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arXiv:2509.07086 · Physical Review A

Scaling bound entanglement through local extensions

Robin Krebs, Mariami Gachechiladze

Published Jun 9, 2026• 0 citations
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Abstract

Entanglement is a central resource in quantum information science, yet its structure in high dimensions remains notoriously difficult to characterize. One of the few general results on high-dimensional entanglement is given by peel-off theorems, which relate the entanglement of a state to that of its lower-dimensional local projections. We build on this idea by introducing , the inverse process to peel-off projections, which provide a systematic way to construct higher-dimensional entangled states from lower-dimensional ones. This dual perspective leads to general bounds on how the Schmidt number can change under projections and extensions, and reveals mechanisms for generating bound entangled states of higher dimensionality. As a concrete application, we construct a positive-partial-transpose state of Schmidt number three in local dimensions 4 × 5 , the smallest system known to host such entanglement. We further extend this approach to identify an elegant family of with increasing Schmidt number, including explicit examples of a 7 × 7 state with Schmidt number 4 and a 9 × 9 state with Schmidt number 5, suggesting ( d + 1 ) / 2 scaling in odd local dimensions d × d . Taken together, our results provide a constructive toolkit for probing the scaling of bound entanglement in high dimensions.

DOI
10.1103/tcq7-q96m

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Entanglement is a central resource in quantum information science, yet its structure in high dimensions remains notoriously difficult to characterize. One of the few general results on high-dimensional entanglement is given by peel-off theorems, which relate the entanglement of a state to that of its lower-dimensional local projections. We build on this idea by introducing , the inverse process to peel-off projections, which provide a systematic way to construct higher-dimensional entangled states from lower-dimensional ones. This dual perspective leads to general bounds on how the Schmidt number can change under projections and extensions, and reveals mechanisms for generating bound entangled states of higher dimensionality. As a concrete application, we construct a positive-partial-transpose state of Schmidt number three in local dimensions 4 × 5 , the smallest system known to host such entanglement. We further extend this approach to identify an elegant family of with increasing Schmidt number, including explicit examples of a 7 × 7 state with Schmidt number 4 and a 9 × 9 state with Schmidt number 5, suggesting ( d + 1 ) / 2 scaling in odd local dimensions d × d . Taken together, our results provide a constructive toolkit for probing the scaling of bound entanglement in high dimensions.
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.
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Entangled states with a positive partial transpose (so-called PPT states) are central to many interesting problems in quantum theory. On the one hand, they are considered to be weakly entangled, since no pure state entanglement can be distilled from them. On the other hand, it has been shown recently that some of these PPT states exhibit genuinely high-dimensional entanglement, i.e., they have a high Schmidt number. Here we investigate the ($d\ifmmode\times\else\texttimes\fi{}d$)-dimensional PPT states for $d\ensuremath{\ge}4$ discussed recently by E. Sindici and M. Piani [Phys. Rev. A 97, 032319 (2018)]. By generalizing their methods to the calculation of Schmidt numbers, we show that a linear d/2d/2d/2 scaling of its Schmidt number in the local dimension ddd can be attained.
Recent progress in quantum optics has led to setups that are able to prepare high-dimensional quantum states for quantum information processing tasks. As such, it is of importance to benchmark the states generated by these setups in terms of their quantum mechanical properties, such as their Schmidt numbers, i.e., the number of entangled degrees of freedom. In this paper, we develop an iterative algorithm that finds Schmidt number witnesses tailored to the measurements available in specific experimental setups. We then apply the algorithm to find a witness that requires the measurement of a number of density matrix elements that scales linearly with the local dimension of the system. As a concrete example, we apply our construction method to an implementation with photonic temporal modes.
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.
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.