Recently, several hybrid approaches to quantum information emerged which utilize both continuous- and discrete-variable methods and resources at the same time. In this work, we investigate the bipartite hybrid entanglement between a finite-dimensional, discrete-variable quantum system and an infinite-dimensional, continuous-variable quantum system. A classification scheme is presented leading to a distinction between pure hybrid entangled states, mixed hybrid entangled states (those effectively supported by an overall finite-dimensional Hilbert space), and so-called truly hybrid entangled states (those which cannot be described in an overall finite-dimensional Hilbert space). Examples for states of each regime are given and entanglement witnessing as well as quantification are discussed. In particular, using the channel map of a thermal photon noise channel, we find that true hybrid entanglement naturally occurs in physically important settings. Finally, extensions from bipartite to multipartite hybrid entanglement are considered.
Recently, several hybrid approaches to quantum information emerged which utilize both continuous- and discrete-variable methods and resources at the same time. In this work, we investigate the bipartite hybrid entanglement between a finite-dimensional, discrete-variable quantum system and an infinite-dimensional, continuous-variable quantum system. A classification scheme is presented leading to a distinction between pure hybrid entangled states, mixed hybrid entangled states (those effectively supported by an overall finite-dimensional Hilbert space), and so-called truly hybrid entangled states (those which cannot be described in an overall finite-dimensional Hilbert space). Examples for states of each regime are given and entanglement witnessing as well as quantification are discussed. In particular, using the channel map of a thermal photon noise channel, we find that true hybrid entanglement naturally occurs in physically important settings. Finally, extensions from bipartite to multipartite hybrid entanglement are considered.
Quantum information theory is a very young area of research offering a lot of challenging open questions to be tackled by ambitious upcoming physicists. One such problem is addressed in this thesis. Recently, several protocols have emerged which exploit both continuous variables and discrete variables. On the one hand, outperforming many of the established pure continuous variable or discrete variable schemes, these hybrid approaches offer new opportunities. However, on the other hand, they also lead to new, intricate, as yet uninvestigated, phenomena. An important ingredient of several of these hybrid protocols is a new kind of entanglement: The hybrid entanglement between continuous variable and discrete variable quantum systems, which is studied in detail in this work. An exhaustive analysis of this kind of entanglement is performed, where the focus is on bipartite entanglement. Nevertheless, also issues regarding multipartite hybrid entanglement are briefly discussed. The quintessence of this thesis is a new classification scheme which distinguishes between effective discrete variable hybrid entanglement and so-called true hybrid entanglement. However, along the way, also other questions are addressed, which have emerged during the studies. For example, entanglement witnessing is discussed not only for hybrid entangled states, but also for fully continuous variable two-mode Schroedinger cat states. Furthermore, subtleties regarding entanglement witnessing in a certain kind of mixed states are examined. Not only theoretical classification and analysis of hybrid entangled states are discussed, but also their generation is presented and a few applications are demonstrated.
Entanglement is a key resource for fundamental tests of physics and emerging quantum technologies. In quantum optics, two perspectives on entanglement coexist. In the continuous-variable framework, entanglement is understood as holding between optical modes. In contrast, discrete-variable quantum optics focuses on quantum correlations in degrees of freedom such as polarization that label fixed numbers of photons. In this paper, we show that entanglement can transcend this separation. Spontaneous parametric down-conversion inherently generates correlations in optical phase space, photon number, and labelling degrees of freedom simultaneously. In polarization, this structure is traditionally described by macroscopic Bell states. Existing witnesses, however, fail to detect the genuine hybrid entanglement of these states, which goes beyond the continuous-discrete-variable categorization. Here, we lay the groundwork for a general framework unifying continuous- and discrete-variable notions of entanglement. In particular, we derive an operational witness providing a sufficient criterion for genuine hybrid number-polarization entanglement and outline its experimental implementation. Finally, we discuss exemplary states which, together with our results on macroscopic Bell states, motivate a broader classification of genuine hybrid quantum correlations.
In the past decades, quantum entanglement has been recognized to be the basic resource in quantum information theory. A fundamental need is then the understanding its qualification and its quantification: Is the quantum state entangled, and if it is, then how much entanglement is carried by that? These questions introduce the topics of separability criteria and entanglement measures, both of which are based on the issue of classification of multipartite entanglement. In this dissertation, after reviewing these three fundamental topics for finite dimensional Hilbert spaces, I present my contribution to knowledge. My main result is the elaboration of the partial separability classification of mixed states of quantum systems composed of arbitrary number of subsystems of Hilbert spaces of arbitrary dimensions. This problem is simple for pure states, however, for mixed states it has not been considered in full detail yet. I give not only the classification but also necessary and sufficient criteria for the classes, which make it possible to determine to which class a mixed state belongs. Moreover, these criteria are given by the vanishing of quantities measuring entanglement. Apart from these, I present some side results related to the entanglement of mixed states. These results are obtained in the learning phase of my studies and give some illustrations and examples.
A complete analysis of entangled bipartite qutrit pure states is carried out based on a simple entanglement measure. An analysis of all possible extremally entangled pure bipartite qutrit states is shown to reduce, with the help of SLOCC transformations, to three distinct types. The analysis and the results should be helpful for finding different entanglement types in multipartite pure state systems. PACS: 03.67.-a; 03.67.Mn; 03.65.Ud; 03.65.Bz Quantum entanglement is an important resource to be implemented in quantum information processing and quantum communication protocols. [1] The interests in studying multi-dimensional entangled states come from both the foundations of quantum mechanics and the development of new protocols in quantum communication. For example, it was shown that maximally entangled states of two quantum systems in a high-dimensional Hilbert space, qudits, violate local realism stronger than qubits, and entangled qudits are less affected by noise than entangled qubits. [2,3] In quantum cryptography [4], the use of entangled qutrits [4,5] or qudits [6,7] instead of qubits is more secure against eavesdropping attacks. Furthermore, the protocols for quantum teleportation or for quantum cryptography, work best with maximally entangled states. All these facts motivate the development of techniques to generate entangled states among quantum systems in higher dimensional Hilbert space with a good quality of entanglement.
The detection and classification of entanglement properties in a two-qubit and a multi-qubit system is a topic of great interest. This topic has been extensively studied, and as a result, we discovered various approaches for detecting and classifying multi-qubit, in particular three-qubit entangled states. The emphasis of this work is on a formalism of methods for the detection and classification of bipartite as well as multipartite quantum systems. We have used the method of structural physical approximation of partially transposed matrix (SPA-PT) for the detection of entangled states in arbitrary dimensional bipartite quantum systems. Also, we have proposed criteria for the classification of all possible stochastic local operations and classical communication (SLOCC) inequivalent classes of a pure and mixed three-qubit state using the SPA-PT map. To quantify entanglement, we have defined a new measure of entanglement based on the method of SPA-PT, which we named as "structured negativity". We have shown that this measure can be used to quantify entanglement for negative partial transposed entangled states (NPTES). Since the methods for detection, classification and quantification of entanglement, defined in this thesis are based on SPA-PT, they may be realized in an experiment.
The wave-particle duality of light has led to two different encodings of optical quantum information. One approach, referred to as the discrete-variable (DV) one, relies on finite dimensional quantum systems, while the other one, referred to as the continuous-variable (CV) one, is based on wave-like states belonging to an infinite dimensional Hilbert space. These two encodings have historically been separated, but recently, new hybrid protocols which aim at combining the advantages of the two encodings have emerged [1] . In that prospect our recent demonstration of the measurement-induced generation of hybrid entanglement between discrete- and continuous-variable quantum states [2] , [3] , located at distant places and connected by a lossy channel, has opened the way to the implementation of hybrid protocols and heterogeneous quantum networks. Following the first protocols based on this resource, such as the remote preparation of continuous-variable qubits [4] , or a violation of an Einstein-Podolsky-Rosen steering inequality [5] , we will here report our latest experimental results on entanglement swapping and quantum teleportation protocols using hybrid entangled states based on different encodings [6] .