Higher-dimensional entanglement is a valuable resource for several quantum information processing tasks, and is often characterized by the Schmidt number and specific classes of entangled states beyond qubit-qubit and qubit-qutrit systems. We propose a criterion to detect higher-dimensional entanglement, focusing on determining the Schmidt number of quantum states and identifying significant classes of positive partial transposition and negative partial transposition entangled states. Our approach relies on evaluating moments of generalized positive maps which can be efficiently simulated in real experiments without the requirement of full-state tomography. We demonstrate the effectiveness of our detection scheme through various illustrative examples. As a direct application, we explore the implications of our moment-based detection schemes in identifying useful quantum channels such as non-Schmidt-number breaking channels and non-entanglement breaking channels. Finally, we present an operational implication of our proposed moment criterion through its manifestation in channel discrimination tasks.
Higher-dimensional entanglement is a valuable resource for several quantum information processing tasks, and is often characterized by the Schmidt number and specific classes of entangled states beyond qubit-qubit and qubit-qutrit systems. We propose a criterion to detect higher-dimensional entanglement, focusing on determining the Schmidt number of quantum states and identifying significant classes of positive partial transposition and negative partial transposition entangled states. Our approach relies on evaluating moments of generalized positive maps which can be efficiently simulated in real experiments without the requirement of full-state tomography. We demonstrate the effectiveness of our detection scheme through various illustrative examples. As a direct application, we explore the implications of our moment-based detection schemes in identifying useful quantum channels such as non-Schmidt-number breaking channels and non-entanglement breaking channels. Finally, we present an operational implication of our proposed moment criterion through its manifestation in channel discrimination tasks.
Transmission of high dimensional entanglement through quantum channels is a significant area of interest in quantum information science. The certification of high dimensional entanglement is usually done through Schmidt numbers, which quantify the entanglement dimensionality of quantum states. States with high Schmidt numbers provide a larger advantage in various quantum information processing tasks compared to quantum states with low Schmidt numbers. However, the action of quantum channels may reduce the Schmidt number of transmitted states, thereby degrading their resourcefulness. Here we present a comprehensive analysis of partially entanglement breaking channels which reduce the Schmidt number of bipartite composite systems. From a resource theoretic perspective, it becomes imperative to identify channels that preserve the Schmidt number. Based on our characterization we lay down prescriptions to identify such channels which are non-resource breaking, i.e., preserve the Schmidt number. Additionally, we introduce a new class of quantum channels, termed partially entanglement annihilating channels which reduce the Schmidt number of a quantum state that is a part of a larger composite system. Finally, we study the connection between entanglement breaking, partially entanglement breaking, and partially entanglement annihilating channels.
The dimensionality of entanglement, quantified by the Schmidt number, is a valuable resource for a wide range of quantum information processing tasks. In this work, we introduce the notion of the absolute Schmidt number, referring to states whose Schmidt number cannot be increased by any global unitary transformation. We provide a characterization of the set of arbitrary-dimensional states whose Schmidt number is invariant under all global unitaries. Our approach enables us to develop both witness-based and moment-based techniques to detect nonabsolute Schmidt number states which could provide significant operational advantages through Schmidt number enhancement by global unitaries. We next formulate two resource-theoretic measures of nonabsolute Schmidt number states, based respectively on Schmidt number witness and robustness, and demonstrate an operational utility of the latter in a channel discrimination task. Finally, we extend our analysis to quantum channels by introducing a new class of channels that possess the absolute Schmidt number property. We derive a necessary and sufficient condition for identifying when a channel has the absolute Schmidt number property, confining our analysis to the class of covariant channels.
Genuine multipartite entanglement (GME) represents the strongest form of entanglement in multipartite systems, providing significant advantages in various quantum information processing tasks. In this work, we propose an experimentally feasible scheme for detecting GME, based on the truncated moments of positive maps. Our method avoids the need for full state tomography, making it scalable for larger systems. We provide illustrative examples of both pure and mixed states to demonstrate the efficacy of our formalism in detecting inequivalent classes of tripartite genuine entanglement. We further demonstrate the detection of quadripartite genuine entanglement, underscoring the effectiveness of our method in identifying entanglement beyond the tripartite case. Finally, we present a proposal for realizing these moments in real experiments.
Detection of entanglement through partial knowledge of the quantum state is a challenge to implement efficiently. Here we propose a separability criterion for detecting bipartite entanglement in arbitrary-dimensional quantum states using partial moments of the realigned density matrix. Our approach enables the detection of both distillable and bound entangled states through a common framework. We illustrate the significance of our method through examples of states belonging to both the above categories, which are not detectable using comparable other schemes relying on partial state information. The formalism of employing partial realigned moments proposed here is further shown to be effective for two-qubit systems too, with a slight modification of our separability criterion.
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.
High-dimensional entanglement offers significant advantages over its low-dimensional counterpart in various information-processing tasks. However, to harness these advantages, it is crucial that the quantum channels used to store or transmit the subsystems of an entangled system not only preserve entanglement but also maintain its dimensionality above a certain threshold. The maximum entanglement dimension that a channel can preserve is referred to as its effective dimensionality, since the channel cannot be used to transmit information of dimension greater than that in a single use. In this work, we present a method to certify whether a quantum channel can preserve entanglement dimension above a given threshold. Unlike existing approaches, our method is faithful, i.e., it can be applied to any channel, and avoids common assumptions such as preparation of entangled states, auxiliary side channels, or perfect measurement devices. Moreover, the method can be extended to faithfully certify other classes of non-resource-breaking channels, such as non-nonpositive-partial-transpose-breaking channels (non-NPT-breaking channels). Finally, we discuss possible experimental realizations of our certification scheme through explicit examples.