Witnessing entanglement without entanglement witness operators

Abstract

Quantum mechanics predicts the existence of correlations between composite systems that, although puzzling to our physical intuition, enable technologies not accessible in a classical world. Notwithstanding, there is still no efficient general method to theoretically quantify and experimentally detect entanglement of many qubits. Here we propose to detect entanglement by measuring the statistical response of a quantum system to an arbitrary nonlocal parametric evolution. We witness entanglement without relying on the tomographic reconstruction of the quantum state, or the realization of witness operators. The protocol requires two collective settings for any number of parties and is robust against noise and decoherence occurring after the implementation of the parametric transformation. To illustrate its user friendliness we demonstrate multipartite entanglement in different experiments with ions and photons by analyzing published data on fidelity visibilities and variances of collective observables.A central problem in quantum technologies is to detect and characterize entanglement among correlated parties (1-3). A most popular approach is based on the implementation of entanglement witness operators (EWs). EW is a Hermitian operator W such that Tr½ρ sep W ≥ 0 for all separable states ρ sep and Tr½ρW < 0 for, at least, one entangled state ρ (4-8). The power of this method relies on the algebraic fact that for each multipartite entangled state there exists (at least) one EW that recognizes it (4). EWs are device-dependent: Experimental mischaracterization may lead to false positives, namely, to the unwitting realization of operators W exp ≠ W such that Tr½ρ sep W exp < 0 also for some separable states, therefore signaling entanglement in states that are only classically correlated (9, 10). The same problem arises, even more dramatically, when trying to detect entanglement via the tomographic reconstruction of the quantum state (10-15).Device-independent entanglement witness operators (DIEWs) can be constructed by exploiting Bell-like inequalities testing the correlations between measurement data. These are obtained for different optimized configurations and demand that the parties be addressed locally and do not interact during operations and measurements. DIEWs recognize an important class of entangled states without relying on any hypothesis about the measurement actually performed (16)(17)(18)(19). The specific configurations required to witness entanglement are only known for particular cases; the extension to an arbitrary state can vary from computationally hard to prohibitive. A recent experiment (20) has exploited Bell-based DIEWs to demonstrate genuine multipartite entanglement up to six trapped ions. Cross-talk among the particles affected the DIEW of larger systems.An alternative approach to detect entanglement that does not require the construction of a witness operator has been proposed (21-23). This exploits the Fisher information (or "statistical speed") quantifying how quickly two slightly differe...

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