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

A New Family of Probability Distributions and Asymptotics of Classical and LOCC Conversions

Wataru Kumagai, Masahito Hayashi

Published Jun 18, 2013• 3 citations
OpenAlex

Abstract

We consider the optimal approximate conversion between multiple copies of pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains a kind of cloning problem with LOCC restriction as a special case. To derive the asymptotic LOCC conversion rate, we consider two kinds of approximate conversions, deterministic conversion and majorization conversion, for probability distributions, and solve their asymptotic conversion rates up to the second order. Then, the asymptotic LOCC conversion rate is obtained via the natural relation between the LOCC conversion and the majorization conversion. To derive these asymptotic rates, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case, and the optimal conversion rate is represented by Rayleigh-normal distribution in a unified manner.

Related papers

Second-Order Asymptotics of Conversions of Distributions and Entangled States Based on Rayleigh-Normal Probability Distributions

Wataru Kumagai, Masahito Hayashi

We discuss the asymptotic behavior of conversions between two independent and identical distributions up to the second-order conversion rate when the conversion is produced by a deterministic function from the input probability space to the output probability space. To derive the second-order conversion rate, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case. Using this family of probability distributions, we represent the asymptotic second-order rates for the distribution conversion. As an application, we also consider the asymptotic behavior of conversions between the multiple copies of two pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains entanglement concentration, entanglement dilution, and a kind of cloning problem with LOCC restriction as special cases.

arXiv:1306.4166•IEEE Transactions on Information Theory•Dec 29, 2016•33 citations

Random Number Conversion and LOCC Conversion via Restricted Storage

Wataru Kumagai, Masahito Hayashi

We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of the first- and second-order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, we can describe the second-order asymptotic behavior of RNC via restricted storage in a unified manner. As an application to quantum information theory, we analyze LOCC conversion via entanglement storage with restricted size. Moreover, we derive the optimal LOCC compression rate under a constraint of conversion accuracy.

arXiv:1401.3781•IEEE Transactions on Information Theory•Jan 24, 2017•12 citations

The Asymptotic Spectrum of LOCC Transformations

Asger Kjærulff Jensen, Péter Vrana

We study the exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in the success probability, known as the converse error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterization of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [1], where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent, the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case, we prove that this bound holds with equality.

arXiv:1807.05130•IEEE Transactions on Information Theory•Jul 16, 2019•17 citations

Asymptotic compatibility between local-operations-and-classical-communication conversion and recovery

Kosuke Ito, Wataru Kumagai, Masahito Hayashi

Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by local operations and classical communication (LOCC) when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies that an error-free reversible conversion is asymptotically possible even between states whose copies can never be locally unitarily equivalent with any finite numbers of copies, although such a conversion is impossible in the finite setting. In fact, we show such an example. Moreover, we establish how to overcome the irreversibility of LOCC conversion in two ways. As for the first method, we evaluate how many copies of the initial state are to be lost to overcome the irreversibility of LOCC conversion. The second method is to add a supplementary state appropriately, which also works for local unitary conversion unlike the first method. Especially, for the qubit system, any nonmaximally pure entangled state can be a universal resource for the asymptotic reversibility when copies of the state are sufficiently many. More interestingly, our analysis implies that far-from-maximally entangled states can be better than nearly maximally entangled states as this type of resource. This fact brings new insight to the resource theory of state conversion.

arXiv:1504.02967•Physical Review A•Nov 6, 2015•8 citations

Stochastic approximate state conversion for entanglement and general quantum resource theories

Tulja Varun Kondra, Chandan Datta, Alexander Streltsov

Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.

arXiv:2111.12646•Quantum•Dec 5, 2025•5 citations

Asymptotic reversibility of LOCC conversions

Kosuke Ito, Wataru Kumagai, Masahito Hayashi

Recently, two of the authors showed that entanglement concentration is irreversible. However, it is still not clear what kind of LOCC conversion is reversible. We derive the necessary and sufficient condition for reversibility of LOCC conversion between two bipartite pure entangled states in an asymptotic setting. Simultaneously, we evaluate how many copies of the initial state is to be lost to overcome irreversibility of LOCC conversion. Our result is useful for designing how to store entangled states via LOCC operations without error.

IEEE International Symposium on Information Theory (ISIT)•Jun 1, 2014•0 citations
We consider the optimal approximate conversion between multiple copies of pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains a kind of cloning problem with LOCC restriction as a special case. To derive the asymptotic LOCC conversion rate, we consider two kinds of approximate conversions, deterministic conversion and majorization conversion, for probability distributions, and solve their asymptotic conversion rates up to the second order. Then, the asymptotic LOCC conversion rate is obtained via the natural relation between the LOCC conversion and the majorization conversion. To derive these asymptotic rates, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case, and the optimal conversion rate is represented by Rayleigh-normal distribution in a unified manner.
We discuss the asymptotic behavior of conversions between two independent and identical distributions up to the second-order conversion rate when the conversion is produced by a deterministic function from the input probability space to the output probability space. To derive the second-order conversion rate, we introduce new probability distributions named Rayleigh-normal distributions. The family of Rayleigh-normal distributions includes a Rayleigh distribution and coincides with the standard normal distribution in the limit case. Using this family of probability distributions, we represent the asymptotic second-order rates for the distribution conversion. As an application, we also consider the asymptotic behavior of conversions between the multiple copies of two pure entangled states in quantum systems when only local operations and classical communications (LOCC) are allowed. This problem contains entanglement concentration, entanglement dilution, and a kind of cloning problem with LOCC restriction as special cases.
We consider random number conversion (RNC) through random number storage with restricted size. We clarify the relation between the performance of RNC and the size of storage in the framework of the first- and second-order asymptotics, and derive their rate regions. Then, we show that the results for RNC with restricted storage recover those for conventional RNC without storage in the limit of storage size. To treat RNC via restricted storage, we introduce a new kind of probability distributions named generalized Rayleigh-normal distributions. Using the generalized Rayleigh-normal distributions, we can describe the second-order asymptotic behavior of RNC via restricted storage in a unified manner. As an application to quantum information theory, we analyze LOCC conversion via entanglement storage with restricted size. Moreover, we derive the optimal LOCC compression rate under a constraint of conversion accuracy.
We study the exact, non-deterministic conversion of multipartite pure quantum states into one-another via local operations and classical communication (LOCC) and asymptotic entanglement transformation under such channels. In particular, we consider the maximal number of copies of any given target state that can be extracted exactly from many copies of any given initial state as a function of the exponential decay in the success probability, known as the converse error exponent. We give a formula for the optimal rate presented as an infimum over the asymptotic spectrum of LOCC conversion. A full understanding of exact asymptotic extraction rates between pure states in the converse regime thus depends on a full understanding of this spectrum. We present a characterization of spectral points and use it to describe the spectrum in the bipartite case. This leads to a full description of the spectrum and thus an explicit formula for the asymptotic extraction rate between pure bipartite states, given a converse error exponent. This extends the result on entanglement concentration in [1], where the target state is fixed as the Bell state. In the limit of vanishing converse error exponent, the rate formula provides an upper bound on the exact asymptotic extraction rate between two states, when the probability of success goes to 1. In the bipartite case, we prove that this bound holds with equality.
Recently, entanglement concentration was explicitly shown to be irreversible. However, it is still not clear what kind of states can be reversibly converted in the asymptotic setting by local operations and classical communication (LOCC) when neither the initial nor the target state is maximally entangled. We derive the necessary and sufficient condition for the reversibility of LOCC conversions between two bipartite pure entangled states in the asymptotic setting. In addition, we show that conversion can be achieved perfectly with only local unitary operation under such condition except for special cases. Interestingly, our result implies that an error-free reversible conversion is asymptotically possible even between states whose copies can never be locally unitarily equivalent with any finite numbers of copies, although such a conversion is impossible in the finite setting. In fact, we show such an example. Moreover, we establish how to overcome the irreversibility of LOCC conversion in two ways. As for the first method, we evaluate how many copies of the initial state are to be lost to overcome the irreversibility of LOCC conversion. The second method is to add a supplementary state appropriately, which also works for local unitary conversion unlike the first method. Especially, for the qubit system, any nonmaximally pure entangled state can be a universal resource for the asymptotic reversibility when copies of the state are sufficiently many. More interestingly, our analysis implies that far-from-maximally entangled states can be better than nearly maximally entangled states as this type of resource. This fact brings new insight to the resource theory of state conversion.
Quantum resource theories provide a mathematically rigorous way of understanding the nature of various quantum resources. An important problem in any quantum resource theory is to determine how quantum states can be converted into each other within the physical constraints of the theory. The standard approach to this problem is to study approximate or probabilistic transformations. Here, we investigate the intermediate regime, providing limits on both, the fidelity and the probability of state transformations. We derive limitations on the transformations, which are valid in all quantum resource theories, by providing bounds on the maximal transformation fidelity for a given transformation probability. As an application, we show that these bounds imply an upper bound on the asymptotic rates for various classes of states under probabilistic transformations. We also show that the deterministic version of the single copy bounds can be applied for drawing limitations on the manipulation of quantum channels, which goes beyond the previously known bounds of channel manipulations. Furthermore, we completely solve the question of stochastic-approximate state conversion via local operations and classical communication in the following two cases: (i) Both initial and target states are pure bipartite entangled states of arbitrary dimensions. (ii) The target state is a two-qubit entangled state and the initial state is a pure bipartite state.
Recently, two of the authors showed that entanglement concentration is irreversible. However, it is still not clear what kind of LOCC conversion is reversible. We derive the necessary and sufficient condition for reversibility of LOCC conversion between two bipartite pure entangled states in an asymptotic setting. Simultaneously, we evaluate how many copies of the initial state is to be lost to overcome irreversibility of LOCC conversion. Our result is useful for designing how to store entangled states via LOCC operations without error.