Classical Branch Structure from Spatial Redundancy in a Many-Body Wave Function

When the wave function of a large quantum system unitarily evolves away from a low-entropy initial state, there is strong circumstantial evidence it develops “branches”: a decomposition into orthogonal components that is indistinguishable from the corresponding incoherent mixture with feasible observations. Is this decomposition unique? Must the number of branches increase with time? These questions are hard to answer because there is no formal definition of branches, and most intuition is based on toy models with arbitrarily preferred degrees of freedom. Here, assuming only the tensor structure associated with spatial locality, I show that branch decompositions are highly constrained just by the requirement that they exhibit redundant local records. The set of all redundantly recorded observables induces a preferred decomposition into simultaneous eigenstates unless their records are highly extended and delicately overlapping, as exemplified by the Shor error-correcting code. A maximum length scale for records is enough to guarantee uniqueness. Speculatively, objective branch decompositions may speed up numerical simulations of nonstationary many-body states, illuminate the thermalization of closed systems, and demote measurement from fundamental primitive in the quantum formalism.

Figure 1

Spatially disjoint regions with the same coloring (e.g., the solid blue regions $\mathcal{F},{\mathcal{F}}^{\prime },\dots$) denote different records for the same observable (e.g., ${\mathrm{\Omega }}_a=\{{\mathrm{\Omega }}_a^{\mathcal{F}},{\mathrm{\Omega }}_a^{{\mathcal{F}}^{\prime }},\dots \}$). (a) The spatial record structure of the Shor-code family of states, which can exhibit arbitrary redundancy (in this case fourfold) for two incompatible observables. (b) The solid orange observable pair-covers the hashed blue observable because the top two orange records overlap all blue records. However, if one of the top two orange records is dropped, then neither observable pair-covers the other, and hence both are compatible, despite many overlaps of individual records. (c) Any spatially bounded set of records can be contained inside a single record of a sufficiently dilated but otherwise identical set of records for an incompatible observable; such a state is given in Eq. (9). (d) Any observable with records satisfying the hypothesis of the corollary for some length $\ell$ cannot pair-cover, or be pair-covered by, any other such observable.