Duplication Experiments and Causal Bounds

A duplication experiment requires one observer to possess an exterior reconstruction of a qubit and also receive the purported interior copy. In a large Schwarzschild black hole, delaying horizon entry by an exterior time $\Delta t$ makes the interior sender’s available proper time shrink exponentially. Localizing a signal in that window requires exponentially growing energy, so Page-time waiting makes the comparison trans-Planckian and scrambling-time waiting already pushes it toward order-one backreaction.

Required background. Quantum Information supplies recovery, no-cloning, and the distinction between a decoded state and a classical record.

Helpful background. Semiclassical Gravity supplies the Schwarzschild evaporation and backreaction scales; Quantum Fields in Curved Spacetime supplies wave-packet localization and local energy; Causality and Global Structure supplies Kruskal coordinates, horizons, and the singularity.

Geometry of the comparison

Take a four-dimensional Schwarzschild black hole of mass $M$ with $GM^2\gg1$, neglecting charge, rotation, and mass loss during the short comparison. In the right exterior $r\gt2GM$, define $r_*=r+2GM\ln(r/(2GM)-1)$ and Kruskal coordinates

$$U=-e^{-(t-r_*)/(4GM)}, \qquad V=e^{(t+r_*)/(4GM)}.$$

Here $U\lt0$ and $V\gt0$. Continue the regular Kruskal coordinates through the future horizon $U=0$ into the black-hole region, where both are positive. They obey $UV=(1-r/(2GM))e^{r/(2GM)}$; thus the future singularity is $UV=1$ in this normalization. The exterior exponential expression for negative $U$ is not an interior Schwarzschild coordinate chart.

Alice crosses the horizon carrying the original qubit. Bob remains outside to collect and decode a radiation representation, then crosses later. If their exterior Schwarzschild-time separation is $\Delta t$, their horizon $V$ coordinates satisfy

$$\frac{V_B}{V_A}=e^{\Delta t/(4GM)}.$$

For a radial signal from Alice to reach Bob before Bob encounters the singularity, it must be emitted at a Kruskal coordinate satisfying approximately $|U_A|V_B\lesssim1$. Hence

$$|U_A|\lesssim V_A^{-1}e^{-\Delta t/(4GM)}.$$

Near Alice’s horizon crossing, proper time is linear in $U$ with scale $GM$. The available emission window therefore scales as

$$\Delta\tau_A\sim GM\,e^{-\Delta t/(4GM)},$$

where the omitted coefficient depends on Alice’s trajectory and the coordinate normalization. This is the central causal result.

Signal-energy bound

A wave packet localized within $\Delta\tau_A$ needs local energy at least of order

$$E_{\mathrm{sig}}\gtrsim\frac{1}{\Delta\tau_A} \sim\frac{1}{GM}e^{\Delta t/(4GM)}.$$

This is an uncertainty and bandwidth estimate, not an exact detector threshold. It assumes the message is encoded in a localized quantum with order-one distinguishability. Multiple soft quanta do not evade the total information, timing, and stress-energy requirements without a separate protocol.

The dimensions check: $GM$ is a time in the global units, so $1/(GM)$ is an energy. At $\Delta t=0$, the bound is the ordinary inverse-horizon scale. Increasing the wait shrinks the window and raises the energy monotonically, as causal intuition demands.

Formally inserting a Page-scale wait of order $G^2M^3$ gives $\Delta t/(4GM)=O(GM^2)$, proportional to the Bekenstein–Hawking entropy, and an exponentially excessive signal energy. A constant-$M$ geometry does not quantitatively control that long wait because the mass changes substantially. The useful conclusion is stronger locally: the signal already reaches order $M$ at the much shorter logarithmic delay derived below, while mass loss is parametrically small. There is therefore no need to trust a fixed geometry for an entire Page time to establish this protocol’s breakdown. This is the operational protection contemplated by the stretched-horizon and complementarity postulates Susskind, Thorlacius, and Uglum 1993, §§ 1 and 3. The four-dimensional Alice–Bob Kruskal calculation is given directly in Hayden and Preskill 2007, § 6, Eqs. (9)–(11).

Scrambling-time boundary

For an old black hole, Hayden–Preskill recovery suggests that new information can emerge after a scrambling delay rather than another Page time. The robust statement is the logarithmic scale $t_{\mathrm{scr}}T=O(\ln S_{\mathrm{BH}})$ Sekino and Susskind 2008, § 2, Eq. (2.4). Adopting the common Schwarzschild normalization gives

$$t_{\mathrm{scr}}\sim\frac{\beta}{2\pi}\ln S_{\mathrm{BH}} =4GM\ln S_{\mathrm{BH}}$$

for Schwarzschild inverse temperature $\beta=8\pi GM$. Substitution gives

$$E_{\mathrm{sig}}\gtrsim\frac{S_{\mathrm{BH}}}{GM}\sim M$$

up to numerical factors because $S_{\mathrm{BH}}=4\pi GM^2$. In the displayed normalization the signal is no longer exponentially beyond the black-hole mass, but it is energetic enough to cause order-one backreaction. The coefficient of the logarithmic delay matters: because the causal relation exponentiates $\Delta t$, an order-one change in that coefficient becomes a power of $S_{\mathrm{BH}}$. The general threshold is therefore the explicit condition $\Delta t\simeq4GM\ln(GM^2)$ for $E_{\mathrm{sig}}\sim M$, not the word “scrambling” by itself. Hayden and Preskill analyze the underlying complementarity bound in Hayden and Preskill 2007, § 6, Eqs. (9)–(11).

Equivalently, imposing an energy budget $E_{\max}$ yields a latest safe exterior delay

$$\Delta t\lesssim4GM\ln(GM E_{\max}).$$

With $E_{\max}\ll M$, this is parametrically below or at the scrambling scale. The precise inequality depends on the allowed backreaction fraction, trajectory, decoding completion time, and message fidelity.

What the bound does and does not prove

The calculation shows that one semiclassical observer cannot execute the standard comparison while keeping the interior signal in a low-energy, negligible-backreaction regime after a long exterior wait. It does not prove that two independent copies never existed in a global Hilbert space. No-cloning remains a constraint on the complete map, and an AMPS-type entropy contradiction does not require the observer to finish this experiment.

Several effects only strengthen the operational obstruction: Bob must collect enough radiation, run a decoder, aim the infall trajectory, and interact with the signal before Planckian curvature. On the other hand, changing the background, using a near-extremal throat, mining modes, or changing the interior operator assignment requires a new causal calculation; the Schwarzschild estimate should not be exported unchanged.

The first omitted corrections are evaporation during waiting, greybody-dependent collection, quantum fluctuations of the geometry, and backreaction from the apparatus. Fixed-background mode theory belongs to Quantum Fields in Curved Spacetime, while the full evaporation history belongs to Semiclassical Gravity. Explicit holographic recovery mechanisms are outside this worked example.

Independent limiting checks

Setting $\Delta t=0$ restores a horizon-scale window and energy. Differentiating the logarithm gives $d\ln E_{\mathrm{sig}}/d\Delta t=1/(4GM)$, the surface gravity, confirming the expected boost rate. Requiring $E_{\mathrm{sig}}\lt M$ gives $\Delta t\lt4GM\ln(GM^2)$ up to order-one factors, consistent with a scrambling-scale boundary rather than a Page-scale one.

These checks test sign, dimensions, and parametric normalization. They do not fix the order-one coefficient, which is trajectory-dependent and irrelevant once the estimate becomes exponentially large.

Exercise

Suppose Bob waits $\Delta t=8GM\ln S_{\mathrm{BH}}$. Estimate the signal energy relative to $1/(GM)$ and to the black-hole mass.

Solution

The exponent is $\Delta t/(4GM)=2\ln S_{\mathrm{BH}}$, so $E_{\mathrm{sig}}\gtrsim S_{\mathrm{BH}}^2/(GM)$. Since $S_{\mathrm{BH}}\sim GM^2$, the ratio to $M$ scales as $S_{\mathrm{BH}}^2/(GM^2)\sim S_{\mathrm{BH}}$, up to constants. The signal exceeds the black-hole mass by a large entropy factor, so treating it as a harmless message is inconsistent.

References

• Hayden, Patrick, and John Preskill. 2007. “Black Holes as Mirrors: Quantum Information in Random Subsystems.” Journal of High Energy Physics 2007 (09): 120. Open text.
• Sekino, Yasuhiro, and Leonard Susskind. 2008. “Fast Scramblers.” Journal of High Energy Physics 2008 (10): 065. Open text.
• Susskind, Leonard, Lárus Thorlacius, and John Uglum. 1993. “The Stretched Horizon and Black Hole Complementarity.” Physical Review D 48: 3743–3761. Open text.