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arXiv:2411.19315 · Journal of Mathematical Physics

On the characterization of partially entanglement breaking and annihilating channels

Bivas Mallick, Nirman Ganguly, A. S. Majumdar

Published Aug 1, 2026• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1063/5.0274921

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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.
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.
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.
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 deep understanding of quantum entanglement is vital for advancing quantum technologies. The strength of entanglement can be quantified by counting the degrees of freedom that are entangled, which results in a quantity called the Schmidt number. A particular challenge is to identify the strength of entanglement in quantum states that remain positive under partial transpose (PPT), otherwise recognized as undistillable states. Finding PPT states with high Schmidt numbers has become a mathematical and computational challenge. In this Letter, we introduce efficient analytical tools for calculating the Schmidt number for a class of bipartite states called grid states. Our methods improve the best-known bounds for PPT states with high Schmidt numbers. Most notably, we construct a Schmidt number 3 PPT state in five-dimensional systems and a family of states with a Schmidt number of (d+1)/2 for odd d-dimensional systems, representing the best-known scaling of the Schmidt number in a local dimension. Additionally, these states possess intriguing geometrical properties, which we utilize to construct indecomposable entanglement witnesses.
There are processes that cannot generate entanglement but may, nevertheless, amplify entanglement already present in a system. Here, we show that a non-entangling operation can increase the Schmidt number of a quantum state only if it can generate entanglement with some non-zero probability. This is in stark contrast to the case where the parties of a quantum network are only able to control their joint state by local operations and classical communication (LOCC). There, being able to apply operations probabilistically (stochastic LOCC) does not increase the Schmidt number. Our findings show that certain non-entangling operations become entangling when selecting on specific measurement outcomes. This naturally leads us to the class of stochastically non-entangling maps, being those that cannot generate entanglement even probabilistically. Intrigued by this finding, we devise a Schmidt number for quantum channels that quantifies whether a channel can generate entanglement probabilistically. Moreover, we show that a channel is non-entangling if and only if its dual map is witness-preserving -- it takes entanglement witnesses to witnesses. Based on this finding, we derive Bell-like inequalities whose violation signals that a process generates entanglement.