State-Independent Uncertainty Relations and Entanglement Detection in Noisy Systems

Quantifying quantum mechanical uncertainty is vital for the increasing number of experiments that reach the uncertainty limited regime. We present a method for computing tight variance uncertainty relations, i.e., the optimal state-independent lower bound for the sum of the variances for any set of two or more measurements. The bounds come with a guaranteed error estimate, so results of preassigned accuracy can be obtained straightforwardly. Our method also works for postive-operator-valued measurements. Therefore, it can be used for detecting entanglement in noisy environments, even in cases where conventional spin squeezing criteria fail because of detector noise.

Figure 1

Minimizing the sum of the variances of two observables $A$ and $B$ can be expressed entirely in terms of the set $\mathcal{C}$ of possible triples $(⟨A{⟩}_{\rho },⟨B{⟩}_{\rho },⟨A^2+B^2{⟩}_{\rho })$ (red solid convex body), namely, as finding that vertical displacement of the surface $z=x^2+y^2$ (green paraboloid) which just touches $\mathcal{C}$ from below. We successively approximate $\mathcal{C}$ by polytopes (blue edges, boxed vertices) from the outside, and perform the minimization on this polytope. This gives a converging sequence of correct state-independent uncertainty relations.

Figure 2

Two dimensional sketch of geometry and the basic algorithm: The set $\mathcal{C}$ (red) with its outer approximation $\mathcal{P}(\mathcal{R})$ (blue and blue dashed lines) and the extremal points $\mathcal{E}(\mathcal{R})$ (white squares). By adding the direction ${\mathbf{r}}^{\prime }$, the polyhedral approximation is refined and the lower bound $c_{-}(\mathcal{R})$ is improved from $\mu ({\mathbf{v}}^{*})$ (dashed green parabola) to $\mu ({\mathbf{v}}^{**})$ (green parabola).

Figure 3

Improving the outer approximation of $\mathcal{C}$ (red convex body) by adding more directions to the set $\mathcal{R}$. Every direction $\mathbf{r}\in \mathcal{R}$ gives a face of $\mathcal{P}(\mathcal{R})$ (blue polytope). New directions are chosen such that the vertex with the lowest value of $\mu$ will be cut off. Example generated from randomly chosen $A,B\in {\mathbb{R}}^{10\times 10}$.

Figure 4

Uncertainty regions for entangled and separable states. Superposition of the graphs for different noise levels $\alpha$: $\text{green}=0$, $\text{blue}=0.2$, $\text{red}=0.5$. In this example we consider local measurements of orthogonal spin-1 components, i.e., $M_i=L_i^A+L_i^B$.