Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:0707.0444

Necessary and Sufficient Conditions for the Trumping Relation

S. Turgut

Published Jul 3, 2007• 5 citations
arXivHTMLPDF

Abstract

Entanglement catalysis allows one to convert certain entangled states into others by the temporary involvement of another entangled state (so-called catalyst), where after the conversion the catalyst is returned to the same state. For bipartite pure entangled states that can be transformed in this way with unit probability, the respective Schmidt coefficients are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. This article provides all necessary and sufficient conditions for the trumping relation in terms of the Schmidt coefficients. The coefficients should satisfy strict inequalities for the entropy of entanglement and for power means excluding the special power 1.

DOI
10.48550/arxiv.0707.0444

Related papers

Catalytic transformations for bipartite pure states

S. Turgut

Entanglement catalysis is a phenomenon that usually enhances the conversion probability in the transformation of entangled states by the temporary involvement of another entangled state (so-called catalyst), where after the process is completed the catalyst is returned to the same state. For some pairs of bipartite pure entangled states, catalysis enables a transformation with unit probability of success, in which case the respective Schmidt coefficients of the states are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. This paper provides all necessary and sufficient conditions for the trumping and two other associated relations. Using these conditions, the least upper bound of conversion probabilities using catalysis is also obtained. Moreover, best conversion ratios achievable with catalysis are found for transformations involving many copies of states.

arXiv:0706.3654•Journal of Physics A Mathematical and Theoretical•Sep 18, 2007•96 citations

Mathematical structure of entanglement catalysis

Sumit Daftuar, M. Klimesh

The majorization relation has been shown to be useful in classifying which transformations of jointly held quantum states are possible using local operations and classical communication. In some cases, a direct transformation between two states is not possible, but it becomes possible in the presence of another state (known as a catalyst); this situation is described mathematically by the trumping relation, an extension of majorization. The structure of the trumping relation is not nearly as well understood as that of majorization. We give an introduction to this subject and derive some results. Most notably, we show that the dimension of the required catalyst is, in general, unbounded; there is no integer k such that it suffices to consider catalysts of dimension k or less in determining which states can be catalyzed into a given state. We also show that almost all bipartite entangled states are potentially useful as catalysts.

arXiv:quant-ph/0104058•Physical Review A•Sep 18, 2001•61 citations

Entanglement Catalysis Beyond the Asymptotic Rate

Alexander Streltsov

Quantum entanglement is a fundamental resource in quantum information science, commonly consumed to perform tasks like quantum teleportation and quantum key distribution. Entanglement concentration converts partially entangled pure states into nearly perfect Bell pairs. While its efficiency is well understood in the asymptotic limit, practical protocols must operate with finite systems. In this finite regime, and at the low error levels relevant in practice, performance is penalized by corrections that reduce the achievable yield. We investigate whether this fundamental finite-size penalty can be mitigated by an entanglement catalyst: an auxiliary state that aids the transformation but is returned exactly and uncorrupted. Although such a catalyst cannot increase the leading-order asymptotic yield, we show that it fundamentally alters the finite-size scaling. For every bipartite pure state with nonuniform Schmidt coefficients, we exactly determine the optimal second-order correction to the yield and show that catalysis completely overcomes this finite-size penalty.

quant-pharXiv:2610.13602•Oct 12, 2026•0 citations

An Equivalence of Entanglement-Assisted Transformation and Multiple-Copy Entanglement Transformation

Runyao Duan, Yuan Feng, Mingsheng Ying

We examine the powers of entanglement-assisted transformation and multiple-copy entanglement transformation. First, we find a sufficient condition of when a given catalyst is useful in producing another specific target state. As an application of this condition, for any non-maximally entangled bipartite pure state and any integer $n$ not less than 4, we are able to explicitly construct a set of $n\times n$ quantum states which can be produced by using the given state as a catalyst. Second, we prove that for any positive integer $k$, entanglement-assisted transformation with $k\times k$-dimensional catalysts is useful in producing a target state if and only if multiple-copy entanglement transformation with $k$ copies of state is useful in producing the same target. Moreover, a necessary and sufficient condition for both of them is obtained in terms of the Schmidt coefficients of the target. This equivalence of entanglement-assisted transformation and multiple-copy entanglement transformation implies many interesting properties of entanglement transformation. Furthermore, these results are generalized to the case of probabilistic entanglement transformations.

arXiv:quant-ph/0404046•Apr 7, 2004•3 citations

When catalysis is useful for probabilistic entanglement transformation

Yuan Feng, Runyao Duan, Mingsheng Ying

We determine all $2\ifmmode\times\else\texttimes\fi{}2$ quantum states that can serve as useful catalysts for a given probabilistic entanglement transformation, in the sense that they can increase the maximal transformation probability. When higher-dimensional catalysts are considered, a sufficient and necessary condition is derived under which a certain probabilistic transformation has useful catalysts.

arXiv:quant-ph/0403170•Physical Review A•Jun 9, 2004•12 citations

Catalytic Transformations of Pure Entangled States

Tulja Varun Kondra, Chandan Datta, Alexander Streltsov

Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We prove that entanglement entropy completely characterizes state transformations in the presence of entangled catalysts. Our results imply that entanglement entropy quantifies the amount of entanglement available in a bipartite pure state to be used for quantum information processing, giving asymptotic results an operational meaning also in the single-copy setup.

arXiv:2102.11136•Physical Review Letters•Oct 5, 2021•46 citations
Entanglement catalysis allows one to convert certain entangled states into others by the temporary involvement of another entangled state (so-called catalyst), where after the conversion the catalyst is returned to the same state. For bipartite pure entangled states that can be transformed in this way with unit probability, the respective Schmidt coefficients are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. This article provides all necessary and sufficient conditions for the trumping relation in terms of the Schmidt coefficients. The coefficients should satisfy strict inequalities for the entropy of entanglement and for power means excluding the special power 1.
Entanglement catalysis is a phenomenon that usually enhances the conversion probability in the transformation of entangled states by the temporary involvement of another entangled state (so-called catalyst), where after the process is completed the catalyst is returned to the same state. For some pairs of bipartite pure entangled states, catalysis enables a transformation with unit probability of success, in which case the respective Schmidt coefficients of the states are said to satisfy the trumping relation, a mathematical relation which is an extension of the majorization relation. This paper provides all necessary and sufficient conditions for the trumping and two other associated relations. Using these conditions, the least upper bound of conversion probabilities using catalysis is also obtained. Moreover, best conversion ratios achievable with catalysis are found for transformations involving many copies of states.
The majorization relation has been shown to be useful in classifying which transformations of jointly held quantum states are possible using local operations and classical communication. In some cases, a direct transformation between two states is not possible, but it becomes possible in the presence of another state (known as a catalyst); this situation is described mathematically by the trumping relation, an extension of majorization. The structure of the trumping relation is not nearly as well understood as that of majorization. We give an introduction to this subject and derive some results. Most notably, we show that the dimension of the required catalyst is, in general, unbounded; there is no integer k such that it suffices to consider catalysts of dimension k or less in determining which states can be catalyzed into a given state. We also show that almost all bipartite entangled states are potentially useful as catalysts.
Quantum entanglement is a fundamental resource in quantum information science, commonly consumed to perform tasks like quantum teleportation and quantum key distribution. Entanglement concentration converts partially entangled pure states into nearly perfect Bell pairs. While its efficiency is well understood in the asymptotic limit, practical protocols must operate with finite systems. In this finite regime, and at the low error levels relevant in practice, performance is penalized by corrections that reduce the achievable yield. We investigate whether this fundamental finite-size penalty can be mitigated by an entanglement catalyst: an auxiliary state that aids the transformation but is returned exactly and uncorrupted. Although such a catalyst cannot increase the leading-order asymptotic yield, we show that it fundamentally alters the finite-size scaling. For every bipartite pure state with nonuniform Schmidt coefficients, we exactly determine the optimal second-order correction to the yield and show that catalysis completely overcomes this finite-size penalty.
We examine the powers of entanglement-assisted transformation and multiple-copy entanglement transformation. First, we find a sufficient condition of when a given catalyst is useful in producing another specific target state. As an application of this condition, for any non-maximally entangled bipartite pure state and any integer nnn not less than 4, we are able to explicitly construct a set of n×nn\times nn×n quantum states which can be produced by using the given state as a catalyst. Second, we prove that for any positive integer kkk, entanglement-assisted transformation with k×kk\times kk×k-dimensional catalysts is useful in producing a target state if and only if multiple-copy entanglement transformation with kkk copies of state is useful in producing the same target. Moreover, a necessary and sufficient condition for both of them is obtained in terms of the Schmidt coefficients of the target. This equivalence of entanglement-assisted transformation and multiple-copy entanglement transformation implies many interesting properties of entanglement transformation. Furthermore, these results are generalized to the case of probabilistic entanglement transformations.
Quantum entanglement of pure states is usually quantified via the entanglement entropy, the von Neumann entropy of the reduced state. Entanglement entropy is closely related to entanglement distillation, a process for converting quantum states into singlets, which can then be used for various quantum technological tasks. The relation between entanglement entropy and entanglement distillation has been known only for the asymptotic setting, and the meaning of entanglement entropy in the single-copy regime has so far remained open. Here we close this gap by considering entanglement catalysis. We prove that entanglement entropy completely characterizes state transformations in the presence of entangled catalysts. Our results imply that entanglement entropy quantifies the amount of entanglement available in a bipartite pure state to be used for quantum information processing, giving asymptotic results an operational meaning also in the single-copy setup.