To understand how entangled states behave under local quantum operations is an open problem in quantum-information theory. The Jamiolkowski isomorphism provides a natural way to study this problem in terms of quantum states. We introduce the Schmidt number for quantum operations by this duality and clarify how the Schmidt number of a quantum state changes under a local quantum operation. Some characterizations of quantum operations with Schmidt number $k$ are also provided.To understand how entangled states behave under local quantum operations is an open problem in quantum-information theory. The Jamiolkowski isomorphism provides a natural way to study this problem in terms of quantum states. We introduce the Schmidt number for quantum operations by this duality and clarify how the Schmidt number of a quantum state changes under a local quantum operation. Some characterizations of quantum operations with Schmidt number $k$ are also provided.

Physical Review AMay 26, 20068 citations

Absolute Schmidt number: characterization, detection and resource-theoretic quantification

Bivas Mallick, Saheli Mukherjee, Nirman Ganguly +1 more

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.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.

quant-pharXiv:2604.02439Physical Review AApr 6, 20260 citations

A Guide to Higher-order quantum operations

Anonymous

PRX QuantumJun 9, 20260 citations

The zero-error entanglement cost is highly non-additive

Qiu-Ling Yue, Eric Chitambar

The Schmidt number is an entanglement measure whose logarithm quantifies the zero-error entanglement cost of generating a given quantum state using local operations and classical communication. In this paper, we show that the Schmidt number is highly nonmultiplicative in the sense that for any integer n, there exist states whose Schmidt number remains constant when taking n copies of the given state. These states also provide a rare instance in which the regularized zero-error entanglement cost can be computed exactly. We then explore the question of increasing the Schmidt number by quantum operations. We describe a class of bipartite quantum operations that preserve the Schmidt number for pure state transformations, and yet they can increase the Schmidt number by an arbitrarily large amount when generating mixed states. Our results are obtained by making connections to the resource theory of quantum coherence and generalizing the class of dephasing-covariant incoherent operations to the bipartite setting.The Schmidt number is an entanglement measure whose logarithm quantifies the zero-error entanglement cost of generating a given quantum state using local operations and classical communication. In this paper, we show that the Schmidt number is highly nonmultiplicative in the sense that for any integer n, there exist states whose Schmidt number remains constant when taking n copies of the given state. These states also provide a rare instance in which the regularized zero-error entanglement cost can be computed exactly. We then explore the question of increasing the Schmidt number by quantum operations. We describe a class of bipartite quantum operations that preserve the Schmidt number for pure state transformations, and yet they can increase the Schmidt number by an arbitrarily large amount when generating mixed states. Our results are obtained by making connections to the resource theory of quantum coherence and generalizing the class of dephasing-covariant incoherent operations to the bipartite setting.

arXiv:1808.10516Journal of Mathematical PhysicsNov 1, 20195 citations

Quantum Computation and Quantum Operations

Stanley Gudder

Advances in mechanics and mathematicsJan 1, 20090 citations

k

are also provided.