One-to-one parametrization of quantum channels
Abstract
In contrast to the many-to-one and quadratic nature of correspondence in the conventional representation of quantum channels, such as the operator-sum or the unitary representation, we present a one-to-one and linear (affine) parametrization of quantum channels. This parametrization enables us to visualize the convex structure of the space of quantum channels. To demonstrate its usefulness, we apply it to the analysis of quantum binary channels.
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The expression of M_Δ given in Eq. (26) is not the most general possible, but it is the correct one for the representation of pure depolarizers with zero polarizance, such as the ones used in Ref. [21].
“Actually, the expression of M ⌬ given in Eq. ͑26͒ is not the most general possible ͓26͔, but it is the correct one for the representation of pure depolarizers with zero polarizance, such as the ones used in Ref. ͓21͔.”
From Linear optics and quantum maps (2007) · cites this paper as [26]
Lu and Chipman have shown that any given Mueller matrix M can be decomposed in the product M = M_D M_B M_Δ, where M_Δ, M_B, and M_D are complex-valued Mueller matrices representing a pure depolarizer, a retarder, and a diattenuator, respectively.
“In Ref. ͓26͔, Lu and Chipman have shown that any given Mueller matrix M can be decomposed in the product M = M D M B M ⌬ ,͑25͒”
From Linear optics and quantum maps (2007) · cites this paper as [26]
The Algoet-Fujiwara condition (11) always holds.
“This implies that the Algoet-Fujiwara condition [1] (λ 1 ± λ 2 ) 2 (1 ± λ 3 ) 2 -t 2 3 (11 )”
From An analysis of completely-positive trace-preserving maps on M2 (2002) · cites this paper as [1]
For bistochastic maps, the extreme points are known to be unitary conjugations and to correspond to the four corners of a tetrahedron in the λk representation.
“For bistochastic maps, the extreme points are known [1,11] to consist of the maps that conjugate by a unitary matrix and in the λ k”
From An analysis of completely-positive trace-preserving maps on M2 (2002) · cites this paper as [1]
The parameters are constrained by the Algoet-Fujiwara inequalities (λ1 ± λ2)^2 ≤ (1 ± λ3)^2.
“It was shown in [1,11] that the λ k 's are limited by the inequalities (λ 1 ± λ 2 ) 2 (1 ± λ 3 ) 2 .”
From An analysis of completely-positive trace-preserving maps on M2 (2002) · cites this paper as [1]
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Quantum mutual information in time
Wu, Z., Parzygnat, A. J., Vedral, V., & Fullwood, J. (2025) · New Journal of Physics · doi:10.1088/1367-2630/adde7d · Cited by 6 · see citing works
Regularized maximal fidelity of the generalized Pauli channels
Siudzińska, K. (2019) · Physical Review A · doi:10.1103/physreva.99.012340 · Cited by 6 · see citing works
Quantum channel-estimation with particle loss: GHZ versus W states
Fraïsse, J. M. E., & Braun, D. (2016) · Quantum Measurements and Quantum Metrology · doi:10.1515/qmetro-2016-0009 · Cited by 6 · see citing works
“Theoretische Physik, 72076 Tübingen, Germany are well known by now [3, 6, 19], and quantum channelestimation amounts hence to estimating these parameters.”
BackgroundBloch Equations and Completely Positive Maps
Daffer, S., Wodkiewicz, K., & McIver, J. K. (2004) · CERN Document Server (European Organization for Nuclear Research) · Cited by 6 · see citing works
Engineering Classical Capacity of Generalized Pauli Channels with Admissible Memory Kernels
Siudzińska, K., Das, A., & Bera, A. (2021) · Entropy · doi:10.3390/e23111382 · Cited by 6 · see citing works
“Indeed, Λ is completely positive if and only if λα satisfies the generalized Fujiwara– Algoet conditions [25,37,38]”
BackgroundNon-Markovian quantum dynamics from symmetric measurements
Siudzińska, K. (2024) · Physical Review A · doi:10.1103/physreva.110.012440 · Cited by 5 · see citing works
A class of Schwarz qubit maps with diagonal unitary and orthogonal symmetries
Chruściński, D., & Bhattacharya, B. (2024) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8121/ad75d6 · Cited by 5 · see citing works
“One has • Φ is completely positive iff Condition (68) was derived by Algoet and Fujiwara [44].”
Universality of sudden death of entanglement
Fanchini, F. F., Mendonça, P. E. M. F., & Napolitano, R. D. J. (2011) · Quantum Information and Computation · doi:10.26421/qic11.7-8-9 · Cited by 5 · see citing works
- Authors
- Fujiwara, A., & Algoet, P.
- Affiliations
- Stanford University (US)
- ORCID
- Fujiwara, Akio
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1999-05-01
- Volume
- 59
- Issue
- 5
- Pages
- 3290-3294
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2023747038
- Cited by
- 156
- References
- 3
- Citations by year
- 2026: 5 · 2025: 3 · 2024: 6 · 2023: 3 · 2022: 6 · 2021: 10 · 2020: 11 · 2019: 8 · 2018: 6 · 2017: 4 · 2016: 6 · 2015: 6 · 2014: 2 · 2013: 6 · 2012: 1
- FWCI
- 6.19
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review A (publishedVersion)
- Funders
- Telecommunications Advancement Foundation
- Topics
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
- Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Keywords
- quantum channels, operator-sum representation, unitary representations
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