scholarly journals Quantum Probability's Algebraic Origin

2020 ◽  
Author(s):  
Gerd Niestegge
Keyword(s):  
Transition Probabilities ◽  
Transition Probability ◽  
Wave Functions ◽  
Quantum Mechanical ◽  
Statistical Interpretation ◽  
Quantum Probabilities ◽  
Mechanical Transition ◽  
Definition Of ◽  
Different Origins ◽  
Quantum Indeterminacy

Max Born's statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. While the lat- ter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the gen- eral definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg's and others' un- certainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.

scholarly journals Quantum Probability’s Algebraic Origin

Entropy ◽  
2020 ◽  
Vol 22 (11) ◽  
pp. 1196
Author(s):  
Gerd Niestegge
Keyword(s):  
Transition Probabilities ◽  
Transition Probability ◽  
Wave Functions ◽  
General Definition ◽  
Uncertainty Relations ◽  
Statistical Interpretation ◽  
Quantum Probabilities ◽  
Mechanical Transition ◽  
Definition Of ◽  
Different Origins

Max Born’s statistical interpretation made probabilities play a major role in quantum theory. Here we show that these quantum probabilities and the classical probabilities have very different origins. Although the latter always result from an assumed probability measure, the first include transition probabilities with a purely algebraic origin. Moreover, the general definition of transition probability introduced here comprises not only the well-known quantum mechanical transition probabilities between pure states or wave functions, but further physically meaningful and experimentally verifiable novel cases. A transition probability that differs from 0 and 1 manifests the typical quantum indeterminacy in a similar way as Heisenberg’s and others’ uncertainty relations and, furthermore, rules out deterministic states in the same way as the Bell-Kochen-Specker theorem. However, the transition probability defined here achieves a lot more beyond that: it demonstrates that the algebraic structure of the Hilbert space quantum logic dictates the precise values of certain probabilities and it provides an unexpected access to these quantum probabilities that does not rely on states or wave functions.

Topology of optimally controlled quantum mechanical transition probability landscapes

2006 ◽  
Vol 74 (1) ◽  
Author(s):  
H. Rabitz ◽  
T.-S. Ho ◽  
M. Hsieh ◽  
R. Kosut ◽  
M. Demiralp
Keyword(s):  
Transition Probability ◽  
Quantum Mechanical ◽  
Mechanical Transition

Exact Reliability Formula for Consecutive-k-Out-of-n Repairable Systems with r Repairmen

2018 ◽  
Vol 15 (2) ◽  
pp. 453-458
Author(s):  
B. Kousalya ◽  
S. Sangeetha ◽  
T. Vasanthi
Keyword(s):  
State Transition ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Repairable Systems ◽  
Reliability Indices ◽  
Laplace Transform Technique ◽  
The Laplace Transform ◽  
The Mean ◽  
Definition Of ◽  
Mean Time

In this paper, consecutive-k-out-of-n repairable systems with r repairmen are studied. The systems are either circular (or) linear. We assume that both the working time, and the repair time of each component are exponentially distributed, and every component after repair is as good as new. Each component is either a key component or an ordinary component so we can adopt a priority repair rule for key components. By using the definition of transition probability, the state transition probabilities of the system are derived. Several fundamental reliability indices (including availability, the mean time to the first failure, reliability and failure rate) of the systems are obtained explicitly by using the Laplace transform technique. Finally, one example is shown to explain the model, and the methodology developed in this paper.

scholarly journals States Transitions Inference of Postpartum Depression Based on Multi-State Markov Model

2021 ◽  
Vol 18 (14) ◽  
pp. 7449
Author(s):  
Juan Xiong ◽  
Qiyu Fang ◽  
Jialing Chen ◽  
Yingxin Li ◽  
Huiyi Li ◽  
...  
Keyword(s):  
Postpartum Depression ◽  
Markov Model ◽  
Normal State ◽  
Sojourn Time ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Public Health Problem ◽  
Mental States ◽  
Hazard Ratio ◽  
Multiple Time

Background: Postpartum depression (PPD) has been recognized as a severe public health problem worldwide due to its high incidence and the detrimental consequences not only for the mother but for the infant and the family. However, the pattern of natural transition trajectories of PPD has rarely been explored. Methods: In this research, a quantitative longitudinal study was conducted to explore the PPD progression process, providing information on the transition probability, hazard ratio, and the mean sojourn time in the three postnatal mental states, namely normal state, mild PPD, and severe PPD. The multi-state Markov model was built based on 912 depression status assessments in 304 Chinese primiparous women over multiple time points of six weeks postpartum, three months postpartum, and six months postpartum. Results: Among the 608 PPD status transitions from one visit to the next visit, 6.2% (38/608) showed deterioration of mental status from the level at the previous visit; while 40.0% (243/608) showed improvement at the next visit. A subject in normal state who does transition then has a probability of 49.8% of worsening to mild PPD, and 50.2% to severe PPD. A subject with mild PPD who does transition has a 20.0% chance of worsening to severe PPD. A subject with severe PPD is more likely to improve to mild PPD than developing to the normal state. On average, the sojourn time in the normal state, mild PPD, and severe PPD was 64.12, 6.29, and 9.37 weeks, respectively. Women in normal state had 6.0%, 8.5%, 8.7%, and 8.8% chances of progress to severe PPD within three months, nine months, one year, and three years, respectively. Increased all kinds of supports were associated with decreased risk of deterioration from normal state to severe PPD (hazard ratio, HR: 0.42–0.65); and increased informational supports, evaluation of support, and maternal age were associated with alleviation from severe PPD to normal state (HR: 1.46–2.27). Conclusions: The PPD state transition probabilities caused more attention and awareness about the regular PPD screening for postnatal women and the timely intervention for women with mild or severe PPD. The preventive actions on PPD should be conducted at the early stages, and three yearly; at least one yearly screening is strongly recommended. Emotional support, material support, informational support, and evaluation of support had significant positive associations with the prevention of PPD progression transitions. The derived transition probabilities and sojourn time can serve as an importance reference for health professionals to make proactive plans and target interventions for PPD.

Further enhancement on H∞ sliding mode control design for singular Markovian jump systems with partly known transition probabilities

2021 ◽  
pp. 107754632198920
Author(s):  
Zeinab Fallah ◽  
Mahdi Baradarannia ◽  
Hamed Kharrati ◽  
Farzad Hashemzadeh
Keyword(s):  
Transition Probabilities ◽  
Sliding Mode ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Linear Matrix ◽  
Sliding Mode Controller ◽  
Sliding Surface ◽  
Markovian Jump ◽  
Markovian Jump System ◽  
Singular Markovian Jump Systems

This study considers the designing of the H ∞ sliding mode controller for a singular Markovian jump system described by discrete-time state-space realization. The system under investigation is subject to both matched and mismatched external disturbances, and the transition probability matrix of the underlying Markov chain is considered to be partly available. A new sufficient condition is developed in terms of linear matrix inequalities to determine the mode-dependent parameter of the proposed quasi-sliding surface such that the stochastic admissibility with a prescribed H ∞ performance of the sliding mode dynamics is guaranteed. Furthermore, the sliding mode controller is designed to assure that the state trajectories of the system will be driven onto the quasi-sliding surface and remain in there afterward. Finally, two numerical examples are given to illustrate the effectiveness of the proposed design algorithms.

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