Information-Theoretic Cosmic Censorship Conjecture

Einstein's equations in general relativity possess various solutions for spacetime. A famous example is the black hole solution. This describes a celestial body accompanied by an event horizon, generated by strong gravity. Inside this black hole, a spacetime singularity appears where the spacetime curvature diverges. Because general relativity fails in its predictive power at this singularity, it provides motivation to consider a deeper level of theory, namely quantum gravity. On the other hand, it was hoped in the last century that classical general relativity itself, even without quantum effects, might be internally consistent as a theory, at least regarding currently observable phenomena. One reflection of that expectation is the "cosmic censorship conjecture."

When a spacetime singularity is not covered by an event horizon, it is called a "naked singularity," and the singularity is fully visible from anywhere in the universe. Since the initial conditions on the singularity cannot be determined, if the singularity scatters with other objects or emits radiation itself, the physical influence of that singularity would extend to a wide region of the subsequent universe. At the present time, without deep knowledge of the physics of singularities, we cannot impose natural boundary conditions on solutions to Einstein's equations. Consequently, it becomes impossible to make reliable predictions of general relativity regarding what will happen in the future universe.

However, the singularities in many solutions are hidden within event horizons. Since physical influences do not travel faster than the speed of light, the physical effects caused by a singularity inside the horizon never reach the outside of the horizon. In other words, the future state outside the horizon can be determined solely by the initial conditions outside the horizon. Therefore, if all singularities in the current universe are hidden inside event horizons, classical general relativity alone can make reliable predictions. And in many known natural solutions to Einstein's equations, singularities indeed only appear inside horizons.

It seems as if the universe itself is practicing self-censorship, covering spacetime singularities—the "villains" that harm future predictions—with event horizons so that their influence does not leak out. Although there is no general proof, this property that singularities are not visible to us is currently called the cosmic censorship conjecture. Research to prove this conjecture is still ongoing worldwide, but a complete proof has not yet been provided.

In this article, we will discuss a quantum information-theoretic extension of this cosmic censorship conjecture. This was proposed by the author and collaborators in 2014, based on research into the black hole firewall hypothesis.

https://journals.aps.org/prd/abstract/10.1103/PhysRevD.89.124023

In quantum measurement theory, the observation process is generally described by measurement operators. For example, the measurement of quantum fluctuations of a field in each spatial region in the vacuum state of a quantum field is also written using these measurement operators. Even in a vacuum state, quantum entanglement exists between the quantum fluctuations at these various points. Therefore, the post-measurement state can deviate from the vacuum state not only at the measurement location but throughout the entire space, becoming an excited state that depends on the measurement result. However, the average value of the energy of quantum fluctuations far from the measurement location remains zero. The average excitation energy after measurement at the measurement location becomes positive. This is supplied to the quantum field at that location in the vacuum state from a physical measuring device corresponding to this measurement operator.

In relativistic field theory, the Reeh–Schlieder theorem is known. It is a theorem stating that through quantum entanglement, the vacuum state contains such rich information that even by manipulating only a local region, one can approach any state in the entire space with arbitrary precision. This means that even for a local operator A(O) definable within a finite spacetime region O, if it is applied to the vacuum state |0〉, A(O)|0〉 is densely distributed in the entire state space. If a measurement operator in O can be constructed from local operators in region O, it suggests that the post-measurement state corresponding to that measurement value can become any state with arbitrary precision. Although probabilistic, if this is true, it would mean that arbitrary excitations with arbitrarily large energy could be created as post-measurement states even in the region outside O. This is a very counterintuitive phenomenon. It is even possible for a local measurement at O to create excitations large enough to form a naked spacetime singularity outside O.

However, in our paper mentioned above, we pointed out that such measurements require enormous measurement energy costs. In a quantum measurement with a post-measurement state that creates a singularity outside O, the measuring device inside O must inject a vast amount of energy into the field in the first place. Of course, in relativistic field theory, this energy distributed inside O does not travel faster than the speed of light to regions far from O. The energy of the naked singularity appearing in the post-measurement state is not what the measuring device at O itself injected. While the average remains zero, fluctuations in positive and negative energy density are decomposed for each measurement result and appear in that remote location. However, the problem lies in the fact that a black hole accompanied by an event horizon forms inside O, where the measurement is performed.

To create an arbitrary excitation containing a naked singularity outside O, the experimenter at O is required to know the measurement result at O. This is because the post-measurement state does not appear unless one knows that measurement result. However, to perform such a measurement, the measuring device must have prepared a vast amount of energy. The energy divided by the square of the speed of light acts as the mass localized at O. Then, the effect of that enormous mass appears through Einstein's equations, the spacetime inside O is greatly distorted, and an event horizon hides the measuring device itself. Therefore, the experimenter outside cannot obtain the measurement result emitted by the measuring device.

For an experimenter who can no longer know the measurement result, the distant naked singularity that was supposed to appear in the post-measurement state also fails to emerge. In other words, it seems that an extended cosmic censorship conjecture holds even for such quantum measurements using quantum entanglement in the vacuum state.

The original cosmic censorship conjecture was discussed strictly within the framework of classical general relativity. However, in reality, even when extended to a quantum information-theoretic framework, the event horizon that swallows the measuring device prevents the appearance of naked singularities that were in danger of emerging in the distance. This also shows one of the non-trivial aspects of the physics of quantum information and spacetime.