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Entanglement of Formation of an Arbitrary State of Two Qubits
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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.
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In the citing paper
“The concurrence, entanglement of formation and tangle are measures of the quantum-coherence properties of a mixed quantum state [29].”
From Measurement of qubits (2001) · cites this paper as [29]
Experimental — accuracy not yet measuredThe source backs part of the claim, not all of it — or backs it for a different population, condition or comparison.
In the citing paper
“As for the concurrence, rigorously, its original Wootters' definition [20] is invalid for two-qudit states with d > 2.”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [20]
Experimental — accuracy not yet measuredThe cited source says what the claim attributes to it.
In the citing paper
“It is also known [19] that in the case of bipartite states with the dimensionality of the oneparticle Hilbert space d = 2, there is a simple algebraic relation between Wootters' concurrence C [20] and the Schmidt entanglement parameter K , owing to which”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [20]
The cited source says what the claim attributes to it.
In the citing paper
“Note that expressions for the concurrence C (equation (5.3) and the last formula of equation (5.4)) can also be derived directly from the original Wootters' definition [20].”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [20]
In the citing paper
“This entanglement, being that of a 2 × 2 system, can also be distilled [7] into pure singlets and used for teleportation.”
From Quantum Communication through an Unmodulated Spin Chain (2003) · cites this paper as [7]
In the citing paper
“Concurrence (CQ) measures the degree of entanglement in a system, where CQ > 0.5 indicates a partially entangled system [73].”
From Mathematical proof of the Fisher-Escolà Q statistical distribution in quantum consciousness modeling (2025) · cites this paper as [73]
In the citing paper
“Only for states of smallest possible dimensions, i.e., for bipartite two-level systems, do a few measures exist that can be evaluated algebraically [27,28,29]”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
In the citing paper
“For these low dimensional systems, the entanglement of a pure state can be expressed as a function of its concurrence, E (Ψ) = E ( c (Ψ)), with [27]”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
In the citing paper
“The infimum of this quantity is known [27] to be given by”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
In the citing paper
“However, concurrence can also be considered as an independent entanglement measure [27].”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
In the citing paper
“The relation between the present and the original approach to concurrence [27,57] becomes apparent with the observation that the matrix τ defined here coincides with that introduced in Eq. (38)”
From Measures and dynamics of entangled states (2005) · cites this paper as [27]
In the citing paper
“The amount of entanglement can also be quantified by concurrence[45], which is directly related to e ij with C ( e ij ) = max { 0 , -1 2 + 3 2 e ij } .”
From Quantum simulation of the wavefunction to probe frustrated Heisenberg spin systems (2011) · cites this paper as [45]
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Metadata
- Authors
- Wootters, W. K.
- Affiliations
- Williams College (US)
- Corresponding author
- Wootters, William K.
- ORCID
- Wootters, William K.
- Venue
- Physical Review Letters
- Publisher
- American Physical Society (APS)
- Published
- 1998-03-09
- Volume
- 80
- Issue
- 10
- Pages
- 2245-2248
- ISSN
- 0031-9007 (ISSN-L)
- 1079-7114
- OpenAlex
- W2011208902
- Cited by
- 8,290
- References
- 21
- Citations by year
- 2026: 320 · 2025: 387 · 2024: 339 · 2023: 340 · 2022: 349 · 2021: 371 · 2020: 328 · 2019: 336 · 2018: 325 · 2017: 301 · 2016: 406 · 2015: 339 · 2014: 381 · 2013: 368 · 2012: 381
- FWCI
- 62.08
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review Letters (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum many-body systems
- Keywords
- entanglement of formation, two-qubit states, mixed states, density matrix, quantum entanglement
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