Affine maps of density matrices
Open accessAbstract
For quantum systems described by finite matrices, linear and affine maps of matrices are shown to provide equivalent descriptions of evolution of density matrices for a subsystem caused by unitary Hamiltonian evolution in a larger system; an affine map can be replaced by a linear map, and a linear map can be replaced by an affine map. There may be significant advantage in using an affine map. The linear map is generally not completely positive, but the linear part of an equivalent affine map can be chosen to be completely positive and related in the simplest possible way to the unitary Hamiltonian evolution in the larger system.
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A map Φ from n×n matrices to m×m matrices is linear if and only if it can be represented in an operator sum representation with left and right operation elements.
“Theorem 1 A map Φ : M n ↦→ M m (where M n is the space of n × n matrices) is linear iff it can be represented as”
From Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps (2009) · cites this paper as [16]
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Cites
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Vanishing Quantum Discord is Necessary and Sufficient for Completely Positive Maps
Shabani, A., & Lidar, D. A. (2009) · Physical Review Letters · doi:10.1103/physrevlett.102.100402 · Cited by 309 · see citing works
- Authors
- Jordan, T. F.
- Affiliations
- University of Minnesota, Duluth (US)
- Corresponding author
- Jordan, Thomas F.
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 2005-03-02
- Volume
- 71
- Issue
- 3
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W1989668101
- Cited by
- 17
- References
- 12
- Citations by year
- 2023: 1 · 2022: 1 · 2020: 1 · 2019: 1 · 2018: 1 · 2015: 1 · 2014: 1
- FWCI
- 1.76
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical Review 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 › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Keywords
- affine transformation, linear maps, density matrix, unitary evolution, completely positive maps, quantum dynamics, finite-dimensional quantum systems, reduced density matrix
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