Measurements without Probabilities in the Final State Proposal

TL;DR

The paper interrogates the Horowitz-Maldacena final-state proposal as a resolution to the black hole information paradox, using the decoherence functional to assign probabilities to measurement histories. It shows that when measuring in the order that verifies interior purification after exterior entanglement, the corresponding histories fail to decohere, and even Gottesman-Preskill refinements with pointers behind the horizon cannot restore well-defined probabilities. Consequently, the final-state approach does not provide a consistent alternative to the firewall scenario for at least some interior experiments. The work highlights fundamental limits of postselected quantum mechanics in gravitational settings and underscores the challenge of reconciling unitarity, horizon smoothness, and interior measurements.

Abstract

The black hole final state proposal reconciles the infalling vacuum with the unitarity of the Hawking radiation, but only for some experiments. We study experiments that first verify the exterior, then the interior purification of the same Hawking particle. (This is the same protocol that renders the firewall paradox operationally meaningful in standard quantum mechanics.) We show that the decoherence functional fails to be diagonal, even upon inclusion of external "pointer" systems. Hence, probabilities for outcomes of these measurements are not defined. We conclude that the final state proposal does not offer a consistent alternative to the firewall hypothesis.

Figures (5)

• Figure 1: The HM Hilbert spaces are defined on the left. The center shows the quantum circuit of the final state proposal gp. In this diagram, diagonal lines meeting at a point represent the maximally entangled state $|\Phi\rangle$. If the lines open upwards, it is a ket vector, and if the lines open downwards, it is a bra. The drawing on the right is an interpretation in terms of path-integral folds similar to those discussed in hmps.
• Figure 2: If we focus on a particular state of the infalling matter, then the final state projects the interior partner of $b$ (called $\tilde{b}$) with the nonlocal interior partner of $r_b$ (called $\tilde{r_b}$).
• Figure 3: The Gottesman-Preskill refinement gp: the diagram at left does not provide a unitary map from $M$ to $out$, due to interactions $U$ between the matter and the in modes. In the right diagram, we add a compensating $U^\dagger$ to the final state. This "undoes" the interaction behind the horizon, up to a modification of the $S$ matrix to $S' = SV$, and results in a unitary circuit.
• Figure 4: Index diagrams as described in the text.
• Figure 5: Diagrams for four different traces discussed in the main text.