Experimentally Accessible Bounds on Distillable Entanglement from Entropic Uncertainty Relations

Entanglement is not only the resource that fuels many quantum technologies but also plays a key role for some of the most profound open questions of fundamental physics. Experiments controlling quantum systems at the single quantum level may shed light on these puzzles. However, measuring, or even bounding, entanglement experimentally has proven to be an outstanding challenge, especially when the prepared quantum states are mixed. We use entropic uncertainty relations for bipartite systems to derive measurable lower bounds on distillable entanglement. We showcase these bounds by applying them to physical models realizable in cold-atom experiments. The derived entanglement bounds rely on measurements in only two different bases and are generically applicable to any quantum simulation platform.

Figure 1

Two particles on a lattice. (a) Illustration of the system Hamiltonian and measurement bases. For the site basis measurement the positions of both particles on the lattice are detected. For the tilted basis measurement the particles are allowed to tunnel independently for time $t$ before their positions are measured. (b) Entanglement entropy of the ground state of two particles with strongly attractive interactions $U/J=-100$ compared to the maximally entangled state. (c) Detectable entanglement using Eq. (3) (solid lines) and Eq. (1) (dashed lines) as a function of the tunneling time $t$.

Figure 2

Collective spin-1 system. (a) Schematic of the system Hamiltonian and local measurement bases. Subsystems $A$ and $B$ are obtained by applying a balanced beam splitter operation to each mode. For the bare mode basis one detects the number of particles in each mode, while for the tilted basis occupation numbers are detected after a noninteracting local unitary was applied. (b) Entanglement generated by the evolution under spin-mixing dynamics out of the polar state $|0,N,0⟩$ ($N=50$, $q=-g(N-1/2)$) (dashed) and state dependent bounds. For short and long-time evolution different tilted bases were used (see text). (c) Entanglement and bounds for the ground state of Hamiltonian (5) for varying $q$ ($N=50$). The dotted vertical lines designate the boundaries between different ground state phases. (d) Scaling of true entanglement entropy and bound using $q_{\mathrm{FSD}}$ deep in the Twin-Fock phase ($q/q_c=-5$). Only even $N$ are shown. For odd $N$, both $-H(A|B)$ and the bound take somewhat larger values but scale in the same way with $N$ [27]. The solid lines are fits to the data points.