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Cited by 3,649Quantum Information and CryptographyQuantum Mechanics and ApplicationsQuantum many-body systems

Separability of mixed states: necessary and sufficient conditions

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Horodecki, Michał, Horodecki, Paweł, Horodecki, Ryszard·1996·Physics Letters A
DOI 10.1016/s0375-9601(96)00706-2
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Crossref · 10.1016/s0375-9601(96)00706-2

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  • The Horodecki family extended PPT to a fully general necessary and sufficient condition for separability, namely that a density matrix after any positive map on one subsystem should remain a density operator.

    “This was extended by Horodecki Family to a fully general necessary and sufficient condition for separability, which is that a density matrix after any positive transformation (map) on one of the subsystems should remain a density operator 9 .”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as Horodecki et al. (1996)

    Theorem 2 (p. 7) gives the fully general necessary and sufficient condition: applying any positive map to one subsystem must leave the operator positive. But the paper defines a density operator (state) as also having trace one (p. 3); Theorem 2 does not require the transformed operator to retain that normalization. (LLM, conf=0.90)

    Experimental — accuracy not yet measured

  • The full set of separable mixed states is a bounded convex set in a multidimensional real space of hermitian operators, and any entangled state is separated from the set of separable states by a hyperplane.

    “The full set of separable mixed states is a bounded convex set in a multidimensional real space of hermitian operators.Thus, any entangled state is separated from the set of separable states by a hyperplane.”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as Horodecki et al., 1996

  • A hermitian operator W is an entanglement witness if for all separable states Tr(W sep) ≥ 0, whereas there exists an entangled state for which Tr(W ent) < 0; thus via measurement of a suitably chosen witness operator one can detect entanglement.

    “A hermitian operator Ŵ is an entanglement witness if for all separable states one has Tr( Ŵ sep ) ≥ 0, whereas there exists an entangled state for which one has Tr( Ŵ ent ) < 0. Thus, via measurement of a suitably chosen witness operator one can detect entanglement.”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as Horodecki et al., 1996

  • Basic structural criteria for deciding whether a density operator represents an entangled state were first given by Peres and in full form by Horodecki et al.

    “Basic structural criteria, which decide whether a given density operator represents an entangled state, were first given by Peres (1996) and in the full form by Horodecki et al. (1996).”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as Horodecki et al. (1996)

  • The Procrustean method requires photon pairs all in a pure non-maximally entangled state with known amplitudes and involves only local filtering operations on single pairs.

    “The first is the so-called Procrustean method (Bennett et al., 1996a).It requires that the photon pairs are all in a pure non-maximally entangled state, say, |Ψ nonmax = α|H |V + β|V |H , where α and β are two known amplitudes.In this case, the scheme only in-volves local filtering operations (Gisin, 1996;Horodecki et al., 1996) on single pairs.”

    From Multiphoton entanglement and interferometry (2012) · cites this paper as Horodecki et al., 1996

    The paper describes separability criteria for mixed states (p. 3), not the Procrustean method. Searches for local filtering, single-pair operations, and known amplitudes found no passage describing those features; the reference to Bennett et al. (References, p. 14) does not establish them. (LLM, conf=0.76)

    Experimental — accuracy not yet measured

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Cites

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Research works printed but not linked: 2

  1. [15][15] K. Kraus, States, Effects and Operations: Fundamental Notions of Quantum Theory , Wiley, New York, 1991. -
  2. [18][18] E. Strømer, Acta. Math. 110 (1963), 233. -

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Authors
Horodecki, M., Horodecki, P., & Horodecki, R.
Affiliations
  • University of Gdańsk (PL)
  • Gdańsk University of Technology (PL)
Venue
Physics Letters A
Publisher
Elsevier BV
Published
1996-11
Volume
223
Issue
1-2
Pages
1-8
ISSN
0375-9601 (ISSN-L)
DOI
10.1016/s0375-9601(96)00706-2
OpenAlex
W2144126354
Cited by
3,649
References
19
Citations by year
2026: 122 · 2025: 206 · 2024: 180 · 2023: 187 · 2022: 168 · 2021: 169 · 2020: 128 · 2019: 146 · 2018: 149 · 2017: 147 · 2016: 140 · 2015: 133 · 2014: 129 · 2013: 144 · 2012: 127
FWCI
23.42
Type
journal article
Language
en
Open access
Licence
www.elsevier.com/tdm/userlicense/1.0/ (tdm)
Locations
  • Physics Letters A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
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
quantum entanglement, necessary and sufficient conditions, partial transposition, separability criteria, quantum states

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