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It is postulated by many cosmologists that at the Big Bang time the universe was in an unusual low entropy state.

Does this claim specifically mean that the entropy of the initial universe was zero?

Is zero-entropy state unique for given physical laws?

Is it possible that entropy was growing always so that only difference in entropy has physical meaning rather than absolute value? Was there ever negative entropy state?

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Zero entropy means unique microstate with given macrostate. Entropy cannot be negative. And absolute value of entropy is meaningful. –  Siyuan Ren Sep 11 '12 at 1:18
    
@Karsus Ren, I see but I conjectured that there can be fractional number of microstates, even the number below 1 because in quantum information theory it is possible to independently manipulate with information quantities below one bit. If a fraction of one bit is possible (which corresponds to the number of microstates between 1 and 2), why there cannot be even negative piece of information in some beyond-quantum theory? –  Anixx Sep 11 '12 at 1:23
    
@Anixx less than one bit is also possible for a classical system. E.g. a 2-state classical system with $p_1 = 1/4$ and $p_2=3/4$ has an entropy of $-(\frac{1}{4}\log_2\frac{1}{4}+ \frac{3}{4}\log_2\frac{3}{4}) \approx 0.811$ bits -- but $<1$ is quite different from $<0$. –  Nathaniel Sep 11 '12 at 8:19
    
@Nathaniel yes, indeed. What I meant is that in classical computing you need an analog computer for that but in quantum computing you can manipulate less-than-bit quantities in a digital manner (without loss). You are completely right though. –  Anixx Sep 11 '12 at 9:00
    
@Nathaniel also I am not sure but is seems to me that to manipulate less-than-bit quantities on a classical analog computer your device should at least support 1 but register. So to store less-than-bit quantity you still need 1 bit of storage. –  Anixx Sep 11 '12 at 9:03

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Whether entropy was zero at the Big Bang or not is very much an open question of physics, in big part due to the fact that we do not yet have a good enough understanding of physics at high energies and high gravitational fields.

But for the zero entropy state this is a bit easier to answer and the answer does depend on laws of physics. Zero entropy state basically depends on how many completely distinguishable states the laws of physics allow. The universe is in a zero entropy state precisely when it is in a single state and it can be known which state it is in. In many situations there are infinitely many different zero entropy states. So the zero entropy state at the beginning of the universe is unique if and only if the laws of physics at that time require that there is a single state in which the universe can be found. Whether they do require that or not is a very big question in physics which everybody would like to know the answer to.

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So based on currently known laws how a zero-entropy state should look like? Infinitely-dense pointlike piece of matter, cosmological horizon of zero radius, vacuum-like state at absolute zero temperature? –  Anixx Sep 11 '12 at 17:27
    
Also is not Dirac delta function-like distribution of density corresponds actually to negatively infinite entropy rather than zero entropy? –  Anixx Sep 11 '12 at 17:29
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Quantum mechanically, a zero entropy state is a pure state $|psi>$, which can be perfectly distinguished out of a complete basis of states. Assuming quantum mechanics can be extended to universal scales (which due to the lack of quantum gravity we are not sure if and how this is done), then a state where every particle in the universe has a definite state, for example, is a zero-entropy state. But it may also be that particles are entangled to one another, which would imply that some particles have positive entropy, while the universe as a whole is still of zero entropy. –  SMeznaric Sep 11 '12 at 17:39
    
On a related note, there is the so called Church of the larger Hilbert space where one makes an assumption that there is a universal state of zero entropy and non-zero entropy states always arise as parts of zero-entropy states. –  SMeznaric Sep 11 '12 at 17:41

The correct answer is we just don't know and never will . This is because all known Physical laws breakdown at about $10^{-42}$ seconds after the Big Bang or if you will Initial Inflation . Quantum calculations will not render an answer , and all hypotheses are not testable thus not provable . This is simply because we cannot roll back the Universe to it's Origin , but only to $10^{-42}$ seconds after it started it's inflation .At this point Science becomes Philosophical . We have absolutely no idea of what Physical Laws drove the Universe into the existence we now observe . Not even Hawking can do better than guess .

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It depends what you mean by big-bang. I consider the big-bang to begin with inflation, not with a singularity, so that the starting point is the inflating universe, making no hypotheses about what came before (if the question even makes sense).

The inflating initial starting state is for all intents and purposes, a perfect deSitter state which is adiabatically growing as the inflaton slides down the potential. At the end of inflation, when the inflaton starts shaking non-thermally, the state is no longer unique, but the semiclassical description of the initial state is by a thermal state inside a deSitter horizon.

The natural entropy to associate with this state is the area of the cosmological horizon in Planck units, and this entropy is far from zero. But it is infinitesimally small compared to the maximum entropy we could squeeze into the universe today, given that the cosmological horizon has grown so much, but past the end of inflation, the growth has been out-of-equilibrium.

So the entropy of the initial state of the universe is about the square of the de-Sitter radius at the end of inflation. I don't know a precise number, but suppose it's a deSitter temperature of $10^{14}$ GeV, that's about a million planck lengths of radius, so a dimensionless entropy of order $10^{12}$. Compare with $10^{135}$, which is the maximum entropy you can squeeze in the modern cosmological horizon, and you can see how low-entropy the initial state was, despite being in thermal equilibrium at the time.

This explanation of the low-entropy initial conditions requires you to consider a single horizon-volume as all there is, and this is the holographic view of inflation promoted by Banks, Fischler, Shenker and Susskind. It was suggested to be the reason for the low-entropy initial condition by Davies in the early 1980s, but it is still not accepted by the astrophysical community, for reasons that I wouldn't be able to properly explain, because I think they are ridiculous.

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I was interested to know about the really initial state not the state at the end of inflation. Was there zero-entropy state ever? –  Anixx Sep 11 '12 at 8:00
    
@Anixx: Doesn't the question above need a metaphysics tag? If you have a thermal state, how can you ask about prior states reasonably? You can do it if you know the scalar potential, but this is a matter of speculation at present. –  Ron Maimon Sep 11 '12 at 8:58
    
I am not asking what state exactly, I am asking about entropy. Was not entropy growing ever? –  Anixx Sep 11 '12 at 9:04
    
Do you think zeo-entropy state actually possible with known physical laws? –  Anixx Sep 11 '12 at 9:06
    
@Anixx: recall that entropy is a statistical concept --- so it's on you to define the macrostate that you want to consider. Ron punts on that question by starting at inflation, so assumes close to equilibrium conditions and there after uses standard thermodynamics (one might quibble about that assumption, but it seems sensible). –  genneth Sep 11 '12 at 10:35

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