Bounding the entanglement of N qubits with only four measurements

Quantum-mechanical systems can be used for real-world applications of significant importance, including quantum information processing [1,2], quantum cryptography [3], and quantum metrology [4]. Many of these applications require entanglement between more than two parties; see, e.g., [1,2]. Such systems also allow us to probe the transition from quantum to classical behaviors in increasingly complex systems [5,6]. These applications are of sufficient importance to have prompted a number of experimental initiatives for generating entanglement between many parties [6–13]. Additionally, achieving quantum systems with scalable architectures continues to motivate much research. In tandem with experimental efforts to create entanglement in many-party systems, there has been a theoretical effort aimed at quantifying the entanglement in many-party systems. Of particular importance is the entanglement existing collectively between all N parties of an N -party system, which we call all-party entanglement [14] and which plays an important role in high-precision metrology as well as other applications [4,15,16]. All-party entanglement can best be identified via its opposite, biseparability. A pure state |ψ〉 is biseparable if it has a pure reduced density matrix, i.e., |ψ〉 = |ψ1〉 ⊗ |ψ2〉 where |ψ1〉 and |ψ2〉 are pure states. A mixed state is biseparable if it can be written as a sum of pure biseparable states; otherwise, the state is all-party entangled [2,17]. Quantifying the all-party entanglement has proved to be a challenging task. Previous studies have produced witnesses and/or lower bounds of the all-party entanglement [18–25]. Many of these results are not well suited for use in experimental settings because they apply only for idealized noise-free states [22–25], require numerical methods only feasible for few-partite systems [21], or require knowledge of the density operator of the state under test [18,19]. The latter requirement is a problem for multiparty systems because, in general, the number of measurements to determine the density operator scales exponentially with the number of parties [26]. Apart from these methods, the measured value of witness operators can be used to find a lower bound for the entanglement realized in an experiment [27–30]. Nonetheless, the efficacy of this approach for producing nontrivial lower bounds is not guaranteed and complementary techniques are desired.