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arXiv:1402.5710 · Physical Review Letters

Experimental Detection of Entanglement with Optimal-Witness Families

Jibo Dai, Yink Loong Len, Yong Siah Teo, Berthold‐Georg Englert, Leonid A. Krivitsky

Published Oct 21, 2014• 23 citations
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Abstract

We report an experiment in which one determines, with least tomographic effort, whether an unknown two-photon polarization state is entangled or separable. The method measures whole families of optimal entanglement witnesses. We introduce adaptive measurement schemes that greatly speed up the entanglement detection. The experiments are performed on states of different ranks, and we find good agreement with results from computer simulations.

DOI
10.1103/physrevlett.113.170402

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We report an experiment in which one determines, with least tomographic effort, whether an unknown two-photon polarization state is entangled or separable. The method measures whole families of optimal entanglement witnesses. We introduce adaptive measurement schemes that greatly speed up the entanglement detection. The experiments are performed on states of different ranks, and we find good agreement with results from computer simulations.
We introduce informationally complete measurements whose outcomes are entanglement witnesses and so answer the question of how many witnesses need to be measured to decide whether an arbitrary state is entangled or not: as many as the dimension of the state space. The witnesses can be measured successively; if all of them give an inconclusive result, one exploits their tomographic completeness for a reconstruction of the quantum state and can then determine its entanglement properties by data processing. There are witnesses that are optimal for this purpose. The optimized witness-based measurement can provide exponential improvement with respect to witness efficiency in high-dimensional Hilbert spaces, at the price of a reduction in the tomographic efficiency. We describe a systematic construction and illustrate the matter with the example of two qubits. For the case of two polarization qubits of photons, we show how existing technology can be used to implement the optimized witnesses in a very efficient way. Owing to the details of the implementation, which actually measures the eigenstate basis of the witness rather than solely determining the expectation value of the witness, one does not need to measure more than six witnesses in this example of a 16-dimensional state space.
In this work, we present a practical and efficient framework for verifying entangled states when only a tomographically incomplete measurement setting is available-specifically, when access to observables is severely limited. We show how the experimental estimation of a small number of observables can be directly exploited to construct a large family of entanglement witnesses, enabling the efficient identification of entangled states. Moreover, we introduce an optimization approach, formulated as a semidefinite program, that systematically searches for those witnesses best suited to reveal entanglement under the given measurement constraints. We demonstrate the practicality of the approach in a proof-of-principle experiment with photon-polarization qubits, where entanglement is certified using only a fraction of the full measurement data. These results reveal the maximal usefulness of incomplete measurement settings for entanglement verification in realistic scenarios.
We address the experimental determination of entanglement for systems made of a pair of polarization qubits. We exploit quantum estimation theory to derive optimal estimators, which are then implemented to achieve ultimate bound to precision. In particular, we present a set of experiments aimed at measuring the amount of entanglement for states belonging to different families of pure and mixed two-qubit two-photon states. Our scheme is based on visibility measurements of quantum correlations and achieves the ultimate precision allowed by quantum mechanics in the limit of Poissonian distribution of coincidence counts. Although optimal estimation of entanglement does not require the full tomography of the states we have also performed state reconstruction using two different sets of tomographic projectors and explicitly shown that they provide a less precise determination of entanglement. The use of optimal estimators also allows us to compare and statistically assess the different noise models used to describe decoherence effects occurring in the generation of entanglement.
Entanglement witnesses are a key instrument in determining the entanglement of unknown bipartite qubit systems without requiring a full tomography. Riccardi et al. (2020) proposed a set of two-qubit witnesses, denoted {W3}\{W^3\}{W3}, which require a set of local measurements in three bases. In this work we propose an additional set of witnesses, denoted {W5}\{W^5\}{W5}, that build on {W3}\{W^3\}{W3} and require measurements in two additional bases. We present an adaptive two-step method that uses the measurement results obtained for {W3}\{W^3\}{W3} to select which W5W^5W5 to evaluate. We computationally demonstrate that our adaptive two-step protocol can identify 72% of entangled states generated according to the methods of Roik et al. (2021) while requiring measurements in only five of the nine bases needed for full quantum tomography. Additionally, we experimentally verify the behavior of both sets of witnesses using polarization-entangled photon pairs.
Detecting and quantifying quantum entanglement of a given unknown state poses problems that are fundamentally important for quantum information processing. Surprisingly, no direct (i.e., without quantum tomography) universal experimental implementation of a necessary and sufficient test of entanglement has been designed even for a general two-qubit state. Here we propose an experimental method for detecting a collective universal witness, which is a necessary and sufficient test of two-photon polarization entanglement. It allows us to detect entanglement for any two-qubit mixed state and to establish tight upper and lower bounds on its amount. A different element of this method is the sequential character of its main components, which allows us to obtain relatively complicated information about quantum correlations with the help of simple linear-optical elements. As such, this proposal realizes a universal two-qubit entanglement test within the present state of the art of quantum optics. We show the optimality of our setup with respect to the minimal number of measured quantities.
We experimentally demonstrate a method for detection of entanglement via construction of entanglement witnesses from a limited fixed set of local measurements ( M ). Such a method does not require a priori knowledge about the form of the entanglement witnesses. It is suitable for a scenario where a full state tomography is not available, but the only resource is a limited set of M . We demonstrate the method on pure two-qubit entangled states and mixed two-qubit entangled states, which emerge from photonic implementation of controllable quantum noisy channels. The states we select are motivated by realistic experimental conditions, and we confirm it works well for both cases. Furthermore, possible generalizations to higher-dimensional bipartite systems have been considered, which can potentially detect both decomposable and indecomposable entanglement witnesses. Our experimental results show perfect validity of the method, which indicates that even a limited set of local measurements can be used for quick entanglement detection and further provide a practical test bed for experiments with entanglement witnesses.