I really can\'t understand what Leonard Susskind means when he says that informa
ID: 1376986 • Letter: I
Question
I really can't understand what Leonard Susskind means when he says that information is indestructible.
Is that information that is lost, through the increase of entropy really recoverable?
He himself said that entropy is hidden information. Then, although the information hidden has measurable effects, I think information lost in an irreversible process cannot be retrieved. However Susskind's claim is quite the opposite. How does one understand the loss of information by an entropy increasing process, and its connection to the statement "information is indestructible".
Black hole physics can be used in answers, but, as he proposes a general law of physics, I would prefer an answer not involving black holes.
Explanation / Answer
How is the claim "information is indestructible" compatible with "information is lost in entropy"?
Let's make things as specific and as simple as possible. Let's forget about quantum physics and unitary dynamics, let's toy with utterly simple reversible cellular automata.
Consider a spacetime consisting of a square lattice of cells with a trinary (3-valued) field defined on it. The values are color-coded such that cells can be yellow, orange, and red. The 'field equations' consist of a set of allowed colorings for each 2x2 block of cells:
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A total of 27 local color patterns are allowed. These are defined such that when three of the four squares are colored, the color of the fourth cell is uniquely defined. (Check this!)
The field equations don't contain a 'direction of evolution'. So how to define a timelike direction? Suppose that when looking "North" or "West" along the lattice directions, you hit a horizon beyond which an infinite sea of yellow squares stretches:
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"North" and "West" we label as 'light rays from the past'. These two rays of cells constitute the 'snapshot' of the universe taken from the spacetime point defined by the intersection of the two rays. Given this 'snapshot', and using the field equations (the allowed 2x2 colorings), we can start reconstructing the past:
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Here, the rule applied to color the cell follows from the square at the bottom of the center column in the overview of the 27 allowed 2x2 squares. This is the only 2x2 pattern out of the 27 that fits the given colors at the right, the bottom, and the bottom-right of the cell being colored. Identifying this 2x2 pattern as uniquely fitting the cell colors provided, the top-left color becomes fixed.
Continuing like this, we obtain the full past of the universe up to any point we desire:
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We notice that we constructed the full past knowing the colorings of 'light ray cells' in the 'snapshot' that, excluding the uniform sea beyond the horizons, count no more than 25 cells. We identify this count as the entropy (number of trits) as observed from the point where the two light rays meet. Notice that at later times the entropy is larger: the second law of thermodynamics is honored by this simple model.
Now we reverse the dynamics, and an interesting thing happens: knowing only 9 color values of light rays to the future (again excluding the uniform sea beyond the horizon):
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We can reconstruct the full future:
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We refer to these 9 trits that define the full evolution of this cellular automata universe as the 'information content' of the universe. Obviously, the 25 trits of entropy do contain the 9 trits of information. This information is present but 'hidden' in the entropy trits. The entropy in this model will keep growing. The 9 trits of information remains constant and hidden in (but recoverable from) an ever larger number of entropy trits.
Note that none of the observations made depend on the details of the 'field equations'. In fact, any set of allowed 2x2 colorings that uniquely define the color of the remaining cell given the colors of three cells, will produce the same observations.
Many more observations can be made based on this toy model. One obvious feature being that the model does not sport a 'big bang' but rather a 'big bounce'. Furthermore, the information content (9 trits in the above example) defining this universe is significantly smaller than the later entropy (which grows without bound). This is a direct consequence of a 'past horizon' being present in the model. Also, despite the 'field equations' in this model being fully reversible, the 'snapshot' taken allows you to reconstruct the full past, but not the future. This 'arrow of time' can be be circumvented by reconstructing the past beyond the 'big bounce' where past and future change roles and a new snapshot can be derived from the reconstruction taken. This snapshot is future oriented and allows you to construct the future beyond the original snapshot.
These observations, however, go well beyond the questions asked.