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Entropy Measures for Transition Matrices in Random Systems
by Zhaohui Chen, Rene Meyer, Zhuo-Yu Xian
Submission summary
| Authors (as registered SciPost users): | Zhaohui Chen · René Meyer · Zhuo-Yu Xian |
| Submission information | |
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| Preprint Link: | scipost_202601_00051v1 (pdf) |
| Date submitted: | Jan. 21, 2026, 12:39 p.m. |
| Submitted by: | Zhuo-Yu Xian |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
The author(s) disclose that the following generative AI tools have been used in the preparation of this submission:
ChatGPT (OpenAI, GPT-5) was used only for improving grammar and readability. The scientific content, results, and analysis are entirely the authors’ own.
Abstract
A transition matrix can be constructed through the partial contraction of two given quantum states. We analyze and compare four different definitions of entropy for transition matrices, including (modified) pseudo entropy, SVD entropy, and ABB entropy. We examine the probabilistic interpretation of each entropy measure and show that only the distillation interpretation of ABB entropy corresponds to the joint success probability of distilling entanglement between the two quantum states used to construct the transition matrix. Combining the transition matrix with preceding measurements and subsequent on-unitary operations, the ABB entropy either decreases or remains unchanged, whereas the pseudo-entropy and SVD entropy may increase or decrease. We further apply these entropy measures to transition matrices constructed from several ensembles: (i) pairs of independent Haar-random states; (ii) bi-orthogonal eigenstates of non-Hermitian random systems; and (iii) bi-orthogonal states in $PT$-symmetric systems near their exceptional points. Across all cases considered, the SVD and ABB entropies of the transition matrix closely mirror the behavior of the subsystem entanglement entropy of a single random state, in contrast to the (modified) pseudo entropy, which can exceed the bound of subsystem size, fail to scale with system size, or even take complex values.
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- Provide a novel and synergetic link between different research areas.
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Author comments upon resubmission
We hereby resubmit our manuscript after revision.
We thank the referees for their careful reading of the manuscript and for their constructive and insightful comments. We have revised the manuscript accordingly. All comments have been addressed in detail in a separate response document, where we provide point-by-point replies and indicate the corresponding changes made in the manuscript.
We believe that these revisions have significantly improved the clarity and quality of the paper, and we hope that the revised version will now be suitable for publication in SciPost Physics.
We appreciate your time and consideration.
Sincerely,
Zhaohui Chen, Rene Meyer, and Zhuo-Yu Xian
List of changes
Due to the substantial number of modifications, we do not provide an exhaustive itemized list here. Instead, all changes in the revised manuscript are clearly indicated by colors:
- Red marks changes made in response to Referee 1,
- Blue marks changes made in response to Referee 2,
- Brown marks additional changes for clarity and completeness.
Detailed point-by-point responses to all referee comments are provided in the attached reply documents.