Fisher information and multiparticle entanglement

Abstract

The Fisher information $F$ gives a limit to the ultimate precision achievable in a phase estimation protocol. It has been shown recently that the Fisher information for a linear two-mode interferometer cannot exceed the number of particles if the input state is separable. As a direct consequence, with such input states the shot-noise limit is the ultimate limit of precision. In this work, we go a step further by deducing bounds on $F$ for several multiparticle entanglement classes. These bounds imply that genuine multiparticle entanglement is needed for reaching the highest sensitivities in quantum interferometry. We further compute similar bounds on the average Fisher information $\bar F$ for collective spin operators, where the average is performed over all possible spin directions. We show that these criteria detect different sets of states and illustrate their strengths by considering several examples, also using experimental data. In particular, the criterion based on $\bar F$ is able to detect certain bound entangled states.Comment: Published version. Notice also the following article [Phys. Rev. A 85, 022322 (2012), DOI: 10.1103/PhysRevA.85.022322] by Geza T\'oth on the same subjec

Electromagnetic shielding in quantum metrology

“…A straightforward way, in addition to increasing the QFI, to enhance the precision when the probe systems are closed is by parallel measurements. Later it was shown that the use of correlated systems such as entangled states can also improve the precision of parameter estimation [5,[9][10][11][12][13][14][15][16][17][18][19][20][21]. On the other hand , however, interaction between a system and an environment is unavoidable in reality, and the quantum decoherence induced by such interactions may decrease the QFI and destroy the quantum entanglement in the probe system exploited to improve the precision.…”

Measuring multipartite entanglement through dynamic susceptibilities

“…In experiments, entanglement becomes measurable via a tomographic determination of the many-particle quantum state [12][13][14][15], and protocols have been developed [16] and implemented in remarkable experiments [17] to measure entanglement entropies in quench dynamics and quantum phase transitions. However, the resources required by these protocols scale exponentially with the system size, and these experimental efforts are thus limited a priori to few-particle systems.To address the problem of detecting and quantifying multipartite entanglement for large systems, we consider below the quantum Fisher information (QFI) as an entanglement witness [18][19][20]. Our key result is thatfor a many-body system at thermal equilibrium at any temperature-the QFI can be determined directly from a measurement of Kubo linear response functions, in particular the dynamic susceptibility (see Fig.…”

“…Background on the quantum Fisher information.-In recent years, the QFI has generated a lot of attention, because it provides a rigorous lower bound for genuinely multipartite entanglement [18,19]. Originally, it was introduced to quantify the maximal precision with which a parameter (a phase) ϑ can be estimated using a given quantum state ρ [7,28].…”

Witnessing Multipartite Entanglement by Detecting Asymmetry

“…There are several entanglement witnesses written in terms of the quantum Fisher information. They relate entanglement to the system speed of response to phase shifts generated by additive spin-1/2 Hamiltonians J N = ∑ N i=1 1/2σ i [32,46,[54][55][56][57][58][59][60]. In particular, a constraint which cannot be satisfied by k-separable states of N qubits is F J N (ρ) ≥ nk 2 + (N − nk) 2 , where n = N k .…”