Entanglement Verification with Finite Data

Suppose an experimentalist wishes to verify that his apparatus produces entangled quantum states. A finite amount of data cannot conclusively demonstrate entanglement, so drawing conclusions from real-world data requires statistical reasoning. We propose a reliable method to quantify the weight of evidence for (or against) entanglement, based on a likelihood ratio test. Our method is universal in that it can be applied to any sort of measurements. We demonstrate the method by applying it to two simulated experiments on two qubits. The first measures a single entanglement witness, while the second performs a tomographically complete measurement.

Figure 1

General schema of a likelihood ratio test. The separable states $\mathcal{S}$ (blue) are a convex subset of all states, surrounded by entangled states (red). Data from an experiment on a state $\rho$ yield a quasiconvex likelihood function [(a)] with a unique maximum (${\widehat{\rho }}_{\mathrm{MLE}}$). ${\widehat{\rho }}_{\mathrm{MLE}}$ is randomly distributed around $\rho$, at a typical length scale $\delta =O(1/\sqrt{N})$. If ${\widehat{\rho }}_{\mathrm{MLE}}$ is separable then there is no evidence for entanglement, but if it is entangled (as shown), then the relative likelihoods of ${\widehat{\rho }}_{\mathrm{MLE}}$ and the most likely separable state determine the weight of evidence. Data are “convincing” if they are very unlikely to have been produced by a borderline separable state. Typical likelihood ratios for such states depend on the shape of $\mathcal{S}$. In (b)–(d) we show three possible cases: in (b) $\mathcal{S}$ is smaller than $\delta$ and behaves like a point; in (c) it is of size $\delta$ and its behavior is hard to characterize; in (d) it is much bigger than $\delta$ and behaves like a half-space.

Figure 2

Log-likelihood ratios ($\lambda$) behave dramatically differently for different states. 1000 independent simulated tomographically complete experiments were performed, on four different Werner states—separable ($q=0.25$), barely separable ($q=1/3$), slightly entangled ($q=0.35$), and highly entangled ($q=0.50$). $\lambda$ is shown for each trial (points), and averaged over all 1000 trials (solid lines). For small $N$ the experiment cannot reliably distinguish them. As $N$ grows, it resolves shorter distances in the state space. For entangled states, typical values of $\lambda$ increase linearly with $N$, whereas the separable state almost certainly yields $\lambda =0$ [not visible in these plots; for ${\rho }_{q=0.25}$ (black), all trials with more than $N\sim {10}^3$ measurements yielded $\lambda =0$, and the average (dashed line) plunges off the graph]. For barely separable states, $\lambda$ behaves as a semi-${\chi }_k^2$ variable with $k=1$ as $N\to \infty$ (see Fig. 3).

Figure 3

Distribution of $\lambda$ for a SIC-POVM experiment. We show the empirical complementary cumulative distribution function of $\lambda$, $\mathrm{CCDF}({\lambda }_c)=\mathrm{Pr}(\lambda >{\lambda }_c)$, for the state ${\rho }_{q=1/3}$ and simulated datasets of size $N=\{10,\dots ,{10}^6\}$. The CCDF is used to compute confidence levels—e.g., to report entanglement at the 95% confidence level, it is necessary to observe $\lambda$ such that $\mathrm{CCDF}(\lambda )<0.05$. For this particular state, the chance of a zero $\lambda$ approaches 50% as $N$ increases. For each $N$, $\mathrm{CCDF}({\lambda }_c)$ was based on roughly ${10}^4$ data points from independent trials, each of which generated a value of $\lambda$ from $N$ tomographically complete measurements on ${\rho }_{q=1/3}$. We also show CCDFs for a semi-${\chi }_1^2$ variable and a ${\chi }_{\dim (\mathcal{M})}^2={\chi }_{15}^2$ variable. The semi-${\chi }_1^2$ ansatz is good for large $N$, but unreliable for small $N$ (yielding too many false positives), while the ${\chi }_{15}^2$ ansatz is very conservative.