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?


1 Answer 1


I think this question is very interesting. As for your final question: I am pretty sure there are no bounds on how fast you can create coherence between your phase space points in classical Quantum Mechanics (QM), since you can move population between them arbitrary quickly with the right Hamiltonian. And if you can move population you can build coherence. In relativistic QM things are probably different.

  • $\begingroup$ Martin, I am assuming a restriction on the locality of the Hamiltonian always, so I think this would rule out the Hamiltonian you are suggesting. I will re-write my question to make this more clear. $\endgroup$ Commented May 18, 2020 at 19:30

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