Citation: Gioia, Lei, and Ryan Thorngren. "W-state is not the unique ground state of any local Hamiltonian." arXiv preprint arXiv:2310.10716 (2023).
Web: https://arxiv.org/abs/2310.10716
Tags:Spin-chains
This paper gives a fascinating result - sometimes the degeneracies implied by long-range entanglement are necessarily exact, even in the presence of disorder! This goes counter to my intuition. In my head, symmetries imply exact degeneracy. Without degeneracy there is always some exponential splitting in the presence of noise. It turns out that this is not the case. This is the case for topologically ordered states, but there are other states with even more exotic long range entanglement for which exact degeneracy is forced.
The setting is as follows. The authors are interested in 1D spin chains. They define a "finite-range 1d Hamiltonian system" to be a sequence of Hilbert spaces at each site, along with a family of terms acting in a finite neighborhood of each site, though the diameter of this finite-radius interaction is allowed to tend towards infinity. A family of states parameterized by system size is called a ground state of the system if it is a ground state for all large enough system size.
The authors define the "W state" to be the state which is the equal superposition over all sites of the state with a single one just at that site. The point is as follows. Suppose that the W-state is the ground state of a finite-range 1D Hamiltonian system. Suppose that it is not exactly degenerate with the 0-state. The only difference between the W-state and the 0-state is having this moving-one. Suppose now we consider the state which is the equal superposition of two ones. Since the two ones are typically far away, the Hamiltonian can not distinguish between the difference of this new state W2 with W and the difference between W and 0. Hence W2 should have even lower energy! There are finite-size effects which stop this argument from going through, but considering the states Wn (equal superposition of n ones) with n tending towards infinity the argument goes though - W and 0 must be degenerate.
It seems like the W-state has all sorts of interesting properties. If one wants to get an idea about what long-range entanglement looks like outside of the nice world of topologically ordered Hamiltonians, it seems like a great place to start. The W-state is in fact uniquely special - it shows up as one of the fundamental classes in the classification of entanglement in three-qubit systems:
> Dür, Wolfgang, Guifre Vidal, and J. Ignacio Cirac. "Three qubits can be entangled in two inequivalent ways." Physical Review A 62.6 (2000): 062314.