I had this conservation with GPT-4 on aspects of black hole quantum
mechanics. I kept it largely non-mathematical. This is interesting, and it
suggests a level of processing that is complex. An AI system was fed
ephemeris data on the appearance of the sun, moon and planets in the sky
and it deduced Kepler's laws. Holy Sh*t Batman --- the world is changing.
Einstein advanced the equivalence principle. In there it is 1) not possible
to distinguish between an inertial frame in flat spacetime from a
sufficiently local inertial frame falling in a gravity field, and 2) the
accelerated frame in flat spacetime is equivalent to the surface
acceleration against gravity on a planet or hard gravitating body. Can it
then be said that two entangled particles in spacetime, whether flat or
with a curvature have that entanglement conserved?
GPT-4 answer: The equivalence principle in general relativity states that
the effects of gravity are indistinguishable from the effects of
acceleration. It does not make any specific statement about the
conservation of entanglement between two particles in spacetime, whether in
flat or curved spacetime.
Entanglement is a quantum mechanical phenomenon that describes the
correlation between two or more particles, even when they are separated by
large distances. The conservation of entanglement is a fundamental property
of quantum mechanics, which holds true regardless of the presence of
gravity or the curvature of spacetime.
Therefore, it can be said that entanglement between two particles is
conserved in both flat and curved spacetime, as long as there is no
external interaction that disturbs or breaks the entanglement. The effects
of gravity, as described by the equivalence principle, do not affect the
conservation of entanglement between two particles.
Good, this is what I concluded as well. Then to take this further. Given an
EPR pair, where Bob remains far from the black hole and Alice is in a
freely falling frame into the black hole, there are then two
interpretations of events. Alice will consider the entanglement with Bob as
continuing. Bob on the other hand will, due to red-shifting on tortoise
coordinates and that Alice’s time is dilated asymptotically to a crawl,
that his EPR pair is entangled with the black hole. The two perspectives
are effectively equivalent to each other.
GPT-4 ans:
Yes, you are correct. According to the equivalence principle, there is no
difference between a uniform acceleration and being in a gravitational
field. Therefore, for Bob, who is far away from the black hole, the
entanglement between his particle and Alice's particle will be affected by
the gravitational field of the black hole. This will cause the entangled
state to evolve differently than it would in a flat spacetime.
On the other hand, for Alice, who is in a freely falling frame into the
black hole, there is no gravitational force acting on her. From her
perspective, the entanglement between her particle and Bob's particle will
continue to evolve according to the laws of quantum mechanics.
These two perspectives are indeed equivalent, even though they appear to be
different. Bob's observations can be explained by the effects of the
gravitational field on his particle's evolution, while Alice's observations
can be explained by the absence of a gravitational force on her particle's
evolution.
It is worth noting that the details of the evolution of the entangled state
in this scenario are highly dependent on the specific setup and initial
conditions. Therefore, the specific outcomes of the two perspectives may
differ in different situations. However, the equivalence principle ensures
that both perspectives are equally valid and consistent with the laws of
physics.
This then means that Bob and Alice have different subjective
interpretations of the same physics. Alice observes no apparent difference
with standard quantum mechanics. Bob on the other hand is uncertain of the
status of the entangled state. The tortoise coordinates with t’ = t – 2m
log|r – 2m|, and the associated red-shifting means that Bob sees this
bipartite entanglement dissolve into an entanglement with all the entangled
states on the Planck region above the event horizon.
GPT-4 ans: Yes, that's correct. Bob and Alice have different subjective
interpretations of the same physics due to their different perspectives in
the presence of the black hole. From Alice's perspective, the entangled
state between her particle and Bob's particle continues to evolve according
to standard quantum mechanics. However, from Bob's perspective, the
entangled state appears to dissolve into an entanglement with the black
hole itself.
As you mentioned, the redshift factor associated with Bob's perspective
causes the entangled state to evolve in a way that appears to dissolve the
entanglement between Alice's particle and Bob's particle. This can be
thought of as an "entanglement swap" where the entanglement between the two
particles is effectively transferred to a new entanglement between Alice's
particle and the black hole.
It's worth noting that while this "entanglement swap" appears to occur from
Bob's perspective, it is still consistent with the laws of quantum
mechanics and the conservation of entanglement. The entangled state between
Alice's particle and Bob's particle is still preserved in the overall
system, even though it may be redistributed among different entangled
states within that system.
Overall, this scenario demonstrates the importance of considering the
subjective perspectives of different observers when interpreting physical
phenomena in the presence of strong gravitational fields.
This begins to get deeper into the big question here. If the black hole has
entropy S = Nk, here assuming a naïve microcanonical partition of N states
and k = Boltzmann’s constant, then from Bob’s perspective the entanglement
has changed from a bipartite entanglement to an N+1 entanglement. Alice
observes nothing of this sort. This appears to suggest a possible crack in
the AMPs argument. I may indicate next what I think that may be.
GPT-4 ans: The scenario you describe, where the entanglement appears to
change from a bipartite entanglement to an N+1 entanglement from Bob's
perspective, does not necessarily indicate a crack in the argument of the
AMPs (Almheiri-Marolf-Polchinski-Sully) paper.
The AMPs argument concerns the black hole information paradox, which arises
from the apparent loss of information when matter falls into a black hole.
The argument suggests that this apparent loss of information can be
resolved by considering the entanglement structure of the Hawking radiation
emitted by the black hole.
In the scenario you describe, the entanglement between Alice's particle and
Bob's particle appears to dissolve into an entanglement with the black
hole, resulting in an N+1 entanglement structure from Bob's perspective.
However, the entanglement structure of the overall system, including the
black hole and the emitted Hawking radiation, remains consistent with the
AMPs argument.
Specifically, the AMPs argument proposes that the entanglement between the
black hole and the Hawking radiation determines the entanglement structure
of the emitted radiation. The N+1 entanglement structure from Bob's
perspective can be seen as a result of this entanglement structure between
the black hole and the emitted radiation.
Therefore, while the scenario you describe may appear to challenge the AMPs
argument, it is still consistent with the overall picture proposed by the
argument. It highlights the importance of considering the subjective
perspectives of different observers in interpreting physical phenomena, but
it does not necessarily imply a fundamental flaw in the AMPs argument
itself.
Let me then rephase that. The black hole information paradox stems from the
fact that Hawking radiation is entangled with a black hole. However, after
the Page time emitted Hawking radiation must be entangled with previously
emitted Hawking radiation plus the black hole. Hence the bipartite
entanglement of the previously emitted radiation and the black hole
transforms into a tripartite entanglement. This is what is a source of the
non-unitary change in information or S = Tr[ρ log(ρ)], which in quantum
mechanics is easily shown to be invariant. Can this then suggest that the
topology of entanglement in curved spacetime may in fact be a matter of
subjective, maybe Bayesian, observation?
GPT-4 ans: The non-unitary evolution of information in the presence of a
black hole is indeed at the heart of the black hole information paradox,
and the tripartite entanglement structure you describe is an important
aspect of this paradox. The AMPs argument attempts to resolve this paradox
by proposing that the entanglement structure of the emitted Hawking
radiation is determined by the entanglement between the black hole and the
radiation, and that this entanglement structure is consistent with unitary
evolution.
Regarding the topology of entanglement in curved spacetime, it is true that
the subjective perspectives of different observers can lead to different
interpretations of entanglement and its evolution. In the scenario you
describe, Bob and Alice have different interpretations of the same physical
system due to their different perspectives in the presence of a black hole.
However, this does not necessarily imply that the topology of entanglement
itself is subjective or Bayesian.
Rather, the topology of entanglement is an objective property of the
physical system that can be described mathematically using the formalism of
quantum mechanics. The subjective perspectives of different observers can
lead to different interpretations of this topology, but this does not
change the objective nature of the topology itself.
In summary, while the subjective perspectives of different observers can
play a role in the interpretation of entanglement in curved spacetime, the
topology of entanglement is an objective property of the physical system
that can be described using the formalism of quantum mechanics.
I would agree if the topology of entanglement is completely accessible to
observations. If Bob has an enormous amount of resources and by some means
can snuggle up to within a Planck length of the horizon, where acceleration
is √{c^7/Għ} ≃ 10^{52}m/sec^2 he could in principle access this
information. The Hawking radiation would gush out of the black hole. The
Shapiro theorem about geodesics with positive energy is turned on its head,
and Bob can access in principle everything. However, in practice this
cannot happen. Bob must remain content with partial access to the topology
of this entanglement. He in fact then does not know whether the
entanglement is equivalent to his original bipartite simple entanglement or
a vastly more complex N-tangle. This appears to be observer dependent, at
least for one observer.
Thank you, you appear to largely agree with some of my assessments ---
though you give some critical changes I need to consider.
GPT-4 ans:
You're welcome! It's always important to carefully consider different
perspectives and arguments in a complex and evolving field like physics. If
you have any further questions or ideas to discuss, feel free to ask!
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