  The co-emergence of locality between the Hamiltonian and initial state of the
universe is studied in a simple toy model. We hypothesize a fundamental loss
functional for the combined Hamiltonian and quantum state and minimize it by
gradient descent. This minimization yields a tensor product structure
simultaneously respected by both the Hamiltonian and the state, suggesting that
locality can emerge by a process analogous to spontaneous symmetry breaking. We
discuss the relevance of this program to the arrow of time problem.
  In our toy model, we interpret the emergence of a tensor factorization as the
appearance of individual degrees of freedom within a previously
undifferentiated (raw) Hilbert space. Earlier work [5, 6] looked at the
emergence of locality in Hamiltonians only, and found strong numerical
confirmation of that raw Hilbert spaces of $\dim = n$ are unstable and prefer
to settle on tensor factorization when $n=pq$ is not prime, and in [6] even
primes were seen to "factor" after first shedding a small summand, e.g.
$7=1+2\cdot 3$. This was found in the context of a rather general potential
functional $F$ on the space of metrics $\{g_{ij}\}$ on $\mathfrak{su}(n)$, the
Lie algebra of symmetries. This emergence of qunits through operator-level
spontaneous symmetry breaking (SSB) may help us understand why the world seems
to consist of myriad interacting degrees of freedom. But understanding why the
universe has an initial Hamiltonian $H_0$ with a many-body structure is of
limited conceptual value unless the initial state, $|\psi_0\rangle$, is also
structured by this tensor decomposition. Here we adapt $F$ to become a
functional on $\{g,|\psi_0\rangle\}=(\text{metrics})\times (\text{initial
states})$, and find SSB now produces a conspiracy between $g$ and
$|\psi_0\rangle$, where they simultaneously attain low entropy by settling on
the same qubit decomposition.
