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:quant-ph/9709029 · Physical Review Letters

Entanglement of Formation of an Arbitrary State of Two Qubits

William K. Wootters

Published Mar 9, 1998• 8,283 citations
arXivHTMLPDF

Abstract

The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state $\ensuremath{\rho}$ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.

DOI
10.1103/physrevlett.80.2245

Related papers

Entanglement of a Pair of Quantum Bits

Scott Hill, William K. Wootters

The ``entanglement of formation'' of a mixed state \ensuremath{\rho} of a bipartite quantum system can be defined as the minimum number of singlets needed to create an ensemble of pure states that represents \ensuremath{\rho}. We find an exact formula for the entanglement of formation for all mixed states of two qubits having no more than two nonzero eigenvalues, and we report evidence suggesting that the formula is valid for all states of this system.

arXiv:quant-ph/9703041•Physical Review Letters•Jun 30, 1997•2,822 citations

Entanglement of formation of pair of quantum dots

S. Sergio Sánchez, José Javier Sánchez Mondragón, J. C. García-Melgarejo +1 more

The Entanglement of quantum systems is a key aspect in order to understand the dynamics and behavior of mixed systems (density matrix) as bipartite systems of quantum bits (q-bits). A quantifiable measure widely used is the "entanglement of formation" of a mixed state, defined as the minimum number of singlets needed to create an ensemble of pure states that represents the density matrix of the system. Considering a double quantum dot system coupled cavity type Jaynes-Cummings investigate the entanglement between two quantum dots, immersed each in its own cavity, showing analytically that entanglement has a very interesting effects such as temporal evolution including the so-called sudden death effect.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Aug 10, 2011•0 citations

A necessary and sufficient condition for optimal decompositions

T. Prager

An important measure of bipartite entanglement is the entanglement of formation, which is defined as the minimum average pure state entanglement of all decompositions realizing a given state. A decomposition which achieves this minimum is called an optimal decomposition. However, as for the entanglement of formation, there is not much known about the structure of such optimal decompositions, except for some special cases, like states of two qubits or isotropic states. Here we present a necessary and sufficient condition for a set of pure states of a finite dimensional bipartite system to form an optimal decomposition. This condition is well suited to treat the question, whether the entanglement of formation is additive or not.

arXiv:quant-ph/0106030•Jun 6, 2001•1 citation

Quantum entanglement of formation between qudits

Seunghwa Ryu, Wei Cai A, Alfredo Caro

We develop a fast algorithm to calculate the entanglement of formation of a mixed state, which is defined as the minimum average entanglement of the pure states that form the mixed state. The algorithm combines conjugat-gradient and steepest-descent algorithms and outperforms both. Using this new algorithm, we obtain the statistics of the entanglement of formation on ensembles of random density matrices of higher dimensions than possible before. The correlation between the entanglement of formation and other quantities that are easier to compute, such as participation ratio and negativity are presented. Our results suggest a higher percentage of zero-entanglement states among zero-negativity states than previously reported.

Physical Review A•May 12, 2008•14 citations

Generalization of the entanglement of formation for multi-party systems

A M Wang

We present the generalization of the entanglement of formation for three-party systems in a pure state. For three qubit system we derive out its explicit and closed expression which is a linear combination of the binary entropy functions with various arguments, and these arguments are clearly determined in terms of the components of state vector of three qubits. As a reasonable measure of entanglement, the main behaviors and elementary properties of this generalized entanglement of formation are showed through discussing some important and interesting examples. Moreover, we propose how to extend our definition to a mixed state in according to the familiar idea. Then, we suggest the generalization of the entanglement of formation for multi-party systems which is consistent with the regular definition for two-party systems and our definition for three-party systems.

arXiv:quant-ph/0011040•Nov 10, 2000•0 citations

Formulas for Schmidt decompositions of mutually orthogonal quantum states in two-qubit systems

Yonghae Lee, Youngho Min, Sunghyun Bae +1 more

Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.

arXiv:2503.06988•Journal of Physics A Mathematical and Theoretical•Oct 20, 2025•1 citation
The entanglement of a pure state of a pair of quantum systems is defined as the entropy of either member of the pair. The entanglement of formation of a mixed state \ensuremathρ\ensuremath{\rho}\ensuremathρ is the minimum average entanglement of an ensemble of pure states that represents \ensuremath{\rho}. An earlier paper conjectured an explicit formula for the entanglement of formation of a pair of binary quantum objects (qubits) as a function of their density matrix, and proved the formula for special states. The present paper extends the proof to arbitrary states of this system and shows how to construct entanglement-minimizing decompositions.
The Entanglement of quantum systems is a key aspect in order to understand the dynamics and behavior of mixed systems (density matrix) as bipartite systems of quantum bits (q-bits). A quantifiable measure widely used is the "entanglement of formation" of a mixed state, defined as the minimum number of singlets needed to create an ensemble of pure states that represents the density matrix of the system. Considering a double quantum dot system coupled cavity type Jaynes-Cummings investigate the entanglement between two quantum dots, immersed each in its own cavity, showing analytically that entanglement has a very interesting effects such as temporal evolution including the so-called sudden death effect.
An important measure of bipartite entanglement is the entanglement of formation, which is defined as the minimum average pure state entanglement of all decompositions realizing a given state. A decomposition which achieves this minimum is called an optimal decomposition. However, as for the entanglement of formation, there is not much known about the structure of such optimal decompositions, except for some special cases, like states of two qubits or isotropic states. Here we present a necessary and sufficient condition for a set of pure states of a finite dimensional bipartite system to form an optimal decomposition. This condition is well suited to treat the question, whether the entanglement of formation is additive or not.
We develop a fast algorithm to calculate the entanglement of formation of a mixed state, which is defined as the minimum average entanglement of the pure states that form the mixed state. The algorithm combines conjugat-gradient and steepest-descent algorithms and outperforms both. Using this new algorithm, we obtain the statistics of the entanglement of formation on ensembles of random density matrices of higher dimensions than possible before. The correlation between the entanglement of formation and other quantities that are easier to compute, such as participation ratio and negativity are presented. Our results suggest a higher percentage of zero-entanglement states among zero-negativity states than previously reported.
We present the generalization of the entanglement of formation for three-party systems in a pure state. For three qubit system we derive out its explicit and closed expression which is a linear combination of the binary entropy functions with various arguments, and these arguments are clearly determined in terms of the components of state vector of three qubits. As a reasonable measure of entanglement, the main behaviors and elementary properties of this generalized entanglement of formation are showed through discussing some important and interesting examples. Moreover, we propose how to extend our definition to a mixed state in according to the familiar idea. Then, we suggest the generalization of the entanglement of formation for multi-party systems which is consistent with the regular definition for two-party systems and our definition for three-party systems.
Abstract We present Schmidt decomposition formulas for mutually orthogonal two-qubit pure states and classify orthonormal sets according to their entanglement structure. We begin by presenting explicit formulas for the Schmidt decomposition of arbitrary two-qubit pure states. Building on this, we construct and analyze all possible configurations of two mutually orthogonal pure states, providing corresponding Schmidt decomposition expressions for each case. For orthonormal sets of three and four states, we derive explicit formulas for selected representative types and discuss the difficulties in obtaining closed-form expressions for more general cases. In particular, we focus on special classes of orthonormal bases that include one or two maximally entangled states. Finally, we prove that no orthonormal basis can contain three product states and one entangled state, revealing a structural constraint that restricts the composition of entangled orthonormal sets.