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.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.

arXiv:2411.19315Journal of Mathematical PhysicsAug 1, 20261 citation

Quantum channels that preserve entanglement

William Arveson

arXiv:0801.2531Mathematische AnnalenSep 23, 200812 citations

The entangling power of non-entangling channels

Julien Pinske, Jan Sperling, Klaus Mølmer

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.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.

quant-pharXiv:2512.14819Physical Review ADec 18, 20250 citations

From local to non local entanglement

Anna Napoli, A. Messina, G. Compagno

Springer eBooksJan 1, 20030 citations

Localization of joint quantum measurements on $\mathbb{C}^d \otimes \mathbb{C}^d$ by entangled resources with Schmidt number at most $d$ Localization of joint quantum measurements on $\mathbb{C}^d \otimes \mathbb{C}^d$ by entangled resources with Schmidt number at most $d$

Seiseki Akibue, Jisho Miyazaki

Localizable measurements are joint quantum measurements that can be implemented using only non-adaptive local operations and shared entanglement. We provide a protocol-independent characterization of localizable projection-valued measures (PVMs) by exploiting algebraic structures that any such measurement must satisfy. We first show that a rank-1 PVM on $\mathbb{C}^d\otimes\mathbb{C}^d$ containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most $d$ if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. This reveals strong limitations imposed by non-adaptive local operations, in contrast to the adaptive setting where any joint measurement is implementable. We then completely characterize two-qubit rank-1 PVMs that can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo, and finally extend our characterization to ideal two-qudit measurements, strengthening earlier results.Localizable measurements are joint quantum measurements that can be implemented using only non-adaptive local operations and shared entanglement. We provide a protocol-independent characterization of localizable projection-valued measures (PVMs) by exploiting algebraic structures that any such measurement must satisfy. We first show that a rank-1 PVM on $\mathbb{C}^d\otimes\mathbb{C}^d$ containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most $d$ if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. This reveals strong limitations imposed by non-adaptive local operations, in contrast to the adaptive setting where any joint measurement is implementable. We then completely characterize two-qubit rank-1 PVMs that can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo, and finally extend our characterization to ideal two-qudit measurements, strengthening earlier results.

quant-pharXiv:2601.02660Physical Review AJan 7, 20260 citations

Efficient Self‐Guided Estimation of Schmidt Number via Local Measurements

Tian‐Ze Yang, Jun Tang, Wen‐Hao Zhang +2 more

ABSTRACT Recent advances in experimental quantum platforms have enabled the development of devices capable of generating high‐dimensional quantum states, facilitating their use in quantum information processing tasks. High‐dimensional entangled states offer significant advantages in areas such as quantum key distribution, quantum communication, and quantum metrology. While the Schmidt rank (SR) characterizes entanglement for pure bipartite states in high‐dimensional spaces, its extension to mixed states is the Schmidt number (SN). However, experimentally determining the SN of unknown mixed states remains a significant challenge. Here, we propose an experimentally feasible method to determine the SN of unknown states, constructing a self‐guided SN witness scheme. Leveraging techniques from quantum state verification (QSV), this method allows SN estimation using only local measurements, thereby avoiding the need for direct projections onto high‐dimensional maximally entangled states. The self‐guided aspect is implemented through a Stochastic Gradient Algorithm (SGA) that systematically explores the space of local unitary operations to maximize a fidelity related to Schmidt number witnesses. We numerically demonstrate that for four‐dimensional entangled states, our scheme accurately estimates the SN, typically converging within approximately 200 iterations. Our results also show that the method is robust to realistic imperfections, highlighting its potential for practical applications in quantum information processing.ABSTRACT Recent advances in experimental quantum platforms have enabled the development of devices capable of generating high‐dimensional quantum states, facilitating their use in quantum information processing tasks. High‐dimensional entangled states offer significant advantages in areas such as quantum key distribution, quantum communication, and quantum metrology. While the Schmidt rank (SR) characterizes entanglement for pure bipartite states in high‐dimensional spaces, its extension to mixed states is the Schmidt number (SN). However, experimentally determining the SN of unknown mixed states remains a significant challenge. Here, we propose an experimentally feasible method to determine the SN of unknown states, constructing a self‐guided SN witness scheme. Leveraging techniques from quantum state verification (QSV), this method allows SN estimation using only local measurements, thereby avoiding the need for direct projections onto high‐dimensional maximally entangled states. The self‐guided aspect is implemented through a Stochastic Gradient Algorithm (SGA) that systematically explores the space of local unitary operations to maximize a fidelity related to Schmidt number witnesses. We numerically demonstrate that for four‐dimensional entangled states, our scheme accurately estimates the SN, typically converging within approximately 200 iterations. Our results also show that the method is robust to realistic imperfections, highlighting its potential for practical applications in quantum information processing.

Annalen der PhysikJan 1, 20260 citations

Localization of joint quantum measurements on

\mathbb{C}^d \otimes \mathbb{C}^d

by entangled resources with Schmidt number at most

d

\mathbb{C}^d\otimes\mathbb{C}^d

containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most

d

if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. This reveals strong limitations imposed by non-adaptive local operations, in contrast to the adaptive setting where any joint measurement is implementable. We then completely characterize two-qubit rank-1 PVMs that can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo, and finally extend our characterization to ideal two-qudit measurements, strengthening earlier results.