Verifying Measures of Quantum Entropy
Abstract
:1. Introduction
2. Proposed Measure
2.1. Characterization of Classical Entropy
2.2. The Effective Quantum Entropy (EQE)
3. Properties of Quantum Entropy
3.1. Fundamental Properties
3.2. Other Properties
3.3. Specific Properties of Composite Operators
4. Numerical Experiments
EYD Algorithm
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Symbol | Description |
---|---|
Schatten p-(pseudo-)norm of matrices | |
concatenation of vectors | |
majorization pre-order of density operators | |
parameters of the EQE, | |
increasing function | |
permutation (component-wise) of vectors | |
effective quantum entropy (EQE) | |
effective (classical) entropy | |
sign (component-wise) of vectors | |
vector (in bold) and its components | |
density operators | |
maximally mixing and pure state, respectively | |
unitary matrices | |
conjugate transpose (or Hermitian transpose) | |
trace of matrices | |
natural logarithm of matrices | |
probability mass function vector | |
component-wise absolute value | |
scalar (real numbers) | |
supremum operator | |
tensor product |
Proposed Properties | Previously Adopted Properties |
---|---|
Fundamental properties: | |
Property 1 | ([21], Axiom B), ([22], Property 6) |
Property 1 (second part) | ([21], Property 1) |
Property 2 | ([20], Property 5), ([21], Axiom C’), ([22], Property 1) |
Property 3 | ([21], Axiom F’) |
Property 4 | |
Property 5 | [22] |
Other properties: | |
Property 6 | [22] |
Property 7 | |
Property 8 | |
Property 9 | ([20], Propositions 1 and 2), ([22], Proposition 2) |
Property 10 | |
Property 11 | ([22], Property 7) |
Property 12 | ([22], Property 4) |
Property 13 | ([22], Proposition 8) |
Property 14 | ([22], Proposition 5) |
Property 15 | |
Specific to composite operators: | |
Property 16 | ([21], Axiom C) |
Property 17 | ([21], Axiom A) |
Property 18 | ([20], Proposition 8), ([22], Proposition 13) |
(# of Measurement Samples) | |||||||||||
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Case | Dimension | L1-Error | |||||||||
2 | |||||||||||
4 | |||||||||||
8 | |||||||||||
2 | |||||||||||
4 | |||||||||||
8 | |||||||||||
2 | |||||||||||
4 | |||||||||||
8 | |||||||||||
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Pastor, G.; Woo, J.-O. Verifying Measures of Quantum Entropy. AppliedMath 2022, 2, 312-325. https://doi.org/10.3390/appliedmath2020019
Pastor G, Woo J-O. Verifying Measures of Quantum Entropy. AppliedMath. 2022; 2(2):312-325. https://doi.org/10.3390/appliedmath2020019
Chicago/Turabian StylePastor, Giancarlo, and Jae-Oh Woo. 2022. "Verifying Measures of Quantum Entropy" AppliedMath 2, no. 2: 312-325. https://doi.org/10.3390/appliedmath2020019