A new paradigm for entanglement certification using noncontextuality inequalities

Abstract

By combining the assumptions of Bell locality with those of generalized noncontextuality, we define classes of noncontextuality inequalities for correlations arising in a bipartite Bell circuit. These classes are distinguished by which subsets of the full set of operational identities are taken as input to the principle of noncontextuality; certain natural subsets form a hierarchy that provides a new way of understanding and classifying quantum correlations, including entanglement, steering, and nonlocality. Each level of this hierarchy gives rise to a corresponding class of noncontextuality inequalities whose violation witnesses one of these forms of bipartite quantum resourcefulness, thereby yielding different sufficient conditions for entanglement. The resulting entanglement certification paradigm requires no prior characterization of the measurements, is independent of tomographic gauge freedom, and can certify any entangled state without auxiliary entangled sources. To illustrate its power, we show that noncontextuality inequalities can certify entanglement for families of two-qubit isotropic states for which Bell or steering inequalities are known to fail. We also show that, compared with the Bell test, this approach certifies a much larger fraction of entangled states, while the associated membership problem is more tractable. On the experimental side, we describe techniques to ensure nontrivial operational identities in the presence of noisy and imperfect implementations. We also identify the key assumption under which these techniques are valid, namely, a particular notion of tomographic completeness, which ensures that the operational identities are gauge-independent. Finally, we provide an experimental demonstration of the superior performance of this entanglement certification technique using polarization-entangled photons.

Figures (6)

• Figure 1: (a) a unipartite prepare-measure (PM) circuit; (b) a bipartite prepare-measure circuit, i.e., a bipartite Bell circuit involving two parties labeled Alice (A) and Bob (B). A classical explanation for a given quantum circuit is given by a linear and diagram-preserving map from the quantum representation to an ontological representation; i.e., to a circuit wherein systems are classical variables and transformations are (sub)stochastic maps on these variables.
• Figure 2: (a): Given a fixed $(\mathsf N, \mathsf M,\mathsf P)$-Bell circuit $\mathcal{B}$, we sketch different NCOM-realizable polytopes in the space of operational statistics, where $\mathcal{NC}(\mathcal{B})=\mathcal{NC}_{\rm aaaaa}(\mathcal{B}) \subseteq \mathcal{NC}_{\rm aattt}(\mathcal{B}) \subseteq \mathcal{NC}_{\rm atttt}(\mathcal{B})\subseteq \mathcal{NC}_{\rm ttttt}(\mathcal{B})=\mathcal{L}(\mathcal{B})\subseteq \mathcal{Q}(\mathcal{B})$. Here, $\mathcal{NC}_{\rm aaaaa}(\mathcal{B})$ is defined by considering operational identities; $\mathcal{NC}_{\rm aattt}(\mathcal{B})$ is defined by considering all nontrivial operational identities on Alice's and on Bob's measurements; $\mathcal{NC}_{\rm atttt}(\mathcal{B})$ is defined by considering all nontrivial operational identities only on Alice's measurements; $\mathcal{NC}_{\rm ttttt}(\mathcal{B})=\mathcal{L}(\mathcal{B})$ is defined by not taking into account any nontrivial operational identities, and $\mathcal{Q}(\mathcal{B})$ is the set of all quantum correlations that is realizable by a Bell circuit with the same cardinality of the setting and outcome variables. (b): a bipartite state being classical, separable, unsteerable or local, is equivalent to it being classical relative to $\mathcal{O}_{\rm aaaaa}(\mathcal{B}^{\rm full})$, $\mathcal{O}_{\rm aattt}(\mathcal{B}^{\rm full})$, $\mathcal{O}_{\rm atttt}(\mathcal{B}^{\rm full})$ or $\mathcal{O}_{\rm ttttt}(\mathcal{B}^{\rm full})$ respectively. Insert: Using a sequence of Bell circuits $\{\mathcal{B}_i= (\mathsf{N}_i,\mathsf{M}_i,\mathsf P)\}_i$ that limit to the measurement-full Bell circuit with $\mathsf{N}_1\subset\cdots\subset \mathsf{N}_n \subset \cdots\subset\mathsf{N}^{\rm full}$ and $\mathsf{M}_1\subset\cdots \subset \mathsf{M}_n\subset \cdots\subset\mathsf{M}^{\rm full}$, the set of states that are classical relative to $\mathcal{O}_{\rm aattt}(\mathcal{B}_{i})$ limits to the set of separable states.
• Figure 3: Geometric configuration of $\mathsf{N}=\{\{N^A_{a|x}\}_{a}\}_{x}$ (Left) and $\mathsf{M}=\{\{M^B_{b|y}\}_{b}\}_{y}$ (Right). (a) Example \ref{['example2']}: with cubic and octahedral symmetry; (b) Example \ref{['example3']}: with icosahedral and dodecahedral symmetry.
• Figure 4: Schematic of the experiment. Polarization-entangled photon pairs are generated via parametric down-conversion in a Sagnac interferometer Kim2006. Polarization controllers (Comp) compensate polarization rotations in the single-mode fibres. A pair of computer-controlled liquid crystal phase retarders (LCR) aims to implement arbitrary strength depolarizing channels on one of the photons. Each photon is then directed to separate polarization analyzers--Alice and Bob--where polarization measurements are performed using half-wave plates (HWPs) and quarter-wave plates (QWPs). Coincidence counts are recorded. PPKTP: periodically-poled potassium titanyl phosphate; PBS: polarizing beamsplitter; $\hat{\mathbb{1}}$, $\hat{\mathbb{X}}$, $\hat{\mathbb{Z}}$: Pauli gates.
• Figure 5: Experimental violation of the noncontextuality (NC) inequality of Eq. \ref{['eq: Inequality']}, i.e., $\mathcal{I}\ge 0$, as a function of the isotropic-state visibility $v$. Our experimental data are shown as points with vertical and horizontal error bars, while the ideal case is shown as the diagonal dashed gray line, and the ideal case under $2\%$ white noise in the measurements is shown as a solid blue line. On top of the diagram, we depict the separable-entangled divide, as well as the parameter regimes certifiable by various techniques: general noncontextuality inequalities, the specific noncontextuality inequality used in this work, and steering inequalities. The region in white contains states that simultaneously i) are entangled, ii) are not certifiable by steering (or Bell) inequalities, and iii) are certifiable by noncontextuality inequalities. Three of our experimental data points lie in this white region, demonstrating the advantage of our approach over these prior approaches. Error bars for both $\mathcal{I}$ and $v$ are obtained from 100 trials of Monte Carlo simulation and are depicted. (See Table \ref{['tab:tomoresults']} for detailed data.)

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Theorem 1
• Proposition 1
• Definition 4
• Example 1
• Example 2
• Example 3
• Example 4