## previous reports

## Research Report (2013-2015)

## General Outline of the Field of Research

In my research group, we study the effects of quantized radiation fields and their propagation in media and interaction with matter. We derive observable nonclassicality conditions to characterize multimode quantum-correlated systems. For the detection of quantum effects, measurement schemes are proposed and rigorously analyzed. Beyond the theoretical identification of quantum effects, we collaborate with experimental groups to implement our methods and we quantify the strength of the quantumness with novel techniques.

## Entanglement Detection

In the growing field of quantum information and communication technologies, entanglement is a key resource. The detection of this nonlocal and nonclassical correlation between two or more quantum systems is, thus, of great interest. We focus on finding entanglement criteria for general systems.

In [1], we introduced a new method to construct optimal witnesses. This technique allows to confirm entanglement in any quantum system, if present. For certain quantum systems, such as the renowned Gaussian states, analytical solutions have been found. We applied this theory to data from the experimental group of C. Fabre and N. Treps in Paris [2]. The results, depicted in Fig. 1, verify the entanglement signature in every combination of subsystems.

## Nonclassicality Quasiprobabilities

Nonclassicality quasiprobabilities provide a universal method to visualize nonclassicality through negativities of non-singular phase-space functions. The underlying idea is the regularization of the Glauber-Sudarshan P function by means of a so-called nonclassicality filter. We have identified optimal filters, which minimize the number of required data points to certify nonclassical effects with high statistical significance [3]. This includes the construction of an analytic filter which preserves the full information of the quantum state.

Using this powerful method, the characterization of multimode correlations can be done beyond other notions of quantum correlations. Furthermore, improvements on traditional measurement procedures can be advised [4]. In particular, we proposed a continuous-in-phase sampling technique, in contrast to standard discrete phase-locked measurements. Our approach allows an unconditional verification of nonclassicality without interpolation errors.

The nonclassicality quasiprobability for both measurement strategies is given in Fig. 2. On the left we see the correctly reconstructed state with our proposal, whereas the right plot in Fig. 2 is deformed due to phase-locking. Experimental data obtained from the group of B. Hage (Rostock) were analyzed and the regularized P function was directly reconstructed. The sampling of nonclassicality quasiprobabilities is a powerful and universal method to visualize quantum effects within arbitrary quantum states of light.

## Click Statistics

In order to verify quantum effects, it is required to formulate a measurement theory and the corresponding quantumness (nonclassicality) tests. For so-called click-counting detectors, we derived such methods in close collaboration with G. S. Agarwal, Oklahoma State University. The theoretical foundation for the identification of all orders of nonclassical correlations with these detection devices was formulated in our work [5]. Based on this approach, we also studied phase-sensitive measurements and reconstructions of phase-space distributions in other recent contributions.

In collaboration with the experimental group of C. Silberhorn, Universitäat Paderborn, we applied our method to uncover quantum correlations [6]. We could demonstrate nonclassical correlations between the two modes of the generated light field, see Fig. 3. With our technique, this was achieved despite high losses and high pump-powers. This renders it possible to access the quantum features of radiations fields for many applications in noisy environments.

## Quantification of Quantumness

A question of fundamental importance is: How strong are the quantum properties of a given quantum system? The answer may be of relevance for applications of quantum effects in novel quantum technologies, for example, in the context of the enhancement of security in quantum communication. However, in our previous work we have shown that a quantification has very different aspects. We distinguish the quantification of the quantum properties of a given state from the quantification of its usefulness for a particular application. The former is based on the convex structure of the quantum state, whereas the latter is merely an operational quantification for certain protocols. Our work concentrates on the universal quantification of quantumness.

In [7] we have unified the quantification of the nonclassicality of a single radiation mode with the quantification of entanglement. A nonclassical light field is combined with vacuum channels by a beam splitter, which yields entangled output beams in general. We have shown that the quantum superposition principle, which is fundamental for the structure of quantum theory, is the key for a unified quantification of nonclassicality and entanglement. We proved that the number of superpositions of coherent states needed for the representation of the single-mode input state is exactly the same as the number of superposition of separable states in the output fields. This yields a unified quantification of nonclassicality and multimode entanglement.

Multimode quantum states have a complex structure, see for example the experimental results for frequency-comb entangled states [2], cf. section *Entanglement Detection*.
Hence, it is of interest to combine the quantification of multipartite entanglement with the problem of uncovering the entanglement structure.
A first attempt to combine these two aspects has been developed in [8].
For this purpose, we derived a modified version of the separability eigenvalue problem to unify the identification of the structuring with the quantification of the entanglement.
This paves the way to a more detailed understanding of complex entangled quantum states.

## References

- J. Sperling and W. Vogel,
*Multipartite Entanglement Witnesses*, Phys. Rev. Lett. 111, 110503 (2013). - S. Gerke, J. Sperling, W. Vogel, Y. Cai, J. Roslund, N. Treps, and C. Fabre,
*Full Multipartite Entanglement of Frequency-Comb Gaussian States*, Phys. Rev. Lett. 114, 050501 (2015). - B. Kühn and W. Vogel,
*Visualizing nonclassical effects in phase space*, Phys. Rev. A 90, 033821 (2014). - E. Agudelo, J. Sperling, W. Vogel, S. Köhnke, M. Mraz, and B. Hage,
*Continuous sampling of the squeezed-state nonclassicality*, Phys. Rev. A 92, 033837 (2015). - J. Sperling, W. Vogel, and G. S. Agarwal,
*Correlation measurements with on-off detectors*, Phys. Rev. A 88, 043821 (2013). - J. Sperling, M. Bohmann, W. Vogel, G. Harder, B. Brecht, V. Ansari, and C. Silberhorn,
*Uncovering Quantum Correlations with Time-Multiplexed Click Detection*, Phys. Rev. Lett. 115, 023601 (2015). - W. Vogel and J. Sperling,
*Unified quantification of nonclassicality and entanglement*, Phys. Rev. A 89, 052302 (2014). - F. Shahandeh, J. Sperling, and W. Vogel,
*Structural Quantification of Entanglement*, Phys. Rev. Lett. 113, 260502 (2014).