The Law of Conservation of Information

Given that all physical laws are time-reversible and computable, information must be conserved as we run a simulation of the Universe forward in time.

Introduction:

Given that all physical laws are time-reversible and computable, information must be conserved as we run a simulation of the Universe forward in time. Cosmological investigations into the origins of the Universe would be founded on an incorrect axiom otherwise.

But, how can we formulate the Law of conservation of Information mathematically?

Unitarity in Everettian Quantum Mechanics:

In the Everettian formulation of Quantum Mechanics, the entire Universe may be identified with a single wave equation that obeys unitarity. That is, the time evolution of a quantum state must conserve probability in the sense that the sum of probabilities is always one. In addition, each Quantum Measurement performed by a conscious observer is a Unitary transformation rather than an event where the wave function collapses.

It follows that a unitary operator describes the time evolution of the state of the Universe.

Conservation of Von Neumann entropy:

If the quantum state of the Universe is given by a positive semi-definite matrix \(\rho\) then the Von Neumann entropy is given by:

\[\begin{equation} S(\rho) = -\text{Tr}(\rho \cdot \ln \rho) \end{equation}\]

which quantifies the total amount of statistical information in the Universe. Now, given that \(\rho\) can only undergo Unitary transformations:

\[\begin{equation} \rho \mapsto U \cdot \rho \cdot U^{*} \end{equation}\]

we may deduce:

\[\begin{equation} S(\rho) = S(U \cdot \rho \cdot U^{*}) \end{equation}\]

since \(S(\cdot)\) only depends on the eigenvalues of \(\rho\).

Proof:

Given that \(\rho\) is positive semi-definite, it is diagonalisable:

\[\begin{equation} \rho = V_{\rho} \lambda_{\rho} V_{\rho}^{-1} \end{equation}\]

where \(\lambda_{\rho}\) is the diagonal matrix of eigenvalues of \(\rho\) and \(V_{\rho}\) is the matrix of eigenvectors of \(\rho\).

Moreover, given that \(\rho^k = V_{\rho} \lambda_{\rho}^k V_{\rho}^{-1}\) we may deduce that the matrix exponential must satisfy:

\[\begin{equation} e^{\rho} = V_{\rho} e^{\lambda_{\rho}} V_{\rho}^{-1} \end{equation}\]

so if we define \(\rho := \log A\) for positive semi-definite \(A\):

\[\begin{equation} \text{eig}(A) = \text{exp}(\text{eig}(\log A)) \end{equation}\]

Now, assuming that any quantum event has strictly non-zero probability of occurring i.e. \(\text{min}(\text{eig}(\rho))>0\), the previous equation generalises as follows:

\[\begin{equation} \log \rho = V_{\rho} \log \lambda_{\rho} V_{\rho}^{-1} \end{equation}\]

and for unitary transformations \(U\) if we define \(Q=V_{\rho} \cdot U\) we have:

\[\begin{equation} \log U\rho U^{*} = Q \log \lambda_{\rho} \cdot Q^{-1} \end{equation}\]

so if we define the diagonal matrix \(\Lambda = \lambda_{\rho} \cdot \log \lambda_{\rho}\), our analysis simplifies to:

\[\begin{equation} \text{Tr}(Q \Lambda Q^{-1}) = \text{Tr}(\Lambda) \end{equation}\]

which concludes our proof.

Discussion:

Whether Everettian Quantum Mechanics is the appropriate theory for a computable Universe or a limit-computable Universe is a question to be addressed by future investigations. A number of scientific results in theoretical physics suggest that the Observable Universe is a computable Universe embedded in a limit-computable Universe.

While the notion of a limit-computable Universe is far from mainstream, it appears to be the main application of Black Hole engineering which is within the scope of Kardashev-III civilisations.

References:

1. Higham, Nicholas J. and Lijing, Lin. Matrix Functions: A Short Course. The University of Manchester. 2013.
2. Edward Witten. A Mini-Introduction To Information Theory. 2019.
3. Von Neumann, John (1932). Mathematische Grundlagen der Quantenmechanik. Berlin: Springer. ISBN 3-540-59207-5.; Von Neumann, John (1955). Mathematical Foundations of Quantum Mechanics. Princeton University Press.