Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information
Belavkin–Staszewski relative entropy can naturally characterize the effects of the possible noncommutativity of quantum states. In this paper, two new conditional entropy terms and four new mutual information terms are first defined by replacing quantum relative entropy with Belavkin–Staszewski rela...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/24/6/837 |
| _version_ | 1827660435088736256 |
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| author | Yuan Zhai Bo Yang Zhengjun Xi |
| author_facet | Yuan Zhai Bo Yang Zhengjun Xi |
| author_sort | Yuan Zhai |
| collection | DOAJ |
| description | Belavkin–Staszewski relative entropy can naturally characterize the effects of the possible noncommutativity of quantum states. In this paper, two new conditional entropy terms and four new mutual information terms are first defined by replacing quantum relative entropy with Belavkin–Staszewski relative entropy. Next, their basic properties are investigated, especially in classical-quantum settings. In particular, we show the weak concavity of the Belavkin–Staszewski conditional entropy and obtain the chain rule for the Belavkin–Staszewski mutual information. Finally, the subadditivity of the Belavkin–Staszewski relative entropy is established, i.e., the Belavkin–Staszewski relative entropy of a joint system is less than the sum of that of its corresponding subsystems with the help of some multiplicative and additive factors. Meanwhile, we also provide a certain subadditivity of the geometric Rényi relative entropy. |
| first_indexed | 2024-03-09T23:51:40Z |
| format | Article |
| id | doaj.art-7f95a4eedeb84f8d9f4410cda88cb068 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-03-09T23:51:40Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-7f95a4eedeb84f8d9f4410cda88cb0682023-11-23T16:33:57ZengMDPI AGEntropy1099-43002022-06-0124683710.3390/e24060837Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual InformationYuan Zhai0Bo Yang1Zhengjun Xi2The College of Computer Science, Shaanxi Normal University, Xi’an 710062, ChinaThe College of Computer Science, Shaanxi Normal University, Xi’an 710062, ChinaThe College of Computer Science, Shaanxi Normal University, Xi’an 710062, ChinaBelavkin–Staszewski relative entropy can naturally characterize the effects of the possible noncommutativity of quantum states. In this paper, two new conditional entropy terms and four new mutual information terms are first defined by replacing quantum relative entropy with Belavkin–Staszewski relative entropy. Next, their basic properties are investigated, especially in classical-quantum settings. In particular, we show the weak concavity of the Belavkin–Staszewski conditional entropy and obtain the chain rule for the Belavkin–Staszewski mutual information. Finally, the subadditivity of the Belavkin–Staszewski relative entropy is established, i.e., the Belavkin–Staszewski relative entropy of a joint system is less than the sum of that of its corresponding subsystems with the help of some multiplicative and additive factors. Meanwhile, we also provide a certain subadditivity of the geometric Rényi relative entropy.https://www.mdpi.com/1099-4300/24/6/837Belavkin–Staszewski relative entropygeometric Rényi relative entropyconditional entropymutual informationclassical-quantum setting |
| spellingShingle | Yuan Zhai Bo Yang Zhengjun Xi Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information Entropy Belavkin–Staszewski relative entropy geometric Rényi relative entropy conditional entropy mutual information classical-quantum setting |
| title | Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information |
| title_full | Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information |
| title_fullStr | Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information |
| title_full_unstemmed | Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information |
| title_short | Belavkin–Staszewski Relative Entropy, Conditional Entropy, and Mutual Information |
| title_sort | belavkin staszewski relative entropy conditional entropy and mutual information |
| topic | Belavkin–Staszewski relative entropy geometric Rényi relative entropy conditional entropy mutual information classical-quantum setting |
| url | https://www.mdpi.com/1099-4300/24/6/837 |
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