Entanglement and sensitivity in precision measurements with states of a fluctuating number of particles

The concepts of separability, entanglement, spin-squeezing and Heisenberg limit are central in the theory of quantum enhanced metrology. In the current literature, these are well established only in the case of linear interferometers operating with input quantum states of a known fixed number of particles. This manuscript generalizes these concepts and extends the quantum phase estimation theory by taking into account classical and quantum fluctuations of the particle number. Our analysis concerns most of the current experiments on precision measurements where the number of particles is known only in average.Comment: Published versio

Relations between Entanglement Witnesses and Bell Inequalities

Bell inequalities, considered within quantum mechanics, can be regarded as non-optimal witness operators. We discuss the relationship between such Bell witnesses and general entanglement witnesses in detail for the Bell inequality derived by Clauser, Horne, Shimony, and Holt (CHSH). We derive bounds on how much an optimal witness has to be shifted by adding the identity operator to make it positive on all states admitting a local hidden variable model. In the opposite direction, we obtain tight bounds for the maximal proportion of the identity operator that can be subtracted from such a CHSH witness, while preserving the witness properties. Finally, we investigate the structure of CHSH witnesses directly by relating their diagonalized form to optimal witnesses of two different classes.Comment: 8 pages, 2 figure

Generation and detection of bound entanglement

We propose a method for the experimental generation of two different families of bound entangled states of three qubits. Our method is based on the explicit construction of a quantum network that produces a purification of the desired state. We also suggest a route for the experimental detection of bound entanglement, by employing a witness operator plus a test of the positivity of the partial transposes

Covariance matrices and the separability problem

We propose a unifying approach to the separability problem using covariance matrices of locally measurable observables. From a practical point of view, our approach leads to strong entanglement criteria that allow to detect the entanglement of many bound entangled states in higher dimensions and which are at the same time necessary and sufficient for two qubits. From a fundamental perspective, our approach leads to insights into the relations between several known entanglement criteria -- such as the computable cross norm and local uncertainty criteria -- as well as their limitations.Comment: 4 pages, no figures; v3: final version to appear in PR

Four-qubit entangled symmetric states with positive partial transpositions

We solve the open question of the existence of four-qubit entangled symmetric states with positive partial transpositions (PPT states). We reach this goal with two different approaches. First, we propose a half-analytical-half-numerical method that allows to construct multipartite PPT entangled symmetric states (PPTESS) from the qubit-qudit PPT entangled states. Second, we adapt the algorithm allowing to search for extremal elements in the convex set of bipartite PPT states [J. M. Leinaas, J. Myrheim, and E. Ovrum, Phys. Rev. A 76, 034304 (2007)] to the multipartite scenario. With its aid we search for extremal four-qubit PPTESS and show that generically they have ranks (5,7,8). Finally, we provide an exhaustive characterization of these states with respect to their separability properties.Comment: 5+4 pages, improved version, title slightly modifie

Differential atom interferometry beyond the standard quantum limit

We analyze methods to go beyond the standard quantum limit for a class of atomic interferometers, where the quantity of interest is the difference of phase shifts obtained by two independent atomic ensembles. An example is given by an atomic Sagnac interferometer, where for two ensembles propagating in opposite directions in the interferometer this phase difference encodes the angular velocity of the experimental setup. We discuss methods of squeezing separately or jointly observables of the two atomic ensembles, and compare in detail advantages and drawbacks of such schemes. In particular we show that the method of joint squeezing may improve the variance by up to a factor of 2. We take into account fluctuations of the number of atoms in both the preparation and the measurement stage, and obtain bounds on the difference of the numbers of atoms in the two ensembles, as well as on the detection efficiency, which have to be fulfilled in order to surpass the standard quantum limit. Under realistic conditions, the performance of both schemes can be improved significantly by reading out the phase difference via a quantum non-demolition (QND) measurement. Finally, we discuss a scheme using macroscopically entangled ensembles.Comment: 10 pages, 5 figures; eq. (3) corrected and other minor change

Parametric amplification of vacuum fluctuations in a spinor condensate

Parametric amplification of vacuum fluctuations is crucial in modern quantum optics, enabling the creation of squeezing and entanglement. We demonstrate the parametric amplification of vacuum fluctuations for matter waves using a spinor F=2 Rb-87 condensate. Interatomic interactions lead to correlated pair creation in the m_F= +/- 1 states from an initial unstable m_F=0 condensate, which acts as a vacuum for m_F unequal 0. Although this pair creation from a pure m_F=0 condensate is ideally triggered by vacuum fluctuations, unavoidable spurious initial m_F= +/- 1 atoms induce a classical seed which may become the dominant triggering mechanism. We show that pair creation is insensitive to a classical seed for sufficiently large magnetic fields, demonstrating the dominant role of vacuum fluctuations. The presented system thus provides a direct path towards the generation of non-classical states of matter on the basis of spinor condensates.Comment: 5 pages, 4 figure

Experimental detection of entanglement via witness operators and local measurements

In this paper we address the problem of detection of entanglement using only few local measurements when some knowledge about the state is given. The idea is based on an optimized decomposition of witness operators into local operators. We discuss two possible ways of optimizing this local decomposition. We present several analytical results and estimates for optimized detection strategies for NPT states of 2x2 and NxM systems, entangled states in 3 qubit systems, and bound entangled states in 3x3 and 2x4 systems.Comment: 24 pages, 2 figures. Contribution to the proceedings of the International Conference on Quantum Information in Oviedo, Spain (July 13-18, 2002). Error in W_W1-witness Eq. (35) corrected as well as minor typos. Reference adde