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arXiv:2004.06412 · Physical Review Research

Quantifying entanglement of parametric down-converted states in all degrees of freedom

Filippus S. Roux

Published May 7, 2020• 11 citations
arXivHTMLPDF

Abstract

The authors compute the Schmidt number, a measure of the entanglement degree, in a parametric-down conversion state using a Wigner functional approach. The result shows that extremely large Schmidt numbers can be obtained under achievable experimental conditions.
DOI
10.1103/physrevresearch.2.023137

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Spontaneous parametric down-conversion (SPDC) is the most widely-used source of high-dimensional entangled two-photon states, and the entanglement in the spatial degree of freedom is considered best suited for harnessing high-dimensional advantages. Although the Schmidt basis provides a natural choice for state characterisation of entangled two-photon states in any degree of freedom, there is currently no technique that can measure the Schmidt basis of an entangled two-photon field. The existing techniques can only reconstruct the Schmidt spectrum when the Schmidt basis is known a priori. In contrast, we present a technique that measures the complete set of spatial Schmidt modes without any prior knowledge. Using this technique, we report measurement of states with over 3000 Schmidt modes -- highest reported yet -- with up to 98%\%% fidelity. We expect our work to significantly advance the harnessing of high-dimensional advantages in SPDC-based systems.
The Schmidt number quantifies the number of modes and is mainly used as a measure for the quality of entanglement. We theoretically compute the photon distribution of type-I spontaneous parametric down conversion (SPDC) with an arbitrary Schmidt number. The photon distribution is used for a novel method to measure the Schmidt number. This method requires only two on–off single-photon detectors with no photon number or temporal resolution. The method works in the strong pumping regime where high photon numbers are non-negligible. We experimentally demonstrate the method for type-II SPDC. The easy and fast measurement of the Schmidt number has a broad range of applications from the calibration of strong pump SPDC and entanglement sources to multi-photon quantum interference and Gaussian boson sampling.
The Schmidt number is an important kind of characterization of quantum entanglement. Quantum states with higher Schmidt numbers demonstrate significant advantages in various quantum information processing tasks. By deriving a class of k-positive linear maps based on symmetric measurements, we present new Schmidt-number witnesses of class (k + 1). By detailed example, we show that our Schmidt number witnesses identify better the Schmidt number of quantum states in high-dimensional systems. Furthermore, we note that the Fedorov ratio, which coincides with the Schmidt number for pure Gaussian states and provides a close approximation in non-Gaussian cases such as spontaneous parametric down-conversion, serves as an experimentally accessible tool for validating the proposed (k +1)-class Schmidt-number witnesses.
A profound comprehension of quantum entanglement is crucial for the progression of quantum technologies. The degree of entanglement can be assessed by enumerating the entangled degrees of freedom, leading to the determination of a parameter known as the Schmidt number. In this paper, we develop an efficient analytical tool for characterizing high Schmidt number witnesses for bipartite quantum states in arbitrary dimensions. Our methods not only offer viable mathematical methods for constructing high-dimensional Schmidt number witnesses in theory but also simplify the quantification of entanglement and dimensionality. Most notably, we develop high-dimensional Schmidt number witnesses within arbitrary-dimensional systems, with our Schmidt witness coefficients relying solely on the operator Schmidt coefficient. Subsequently, we demonstrate our theoretical advancements and computational superiority by constructing Schmidt number witnesses in arbitrary dimensional bipartite quantum systems with Schmidt numbers four and five.
The dimensionality of entanglement in high-dimensional quantum systems is quantified by the Schmidt number, whose large value signals potential advantages for quantum communication and quantum computation. A commonly used approach to practical Schmidt-number detection in finite-dimensional systems is based on the estimation of fidelities to pure states with large entanglement dimensionality or lower bounds to these fidelities. However, for mode-entangled continuous-variable systems, the corresponding measurements are typically difficult to realize. Here, we introduce and assess fidelity bounds for Schmidt-number detection based on practically more accessible measurements of the first and second statistical moments of arbitrary continuous-variable quantum states. We provide analytical formulas that allow evaluating these bounds and test them for families of Gaussian and non-Gaussian states, including two-mode squeezed thermal states and photon-added displaced two-mode squeezed vacuum states. We compare our bounds to other methods for the certification of high-dimensional entanglement based on the same measurements.