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Conference on Laboratory Instruction Beyond the First Year (BFY)

Witnessing Entanglement

Marisol N. Beck, M. Beck

Published Nov 17, 2015• 0 citations
Publisher

Abstract

An entangled state of a two-particle system is a quantum state that cannot be separated-it cannot be written as the product of states of the individual particles.One way to tell if a system is entangled is to use it to violate a Bell inequality (such as the Clauser-Horne-Shimony-Holt, CHSH, inequality), because entanglement is necessary to violate these inequalities.However, there are other, more efficient measurements that determine whether or not a system is entangled; an operator that corresponds to such a measurement is referred to as an entanglement witness.We present the theory of witness operators, and an undergraduate experiment that measures an entanglement witness for the joint polarization state of two photons.We are able to produce states for which the expectation value of the witness operator is entangled by more than 160 standard deviations.

DOI
10.1119/bfy.2015.pr.003

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An entangled state of a two-particle system is a quantum state that cannot be separated-it cannot be written as the product of states of the individual particles.One way to tell if a system is entangled is to use it to violate a Bell inequality (such as the Clauser-Horne-Shimony-Holt, CHSH, inequality), because entanglement is necessary to violate these inequalities.However, there are other, more efficient measurements that determine whether or not a system is entangled; an operator that corresponds to such a measurement is referred to as an entanglement witness.We present the theory of witness operators, and an undergraduate experiment that measures an entanglement witness for the joint polarization state of two photons.We are able to produce states for which the expectation value of the witness operator is entangled by more than 160 standard deviations.
An entangled state of a two-particle system is a quantum state that cannot be separated, meaning it cannot be written as the product of states of the individual particles. One way to tell if a system is entangled is to use it to violate a Bell inequality (such as the Clauser-Horne-Shimony-Holt, CHSH, inequality), because entanglement is necessary for such a violation. However, there are other, easier-to-perform measurements that determine whether or not a system is entangled. An operator that corresponds to such a measurement is referred to as an entanglement witness. Here, we present the theory of witness operators and an undergraduate experiment that measures entanglement witnesses for the joint polarization state of two photons. We are able to produce states for which the expectation value of a witness operator is entangled by more than 300 standard deviations. In order to further examine the performance of these witness operators, we present a simple way to generate states that closely approximate Werner states, which have a controllable degree of entanglement.
All previous tests of local realism have studied correlations between single-particle measurements. In the present experiment, we have performed a Bell experiment on three particles in which one of the measurements corresponds to a projection onto a maximally entangled state. We show theoretically and experimentally that correlations between these entangled measurements and single-particle measurements are too strong for any local-realistic theory and are experimentally exploited to violate a Clauser-Horne-Shimony-Holt-Bell inequality by more than 5 standard deviations. We refer to this possibility as "entangled entanglement."
Abstract Quantum entanglement is a key resource in quantum information. Teleportation, dense coding, and many other quantum protocols rely on the existence of entanglement, and would not be possible with just classical correlations. Furthermore, using entanglement between qubits that support computation, quantum computers can solve problems faster than classical computers. Bell's inequalities can sometimes help in discriminating between entangled and separable states. However, these inequalities are not always reliable — this is true in the sense that they may fail to detect genuinely entangled states. This question is addressed in the present chapter by focusing on an entanglement witness, a Hermitian operator winch helps us to decide whether a state is entangled or not. The basic idea is that the expectation value of the witness will be different for separable and entangled states. This chapter discusses entanglement witnesses, the Jamiolkowski isomorphism, and the Peres Horodecki criterion. More examples of entanglement witnesses are given.
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.
In a future quantum network, two spatially separated individuals sharing a polarization entanglement source may need to assess the entanglement quality of the source without the presence of classical auxiliary signals. When the two are separated by a single-mode fiber, automated methods are necessary for the validation and estimation of polarization entanglement. We experimentally examine real-time iterative methods to search for maximal Bell violations, specifically the Clauser–Horne–Shimony–Holt (CHSH) inequality, between two observers sharing polarization entangled photons to evaluate the entanglement quality of the source. Our source allows us to tune the degree of entanglement of their shared photon pair by changing the temporal overlap of the two photons at an entangling 50/50 beam splitter so that we can compare the CHSH parameter these methods find to the theoretical values our source produces. The iterative methods used within our experiment are the Nelder–Mead optimization method, stochastic gradient descent, and Bayesian optimization. This is the first feedback experiment to study automated Bell violations in fiber and the first to compare all three of these iterative methods to one another in a quantum polarization control experiment. In our experiment, all methods are able to find Bell violations, but the Nelder–Mead method performed the best in terms of the speed and accuracy in finding the maximal violation.
We show that it is possible to find maximal violations of the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality using only position measurements on a pair of entangled nonrelativistic free particles. The device settings required in the CHSH inequality are done by choosing one of two times at which position is measured. For different assignments of the “+” outcome to positions, namely, to an interval, to a half-line, or to a periodic set, we determine violations of the inequalities and states where they are attained. These results have consequences for the hidden variable theories of Bohm and Nelson, in which the two-time correlations between distant particle trajectories have a joint distribution, and hence cannot violate any Bell inequality.