# Quantum States of a Time-Asymmetric Universe: Wave Function, Density Matrix, and Empirical Equivalence

## TL;DR

This paper explores the quantum state of the universe, proposing that a density matrix derived from the Past Hypothesis offers a natural, empirically equivalent alternative to wave functions, with implications for understanding time asymmetry.

## Contribution

It introduces a density matrix approach based on the Past Hypothesis, classifies quantum theories by fundamental state type, and analyzes empirical equivalence with wave function theories.

## Key findings

- Density matrix derived from Past Hypothesis is natural and well-defined.
- Quantum theories with density matrices and wave functions can be empirically indistinguishable.
- Subsystem analysis in Bohmian theories supports empirical equivalence.

## Abstract

What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wave function (representing a pure state) but by a density matrix (representing a mixed state). I begin by classifying quantum theories into two types: theories with a fundamental wave function and theories with a fundamental density matrix. The Past Hypothesis is compatible with infinitely many initial wave functions, none of which seems to be particularly natural. However, once we turn to density matrices, the Past Hypothesis provides a natural choice---the normalized projection onto the Past Hypothesis subspace in the Hilbert space. Nevertheless, the two types of theories can be empirically equivalent. To provide a concrete understanding of the empirical equivalence, I provide a novel subsystem analysis in the context of Bohmian theories. Given the empirical equivalence, it seems empirically underdetermined whether the universe is in a pure state or a mixed state. Finally, I discuss some theoretical payoffs of the density-matrix theories and present some open problems for future research.