Concurrence-based entanglement measures for isotropic states
Open accessAbstract
We discuss properties of entanglement measures called I-concurrence and tangle. For a bipartite pure state, I-concurrence and tangle are simply related to the purity of the marginal density operators. The I-concurrence (tangle) of a bipartite mixed state is the minimum average I-concurrence (tangle) of ensemble decompositions of pure states of the joint density operator. Terhal and Vollbrecht [Phys. Rev. Lett. 85, 2625 (2000)] have given an explicit formula for the entanglement of formation of isotropic states in arbitrary dimensions. We use their formalism to derive comparable expressions for the I-concurrence and tangle of isotropic states.
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The I-concurrence is defined via the Schmidt entanglement parameter K as C_I = 2(1 - K^{-1}).
“But instead, one can use the so-called 'I-concurrence' [19], by definition, determined via the Schmidt entanglement parameter K as C I = 2(1 -K -1 ).”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [19]
Section 2 (p. 4) identifies the expression in terms of Schmidt coefficients as the tangle, or squared I-concurrence, C², not I-concurrence C. The same section distinguishes C from C² by taking the square root. The displayed formulas are not decoded in our copy, so the precise expression involving K cannot be checked, but the claim's identification of the unsquared I-concurrence is contrary to the text. (LLM, conf=0.84)
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Experimental — accuracy not yet measuredFor bipartite states with one-particle Hilbert space dimension d=2, there is a simple algebraic relation between Wootters' concurrence C and the Schmidt entanglement parameter K, giving equation (5.3).
“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 [19]
Section 2 (p. 4) states that, for pure bipartite states, squared concurrence is determined by the Schmidt coefficients; the Introduction (p. 3) also relates pure-state I-concurrence to marginal purity. Our copy does not name or define the Schmidt entanglement parameter K, so it does not establish the claimed algebraic relation to K or its extension to bipartite states generally. (LLM, conf=0.78)
Experimental — accuracy not yet measuredFor pure two-particle states, the single-particle Hilbert-space dimensionality d directly determines the maximal achievable Schmidt entanglement parameter, Kmax = d.
“As is known [19], in the case of pure two-particle states the dimensionality of the single-particle Hilbert space d determines directly the maximal achievable value of the Schmidt entanglement parameter, K max = d.”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [19]
Section 2 (p. 4) describes the Schmidt decomposition of a bipartite pure state using d = min(d_A, d_B) coefficients, supporting the dimension-dependent limit on entanglement. But our copy gives the maximum of the tangle as 2(d−1)/d; it does not define a Schmidt entanglement parameter K or state Kmax = d. (LLM, conf=0.83)
Experimental — accuracy not yet measuredThe maximal achievable values of K and C_I for ququarts are Kmax = 4 and C_I max = √3/2, in agreement with general expectations for qudits.
“As follows from equations (7.16) and (7.17), the maximal achievable values of K and C I for ququarts are K max = 4 and C I max = √ 3/2, in agreement with the general expectations for qudits K max = d and C I max = 2(1 -d -1 ) [19].”
From Entanglement of biphoton states: qutrits and ququarts (2011) · cites this paper as [19]
Section 2 (p. 4) states that the squared I-concurrence reaches 2(d−1)/d for maximally entangled states, giving C_I,max = √(3/2) when d = 4. The cited paper does not state the claimed K_max = d relation or identify a measure K, so it supports only the I-concurrence part of the general expectation. (LLM, conf=0.85)
Experimental — accuracy not yet measured
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Cites
<i>Quantum Computation and Quantum Information</i>
Nielsen, M. A., Chuang, I., & Grover, L. K. (2002) · American Journal of Physics · doi:10.1119/1.1463744 · Cited by 21,895 · see citing works
Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?
Einstein, A., Podolsky, B., & Rosen, N. (1935) · Physical Review · doi:10.1103/physrev.47.777 · Cited by 16,654 · see citing works
Entanglement of Formation of an Arbitrary State of Two Qubits
Wootters, W. K. (1998) · Physical Review Letters · doi:10.1103/physrevlett.80.2245 · Cited by 8,290 · see citing works
Entanglement of a pair of quantum bits
Hill, S., & Wootters, W. K. (1997) · Physical Review Letters · doi:10.1103/physrevlett.78.5022 · Cited by 2,823 · see citing works
Entanglement measures and purification procedures
Vedral, V., & Plenio, M. B. (1998) · Physical Review A · doi:10.1103/physreva.57.1619 · Cited by 1,419 · see citing works
Entanglement of formation for isotropic states
Terhal, B. M., & Vollbrecht, K. G. H. (2000) · Physical Review Letters · doi:10.1103/physrevlett.85.2625 · Cited by 254 · see citing works
Flagged first, then by citations
Research works printed but not linked: 2
- [4]Introduction to Quantum Computation and Information , edited by H.-K. Lo, S. Popescu, and T. Spiller (World Scientific, Singapore, 1998).
- [26]P. Rungta, W. J. Munro, K. Nemoto, P. Deuar, G. J. Milburn, and C. M. Caves, in Directions in Quantum Optics: A Collection of Papers Dedicated to the Memory of Dan Walls, edited by H. J. Carmichael, R. J. Glauber, and M. O. Scully (Springer, Berlin, 2001), p. 149.
Cited by
Sorted by citations
Entanglement of biphoton states: qutrits and ququarts
Fedorov, M. V., Volkov, P. A., Mikhailova, J. M., Straupe, S. S., & Kulik, S. P. (2011) · New Journal of Physics · doi:10.1088/1367-2630/13/8/083004 · Cited by 37 · see citing works
The Physical Meaning of the Holographic Principle
Fields, C., Glazebrook, J. F., & Marcianò, A. (2022) · Quanta · doi:10.12743/quanta.v11i1.206 · Cited by 37 · see citing works
- Authors
- Rungta, P., & Caves, C. M.
- Affiliations
- University of New Mexico (US)
- ORCID
- Caves, Carlton M.
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2003-01-14
- Volume
- 67
- Issue
- 1
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W1686382731
- Cited by
- 206
- References
- 6
- Citations by year
- 2026: 3 · 2025: 12 · 2024: 9 · 2023: 4 · 2022: 7 · 2021: 4 · 2020: 9 · 2019: 9 · 2018: 5 · 2017: 6 · 2016: 11 · 2015: 11 · 2014: 8 · 2013: 12 · 2012: 17
- FWCI
- 4.26
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Office of Naval Research
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › 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 measures, tangles, isotropic states, concurrence, entanglement of formation, bipartite mixed states, higher-dimensional quantum systems
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