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:1310.4473

Local Records and Global Entanglement: A Unique Multi-Partite Generalization of the Schmidt Decomposition

Riedel, C. Jess

Published Oct 16, 2013• 1 citation
arXivHTMLPDF

Abstract

We show that there is a unique maximal decomposition of a pure multi-partite (N>2) quantum state into a sum of states which are "locally orthogonal" in the sense that the local reduced state for a term in the sum lives in its own orthogonal subspace for each subsystem. Observers can make local measurements on any subsystem and determine which "branch" they are on. The Shannon entropy of the resulting branch weights defines a new measure of global, GHZ-like entanglement, which is insensitive to local pairwise entangling operations and vanishes when there is no piece of information recorded at every subsystem. In the bi-partite (N=2) case, this decomposition reduces to the (not necessarily unique) Schmidt decomposition and the entropy reduces to the entropy of entanglement

DOI
10.48550/arxiv.1310.4473

Related papers

Generalized Schmidt Decomposition and Classification of Three-Quantum-Bit States

A. Acín, Alexander A. Andrianov, Laura Costa +3 more

We prove for any pure three-quantum-bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the three-quantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties.

arXiv:quant-ph/0003050•Physical Review Letters•Aug 14, 2000•521 citations

Schmidt decomposition and classification of three-quantum-bit states

A. Acín, Alexander A. Andrianov, Laura Costa +3 more

We prove for any pure three-quantum-bit state the existence of local bases which allow to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the three-quantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties.

•Mar 14, 2000•4 citations

Morphology of Two-Partite Entanglement in some, Highly Entangled, Small Qubit Systems

Roman Gieralak, Kozik, Wojciech

The most popular, bipartite methods approach to the analysis of entanglement included in genuine entangled quantum states, describing n-partite, n > 2 quantum systems are based on the notion of entropy of the corresponding reduced density matrices or/and the use of Negativity measure. However the equivalence of these two different measures is not clear. In particular, the equivalence of these notions of being maximally entangled in the sense of amount of entanglement included is not known. In the case of special, maximally entangled, with respect to the Negativity measure, pure states, describing 5 and 6 qubit systems, the equivalence to the entropy measure based definitions was shown by the straightforward calculations based on Schmidt’s decomposition method.

Studia Informatica System and information technology•Jun 8, 2016•1 citation

Quantifying the dimensionality of multiparticle entanglement via partition rank

Sophia Denker, Ismaël Septembre, Robin Krebs +1 more

The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.

quant-pharXiv:2608.23399•Aug 25, 2026•0 citations

On Schmidt Decomposition: Approach Based on Correlation Operator as Bipartite Entanglement Entity

Fedor Herbut

An elaborated review with proofs of Schmidt canonical decomposition of any bipartite state vector is approached through general subsystem basis expansion. The upgraded forms of Schmidt decomposition in terms of correlation operator and twin observables are presented in detail. The discussion is extended to distant measurement, Einstein–Podolsky–Rosen states and Schrödinger's steering. All claims and proofs are given in standard form unlike in the previous articles of the author where all results were obtained utilizing the very rarely used antilinear Hilbert–Schmidt maps of one subsystem state space into the other. For practical reasons the formalism of partial traces with their rules and reduced density operators together with correlation operator are used. Quanta 2018; 7: 19–39.

Quanta•Feb 20, 2018•2 citations

Universal Measure of Entanglement

M. Hossein Partovi

A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.

arXiv:quant-ph/0308051•Physical Review Letters•Feb 20, 2004•23 citations
We show that there is a unique maximal decomposition of a pure multi-partite (N>2) quantum state into a sum of states which are "locally orthogonal" in the sense that the local reduced state for a term in the sum lives in its own orthogonal subspace for each subsystem. Observers can make local measurements on any subsystem and determine which "branch" they are on. The Shannon entropy of the resulting branch weights defines a new measure of global, GHZ-like entanglement, which is insensitive to local pairwise entangling operations and vanishes when there is no piece of information recorded at every subsystem. In the bi-partite (N=2) case, this decomposition reduces to the (not necessarily unique) Schmidt decomposition and the entropy reduces to the entropy of entanglement
We prove for any pure three-quantum-bit state the existence of local bases which allow one to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the three-quantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties.
We prove for any pure three-quantum-bit state the existence of local bases which allow to build a set of five orthogonal product states in terms of which the state can be written in a unique form. This leads to a canonical form which generalizes the two-quantum-bit Schmidt decomposition. It is uniquely characterized by the five entanglement parameters. It leads to a complete classification of the three-quantum-bit states. It shows that the right outcome of an adequate local measurement always erases all entanglement between the other two parties.
The most popular, bipartite methods approach to the analysis of entanglement included in genuine entangled quantum states, describing n-partite, n > 2 quantum systems are based on the notion of entropy of the corresponding reduced density matrices or/and the use of Negativity measure. However the equivalence of these two different measures is not clear. In particular, the equivalence of these notions of being maximally entangled in the sense of amount of entanglement included is not known. In the case of special, maximally entangled, with respect to the Negativity measure, pure states, describing 5 and 6 qubit systems, the equivalence to the entropy measure based definitions was shown by the straightforward calculations based on Schmidt’s decomposition method.
The usefulness of entanglement as a resource in quantum technologies increases for larger systems, that is, if more particles or higher-dimensional quantum systems are considered. Yet, the interplay between dimensionality and multiparticle entanglement is not well understood. Only for two-particle systems an unambiguous and coherent notion of entanglement dimensionality, based on the Schmidt decomposition, is known. We introduce a concept to characterize the entanglement dimensionality of multiparticle states based on decompositions of pure states into superpositions of states without genuine multiparticle entanglement. We provide constructive methods to characterize the resulting partition rank for pure and mixed states. This allows the identification of novel maximally correlated states as well as a discrete classification of quantum states under stochastic local operations and classical communication. From a mathematical perspective, our approach can be formulated in terms of the slice rank and partition rank of tensors and our results allow to characterize these by connecting them to a generalized injective tensor norm.
An elaborated review with proofs of Schmidt canonical decomposition of any bipartite state vector is approached through general subsystem basis expansion. The upgraded forms of Schmidt decomposition in terms of correlation operator and twin observables are presented in detail. The discussion is extended to distant measurement, Einstein–Podolsky–Rosen states and Schrödinger's steering. All claims and proofs are given in standard form unlike in the previous articles of the author where all results were obtained utilizing the very rarely used antilinear Hilbert–Schmidt maps of one subsystem state space into the other. For practical reasons the formalism of partial traces with their rules and reduced density operators together with correlation operator are used. Quanta 2018; 7: 19–39.
A general framework is developed for separating classical and quantum correlations in a multipartite system. Entanglement is defined as the difference in the correlation information encoded by the state of a system and a suitably defined separable state with the same marginals. A generalization of the Schmidt decomposition is developed to implement the separation of correlations for any pure, multipartite state. The measure based on this decomposition is a generalization of the entanglement of formation to multipartite systems, provides an upper bound for the relative entropy of entanglement, and is directly computable on pure states. The example of pure three-qubit states is analyzed in detail, and a classification based on minimal, four-term decompositions is developed.