation time; estimating the latter can require exponentially many independent copies. Exact degeneracy, near-degeneracy, and restricted observable indistinguishability must therefore be treated separately.

Required background. Quantum Information supplies trace distance, optimal state discrimination, and sampling error; Gauge Theory and String-Theory Preparation supplies the constrained microscopic sectors and large-entropy setting.

Helpful background. Semiclassical Gravity supplies black-hole entropy and macroscopic energy scales; Quantum Fields in Curved Spacetime supplies the exterior probes whose resolving power is assessed.

Three notions of distinguishability

For two equally likely states $\rho$ and $\sigma$, unrestricted one-copy discrimination has minimum error

$$p_{\rm err}^{\rm opt} =\frac12\left(1-D(\rho,\sigma)\right), \qquad D(\rho,\sigma)=\frac12\lVert\rho-\sigma\rVert_1.$$

Orthogonal pure states have $D=1$ and zero optimal error. The optimal projector may nevertheless be exponentially complex or unavailable in a gravitationally dressed exterior algebra. For a restricted observable set $\mathcal A$, define

$$D_{\mathcal A}(\rho,\sigma) =\frac12\sup_{A\in\mathcal A,\,\lVert A\rVert_\infty\le1} \left|\operatorname{Tr}A(\rho-\sigma)\right|.$$

$D_{\mathcal A}$ can be exponentially small even when $D=1$. This is operational indistinguishability relative to $\mathcal A$, not equality of states.

Watrous gives the trace-distance discrimination theorem and its measurement interpretation in Watrous 2018, PDF § 3.1.2, pp. 84–88. A black-hole application must add the detector, time, energy, and gravitational access constraints.

Level spacing and the time–energy requirement

Suppose $d=e^S$ nondegenerate levels lie in a microcanonical bandwidth $W$ and are sufficiently evenly distributed for a mean-spacing estimate. Then

$$\delta E_{\rm mean}\simeq\frac{W}{d}=We^{-S}.$$

Resolving two frequencies separated by $\delta E$ in an observation of duration $T$ requires a Fourier width no larger than the separation,

$$T\,\delta E\gtrsim1, \qquad T\gtrsim\frac{e^S}{W}.$$

The dimensions check: $W$ has units of energy and $T$ has units of inverse energy in $\hbar=1$ units. Doubling the bandwidth at fixed state count halves the required time, while increasing $S$ by $\log2$ doubles it.

This is a mean-spacing estimate, not a lower bound for every pair. Level repulsion, degeneracies, symmetry sectors, and spectral edges change local spacings. If two states are exactly degenerate, no duration of energy measurement distinguishes them; another commuting observable or a more detailed probe is required.

Simple-observable precision and copy count

Under the ETH ansatz, two nearby eigenstates can differ in a normalized simple observable by a scale

$$\delta O\sim e^{-S/2}.$$

Assume independent copies and a measurement outcome confined to an interval of fixed order-one width. Hoeffding concentration for the empirical-mean estimator gives the sufficient sample-size scaling

$$N\gtrsim\frac{1}{(\delta O)^2}\log\frac{2}{\alpha} \sim e^S\log\frac{2}{\alpha}$$

to resolve the difference with failure probability $\alpha$. The exponential copy cost follows from the assumed signal size, not from orthogonality alone. The ETH ansatz and its observable and energy-window restrictions are reviewed in D’Alessio et al. 2016, §§ 2–4 and 6.

Typical-state correlator variances can likewise be entropy suppressed, while analytic continuation or sufficiently detailed probes amplify distinctions; see Balasubramanian et al. 2008, §§ 2–4. Such an amplification is useful only if the required complex-time information or high-point observable corresponds to an implementable measurement.

Independent checks and limits

At $S=0$, the sector has one state and there is no adjacent pair to resolve; the mean-spacing approximation is intended for many levels. At large $S$, the estimates become severe but remain conditional on a bandwidth and signal model. A specially chosen conserved charge can distinguish states in one shot even when their simple local observables agree; conversely, two states in the same exact charge and energy eigenspace defeat energy-and-charge measurements at any precision.

Collective measurements on $N$ copies can outperform estimating one observable sample mean, so $N\sim e^S$ is not a universal quantum discrimination lower bound. The correct lower bound depends on the restricted states and allowed measurements. Likewise, a single-copy state-specific projector can distinguish orthogonal states in principle, but synthesizing and gravitationally dressing it may require the microscopic description one hoped to test.

Generic quantum-information theory, stable thermodynamics, and explicit string constructions remain outside this derivation. Its bounded answer is that typical black-hole microstates can require relative energy precision $e^{-S}$ or simple-observable precision $e^{-S/2}$ under the stated sector and ETH assumptions, while an unrestricted ideal measurement obeys the full trace distance instead.

Exercises

Take $S=50$ and bandwidth $W$. Estimate $\delta E/W$, the dimensionless time $TW$ needed to resolve a mean spacing, and the copy scaling for a simple signal $\delta O=e^{-S/2}$ at fixed confidence.

Solution

The mean relative spacing is $e^{-50}\simeq1.93\times10^{-22}$, so Fourier resolution requires $TW\gtrsim e^{50}\simeq5.18\times10^{21}$. The simple signal has magnitude $e^{-25}$, and inverse-square sampling gives $N\sim e^{50}$ up to the confidence logarithm. The equal exponents arise from different mechanisms—frequency resolution and sampling variance—and should not be combined into one universal resource theorem.

References

• Balasubramanian, Vijay, Bartłomiej Czech, Veronika E. Hubeny, Klaus Larjo, Mukund Rangamani, and Joan Simón. 2008. “Typicality versus Thermality: An Analytic Distinction.” General Relativity and Gravitation 40, 1863–1890. arXiv:hep-th/0701122.
• D’Alessio, Luca, Yariv Kafri, Anatoli Polkovnikov, and Marcos Rigol. 2016. “From Quantum Chaos and Eigenstate Thermalization to Statistical Mechanics and Thermodynamics.” Advances in Physics 65, 239–362. arXiv:1509.06411.
• Watrous, John. 2018. The Theory of Quantum Information. Cambridge: Cambridge University Press. Open PDF.