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When I began to fully understand entanglement of quantum systems and what this implies, I was in particular excited by the fact that:

a composite quantum system say the composite of two factor systems A and B, can have less entropy than the sum of the entropies of the factor systems by virtue of entanglement.

A typical example is where A and B are two single particle systems considered as factor systems of a possibly entangled particle pair. The whole pair is sharply defined and has zero entropy while each piece will have no sharp description and hence have non-zero entropies.

Mathematical Details:

Working with the density matrices (co-operators) we have that the total entropy is:

[tex] S = -k\mathop{Trace}(\rho \log \rho) [/tex]

In the case where the system has been sharply defined i.e. has a specific "wave function" then the density operator is of the form:

[tex] \rho = \psi\otimes \psi^\dag[/tex]

and the entropy is zero. (Likewise zero entropy implies the density (co)operator is a projection operator and thus expressible in terms of a single "wave function".)

Now consider a product system with Hilbert space:

[tex] \mathcal{H} = \mathcal{H}_A \otimes \mathcal{H}_B[/tex]

and density operator space:

[tex] \mathcal{H}\otimes \mathcal{H}^\dag[/tex]

The two factor systems are not entangled if:

(a.) the system has zero entropy and its wave function is a single product:

[tex] \psi = \psi_A \otimes \psi_B[/tex]

or more generally if:

(b.) the composite system's density operator factors:

[tex] \rho = \rho_A \otimes \rho_B[/tex]

Given the composite density operator we may resolve the density operators of the component systems by partial traces:

[tex] \rho_A = {\mathop{Tr}\nolimits}_B(\rho)[/tex] et vis versa.

The "not-entangled" condition then becomes the condition of additive entropy:

[tex] S = S_A + S_B[/tex]

i.e.

[tex] -k\mathop{Tr}(\rho \log \rho) = -k(\mathop{Tr}\nolimits _A (\rho_A \log \rho_A) + \mathop{Tr}\nolimits _B(\rho_B\log\rho_B)[/tex]

and the systems are entangled if:

[tex] S < S_A + S_B[/tex]

(Note that this condition is dependent on the way we choose to factor the system and thus (in part?) entanglement a relative quality.)

Pardon the review but I need to refer to it for what comes next

I found this fascinating. The more you combine pieces of the universe into a single quantum system the lower the entropy. It then occurred to me that:

(a.)For all we know the entropy of the universe as a whole quantum system could be considered to be zero!

and

(b.)We could interpret all entropy as entanglement of the system with its environment!

One might immediately object to this given the obvious thermal behavior of the observed universe, but first let me point out that we are in such cases adding up entropies of pieces of the universe. As far as trying to write down a wave function or density operator for the "Universe as a whole" we must in being operational admit that the "Universe as a whole" has only one empirically measurable property: existence=1 vs. non-existence=0 and it is constantly observed to be in the sharply defined 1 state.

Well this reasoning is admittedly fuzzy but so is anyscientificreasoning about "the universe as a whole", as we cannot execute repeatable experiments on such a conceptual object. So I will also admit that the idea of the "entropy of the universe is zero" is likewise somewhat nonsensical as it stands but it might be considered as a defining condition of what we mean by "the universe" or possibly a sort of "gauge condition".

The main point however is that such an assumption nicely resolves some of the cosmological entropy questions. Informationisconserved and the entropy oftheuniverseasawholeremains constant (whether you believe it to be zero or not) and it is only the entanglement of causally separated pieces of the universe which increases...of necessity due to interactions along the boundary of said pieces.

At least I find it intellectually satisfying to identify entropy of a system as entanglement with its environment.

Anyway I thought I would share this provocative thought with everyone.

Regards,

James Baugh

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# Is the Entropy of the Universe Zero? Entropy as Entanglement)

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