Optimal Verification of Entangled States with Local Measurements

Consider the task of verifying that a given quantum device, designed to produce a particular entangled state, does indeed produce that state. One natural approach would be to characterize the output state by quantum state tomography, or alternatively, to perform some kind of Bell test, tailored to the state of interest. We show here that neither approach is optimal among local verification strategies for 2-qubit states. We find the optimal strategy in this case and show that quadratically fewer total measurements are needed to verify to within a given fidelity than in published results for quantum state tomography, Bell test, or fidelity estimation protocols. We also give efficient verification protocols for any stabilizer state. Additionally, we show that requiring that the strategy be constructed from local, nonadaptive, and noncollective measurements only incurs a constant-factor penalty over a strategy without these restrictions.

Figure 1

The number of measurements needed to verify the state $|{\psi }_{\theta }⟩=\sin \theta |00⟩+\cos \theta |11⟩$, as a function of $\theta$, using the optimal strategy. [See Eq. (10).] Here, $1-\epsilon =0.99$ and $1-\delta =0.9$.

Figure 2

A comparison of the total number of measurements required to verify to fidelity $1-\epsilon$ for the strategy derived here versus the known bounds for estimation up to fidelity $1-\epsilon$ using nonadaptive tomography in [12], the fidelity estimation protocol in [13], and the globally optimal strategy given by projecting onto $|\psi ⟩$. Here, $1-\delta =0.9$ and $\theta =(\pi /8)$.