Entanglement detection in quantum many-body systems using entropic uncertainty relations

We study experimentally accessible lower bounds on entanglement measures based on entropic uncertainty relations. Experimentally quantifying entanglement is highly desired for applications of quantum simulation experiments to fundamental questions, e.g., in quantum statistical mechanics and condensed-matter physics. At the same time it poses a significant challenge because the evaluation of entanglement measures typically requires the full reconstruction of the quantum state, which is extremely costly in terms of measurement statistics. We derive an improved entanglement bound for bipartite systems, which requires measuring joint probability distributions in only two different measurement settings per subsystem, and demonstrate its power by applying it to currently operational experimental setups for quantum simulation with cold atoms. Examining the tightness of the derived entanglement bound, we find that the set of pure states for which our relation is tight is strongly restricted. We show that, for measurements in mutually unbiased bases, the only pure states that saturate the bound are maximally entangled states on a subspace of the bipartite Hilbert space (this includes product states). We further show that our relation can also be employed for entanglement detection using generalized measurements, i.e., when not all measurement outcomes can be resolved individually by the detector. In addition, the impact of local conserved quantities on the detectable entanglement is discussed.

Figure 1

Entanglement entropy between two particles in the ground state of a 1D Hubbard model in the highly attractive regime ($\left|U\right|\gg J$ and $U<0$), compared with the entanglement entropy of a maximally entangled state on the same Hilbert space.

Figure 2

Normalized complementarity factor $-log\left({max}_{x,z}{c}_{xz}\right)$ for a measurement $X$ in the site basis and a second measurement $Z$, for varying number of lattice sites $L$. The second measurement is implemented by letting the particles evolve independently under the hopping Hamiltonian for time $t$ before measuring their positions on the lattice. The normalization is chosen such that a value of one corresponds to measuring in MUBs.

Figure 3

Histogram of the individual overlap elements $c_{xz}$ for $L=30$ lattice sites at tunneling time $t=0.5L$.

Figure 4

Detectable entanglement in the ground state for attractive interactions ($U/J=-100$) using the fully-state-dependent relation (28) and independent tunneling as the transformation to the second measurement basis. The abscissa shows tunneling time over the number of lattice sites, which parametrizes all such transformations. Even though the true entanglement entropy of the considered ground state grows as $-H\left(A\right|B)\approx log(L)$, the detectable entanglement using this method is roughly constant.

Figure 5

Similar to Fig. 4 but using the state-independent uncertainty relation (18). The detectable entanglement is significantly lower and for high number of sites no entanglement can be detected.

Figure 6

Detectable entanglement for the time-evolved state of a spin-1 BEC. Bounds on the entanglement entropy using a numerically optimized first measurement and its single-particle Fourier transform. Numerical optimization has been applied to maximize detection at $r=0.5$ for $N=15$. The optimal basis choice was used to produce the data for $N=50$ shown in the figure.

Figure 7

Bounds on the entanglement entropy using a measurement in the bare mode basis and its single-particle Fourier transformation. The prepared state is the same as in Fig. 6.

Figure 8

Bounds on $-H\left(A\right|B)$ in the ground state of the spin-1 BEC using multiple complementarity factors as a function of the Hamiltonian parameter $q$ for $N=50$ particles. Measurements are performed in the bare mode basis and its single-particle Fourier transform.