# Bounds on the locality of single-particle coherence

It's well known in the multi-partite setting that entanglement can't be generated by local operations and classical communication; indeed, this is often taken as one of the defining properties of entanglement measures. I am interested in what bounds have been proven about a single-particle analog.

Let's concentrate on the density matrix of a single spin-less degree of freedom (particle on a line) or, better yet, the corresponding Wigner function. Consider the pure state $$\rho = |\psi\rangle\langle\psi|$$ with $$|\psi\rangle = |\psi_1\rangle+|\psi_2\rangle$$ where $$|\psi_i\rangle$$ are wavefunctions localized around distantly separated points in phase space $$\alpha_i = (x_i,p_i)$$, e.g., Gaussian wavepackets with position and momentum variance much smaller than the distance between the wavepackets. This state has long-distance coherence across the phase-space displacement $$\Delta \alpha = \alpha_1-\alpha_2$$. This can be quantified with a measure like $$C(\alpha,\beta) = \frac{|\langle \alpha|\rho|\beta\rangle|^2}{\langle \alpha|\rho|\alpha\rangle\langle \beta|\rho|\beta\rangle}.$$

Now, if this is an open system, this coherence could be destroyed by decoherence. However, in the semiclassical limit, and assuming close-system or Markovian dynamics, generating such coherence from a different initial state would be "hard". I'm not quite sure how I want to formalize the semiclassical limit, but for concreteness let's say we assume the Hamiltonian is local in this sense: when Taylor expanding the Hamiltonian $$\hat{H}$$ in $$\hat{x}$$ and $$\hat{p}$$, most of the weight is on the lower-order terms: $$\hat{H} = h_0 + h_x \hat{x} + h_p \hat{x} + h_{xx} \hat{x}^2 + h_{pp} \hat{p}^2+h_{xp} (\hat{x}\hat{p}+\hat{p}\hat{x}) + \ldots,$$ where the $$h$$ coefficients get small rapidly for higher powers.

Intuitively, with a quasiclassical Hamiltonian, if we have a state that doesn't have coherence between two phase-space points $$\beta_1$$ and $$\beta_2$$, there's only two ways for it to be created:

1. The Hamiltonian dynamics "generates" long-range coherence by starting with short-range coherence (e.g., the minimal length scale coherence of a wavepacket) and "stretching" it over the longer scale $$\Delta\beta = \beta_1-\beta_2$$.
2. The state already has long-range coherence between two different phase-space points and the Hamiltonian dynamics "carry" those end points to the new end points $$\beta_1$$ and $$\beta_2$$.

(It's easy to see that some restriction on the Hamiltonian is necessary. For an arbitrary Hamiltonian the system can evolve from a short-range-coherence state to a long-range-coherence state directly without the coherence needing to "move" through intermediate-range-coherence states as described in (1) and (2) above.)

So: What bounds of this nature are known to exist on the phase-space coherence of a single particle, assuming some restriction on the Hamiltonian to be local/quasiclassical?