# The Law of conservation of Information

## Motivation:

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. We would not be able to run the simulation backwards in time otherwise.

But, how can we formulate the law of conservation of information mathematically?

## Unitarity in the Everettian picture:

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.

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 \rho U^{*} \end{equation}

we may deduce:

\begin{equation} S(\rho) = S(U \rho 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:

\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:

The law of conservation of quantum information is actually consistent with the Second Law as the Second Law merely states that information is lost with respect to local observers. The Second Law does not even rule out a Universe with finite PoincarĂ© recurrence time.

In fact, it is worth adding that the Second Law is hardly applicable to systems that are far from equilibrium so it may not even be a relevant counter-argument.

## References:

- Higham, Nicholas J. and Lijing, Lin. Matrix Functions: A Short Course. The University of Manchester. 2013.
- Edward Witten. A Mini-Introduction To Information Theory. 2019.
- 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.