Schmidt number for density matrices
Open accessAbstract
We introduce the notion of a Schmidt number of a bipartite density matrix. We show that k-positive maps witness the Schmidt number, in the same way that positive maps witness entanglement. We determine the Schmidt number of the family of states that is made from mixing the completely mixed state and a maximally entangled state. We show that the Schmidt number does not necessarily increase when taking tensor copies of a density matrix $\ensuremath{\rho};$ we give an example of a density matrix for which the Schmidt numbers of $\ensuremath{\rho}$ and $\ensuremath{\rho}\ensuremath{\bigotimes}\ensuremath{\rho}$ are both $2.$
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Another experimental approach to determine entanglement without QST can be based on measuring a bipartite Schmidt number, which satisfies conditions of a good entanglement measure and can be determined experimentally via a witnessing approach.
“Another experimental approach to determine entanglement of a given state without QST can be based on measuring a bipartite Schmidt number, which satisfies various conditions of a good entanglement measure44,45 and can be determined experimentally via a witnessing approach46.”
From Experimental hierarchy of two-qubit quantum correlations without state tomography (2023) · cites this paper as [44]
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Research works printed but not linked: 3
- [14]M.-D. Choi. Positive linear maps on C ∗ -algebras. Canadian Journal of Mathematics , XXIV, No.3:520-529, 1972.
- [15]M. Horodecki and P. Horodecki. Reduction criterion of separability and limits for a class of distillation protocols. Physical Review A , 59:4206-4216, 1999, quantph/9708015.
- [18]D. Jonathan and M. Plenio. Entanglement-assisted local manipulation of pure quantum states. Physical Review Letters , 83:3566-3569, 1999, quant-ph/9905071. · arXiv
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“Another experimental approach to determine entanglement of a given state without QST can be based on measuring a bipartite Schmidt number, which satisfies various conditions of a good entanglement measure [43, 44] and can be determined experimentally via a witnessing approach [45].”
Background
- Authors
- Terhal, B. M., & Horodecki, P.
- Affiliations
- IBM Research - Thomas J. Watson Research Center (US)
- Leibniz University Hannover (DE)
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2000-03-06
- Volume
- 61
- Issue
- 4
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W1993828178
- Cited by
- 427
- References
- 0
- Citations by year
- 2026: 24 · 2025: 28 · 2024: 33 · 2023: 53 · 2022: 20 · 2021: 18 · 2020: 14 · 2019: 19 · 2018: 21 · 2017: 15 · 2016: 13 · 2015: 16 · 2014: 17 · 2013: 20 · 2012: 15
- FWCI
- 8.74
- 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)
- 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
- Schmidt number, positive maps, entanglement, maximally entangled states, tensor powers, entanglement measures
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